Scale effects modeling in bubbles and trusses using first strain gradient elasticity

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1 UNIVERSITY OF THESSALY SCHOOL OF ENGINEERING DEPARMENT OF CIVIL ENGINEERING Diplom Thesis Scle effects modeling in ules nd tusses using fist stin gdient elsticity y APOSTOLOS NASIAS Supeviso: D. Antonios. E. Ginnkopoulos, Pofesso Sumitted in ptil fulfillment of the equiements fo the Diplom of Civil Engineeing Volos, 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

2 ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ Διπλωματική Εργασία Μοντελοποίηση φαινομένων κλίμακας σε φυσαλίδες και δικτυώματα μέσω βαθμωτής ελαστικότητας υπό ΑΠΟΣΤΟΛΟΥ ΝΑΣΙΚΑ Επιβλέπων: Δρ. Αντώνιος. Ε. Γιαννακόπουλος, Καθηγητής Υπεβλήθη για την εκπλήρωση μέρους των απαιτήσεων για την απόκτηση του Διπλώματος Πολιτικού Μηχανικού Βόλος, 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

3 Appovl of this diplom thesis y the Deptment of Civil Engineeing, School of Engineeing, Univesity of Thessly, does not constitute in ny wy n cceptnce of the views of the utho y the sid cdemic ogniztion L. 54/, t.,. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

4 Exmintion Committee: D. Antonios E. Ginnkopoulos Supeviso, Pofesso, Deptment of Civil Engineeing, Univesity of Thessly D. Peleksis Nikolos, Pofesso, Deptment of Mechnicl Engineeing, Univesity of Thessly D. ksidis Theodoos, Associte Pofesso, Deptment of Civil Engineeing, Univesity of Thessly Εξεταστική Επιτροπή: Δρ. Αντώνιος Ε. Γιαννακόπουλος Επιβλέπων Καθηγητής, Τμήμα Πολιτικών Μηχανικών, Πανεπιστήμιο Θεσσαλίας Δρ. Πελεκάσης Νικόλαος Καθηγητής, Τμήμα Μηχανολόγων Μηχανικών, Πανεπιστήμιο Θεσσαλίας Δρ. Καρακασίδης Θεόδωρος Αναπληρωτής Καθηγητής, Τμήμα Πολιτικών Μηχανικών, Πανεπιστήμιο Θεσσαλίας 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

5 Scle effects modeling in ules nd tusses using fist stin gdient elsticity Apostolos Nsiks Univesity of Thessly, Deptment of Civil Engineeing,5 Supeviso: D. Antonios. E. Ginnkopoulos, Pofesso.. Astct In tody s technology pplictions in most scientific nches, stuctues of mico nd nno scle e often used in ttempt to minimize the volume of the ojects, the mteil use nd to optimize the mteil s popeties. Expeimentl studies, though, indicte tht the ehvio of stuctues of such scle cnnot lwys e descied using the clssicl theoy of elsticity. Scle effect ppe esulting to significntly stiffe ehvios in mny cses. Tht diected esech to ppoch smll scle polems using non clssicl non locl theoies with ext length pmetes in ode to model scle effects. Vious such theoies hve een developed, nd one of the simplest ones is the Aifntis Aifntis, 99 Altn & Aifntis, 99 modifiction of Mindlins Mindlin & Tiesen, 964 Mindlin, 964 stin gdient theoy of elsticity, which is used in this wok. Sevel oundy condition BC polems hve een ddessed nd solved the pst decdes using stin gdient elsticity nd othe non locl theoies. Howeve, due to the complexity of esulting fouth ode diffeentil equtions, only polems of simple geomety hve een solved. This fct ws the inspition fo the pesent thesis. This thesis is sed on gdient polems ledy solved, especilly those y Tsepou, et l., nd Polyzos, et l.,, nd ttempts to model the ehvio of stuctues of moe complex geomety. Such e stuctues consisting of multiple stuctul elements, possily of diffeent mechnicl popeties. Following this, need emeged to etun to the 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

6 fundmentl theoy nd then mke n extensive esech in the ville litetue in ode to investigte the continuity nd oundy conditions etween the diffeent memes of the composite odies tht e eing studied. Τhe pesent wok is divided in two min pts. Fist, is pesented the simplified stin gdient theoy fo two nd thee dimensionl D nd D polems suggested y Aifntis long with the espective clssicl one. Then, its fom is otined fo odies of spheicl geomety, sujected to dil lods only. Such e the cses of solid sphee nd spheicl cvity in n infinite gdient elstic spce unde vious loding cses. Although, some loding cses hve ledy een ddessed y Tsepou, et l.,, they e lso pesented in the pesent wok, long with some new oundy vlue polems, fo completeness. All esults e comped to the esults of espective clssicl elsticity polems. Next, thin wlled spheicl shell polems e eing studied, tiggeed y n expeimentl investigtion y Glynos & outsos, 9. It suggests stiffening of micoules which my e ttiuted to scle effects. A thin wlled spheicl shell theoy is developed tht spies to model the scle effects in micoules -o micosphees which is the tem used in the ticle-. Hving modeled the single gdient elstic ule, composite ule is modeled. It is the cse of composite thin wlled spheicl shell, consisting of two gdient elstic mteils tht she fully elstic intefce doule-lyeed micoule. The shell s ehvio is found nd comped with ehvio descied y the espective clssicl polems. In the second pt of this thesis, the gol is to fom n lgoithm fo solving gdient elstic D tuss stuctues. This pt egins y ddessing the simplest polem in gdient elsticity. Tht is the cse of one dimensionl sujected to unixil loding. Sevel oundy vlue polems fo the e pesented in ode to otin good undestnding of ech BC s effect on the s ehvio. As fist step to solving D stuctues, D composite stuctues of multiple colline s unde vious lod cse e solved. In ode to investigte the intection of diffeent colline elements, the connecting nodes popeties needed to e detemined. Two types of node 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

7 whee consideed; the fully elstic node, which woks s n intefce etween the elements, hs no stiffness of its own nd imposes continuous stin to the s ends, nd the igid node tht estins the stin t the s ends. The ppopite D stiffness mtix is otined, nd using the stiffness method, multiple polems of non stndd loding nd geomety e solved nd comped to the espective nlyticl gdient solutions nd, oth nlyticl nd numeicl, clssicl solutions. In this wy, it is show tht the elements cn e used s finite elements, too, in D polems nd descie the s scle effects. Finlly, the D tuss polem is ddessed. The function of the node is discussed in the D cse, nd the cse of two tuss is pesented s simplest exmple of D stuctue. Also, α simple indeteminte stuctue with clssicl loding is solved, in ode to find how indeteminte gdient tusses ehve. 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

8 Μοντελοποίηση φαινομένων κλίμακας σε φυσαλίδες και δικτυώματα μέσω βαθμωτής ελαστικότητας Απόστολος Νασίκας Πανεπιστήμιο Θεσσαλίας, Τμήμα Πολιτικών Μηχανικών, 5 Επιβλέπων: Δρ. Αντώνιος. Ε. Γιαννακόπουλος, Καθηγητής.. Περίληψη Στις σύγχρονες τεχνολογικές εφαρμογές, στους περισσότερους επιστημονικούς κλάδους, συχνά χρησιμοποιούνται κατασκευές μικρό ή νάνο κλίμακας στην προσπάθεια να ελαχιστοποιηθεί ο όγκος και η μάζα των τελικών αντικειμένων, να βελτιστοποιηθούν η χρήση υλικών και οι ιδιότητες αυτών. Πειραματικές μελέτες δείχνουν ότι η συμπεριφορά μιας κατασκευής τέτοιας κλίμακας δε μπορεί πάντα να περιγραφεί μέσω της κλασσικής θεωρίας ελαστικότητας, διότι φαινόμενα κλίμακας παρουσιάζονται, προκαλώντας συχνά κράτυνση αυτών των μελών. Αυτό οδήγησε την επιστημονική έρευνα στο να προσεγγίσει μικρής κλίμακας προβλήματα μέσω μη κλασικών, μη τοπικών non locl θεωριών ελαστικότητας με επιπλέον παραμέτρους μήκους, οι οποίες μπορούν να μοντελοποιήσουν φαινόμενα κλίμακας. Διάφορες non locl θεωρίες έχουν αναπτυχθεί. Μια από τι απλούστερες, η οποία χρησιμοποιείται στην παρούσα εργασία είναι η θεωρία της βαθμωτής ελαστικότητας του Mindlin Mindlin & Tiesen, 964 Mindlin, 964 που στην απλοποιημένη της μορφή προτάθηκε από τον Αϋφαντή Aifntis, 99 Altn & Aifntis, 99. Τις τελευταίες δεκαετίες, αρκετά προβλήματα συνοριακών συνθηκών έχουν επιλυθεί αναλυτικά χρησιμοποιώντας βαθμωτή ελαστικότητα. Λόγω, όμως, της πολυπλοκότητας που επιφέρει η αύξηση της τάξης των διαφορικών εξισώσεων, και η αβεβαιότητα γύρω από την φυσική σημασία των μη κλασικών συνοριακών συνθηκών, αυτά περιορίζονται σε σώματα πολύ απλής γεωμετρίας και περιορισμένα είδη φορτίσεων. Αυτή 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

9 η παρατήρηση αποτέλεσε το έναυσμα για την παρούσα διπλωματική εργασία. Αυτή η δουλειά, βασιζόμενη σε ήδη λυμένα απλά προβλήματα, κυρίως αυτά των Τσεπούρα και συνεργάτών Tsepou, et l.,, και Πολύζου και συνεργατών Polyzos, et l.,, στοχεύει να μοντελοποιήσει τη συμπεριφορά πιο σύνθετων κατασκευών. Τέτοιες κατασκευές αποτελούνται από περισσότερα του ενός μέλη τα οποία αλληλεπιδρούν και ενδεχομένως έχουν μεταξύ τους διαφορετικές μηχανικές ιδιότητες. Για την μελέτη της αλληλεπίδρασης των μελών, χρειάστηκε αρχικά αναδρομή στην βασική θεωρία και στη συνέχεια εκτεταμένη βιβλιογραφική έρευνα, ώστε να επιλεγούν οι κατάλληλες συνθήκες συνέχειας και ισορροπίας μεταξύ των μελών του σύνθετου σώματος. Η παρούσα εργασία χωρίζεται σε δυο κύρια μέρη. Αρχικά, παρουσιάζεται η απλοποιημένη θεωρία βαθμωτής ελαστικότητας του Αϋφαντή, παράλληλα με την αντίστοιχη κλασική, ώστε είναι διακριτές οι ομοιότητες και οι διαφορές τους. Στην συνέχεια, βρίσκεται η μορφή που αυτή παίρνει για σώματα σφαιρικώς συμμετρικά, στα οποία ασκούνται μόνο ακτινικά φορτία. Τέτοιες περιπτώσεις είναι αυτή μίας συμπαγούς σφαίρας και αυτή μιας σφαιρικής κοιλότητας σε έναν άπειρα εκτεινόμενο βαθμωτά ελαστικό χώρο, υπό διάφορους συνδυασμούς φορτίσεων. Κάποιες φορτίσεις συνοριακές συνθήκες έχουν ήδη εξεταστεί από τους Πολύζο και συνεργάτες Polyzos, et l.,, ωστόσο και αυτά μεταξύ άλλων προβλημάτων συνοριακών συνθηκών παρουσιάζονται, για λόγους πληρότητας. Όλα τα αποτελέσματα συγκρίνονται τελικά με τα αντίστοιχα της κλασσικής ελαστικότητας. Στη συνέχεια, ένα λεπτότοιχο σφαιρικό κέλυφος μελετάται, με αφορμή μια πειραματική εργασία των Γλυνού και Κουτσού Glynos & outsos, 9. Σε αυτήν παρατηρήθηκε κράτυνση μικροφυσαλίδων, η οποία θεωρήθηκε ότι ίσως οφείλεται σε φαινόμενα κλίμακας. Μια θεωρία λεπτότοιχων σφαιρικών κελυφών αναπτύσσεται με στόχο τη μοντελοποίηση φαινομένων κλίμακας στις μικροφυσαλίδες. Έχοντας μελετήσει την απλή φυσαλίδα, ακλουθεί μια σύνθετη, αποτελούμενη από δυο διαφορετικά υλικά που μοιράζονται μια επιφάνεια διεπιφάνεια. Η συμπεριφορά της σύνθετης φυσαλίδας, συγκρίνεται με την συμπεριφορά που περιγράφεται στο αντίστοιχο κλασικό μοντέλο, το οποίο επίσης παρουσιάζεται. 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

10 Τα παρακάτω συμπεράσματα εξάγονται από την μελέτη αυτών των προβλημάτων: Για τη συμπαγή σφαίρα: Όταν δεσμεύεται η παραμόρφωση στο σύνορο της σφαίρας, αυτή κρατύνεται, οπότε προκύπτουν μικρότερες παραμορφώσεις. Ανεξάρτητα από το μέγεθος της σφαίρας και της μικροδομής, όταν μόνο κλασικά φορτία ασκούνται σε αυτή, τότε η απόκριση της είναι η κλασσική. Για την ανάπτυξη φαινομένων κλίμακας, πρέπει να ασκηθούν και διπλές τάσεις. Για μικρές τιμές του λόγου του εσωτερικού μήκους του υλικού g προς την ακτίνα, η συμπεριφορά της σφαίρας πρακτικά δεν αποκλίνει από την αντίστοιχη κλασική. Για τη σφαιρική κοιλότητα: Η μικροδομή προκαλεί κράτυνση όταν δεν είναι πολύ μικρή, ανεξάρτητα από την άσκηση διπλών τάσεων, ακόμα και με την άσκηση μόνο πίεσης. Σημαντική είναι και η επιρροή του λόγου Poisson. Οι διπλές τάσεις δεν είναι αμελητέες μόνο κοντά στο όριο της κοιλότητας. Πάντα η λύση απλοποιείται στην κλασική όταν η ακτίνα είναι πολύ μεγαλύτερη του μήκους g της μικροδομής. Για το σφαιρικό κέλυφος φυσαλίδα: Μόνο το πρόβλημα όπου οι ασκούμενες κλασικές και μη κλασικές δυνάμεις έδωσε χρήσιμα αποτελέσματα. Ακόμα και όταν δεν ασκούνται διπλές τάσεις οι μετατοπίσεις είναι μικρότερες από τις κλασικές. Όταν η μικροδομή είναι ασήμαντη, τα πεδία των μετακινήσεων και των παραμορφώσεων απλοποιούνται στα κλασικά, όχι όμως και το πεδίο των καμπυλοτήτων. Για το διπλό σφαιρικό κέλυφος: Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

11 Μόνο το πρόβλημα όπου οι ασκούμενες κλασικές και μη κλασικές δυνάμεις έδωσε χρήσιμα αποτελέσματα. Επειδή οι παράμετροι είναι πολλές, δύσκολα εξάγονται αποτελέσματα αν δεν επιλεχθούν πιο στοχευόμενα κάποιες περιπτώσεις. Όταν θεωρηθεί ότι τα δυο υλικά που το αποτελούν είναι ίδια, το μοντέλο αυτό απλοποιείται στο μοντέλου του απλού σφαιρικού κελύφους Όταν μηδενιστεί η μικροδομή των δυο υλικών, η συμπεριφορά που το μοντέλο προβλέπει, δεν είναι ίδια με την κλασική. Στο δεύτερο μέρος της παρούσας διπλωματικής εργασίας, στόχος είναι η εύρεση ενός αλγορίθμου για την επίλυση επίπεδων δικτυωμάτων στα πλαίσια της βαθμωτής ελαστικότητας, για κάθε είδος φόρτισης. Αυτό ξεκινά από το απλούστερο πρόβλημα που μπορεί να λυθεί μέσω βαθμωτής ελαστικότητας, αυτό της μονοδιάστατης ράβδου, σε μονοαξονική φόρτιση. Αρκετά προβλήματα συνοριακών συνθηκών για την ράβδο παρουσιάζονται αναλυτικά με στόχο την πλήρη κατανόηση της επιρροής κάθε συνοριακής συνθήκης στην συμπεριφορά της ράβδου. Τα επόμενα γενικά συμπεράσματα μπορούν να εξαχθούν για τη συμπεριφορά της ράβδου με μικροδομή Ορίζοντας τις μετακινήσεις και τις τροπές των άκρων μιας ράβδου, στη γενική περίπτωση έχει σαν αποτέλεσμα την μη ομογενή απόκριση αυτής. Δεσμεύοντας τις τροπές στα άκρα των ράβδων, μικρότερες από τις κλασικές μετακινήσεις προκύπτουν για οποιαδήποτε φόρτιση και το πεδίο των μετακινήσεων είναι πάντα μη ομογενές, ανεξάρτητα από το μέγεθος της ράβδου και τη μικροδομή. Όταν αντί να δεσμευθεί η παραμόρφωση στα άκρα της ράβδου, ασκούνται μηδενικές διπλές δυνάμεις, τότε ανεξαρτήτως της μικροδομής, η συμπεριφορά της ταυτίζεται με την κλασική. Η παράμετρος l επηρεάζει την ράβδο σε κάθε περίπτωση με τρόπο που να τείνει να μειώσει το συνολικό μήκος αυτής. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

12 Οποιοδήποτε πρόβλημα συνοριακών συνθηκών μπορεί να αναχθεί στο αντίστοιχο κλασικό όταν ασκηθεί κατάλληλος συνδυασμός συνοριακών συνθηκών. Η συμπεριφορά της ράβδου υπό τις περισσότερες συνθήκες φόρτισης ταυτίζεται με τις αντίστοιχες κλασικές συμπεριφορές, όταν το χαρακτηριστικό μήκος της μικροδομής είναι ασήμαντο σε σχέση με το μήκος της ράβδου. Σαν πρώτο βήμα για την επίλυση σύνθετων ραβδωτών κατασκευών, επιλύεται μια σειρά από μονοδιάστατα προβλήματα, σύνθετων κατασκευών αποτελούμενων από ομοαξονικές ράβδους που αλληλεπιδρούν, υπό διάφορους συνδυασμούς φορτίσεων. Για να περιγραφεί η αλληλεπίδραση των ράβδων, χρειάζεται πρώτα να διερευνηθούν οι ιδιότητες των κόμβων που τις συνδέουν. Δυο είδη κόμβων επιλέγονται. Ο ένας είναι πλήρως ελαστικός, λειτουργεί σαν διεπιφάνεια μεταξύ των ράβδων, και δεν έχει καμία δική του αντοχή, αλλά απαιτεί συνέχεια μετακινήσεων και των παραμορφώσεων των συνδεόμενων σε αυτόν άκρων ράβδων. Ο δεύτερος είναι πλήρως απαραμόρφωτος και δεσμεύει την παραμόρφωση κάθε συνδεδεμένου σε αυτόν άκρου ράβδου. Στην συνέχεια, ο πίνακας δυσκαμψίας της ράβδου, στα πλαίσια της βαθμωτής ελαστικότητας, εξάγεται και χρησιμοποιώντας μια γενικευμένη μέθοδο μετακινήσεων, επιλύονται διάφορα μονοαξονικά προβλήματα. Τα αποτελέσματά τους συγκρίνονται με αναλυτικές, βαθμωτές και αντίστοιχες κλασικές λύσεις. Με αυτόν τον τρόπο αποδεικνύεται ότι η ράβδος αυτή μπορεί να χρησιμοποιηθεί και σαν πεπερασμένο στοιχείο τουλάχιστον σε μονοδιάστατα προβλήματα. Καταλήγοντας, εξετάζεται το πρόβλημα του επίπεδου δικτυώματος. Η λειτουργία του κόμβου στις δύο διαστάσεις συζητείται και εξετάζεται το απλό παράδειγμα του τριαρθρωτού τόξου συγκριτικά με την κλασική περίπτωση. Τέλος, εξετάζεται η περίπτωση ενός στατικά αόριστου δικτυώματος, με την κλασική έννοια του όρου, με στόχο να φανεί αν και πώς η μικροδομή επηρεάζει τη λειτουργία και στατικά αορίστων δικτυωμάτων. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

13 . Tle of Contents.. Astct Περίληψη Tle of Contents.... Intoduction Conventions Equiliium eqution nd oundy conditions Clssicl Theoy of Elsticity Gdient Theoy of Elsticity I. PART I: SPHERICAL GEOMETRY PROBLEMS... 4.I.. Spheicl ody pplictions... 4.I.. Spheicl Symmety Polems Equiliium... 4.I..i. Clssicl Theoy... 4.I..ii. Non Clssicl / Gdient Theoy I.. Boundy Condition Polems I... The solid sphee I..ii. The spheicl cvity I..iii. The Spheicl Shell I..iv. The doule-lye shell Clssicl theoy Clssicl theoy I.4. Expeimentl dt tetment I.5. Conclusions II. PART II: D PROBLEMS & TRUSSES II.. Tuss modeling though gdient elsticity II.. The single ehvio... 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

14 4.II..i. Equiliium eqution nd oundy conditions in D polems II..ii. Boundy Condition Polems II..iii. Conclusions II. -D composite stuctues... 4.II..i. Node function in composite D stuctues... 4.II..ii. -D Stiffness Mtix II..iii. -D Applictions II.4. -D B Stuctues - Tusses II.4.i. Node function in composite D stuctues II.4.ii. -D stiffness mtix II.4.iii. -D Applictions II.5. Conclusions Appendices I. Appendix I: Clssicl tuss node function investigtion II. Appendix II: Bsic Spheicl Coodinte System Identities III. Appendix III: Hypeolic Function Identities IV. Appendix IV: The petwisted em nlogy y odolemis Refeences Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

15 . Intoduction The clssicl theoy of elsticity is quite sufficient fo most pplictions, minly of mcoscopic scle, since it s ssocited with the concepts of homogeneity nd loclity of stess. Expeimentl investigtions, though, indicte tht stuctues of mico nd nno scle, such s the ems, s, pltes nd shells used it moden technology pplictions, exhiit non homogenous ehvio nd significnt micostuctue effects. The clssic theoy fils to descie dequtely such size dependnt mechnicl ehvios of smll scled, line elstic stuctues due to thei dependence of the mteils micostuctue. The clssic theoy lso cnnot descie the ehvio of mteils with significnt micostuctue effects like polymes, polycystls, gnul nd textile mteils. In these cses the stte of stess needs to e descied in non-locl mnne. This cn e chieved y using mny diffeent types of size dependent theoies s highe ode stin gdient theoies, o couple stess theoies. Such theoies wee developed y Mindlin nd co-wokes Mindlin, 965 Mindlin, 964 Mindlin & Eshel, 968, Aifntis nd cowokes Aifntis, 99 Aifntis, Altn, et l., 996 Altn & Aifntis, 99 nd Vdoulkis nd co-wokes Exdktylos & Vdoulkis,,in connection with highe-ode stin gdient theoies, nd Cosset, Mindlin nd Tiesen Mindlin & Tiesten, 96 nd Toupin in connection with couple-stess theoies. Fom the ove theoies, the most genel nd compehensive is the one developed y Mindlin nd co-wokes, ut the simplest one is the one y Aifntis nd co-wokes, ccoding to Tsepou []!!!!! The pst decdes, these theoies hve een used, mostly in simplified foms, to solve mny oundy vlue polems of oth sttic nd dynmic elsticity. One cn mention sttic polems deling with disloctions, fctue mechnics, the hlfspce unde vious sufce lods, oehole unde pessue, unde pessue nd em in ending. It hs een found tht when using such non locl theoies, singulities nd discontinuities of clssicl elstic theoy disppe. Also, size effects e usully cptued nd wve dispesion effects e oseved in cses whee it ws not possile in clssicl line elsticity. 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

16 .. Conventions Stndd nottion is used though the entie ppe. Boldfce symols denote tensos whose odes e indicted y the context. All tensos components e witten with espect to fixed Ctesin coodinte system with se vectos ˆε i. A supescipt T fte second ode tenso indictes its tnspose Let,, c, k, l, m e vectos nd A second ode tenso. The following poducts e used in the text: Inne: i i Oute: ijk i εˆ whee ε ijk is the ltento, e.i. Dyd: i k i j ijk εˆ εˆ i j,,, i j if if εˆ Dyd inne: : c d c d i, j,k i, j,k e cyclic if ny two of i εˆ j e nticyclic i, j,k. Tid inne: c : k l m c k l m Also c c the gdient opeto: the Lplcin opeto: ε ˆ i x i εˆ Let S e sufce nd nˆ the unit noml vecto on S, S I nˆ nˆ A A i i ii e equl When the elements of ow vecto e pesented, they e eithe septed y spces o coms. The elements of column vecto e septed y semicolons, i.e., ] [ ] nd ] [ T [ ; 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

17 .. Equiliium eqution nd oundy conditions In this section, the equiliium eqution nd the coesponding oundy conditions tht should e stisfied y ny line elstic mteil e deived. Fist, the cse of clssicl elstic mteil is investigted nd the clssicl theoy of elsticity fo D nd D odies is pesented. It is followed y the cse of the line elstic mteil with micostuctue gdient elstic mteil, investigted in the fmewok of Mindlin s fist stin gdient elsticity theoy, using the constitutive eqution poposed y Aifntis. The ltte s specil cse of Mindlin s stin gdient theoy, llows the deivtion of oth, the equiliium eqution nd the oundy conditions y fist tking the vition of the stin enegy defined y Mindlin nd then inseting the given constitutive eqution.... Clssicl Theoy of Elsticity Conside line elstic ody of volume V suounded y sufce S. Let nˆ e the unit noml vecto on S, nd Ctesin coodinte system with its oigin locted inteio to V. Accoding to the clssicl theoy of elsticity the stin enegy depends upon the stin, s follows: : e dv e dv U V τ ij ij, V whee τ nd e e the second ode stess nd stin tensos espectively. Since e u u, the vition of the odies stin enegy cn e witten in tems of the displcement u s, U V τ : udv Afte some diffeentil clculus, using the divegence theoem, the eqution ove tkes the fom: U V τ udv nˆ τ uds S Menwhile, the vition done y extenl foces to V is: 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

18 W f udv V S P uds, f, P eing espectively ody foces cting to the ody nd extenl sufce tctions. Since U W the equtions ove imply tht the equiliium eqution fo clssicl elstic ody is: τ f nd the coesponding clssicl oundy conditions BCs e: P x nˆ τ P nd/o u u, whee P, u denote pescied vlues. In Hooke s Lw the stess nd stin hve the following eltion τ e ui. So, the equiliium eqution fo n elstic continuum cn e otined in tems of the displcement fieldu : u λ u f... Gdient Theoy of Elsticity Conside line elstic ody of volume V suounded y sufce S, which is chcteized y micostuctue modeled mcoscopiclly y the gdient of the defomtion. Let nˆ e the unit noml vecto on S, nd Ctesin coodinte system with its oigin locted inteio to V. Accoding to Mindlin s stin gdient theoy the stin enegy depends upon oth the stin nd the stin s gdient: U V τ : e μ. : e dv V ij e ij ijk e i jk dv,whee τ nd e e the clssicl second ode stess nd stin tensos espectively, μ is the thid ode tenso with 7 components μ ijk epesenting doule foces pe unit e. The fist suscipt indictes the noml vecto of the sufce the second of the foces leve nd the thid the diection of the foces. It should e noted tht the doule stesses contiute only to the potentil enegy nd to the oundy conditions of 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

19 the polem, without giving ny esultnt stess of couple vecto t ny sufce of the studied ody. Since e u u, the vition of the odies stin enegy cn e witten in tems of the displcement u s, U : V τ u μ. : udv Using the symmety eltion μ μ, nd fte some line lge nd diffeentil clculus, which e pesented in detil in tsepou[], the vition of the stin enegy tkes the following fom: U τ μ udv ˆ ˆ ˆ ds V S n μ n n u μ ˆ ˆ ˆ : ˆ ˆ S n τ n n n S S μ n μ u nˆ S n ˆ n ˆ nˆ : μ Snˆ : μ u ds mˆ nˆ : S ds C C μ u dc whee fo non-smooth oundies C e the edge lines fomed y the intesection of two sufce potions S i nd S j of S, mˆ sˆ nˆ,with ŝ eing the tngentil vecto to C, nd the ckets indicte tht the enclosed quntity is the diffeence etween the vlues on the sufce potions S i nd S j. Fo smooth D oundies nd oth smooth nd non-smooth D oundies the lst tem is lwys equl to zeo. Menwhile, the vition done y extenl foces to V is: W V f udv S R nˆ u ds P uds E u, f, P, R, Eeing espectively the clssicl ody foces cting to the ody, the clssicl extenl sufce tctions, non clssicl tction-like vecto of sufce doule stesses nd non clssicl sufce jump stesses cting on the non smooth odies sufce. Since U W the equtions ove imply tht the equiliium eqution fo D o D gdient elstic ody is τ μ f S C C dc 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

20 And the coesponding clssicl oundy conditions e μ P x nˆ τ ˆ n nˆ : nˆ nˆ μ nˆ μ S nˆ nˆ nˆ : μ S nˆ : μ P And the non-clssicl ones u R nˆ μ nˆ R nd/o q, nˆ S S nd/o u u, E mˆ nˆ : μ E, whee P, R, q, E denote pescied vlues. Mindin Mindlin, 965 poposed modifiction of Hooke s Lw expessed y the following eltions: σ τ s τ e ui e u u s [ c e c I u c ] u σ eing the totl stess tenso, coelted to the stins nd thei gdients though five independent mteil constnts,,,c,c, cthe fist two eing the Lme constnts. Aifntis Aifntis, 99 poposed the following modifiction: μ g τ s μ g τ g eing the volumetic stin gdient enegy coefficient, the unique constnt elted to the mteil s micostuctue. The equiliium eqution fo gdient elstic continuum in tems of the displcement field u tkes the fom: u λ u g u u f Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

21 4.I. PART I: SPHERICAL GEOMETRY PROBLEMS Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

22 4.I.. Spheicl ody pplictions The simplest thee dimensionl odies tht cn e consideed in tems of geomety e those of spheicl symmety. Howeve, pefectly spheicl odies ely ppe in eveydy s life noml scled pplictions. Especilly in cses of only dil loding nd no ody foces, which e ddessed in this wok, spheicl odies do not hve mny mco scle pplictions. Howeve, when smll scled odies e consideed, the ssumption of only dil lods is not s ity, while the ody foces might e vey smll comped to the dil lods. So, s f s smll scled ppliction cn e consideed, spheicl odies cn e found nd in sevel cses o e used s n ppoximtion to the tue geomety of the ody. The most common spheicl symmeticl ody used in smll scle pplictions is the spheicl shell. Tht is the cse of micoules used in medicine. Two pimy pplictions of micoules e consideed to e in the contst enhnced ultsoundceus, which is the ppliction of n ultsound contst medium to tditionl medicl sonogphy, nd tgeted dug delivey. Thee e sevel dvntges to using micoules insted of using ltentive methods, since thei use is consideed to e cost effective nd sfe fo the ptients, since no dition is pplied nd smlle dug mounts e used. These dvntges mke thei use vey ttctive, nd thus thei ehvio needs to e modeled nd studied. Since thei dimensions e vey smll, scle effects should not e ignoed, since they might e significnt, nd tht is the eson tht it is ttempted to medl them using gdient elsticity. Of couse, in these pplictions the loding nd the geomety of the ules is much moe complex thn the ones studied in this wok. Howeve, in ode to undestnd the ules ehvio, one must stt fom the sics, so oth geomety nd loding simplifictions e mde. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

23 4.I.. Spheicl Symmety Polems Equiliium In the pevious section, genel theoy hs een pesented fo odies of ny geomety, sujected to vious lod comintions. In ode to study spheiclly symmetic odies, the equiliium nd the oundy conditions need to e otined, using spheicl coodintes. In the following pgphs the equiliium nd the BCs fo spheicl odies e otined in spheicl coodintes, in oth the clssicl nd the gdient theoy of elsticity 4.I..i. Clssicl Theoy Conside ny ody chcteized y spheicl symmety, fo exmple solid sphee, spheicl cvity, spheicl shell, ny nume of spheicl shells the one inside the othe nd ny comintion of the ove. In spheicl coodintes the displcement vecto tkes the fom u u,, ˆ u,, ˆ θ u,, φˆ The ody is sujected only to dil lods nd displcements of spheicl symmety nd zeo ody foces e ssumed. Unde these ssumptions, the odies displcement vecto is simplified tou ˆ, u u, u,,, u,, whee u is the dil displcement nd the distnce of the cente of the ody. The clssicl equiliium eqution tkes the following fom: u λ u λ u u u,., since λ is non-zeo mteil constnt. Even though D polem is ddessed, the loding chosen educed the equiliium eqution to n odiny diffeentil eqution. It s genel solution is: u Ciiˆ i, C C ˆ C C ˆ u Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

24 i.e, the O.D. s fundmentl solutions e: nd / In the following clssicl polems, the pseudo vectos ρ u nd C e used. The fist one is vecto whose elements e the O.D.s fundmentl solutions. The second is the vecto of the solution s constnts. They e used in ode to optimize the disply of the esults. Fo exmple, the displcement field cn e witten s T C ρ ˆ C ˆ u i i.in the sme wy e define the pseudo vectos of the fundmentl solutions ν th deivtive ρ nd the stess functions pseudo vecto P Pˆ n. ρ [, ], nd 4 i P [P,P ],, P ' i ' i etc. Due to spheicl symmety the sufce noml vecto nˆ cn e eithe ˆ o ˆ thus the oundy conditions tke the following fom C P nˆ C P ˆ n, P nˆ nˆ τ nˆ ˆ, thus P ˆ Pˆ i i i i 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

25 5 4.I..ii. Non Clssicl / Gdient Theoy Conside ny ody chcteized y spheicl symmety with considele micostuctue tht cn e modeled y dding n dditionl length pmete g. In spheicl coodintes the displcement vecto tkes the fom φ θ u ˆ,, u ˆ,, u ˆ,, u Once gin, the ody is sujected only to dil lods nd displcements of spheicl symmety nd y the ssumption of zeo ody foces the displcement vecto is simplified to u ˆ u, u u while,, u nd,, u, u eing the dil displcement nd the distnce of the cente of the ody. The equiliium eqution cn e simplified to: f u u u u g λ u u g g λ The genel solution fo the displcement field is: ˆ C C C C C i i i u u ˆ / / / cosh / sinh / 4 g g C g g g g C C C i.e., the O.D. s fundmentl solutions e:, / / / / / cosh / sinh / / g g g g g g g g g g e e, / 4 / / / / / g g g g g e, / eing the modified Bessel Function n with n=/. The C i constnts e detemined y the BCs. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

26 Due to spheicl symmety of the ody the sufce noml vecto nˆ cn e eithe ˆ o ˆ thus the oundy conditions tke the following fom The tction vecto: μ Pˆ n nˆ τ ˆ n nˆ : nˆ S S nˆ Ci Pi nˆ, i,4 i '' i ' i Pi i ' g i ''' 6 6 i i ' g μ nˆ μ nˆ n ˆ nˆ : μ nˆ i '' i ' i i ''' 4 4, ' thus P ˆ Pˆ The sufce doule tction vecto: R n n μ ˆ ˆ nˆ i,4 CiR i ˆ, fo oth nˆ ˆ, S i S i etc. : μ R i i ' ' i i '' i g ' And the noml displcement gdient u qˆ n Ciqi nˆ, whee qi i'. Thus, the noml displcement nˆ i,4 gdient vecto q ˆ qˆ In the following non-clssicl polems the pseudo vectos ρ u, q qˆ n, P Pˆ n nd R Rˆ n e used to indicte espectively the ow vectos of the O.D.s fundmentl solutions, thei deivtives, the functions of the tction nd doule tction tem of ech fundmentl solution. i.e. ρ ˆ ε [,,, ], i i 4 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

27 q 'ˆ ε [ ', ', ', '], i i 4 P Pi εˆ i [P,P,P,P4 ] sinh / g cosh / g sinh / g sinh / g cosh / g g g / g / g / g / g / g 4 / g / g e g 5 4 / g / g / g / g / g T R Riεˆ i [R,R,R,R4] sinh / g cosh / g sinh / g cosh / g sinh / g cosh / g / g / g / g / g / g / g 4 g / g / g 6 6 e 4 / g / g / g / g / g / g T 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

28 4.I.. Boundy Condition Polems In the following pges numeous polems will e ddessed using oth the clssicl nd the non -clssicl gdient line elsticity, ll chcteized y spheicl symmety. The fist two polems hve oiginlly een ptly solved y Polyzos, et l.,, ut fo completeness those polems e lso pesented hee. 4.I... The solid sphee Conside solid sphee of dius, whose cente O coincides with the oigin of oth the Ctesin nd the spheicl coodinte system. The point is mteil point nd themodynmics dictte tht its displcement cnnot e infinite. In the clssicl polem the displcement field tkes the fomu C ˆ, eing the dil coodinte. C Since lim lim, the C tem needs to e zeo in ode to mintin finite displcement t the cente of the sphee. In the gdient polem, the displcement field tkes the fom u ˆ C C C C ˆ. Fig. x-y plne section of solid sphee of dius. sujected to dil displcement U u 4 4 Since oth lim nd lim / g lim e, fo g, /g / g oth tems C nd C 4 e zeo due to the themodynmics, nd the constnts to e detemined e two. Clssicl theoy In this fist polem, the sphee oundy is sujected to dil displcement U, i.e the BC is u U ˆ 4 C ˆ U ˆ C U U u ˆ u U Gdient theoy 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

29 In the espective gdient polem, the sphee is mde of mteil with micostuctue nd eing sujected to dil displcement U clssicl BC, while the noml displcement gdient t the oundy is q nonclssicl BC. The BCs nd the displcement field e given elow in vecto foms. u Uˆ, u q ˆ u q ˆ uˆ C C ˆ Fig. x-y plne section of solid sphee of dius. sujected to pescied dil displcement u Uˆ nd noml displcement gdient q q ˆ Cying though with the line lge, the unique exct solution of this polem is otined. It ode to optimize the esults disply, the pmete c / g dius to intenl length pmete tio is used. C c c sinh c ccoshc sinh c U sinh c ccoshc sinh c c ccoshc sinh c q sinh c c coshc sinh c C c U c sinh c ccoshc sinh c c c q sinh c c coshc sinh c This solution cn e divided into two simple ones. In the fist, the noml displcement gdient is estined nd so it vnishes t the oundy q = while the sphee is sujected to dil displcement U. In the second oundy s displcement is estined U = while the noml displcement gdient on the oundy is pescied q. Oviously, ny othe polem cn e ddessed s line comintion of the two polems ove, s esult of the supeposition pinciple. Heeupon, the solutions of these two polems e pesented 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

30 Fig. Nomlized dil displcement u U vesus nomlized dil distnce of the solid sphee of dius, fo vious g tios. The clssicl oundy condition is u U ˆ nd the non-clssicl one q Fig. 4 Nomlized dil defomtion e U / vesus nomlized dil distnce of the solid sphee of dius, fo vious g tios. The clssicl oundy condition is u U ˆ nd the non-clssicl one q Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

31 Fig. Nomlized doule stesses fo vious / EU vesus nomlized dil distnce of the solid sphee of dius, g tios. The clssicl oundy condition is u U ˆ nd the non-clssicl one q Some peliminy conclusions cn e dwn t this point. By estining the sphees dil defomtion t it s oundy, non clssicl ehvio is otined, whee the sphees displcement field long it s dius is gete thn the one of the clssicl theoy. Even in the cses tht the c=l/g tio is get c=, c=, the divegence fom the clssicl solution is discenile. Also, the defomtion field ppoches the clssicl defomtion field s the c tio inceses. Fo get c tio vlues, only close to the sphee s oundy the stin diveges fom the clssicl, while long the est of it s dius, the stins e pcticlly those descied y the clssicl theoy. The dil doule stesses distiution long the s dius tkes gete vlues fo smll c tios, with it s mximum lwys eing t the sphees sufce, nd zeo doule stesses t it s cente. Fo get c vlues, the doule stesses e significnt only in smll pt of the sphees dius, close to its sufce. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

32 Fig. 5 Nomlized dil displcement dius, fo vious u q vesus nomlized dil distnce of the solid sphee of g tios. The clssicl oundy condition is u nd the non-clssicl one q q ˆ Fig. 4 Nomlized dil defomtion e q vesus nomlized dil distnce of the solid sphee of dius, fo vious g ˆ tios. The clssicl oundy condition is u nd the non-clssicl one q q Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

33 Fig. 6 Nomlized doule stesses, fo vious g / Eq vesus nomlized dil distnce of the solid sphee of dius tios. The clssicl oundy condition is u nd the non-clssicl one q q ˆ In the figues ove is clely depicted the fct tht the effect of the non clssicl BC deceses fo get c tio vlues, nd it is limited only close to the sphees oundy. It is noted tht pplying cetin comintions of U nd q the clssicl o the gdient fundmentl solution cn e eliminted. Hence, evey sphee even one with significnt micostuctue sujected to the ight comintion of lods q=u/ ehves exctly s the clssicl theoy of elsticity dicttes. Futhemoe, y pplying diffeent set of U, q, U c q ccothc the clssicl tem fom the sphees displcement field is eliminted nd completely non-clssicl field is cquied. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

34 Clssicl theoy In this second polem non holonomic kind of loding is discussed, so the sphee is sujected to dil tensile stess P in the common cse of compessive pessue P<. The oundy condition is P P ˆ nd the displcement field is otined s follows Fig. 7 x-y plne section of solid sphee of dius sujecte to dil stess P P ˆ P ˆ ˆ C P C P u ˆ Gdient theoy Fig. 8 x-y plne section of solid sphee of dius, sujected to tesile stess P P ˆ nd sufce doule stesses R R ˆ In the cse of the gdient elstic sphee unde dil tensile stess P clssicl BC, while the sufce doule stesses t the oundy ed R non-clssicl BC, i.e. P P ˆ nd R R ˆ, the unique solution otined is the following u u ˆ C C Boundy ˆ Conditions C P C P [c [c [c c6]coshc [c 6]sinh c R c]coshc [c 6 ]sinh c c]coshc [c c 4 R 6 ]sinh c 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

35 The solution ove denotes tht when no doule stesses e pplied t the sphees sufce R =, while P stess is pplied, the sphee ehves exctly s the clssicl theoy descies, no mtte how smll it might e o how significnt it s micostuctue g length. The following figues pesent the sphee s displcement, stin nd doule stess dil distiution. In ode to nomlize this solution the Lme constnts hve e sustituted with thei equivlents in tems of the Young s modulus E E E nd the Poisson tio ν, i.e. nd. In the pessue polem the length g is eliminted fom this solution, nd the pmetes left in its nomlized fom e the / tio nd the Poisson constnt. Thus the next figues e plotted it tems of these pmetes. Fig. 9 Nomlized dil displcement dius, fo vious u P / E vesus nomlized dil distnce of the solid sphee of g tios. The clssicl oundy condition is P P ˆ nd the non-clssicl one R 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

36 Fig. Nomlized dil defomtion dius, fo vious e P / E vesus nomlized dil distnce of the solid sphee of g tios. The clssicl oundy condition is P P ˆ nd the non-clssicl one R When only doule stesses e pplied the sphee s ehvio is size dependent, i.e. the g pmete is not eliminted s expected - nd fte the nomliztion of the solution, thee pmetes e left, the tios /, /g nd ν. The displcement, stin nd doule foces fields e plotted with espect to the / nd /g tios, fo the vlues of the Poisson tio ν=. nd ν=., which e consideed to e typicl. It should e noticed tht the dil doule stesses ffect oth the displcement nd the stin field so when estining of pesciing eithe of them the fist polem doule stesses need to e pplied. Also, in the cse tht the poisons tio is ν=., fo get c=l/g vlues, these fields tend to e educed to the clssicl ones zeo fields, since this BC is not tkes into ccount it the clssicl elsticity. Howeve, fo othe ν vlues the solution is not educed to zeo field fo get c vlues, mening tht even lge scle sphees will ehve non clssiclly when popely loded. 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

37 Fig. Nomlized dil displcement dius, fo vious nd the nonclssicl one R R ˆ u R / E vesus nomlized dil distnce g tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of Fig. Nomlized dil defomtion dius, fo vious nd the nonclssicl one R R ˆ e R / E vesus nomlized dil distnce g tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

38 Fig. Nomlized doule stesses R vesus nomlized dil distnce, fo vious g one R R ˆ tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of dius nd the non-clssicl Fig. 4 Nomlized dil displcement dius, fo vious nd the nonclssicl one R R ˆ u R / E vesus nomlized dil distnce g tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

39 Fig. 5 Nomlized dil defomtion dius, fo vious nd the nonclssicl one R R ˆ e R / E vesus nomlized dil distnce g tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of Fig. 6 Nomlized doule stesses R vesus nomlized dil distnce, fo vious g one R R ˆ tios fo Poisson tio ν=.. The clssicl oundy condition is P of the solid sphee of dius nd the non-clssicl 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

40 Thee e two moe possile cses of loding comintions. The fist is the one tht the displcement nd the doule stesses t the sphees sufce e pescied, i.e. the BCs e u U ˆ nd R R ˆ, while the second one, is the cse in which the stess nd the stin t the sphee s oundy is pescied, i.e. P P ˆ nd, q u ˆ qˆ. These cses cn e ddessed though the ones ledy pesented y sustituting eithe the degee of feedom BC with its wok conjugte, o the genelized stess BC with its espective degee of feedom BC, using the following eltions: U q P R P R U q 4 4c c coshc c c 8 c c coshc c 6 sinhc sinhc 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

41 4.I..ii. The spheicl cvity Conside n infinite elstic D spce with spheicl cvity emedded in it t point which is the oigin of oth the spheicl nd the Ctesin coodinte system used. Since the spce exceeds infinitely due to themodynmics the displcement of the mteil point t infinity needs to e finite.. In the clssicl polem the displcement field tkes the fomu C ˆ. Since lim, the C C lim tem needs to e zeo. In the gdient polem, the displcement field is u uˆ C C C C4 4 ˆ. Both limits lim nd lim, thus the tems C nd C e oth zeo in this cse, fo finite displcement t infinity. Clssicl theoy The spheicl cvity is sujected to dil displcement U. The BC isu U ˆ C ˆ U ˆ C Fig. 7 Schemtic epesenttion of spheicl cvity x-y section U u U ˆ Fig. 8 x-y plne section of the spheicl cvitysujected to dil displcement U Gdient theoy u U In this cse, the infinite spce is mde of mteil with micostuctue nd the cvity is sujected to dil displcement U clssicl BC, while the noml displcement gdient t the oundy is q non-clssicl BC, i.e. u u Uˆ, q qˆ ˆ Fig. 9 x-y plne section of the spheicl cvity sujected to pescied dil displcement u U ˆ nd defomtion q q ˆ 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

42 This polems exct nd unique solution is found: u C ˆ, C U c c 4 4 e c U C 4 4 c q C e c q whee c is the tio of the spheicl cvity dius to the chcteistic mteil length g. Dividing this polem into two simple ones, one tht the noml displcement gdient vnishes on the oundy q = while the sphee is sujected to dil displcement U, nd one tht one tht the oundies displcement is zeo U = while the noml displcement gdient on the oundy is detemined q, ette undestnding of the cvities ehvio cn e otined. The following figues pesent this ehvio nomlized ppopitely in ech cse, fo vious vlues of the polems pmetes. Any othe polem cn e ddessed s line comintion of the two polems ove, sed on the supeposition pinciple. c Fig. Nomlized dil displcement u U vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is u U oundy condition is q ˆ nd the non clssicl 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

43 Fig. Nomlized dil defomtion e U / vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is clssicl oundy condition is q u U ˆ nd the non Fig. Nomlized doule stesses UE vesus nomlized dil distnce / of solid sphee of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is u Uˆ nd the non clssicl oundy condition is q 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

44 Fig. Nomlized dil displcement u q vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is u nd the non clssicl oundy condition is q q ˆ Fig. 4 Nomlized dil defomtion e q vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is u nd the non clssicl oundy condition is q q ˆ 44 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

45 Fig. 5 Nomlized doule stesses q E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius. The clssicl oundy condition is u nd the non clssicl oundy condition is q q ˆ When the defomtion t the cvity s oundy is detemined, the displcement distiution in the mteil is gete thn the one descied y the clssicl theoy. The doule foces e significnt only ne the oundy, nd s the c tio inceses thei nge deceses, while the stin t the oundy ppoches the clssicl one. Also, the ppliction of cetin comintions of BCs eithe the clssicl o the gdient pt of the solution cn e eliminted, exctly like the cse of the solid sphee. Hence, spheicl cvity, too, my ehve exctly like the clssicl theoy pedicts even one with significnt micostuctue, when sujected to the ight comintion of lods q=-u/. If the stin is pescied s U c q ccothc the clssicl pt of the solution is eliminted nd completely non-clssicl ehvio is descied fo the spheicl cvity. 45 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

46 Clssicl theoy When the spheicl cvity is sujected to dil compessive stess - pessue P, i.e. the BC is P P ˆ, the nlyticl displcement field cquied is: Fig. 6 x-y plne section of the spheicl cvity with dil pessue pplied t the oundy P P ˆ Fig. 7 x-y plne section of the spheicl cvity. The oundy conditions e R R ˆ P P ˆ nd P u 4 ˆ Gdient theoy When gdient elstic spheicl cvity is sujected to dil compessive stess P clssicl BC, while the sufce doule stesses t the oundy e R nonclssicl BC,i.e. P P ˆ nd R R ˆ, the displcement field tkes the following fom:. u ˆ C C ˆ u 4 4 Whee c=/g the spheicl cvity dius to g length tio nd C nd C 4 expessions e given elow: C C4 c 6 c c P 4 c 6 c 9 c 9 c c 6 c 6 R c 6 c 9 c 9 c c c 6 c e P 6 c 9 c 9 4 c c 6 c e R 6 c 9 c 9 Fist, the cse whee pessue P is pplied nd no doule stessesr=. 46 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

47 Fig. 8 Nomlized dil displcement u P / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius.,fo o poissons tio ν=. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R Fig. 9 Nomlized dil defomtion e P / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio ν=.. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R 47 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

48 Fig. Nomlized doule stesses P / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio ν=.. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R When nomlizing the fields in this polem, exctly like in the cse of the solid sphee, the Poisson s tio could not e eliminted, so diffeent plots e een given fo typicl poisson tio vlues. Aove the cvities ehvio is pesented in the cse tht ν=, nd the cse tht ν=, is pesented next. It should e noted tht the cvilty s ehvio is stiffe, when the mteil is gdient elstic, mening tht smlle displcements nd defomtions e nticipted, thn the ones the clssicl theoy descies. In the cse tht the c=/g tio is eltively smll, especilly when it is close to unity, this ehvio is considely stiffe. Futhemoe, doule stesses ppe to the mteil even when no doulestesses e pplied t the cvitys sufce, which e smlle in nge s the c tio inceses. 48 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

49 Fig. Nomlized dil displcement u P / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R Fig. Nomlized dil stin e P / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio ν=.. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

50 Fig. Nomlized doule stesses P vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio ν=.. The clssicl oundy condition is P P ˆ nd the non clssicl oundy condition is R A vey inteesting emk on the doule stesses is tht in the cse in which the poisson tio is not zeo the dil doule stesses t the cvities sufce e not zeo, no mtte the fct tht the totl doule stesses R e zeo. This cn e ttiuted to the fct tht due to the poisson tio the non dil defomtion nd its wok conjugte non dil doule stesses contiute to the totl doule stess, so imposing the sufce doule stesses e zeo does not men tht the dil ones tht e plotted ove e imposed to e zeo. Aout the diplcement nd defomtion fields, the sme osevtion s in the cse cn e mde. The following figues depict the cvity s ehvio when only doule stesses e pplied to its sufce nd no pessue. 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

51 Fig. 4 Nomlized dil displcement u R / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ Fig. 5 Nomlized dil defomtion u R / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

52 Fig. 6 Nomlized doule stesses / E vesus nomlized dil distnce / of spheicl cvity of dius R, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ As pictued ove, even to this non clssicl BC, the mteils ehvio is stiffe when the c=/g tio gets smlle, nd when it ppoches unity, this diffeence is significnt. Also, in the gph ove, the locl chcte of the doule stesses is depicted. It is shown tht the doule stesses e pesent nd significnt only ne the ody s oundy nd vnish gdully when moving wy fom it. Next, the cvity s ehvio is displyed when the poisson tio is ν=,, in which cse the sme conclusion cn e dwn. 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

53 Fig. 7 Nomlized dil displcement u R / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ Fig. 8 Nomlized dil defomtion e R / E vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

54 R Fig. 9 Nomlized dil doule stesses vesus nomlized dil distnce / of spheicl cvity of dius, fo vious tios /g, fo ny dius, fo mteils poisson tio n=.. The clssicl oundy condition is P nd the non clssicl oundy condition is R R ˆ Any othe cse of loding, i.e. the cses tht t the oundy one degee of feedom nd one genelized foce e pescied cn e ddessed y conveting the one BC to the othe kind using the eltions tht follow. The new polem will e one of the cses ove, nd hs ledy een studied. U q T 4T T T P T / R T T T T T c c 4c 4 c c 6 c 6 c c c 4 c 9 c 9 T 4 c 54 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

55 4.I..iii. The Spheicl Shell Conside spheicl shell consisting of solid sphee of dius nd concentic spheicl cvity of dius. Let S e the intenl nd S the oute sufce of the shell. Fig. 4 x-y plne section of the shell. The oundy conditions e u U ˆ nd u U ˆ u [ ] C Clssicl theoy In the fist polem oth sufces e sujected to dil displcements U nd U on S nd S espectively clssicl BCs, i.e u Uˆ C C U T U U C C U, u U C C U U U ˆ U U C M U Note tht in the equtions ove, the [] function denotes hoizontl vecto, with elements the two fundmentl solutions of clssicl elsticity spheicl polem. This symolism is used in ode to optimize the pesenttion of the esults. The mtix M is singul only when the two oundies coincide, i.e. the inne nd the oute dius of the shell tend to e the sme, which is the cse of the thin wlled spheicl shell. In this cse the singulity cn e eliminted y using fist ode Tylo expnsions of the displcement functions, i.e. T', T. Since i i i =>, so the displcement tkes the fom, which mens tht the solution collpses/cushes. u Fig. 4 Schemtic epesenttion of spheicl shell x-y section D 55 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

56 56 Gdient Theoy In this polem oth sufces e sujected to dil displcements U nd U on S nd S espectively clssicl BCs while the noml displcement gdient t ech oundy is pescied, espectively q nd q non-clssicl BCs, i.e. u ˆ U, u ˆ U u n u q ˆ ˆ q, u n u q ˆ ˆ q These BCs e tnslted to the following eqution: D M C 4 4 q U q U ' ' C C C C q U q U C C C C ' ' ' ' q ' q ' U ' ' U ' ' u C C C C u 4 T 4 This time, the ] [ function denotes hoizontl vecto, with elements the fou fundmentl solutions of the spheicl gdient polem. It cn e poved tht this solution is educed to the clssicl one when the micostuctue is insignificnt comped to the shells dimensions i.e. g / o /g, hence, the solutions ility of eing educed to the clssicl one when the scles e get is not lost in this moe complex polem. Fig. 4 x-y plne section of the shell. The oundy conditions e u ˆ U q ˆ q, u ˆ U nd q ˆ q Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

57 The mtix M is singul only when the inne nd the oute dius of the shell e vey close to one nothe, i.e. the cse of thin wlled spheicl shell. Let T=- e the shells thickness. The singulity cn e eliminted using second ode Tylo expnsions of the displcement functions T T '' / nd its deivtive. i i i' ' ' T'' T ''' /. i i i' The displcement field then tkes the following fom: u ' T U ' ' T U ' 4 T, T u=u = i ' i T ' ' ' ' '' q ' T q ' So the gdient solution cushes, too, exctly the clssicl one does. This my e intepeted physiclly s: on thin wlled spheicl shell oth oundies e pcticlly one thus diffeent displcement cnnot e imposed on them without the ody getting destoyed/filing., 57 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

58 Clssicl Theoy In this polem the sufce S is sujected to dil displcements U while on the sufce S tensile stess P is pplied, i.e. u u U ˆ, P P ˆ U P P P 4 U P 4 4 Fig. 4 x-y plne section of the shell. The oundy conditions e nd P P ˆ U P u U In the cse of thin wlled spheicl shell, of thickness T=- the polem doesn t ecome singul nd the solution gets vey simple, u u ˆ U ˆ, P ˆ, this thought doesn t men tht ny P displcement U cn e sujected y pplying ny stess P. The eltion etween them must e pefixed nd it is otined though polem 4. Thus, this is not polem of get pcticl inteest. Gdient Theoy ˆ Fig. 44 x-y plne section of the shell. The oundy conditions e u U ˆ q q ˆ, P P ˆ nd R R ˆ In the espective gdient elsticity polem the sufce S is sujected to dil displcements U clssicl BC while the noml displcement gdient on oundy is pedetemined q nonclssicl BC nd on the sufce S tensile stess P clssicl BC nd sufce doule stesses R non-clssicl BC e pplied, i.e. u U ˆ, P P ˆ q u q ˆ, R R ˆ ˆ The displcement field tkes the following fom: 58 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

59 u 4 T C C C C 4 u ' U P R P ' R ' q P R R R ' P When the micostuctue ecomes insignificnt i.e /g the solution is educed to the one otined using the clssicl theoy. This solution is neve singul not even in the cse tht, the cse of thin wlled spheicl shell. In tht cse the solution is simplified to: u u ˆ U ˆ, q ˆ, P ˆ, R ˆ q P This, too, is solution of no pcticl inteest, thought, one cn deduce tht the displcement nd its noml sufce gdient of thin wlled spheicl shell e ppoximtely unifom thoughout its ody. Also the extenl lod P needs to e the totl extenl lod fom oth sufces, nd this is esult of the smll shell s thickness, due to which we cnnot e pecise out whee exctly the extenl lods e pplied. The sme emk pplies on the sufce doule stesses. Once gin, the displcement nd its noml gdient cnnot e independent of the extenl lods P nd R, so this solution isn t vey useful, since it doesn t pesent this eltion. The next polem to e studied is the symmeticl of this one, mening tht the BCs pplied t ech oundy of the shell e pplied in the othe one in tht polem. R 59 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

60 u Fig. 45 x-y plne section of the shell. The oundy conditions e u U Clssicl Theoy In this polem on the sufce S compessive stess-pessue P is pplied, while the sufce S is sujected to dil displcement U, i.e. P u P ˆ nd U ˆ, P U P P 4 U P U P 4 4 In the cse of thin wlled spheicl shell, of thickness T=- the polem doesn t ecome singul nd the solution gets vey simple exctly like in polem. No new inteesting conclusion out this solution cn e dwn. u u ˆ U ˆ, P ˆ P R ˆ P Gdient theoy P ˆ, u U ˆ In this gdient polem on the sufce S compessive stess/pessue P clssicl BC nd sufce doule stesses R non-clssicl BC e pplied, while the sufce S is sujected to dil displcement U clssicl BC nd the noml displcement gdient on oundy is q BC, i.e. u u R ˆ, q q ˆ n ˆ P P ˆ nd Fig. 46 x-y plne section of the shell. The oundy conditions e P P ˆ, R R ˆ, u U ˆ nd q q ˆ Thus the displcement field is:, 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

61 R P P P T P U q R C R R C ' ' ' u u C P C 4 4 R ' nd when the micostuctue is insignificnt i.e. /g, it is educed to the field otined using the clssicl theoy. In the cse of thin wlled spheicl shell, of thickness T=- the polem doesn t ecome singul nd the solution gets vey simple, ut isn t vey useful, i.e. u u ˆ U ˆ, q ˆ, P ˆ, R ˆ, exctly like the pevious polem. This solution is vey simil to the solution of the second polem nd the pevious conclusions out the shell cn e dwn fom this cse too. The following polem is the one of the most inteest nd pcticl ppliction. q P 4Clssicl theoy R In this polem on the sufces S to compessive stess-pessue P is pplied, while the sufce S is sujected to tensile stess, i.e. Fig. 47 x-y plne section of the shell. The oundy conditions e P P ˆ P P ˆ nd P P ˆ, P P ˆ The displcement field is found to e the following: u P P P P P P P P P P 4 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

62 In the cse of thin wlled spheicl shell, of thickness T=-, the system ecomes singul. Any singulity cn e eliminted using fist ode Tylo expnsions of the functions of the extenl dius, i.e. P P TP' '. The displcement, defomtion nd cuvtue e: i i Fig. 48 x-y plne section of the shell. The i u P P T 4 u ' P P T u '' P P T 4 Gdient theoy P P ˆ, On the sufces S of the shell R ˆ, P P ˆ nd R R ˆ compessive stess-pessue P clssicl BC is pplied nd sufce doule stesses R non-clssicl BC, while the sufce S is sujected tensile stess P clssicl BC nd sufce doule stesses R non-clssicl BC, i.e. oundy conditions e R P P ˆ, P P ˆ R R ˆ, R R ˆ The solution tkes the following fom. u 4 T C C C C 4 u R P P R P P R P R P R R P R P R P R R P Note tht this solution too, when the micostuctue is insignificnt, i.e. /g is educed to the clssicl one. 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

63 6 In the cse of the thin wlled spheicl shell, of thickness T=-, this solution ecomes singul. The singulity is eliminted using fist ode Tylo expnsions of the functions of the extenl dius, i.e. ' ' TP P P i i i, ' ' TR R R i i i So the solution finlly tkes the following fom: ' ' ' ' 4 R R P P T R R R P P P P R R P u D B C A u T, T, This fom of the solution cn e deivted nd ny comintion of the deivtives of u cn e esily e otined. Tht isn t the cse with the following simplified foms of it, which thought e moe lucid. 5 6 / 5 6 / 4 4 P P g g T u 5 6 / 5 6 / R R g g T 5 6 / 5 6 / ' P P g g T u 5 6 / 5 6 / R R g g T 5 6 / ' ' P P g T u 5 6 / R R g T It comes with supise tht when the tio /g, oth the displcement nd the defomtion functions e educed to the espective Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

64 clssicl functions, while the highe ode deivtives of the displcement don t. In ode to exmine the effect of ech BC to the displcement nd defomtion fields of this solution, the sme pocedue s efoe is followed. The polem is divided into two simple, i one tht the totl pplied stess is non zeo while the sufce doule stesses t oth sufces e equl i.e. P P nd R R nd ii one tht the totl pplied stess is zeo while the sufce doule stesses t oth sufces en t equl i.e. P P while R R. Any polem cn e ddessed s supeposition of these two polems. It is noted tht fo smll c=/g vlues, the displcement of the shell is considely smlle thn the espective clssicl one nd get dil doule stesses e nticipted fo such vlues, with the getest ones ppeing in the mteils tht Poisson s tio is the getest. Fo get c vlues, oth the displcement nd defomtion fields e educed to the clssicl ones, while the doule stesses tend to disppe. Fig. 49 Nomlized dil displcement u T P P vesus the tio /g fo diffeent vlues of the poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R / E R ˆ with R R 64 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

65 Fig. 5 Nomlized dil defomtion e T P P / E vesus the tio /g fo diffeent vlues of the poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R R ˆ with R R Fig. 5 Nomlized doule stesses T P P vesus the tio /g fo diffeent vlues of the poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R R ˆ with R R 65 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

66 Fig. 5 Nomlized dil displcement u T R R / E vesus the tio /g fo diffeent vlues of the poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R R ˆ with P P Fig. 5 Nomlized dil defomtion e T R R E vesus the tio /g fo diffeent vlues of the / poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R R ˆ with P P 66 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

67 Fig. 54 Nomlized doule stesses T R R vesus the tio /g fo diffeent vlues of the poisons tio ν. The clssicl oundy conditions e P P ˆ nd P P ˆ nd the non clssicl ones R R ˆ nd R R ˆ with P P When dil doule stesses e pplied to the shell s oundy, dditionl displcements nd defomtions nd doule stesses e imposed. In this cse too, it should e noted, the clssicl ehvio fom the ule unde pessue P P cn e otined then the ight BCs e clssicl pplied, i.e. P P gdient P P clssicl nd R R P P clssicl. Hence, independent of the 4 micostuctue, in ode to otin the clssicl ehvio, oth gete thn the clssicl pessue needs to e pplied, nd cetin doule stesses. Consequently, the gdient elstic ule is considely stiffe. 67 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

68 4.I..iv. The doule-lye shell Hving ledy consideed the cse of thin wlled spheicl shell, the next step of this study is to conside the cse of composite thin wlled spheicl shell. The composite thin wlled spheicl shell is consideed to e of dius d consisting of two tngent spheicl shells of thicknesses T nd T espectively, ech one mde of mteils, with Lme s constnts μ,λ nd μ,λ nd chcteistic lengths g nd g espectively. Let S e the intenl nd S the oute sufce of the shell. In the clssicl theoy displcement continuity nd equiliium of the intefce =d dictte tht t oth its ends displcement nd the stess must e equl. In the gdient theoy, these conditions e not sufficient. Two moe need to e chosen. Following the woks of Weitsmn Weitsmn, 965 who ddesses the polem of the intefce though couple-stess theoy nd Yueqiu Li Li, et l., 5 who consides gdient elstic intefce, esides the two clssicl oundy conditions, the non clssicl ones e chosen to e the continuity of the displcement s noml gdient nd the sufce doule stesses t the intefce. These BCs epesent the continuity of the second degee of feedom of the gdient elstic shell nd its wok conjugte. These e consideed to e the BC tht llow stin enegy to e feely tnsmitted though the intefce, s Wetsmn Weitsmn, 965 notes in the espective theoy used in tht ppe. Clssicl theoy Gdient theoy u u, P P d d d d c c u u, P P d d q q, R R d d Also, the ow vectos depicted in the fist nd the second column of the following expessions e functions of the fist nd the second mteil s pmetes μ,λ, g nd μ,λ, g espectively. Note tht in the clssicl polems, [,,, ] 4 d d [, ] etc while in the gdient ones etc. while O is the zeo vecto with nd 4 in ech theoy elements espectively. In ode to otin the solution to ny 68 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

69 clssicl doule lyeed shell BC polem fou constnts need to e detemined two fo ech lye of the shell. These constnts e nmed using the following convention: C ij, i=, fo the fist nd second mteil espectively, j=, nd e multiplied with the j th fundmentl clssicl solution,, /. In the gdient polems, fou constnts e necessy fo ech lye, so the totl unknowns e eight. The sme convention is used fo nming them: C ij, i=, fo the fist nd second lye espectively, j=,,,4 nd e multiplied with the j th gdient fundmentl solution Clssicl theoy =d-t =d+t d Fig. 55 x-y plne section of the doule-lye shell. The oundy conditions e u U ˆ nd u U ˆ In this polem oth sufces e sujected to dil displcements U nd U on S nd S espectively, i.e. the BCs e: u Uˆ, u Uˆ Thin wlled d T' d C d T 'd C d d C Pd Pd C U U u C [ d] C C [ d] C u UT u d T UT T In the cse of i lyeed shell, the solution does not cush, s it hppens into the single lyeed one. The displcement field otined is pcticlly weighted men vlue of the displcements of the two oundies. The eson fo such diffeence is tht in this cse the intefce is mthemticlly modeled s two diffeent sufces. 69 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

70 7 Gdient theoy In the espective gdient polem, oth sufces e sujected to dil displcements U nd U on S nd S espectively clssicl BCs while the noml displcement gdient t ech oundy e pescied, espectively q nd q non-clssicl BC, i.e. u ˆ U, u ˆ U u n u q ˆ ˆ q u n u q ˆ ˆ q The oundy vlue polem tkes the djcent mtix eqution fom. Using second ode Tylo expnsions ound the dius d fo the functions of nd, i.e. d/ '' T 'd T d i i i i, T = d-. d / '' T 'd T d i i i i, T = -d, the following solution fo the shell s displcement is otined. ] [ T T T T T T g g T T T g T g U T U T g g T T U T λ μ g U T λ μ g T g T g u ] [ T T T T T T g g T T T g T g q T g q T g T g T g T T q U q U C C C C C C C C Rd Rd Pd Pd d ' d ' d d ' O O O ' O 4 4 d =d-t =d+t Fig. 56 x-y plne section of the doulelye shell. The oundy conditions e u ˆ U, q ˆ q, u ˆ U nd q ˆ q Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

71 7 Clssicl polem In this polem the sufce S is sujected to dil displcements U while on the sufce S tensile stess P is pplied, i.e. u ˆ U, P ˆ P Thin wlled Since the polem doesn t ecome singul fo ==c, nd its solution is u ˆ U ˆ u, Gdient theoy In the gdient polem, the sufce S is sujected to dil displcements U while the noml displcement gdient on oundy is q nd on the sufce S tensile stess P clssicl oundy conditions nd sufce doule stesses R non-clssicl oundy conditions e pplied, i.e. u ˆ U, P ˆ P n u q ˆ q, R ˆ R Consequently, the oundy condition polem tkes the following fom: Now, once gin, this polem even with ==d doesn t ecome singul, thus the ' ' ' 4 4 R P q U C C C C C C C C d R d R d P d P d d d d R O P O O O Fig. 58 x-y plne section of the doulelye shell. The oundy conditions e u ˆ U, q ˆ q, P ˆ P nd R ˆ R Fig. 57 x-y plne section of the doulelye shell. The oundy conditions e u ˆ U nd P ˆ P P U C C C C Pd d Pd Pd d d P U C C C C Pd Pd d d P O O d d =d-t =d-t =d+t =d+t Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

72 7 solution one otins is exctly the sme otined fo the single lye shell, i.e. u ˆ U ˆ u, nd lso q ˆ q, P ˆ P nd R ˆ R Clssicl theoy In this polem on the sufce S is pplied to compessive stess-pessue P while on the sufce S is sujected to displcement U, i.e. P ˆ P, u ˆ U The polem doesn t ecome singul fo ==d, nd its solution is u ˆ ˆ U u, while P ˆ P Gdient theoy In this polem on the sufce S compessive stess-pessue P clssicl BC nd sufce doule stesses R nonclssicl BC e pplied, while the sufce S is sujected to dil displcement U clssicl BC while the noml displcement gdient on oundy is q Non-clssicl BC, i.e. P ˆ P, u ˆ U R ˆ R, u q ˆ q ˆ Consequently, the oundy condition polem tkes the djcent fom: This solution even in the cse of thin wlled ' ' ' 4 4 q U R P C C C C C C C C d R d R d P d P d d d d O O O R O P Fig. 59 x-y plne section of the doulelye shell. The oundy conditions e P ˆ P nd u ˆ U Fig. 6 x-y plne section of the doule-lye shell. The oundy conditions e P ˆ P, R ˆ R, u ˆ U U P C C C C Pd Pd d d O O P d d =d-t =d-t =d+t =d+t Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

73 spheicl shell ==d is not singul nd the solution otined is exctly the sme otined y the single lye shell, i.e. u u ˆ U ˆ, nd lso q ˆ, P ˆ nd R ˆ. R q P =d-t =d+t Fig. 6 x-y plne section of the doulelye shell. The oundy conditions e P P ˆ nd P P ˆ d 4Clssicl theoy In this polem on the sufces S is sujected to compessive stesspessue P nd the sufce S is sujected tensile stess P, i.e. P P ˆ, u P ˆ Pd T P' d C Rd T R' d C d d C Pd Pd C C C u [ d] [ d] C C P P, u u c c 4 P P T T 4dGdient theoy =d-t =d+t Fig. 6 x-y plne section of the shell. The oundy conditions e P P ˆ, R R ˆ, P P ˆ nd R R ˆ d On the gdient elstic ilye shell s sufces S is pplied compessive stess-pessue P clssicl BC nd sufce doule stesses R non-clssicl BC, while the sufce S is sujected to tensile stess P clssicl BC nd sufce doule stesses R non-clssicl BC, i.e. P R P ˆ, P P ˆ R ˆ, R R ˆ 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

74 74 R P R P C C C C C C C C Rd Rd Pd Pd 'd 'd d d T R'd Rd O T P'd Pd O O T R'd Rd O T P'd Pd 4 4 Fo thin wlled doule-lye shell, using fist ode Tylo expnsions ound d, this BC polem tkes the fom ove, which finlly leds to the next solution 5 6 d / g 5 6 d / g T T 5 6 d / g T 5 6 d / g T 5 6 d / 4g T 5 6 d / 4g T P P 4 d u 5 6 d / g 5 6 d / g T T 5 6 d / g T 5 6 d / g T 5 6 d / g T 5 6 d / g T R R d Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

75 u d ' P P T g / c 6 5 T g / c 6 5 T g / d 6 5 T g / d 6 5 g / d 6 5 TT g / d 6 5 d R R T g / d 6 5 T g / d 6 5 T g / d 6 5 T g / d 6 5 g / d 6 5 TT g / d 6 5 Inteesting popety of this solution is tht if ny of the of T nd T is zeo o the mteil is the sme in oth lyes, thus the shell isn t composite, then the solution degenetes is educed to the one otin fo single lye shell. If oth mteils micostuctues e ssumed to e insignificnt g nd g, this solution doesn t tke the fom of the clssicl solution. Howeve, this does not men tht it is wong. In the litetue, cses cn e found tht the solutions otined, when the length pmete g is ssumed zeo, e not educed to the clssicl espective polem solutions. In this pticul cse this esult might lso e ttiuted to the fct tht only the lowest ode thickness odes of the solution e kept, nd the othes e ssumed to e insignificnt comined with the ssumption tht ws mde tht T / d g / d, T, g, which is the cse of the micosphees studied y Glynos & outsos, 9 75 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

76 4.I.4. Expeimentl dt tetment Glynos,E. nd outsos,v. Glynos & outsos, 9 investigted expeimentlly the mechnicl ehvio of mico scle spheicl shells, they used the tem MS Mico Sphee, using nnocompession testing with tipless cntileves. The MSs used consisted of thin shell of stiff stuctul polyme polylctide, suounded y coss linked lium oute lye nd incpsulted nitogen gs t tmospheic pessue. Fig. 6 Young's modulus of MS stuctul shell vesus MS shell thickness Glynos & outsos, 9 The MS s thickness to dius tio ws consideed to e constnt T/=.5x - nd the polyme s Poisson s tio ws ssumed to e.4. The clssicl theoy of elsticity clcultes the MS s stiffness, k, unde this type of loding using the following eqution. k 4 T E 5.7x 4 E Tht indictes tht igge MS hve e stiffe thn smlle ones, then the thickness to dius tio is constnt. The expeiment s findings, though, indicte tht smll MSs e stiffe thn lge, which comes in contdiction with the clssicl theoy. Relxing the ssumption fo constnt Young s modulus, Glynos otined the figue 6 pesenting the eltion fo the Young s modulus to the MS s thickness. Using the spheicl shell model tht ws developed, it is ttempted to intepet those esults. Thee isn t enough dt ville to ddess the polem s multiple lye shell s it is, so it will e teted s single lyeed one. Also the MSs loding is not the one studied in the model. Howeve, in hope tht the sme stiffening effects might e oseved in oth loding cses, this ttempt of fitting the model to the expeimentl esults is eing mde. 76 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

77 The following eltion cn e otined fo its mteils Young s modulus s function of the shell s dius, it s mteils chcteistic length g nd Poisson tio ν: g / g / / g E E E,E o eing the Young s modulus computed using the clssicl theoy fo the MS in the cse tht it is loded only y pessue. Sustituting the Poisson tioν=.4 nd the thickness to dius tio T/=.5x -,, the following eqution is otined Fig. 64 Expeimentl dt comped to the Eo=.6GP, g=6nm stiffness cuve descied y the model of thin wlled spheicl shell E E / g E T / g This does indicte tht fo smlle shells the Young s modulus does incese. Howeve, in the model, sole the eduction of the shells size cnnot induce such dmtic incese of its stiffness, s the one descied y the expeiment s esult. This does not men necessily tht the model fils to cptue these stiffening effects. The genel stiffening tend is cptued, nd thee cn e ette fitting of the model to the expeimentl esults if the doule stesses tken into ccount. Howeve, mny ssumptions hve ledy een mde, so it ws chosen not to poceed to this ttempt without moe dt. In figue 66 is depicted, esides the expeiments esult, cuve fo. E =.6 while the chcteistic length g is 6 nm, which is one of the cuves tht pedict the getest incese fo the Young s modulus 77 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

78 4.I.5. Conclusions Solid Sphee The gdient theoy does cptue size effects in the sphees ehvio. When the sufce noml displcement gdient is estined, the sphee hs stiffe ehvio. When only clssicl lods e pplied to the sphee, its esponse is clssicl, no mtte how significnt the micostuctue might e. Doule stesses need to e pplied fo size effects to ppe nd stiffe esponses to e otined. When the micostuctue is smll comped to the sphees dius, the effect of the non clssicl BCs is vey smll nd the sphees ehvio is educed to the clssicl one. Spheicl Cvity Significntly stiffe thn clssicl ehvio even when no doule stesses e pplied, fo get g length to dius tios. Significnt effect of the poisson s tio to its ehvio The doule stesses e significnt only close to the oundy The solution is lwys educed to the clssicl one when the micostuctue is smll. Spheicl Shell Only the polem tht the extenl lods e known povides useful esults. Its ehvio is lwys stiffe thn the clssicl, even when no doule stesses e pplied. When the micostuctue is smll, the displcement nd stin fields e educed to the clssicl ones, while the cuvtue field is not. The Poisson s tio ffects the s ehvio, s it does in the clssicl cse 78 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

79 Doule Lye Spheicl Shell Only the polem tht the extenl lods e known povides useful esults. In this cse too mny pmetes e consideed, which mkes the extction of conclusions difficult. Genelly, stiffe thn clssicl ehvio is otined fo the shell When the two mteil pmetes e the sme, the solution is educed to the simple spheicl shell gdient solution When oth mteils length pmetes e smll comped to the shell s dius, the solution otined is not the clssicl one 79 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

80 4.II. PART II: D PROBLEMS & TRUSSES 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

81 4.II.. Tuss modeling though gdient elsticity. Tusses e vey common nd useful nd common kind of stuctue. Thee e sevel el life pplictions fo tusses. Fo sttes, in stuctul mechnics, they e commonly used fo idges nd oofs. In vscul sugey, the STENTs used to incese lood flow in es locked y plque e thee dimensionl tusses. In mteil science, in esech fo vey light mteils, micoscle D lttice ws developed, with the intent to e possily used s stuctul mteil. Tusses, lso, hve moe theoeticl use. Fo exmple, they e vey simple wy of modeling complex odies unde vious kinds of loding, y discetizing continuum in netwok of tuss elements in ode to solve mny elsticity polems. Fom geometicl pespective, the simplest finite elements is the one dimensionl element, which, in the clssicl elsticity fmewok, cn e sujected only to xil lods, eithe tensile o compessive. These elements, though, cn e used not only in one ut lso in two o thee sptil dimensions y tnsfoming thei locl coodinte system to glol one. A tuss elements cn model stisfctoily even complex geometies nd give good stess nd displcement esults, using vey few nd simple equtions, especilly when used in ppopite polems, they give vey cost effective solution. Tuss polems e mostly solved the stiffness method. This method stts y otining the tuss elements stiffness mtices, ssemling them in glol coodinte system nd otining the glol stiffness mtix of the stuctue. Then, the glol lod vecto is defined nd it s eltion to the displcement vecto is otined fo the whole stuctue nd the chosen oundy conditions BCs e imposed. 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

82 The following pt of this wok ddesses the ehvio of gdient elstic, s stt, in ode to conclude out the ovell ehvio of composite gdient stuctue- tuss. Fist, the equiliium eqution nd the coesponding oundy conditions of gdient elstic unde sttic loding e deived using the vition pinciple. Second, vious BC polems of the e ddessed in ode to otin good undestnding of the effect tht ech BC hs on the s displcement, stin nd doule stesses fields, nd the effect of the sufce enegy pmete l, which is not consideed pmete in the genel stin gdient elsticity theoy y Aifntis nd is not tken into considetion in the genel D polem pesented in the fist pt of this wok. Thid, the function- ehvio of elstic nd holonomic node is discussed in the fmewok of gdient elsticity in D polems nd the non clssicl BCs tht should e pplied e found. The D stiffness mtix of the element is otined nd seies of D polems e solved in oth thei gdient nd clssicl cses using the stiffness method nd comped to thei espective nlyticl solutions, in ode to cetify tht the gdient elements cn e used s D FEMs. Finlly, the D tuss polem is ddessed diectly. The function of node in dimensions is discussed. The Glol D element stiffness mtix is otined nd some D exmples e pesented. Lst, nothe model fo D node ehvio is poposed, nd n exmple of its ppliction is given. 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

83 4.II.. The single ehvio 4.II..i. Equiliium eqution nd oundy conditions in D polems The polem of the gdient elstic in unixil loding is the simplest one dimensionl polem tht cn e ddessed though fist stin gdient elsticity. It hs een ddessed sevel times in the pst yes, not only s sttic polem, s it will e hee, ut lso s dynmic one. This is the cse of the woks y Aifntis Altn, et l., 996, Tsepoy Tsepou, et l.,, Polizzotto Polizzotto,, Mustph Mustph & Run, 5. This polem hs lso een ddessed though othe non locl elsticity theoies, s shown in Auo Angel Pisno nd E. Benvenuti. Following, ssuming mteil tht oeys the line gdient elsticity theoy of Aifntis, the equiliium eqution nd the coesponding oundy conditions e pesented s otined y Tsepoy y ppliction of the vition pinciple. Conside stight pismtic slende, with constnt coss sectionl nd elstic popeties long its xis sujected to unixil tensile stess σ x x esulting to n xil displcement u=u x x, long its longitudinl xis x. The displcements u y nd u z e ssumed to e zeo, thus the stin du field e x e dx Following the one dimensionl gdient elsticity theoy with sufce enegy y Aifntis the stin enegy U of the in xil loding is defined s U e edv A e e V L de Whee A is the e of coss section, e is the stins gdient, dx nd, denote the Cuchy stesses nd doule stesses, espectively nd dx e given y the following constitutive eltions 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

84 u lu lu g u Whee the constnts l, g epesent,mteil lengths elted to the sufce nd volumetic elstic stin enegy espectively with l g, E is the Young modulus nd d dx, The s stin enegy nd its vition e given elow U L L EA u' U EA u" g g u iv u" u dx lu' u" dx L EA u' g u''' u L EA g u" lu' u' The vition of the wok done y extenl clssicl foces q nd P nd non-clssicl doule foces R ed L W qu dx L Pu L Ru' The vition eqution U W tkes the following fom U L EA u" g u iv q u dx L P EA u' g u''' u L R EA g u" lu' u' The eqution ove implies tht ech tem must e equl to zeo, thus the govening eqution of the nd its oundy conditions BCs e extcted. u"x g u iv And t ech end x qx EA i. Eithe the displcement is known u u o the pplied xil foce is known u' g u''' P EA 84 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

85 ii. Eithe the stin t ech end is known u' u' o the pplied xil doule foces e known lu' g u" Whee the dshed supescipt denotes pescied vlues. R EA The O.D.s genel solution is u u hom u ptil C i i u ptil g x t x t u Cx C C coshx /g C4 sinhx /g sinh qt dt EA g g x The foce vecto is: P P xˆ, P [ ] C i i And the doule foce vecto is R C R xˆ i i, R [ coshx /g l sinhx /g g sinhx /g l coshx /g] g 85 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

86 4.II..ii. Boundy Condition Polems In the following section the effect in the ehvio of gdient elstic of sevel oundy conditions will e pesented in extend. The ehvio pedicted will e comped to the one pedicted y the espective clssicl elstic. The effect of the ext sufce enegy pmete l will e discussed septely. Conside of length L, modulus of elsticity E nd coss-sectionl, two moe e A. Besides the oiginl coodinte system x,l coodintes systems e intoduced in ode to simplify the esults nd optimize thei disply. The fist is the nomlized xil coodinte system x / L,, nd the second one is nomlized coodinte system with oigin t the end of the i.e. x L * nd diection opposite to pevious systems, * L x/ L, *, polems the non dimensionl pmetes. In the following gdient c L/ g nd l/ g e used fo the sme eson.. Fig. Fig. 65 Repesenttion of the coodinte systems x, ξ, ξ* tht e eing used it ode to simplify the following polems nd used the symmety of the solutions The fixed-fixed Clssicl theoy In fixed-fixed s ends e sujected to 68 Fixed-Fixed sujected to xil displcements U nd U espectively, U xˆ u U xˆ nd the BCs e u Und u U. The following displcement field is vey simple, well known nd esily otined. displcements u u U U U U* U 86 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

87 Oviously the cse in which the displcement of one end of the is estined nd the othe end is sujected to n xil displcement is otined y zeoing the espective displcement. Gdient theoy Fo the gdient polem the is mde of mteil with micostuctue. The clssicl BCs e the sme s in the clssicl theoy. Two sets of non-clssicl BCs need to e studied next. In the fist su cse the stin t oth ends is pescied, ε nd ε espectively, i.e. the BCs e u U, u U, u' nd u'. This is symmeticl polem, mening tht the BCs, u' t oth sides e of the sme type. Thus the displcement field is expected to e symmeticl too. This mens tht the effect of ech BC on the field must e symmeticl to the effect of its symmeticl BC, o moe mthemticlly, the displcement field due to one BC while the othes e zeo cn e otined y sustituting the ξ pmete with ξ*=- ξ symm, nd the BC i with BC i i.e Fig. 69 Fixed-Fixed. The s ends displcements nd stins e pescied u U xˆ, u U xˆ nd u' If u, u t BC u, symm u t * BC BCi i i BCi i symm i The displcement field of this polem tkes the following fom c sinh [ ] sinhc / u U tnhc / U c tnhc / * tnhc / [ c sinh * *] sinhc / c tnhc / c c cosh cosh * coshc coshc * coshc / coshc* *coshc coshc / L L sinhc c tnhc / c c tnhc / sinhc c tnhc / c c tnhc / 87 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

88 It is eminded tht the pmete c is non dimentionl, nd defined s c=l/g in the polems. As expected the solution is symmeticl thus the effect of only one of the symmeticl BCs - solutions will e discussed. The λ pmete does not effect the s displcement o stin in this cse. This fct cn e ttiuted to the use of only holonomic BCs, no dynmic ones. In ode to fully dess evey polem possile, the solution is divided in two simple ones, s done in the pevious pt of this wok. In the fist, the is fixed t one end while the othe is sujected to n xil displcement U nd the stin t oth ends is estined, i.e. u, u nd u' u' U. The nomlized displcement nd the stin vesus the nomlized xil distnce of the e given elow. Also, the nomlized doule stesses distiutions, fo λ= nd in the cse tht c=4, fo diffeent λ vlues e pesented. Fig. 7 Nomlized xil displcement u / U vesus nomlized xil distnce x / L of Fixed-Fixed. The clssicl BCs e u, u U, nd the non clssicl ones u', u' 88 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

89 Fig. 7 Nomlized xil stin u' /U / L vesus nomlized xil distnce x / L of Fixed- Fixed. The clssicl BCs e u, u U, nd the non clssicl ones u', u' Fig. 7 Nomlized xil doule stesses /U E vesus nomlized xil distnce xxx x / L of Fixed-Fixed. The clssicl BCs e u, u U, nd the non clssicl ones u', u' 89 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

90 Fig. 7 Nomlized xil doule stesses /U E vesus nomlized xil distnce xxx x / L of Fixed-Fixed, fo vious λ vlues, nd c=l/g=4. The clssicl BCs e u, u U, nd the non clssicl ones u', u' The following conclusions cn e dwn. Both the displcement nd the stin fields depend on the L/g tio, nd s this tio inceses the s ehvio is educed to the clssicl one. Due to the holonomic non-clssicl BC s, the stin t the s ends is zeo egdless of how get the c pmete might e, ut the devition fom the clssicl solution deceses significntly s the c pmete tkes get vlues nd only close to the s ends the devition fom the clssicl solution is significnt. In othe wods, s gets smlle, gete pt of its ody is ffected y the end effects/non clssicl BCs nd its ehvio devites fom the clssicl one, oth displcement-wise nd stin-wise. The doule stesses μ xxx e stongly dependent to oth the c nd λ tios. Incesing the c tio induces get eduction to oth the getest doule-stess vlue, nd the nge of the s length tht these stesses e significnt. Gete λ vlues sufce stin enegy esult to gete 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

91 doule stesses in the gete pt of the, nd t the sme time eliminte the ntisymmety fom the μ xx distiution. The λ pmete does not ffect the displcement o the stin field in this polem, since ll degees of feedom e confined. It should e undelined though tht the pplied BCs - u = - e vey invsive in ode to otin vey non homogenous ehvio fom the. Howeve, mthemticlly, y pplying othe stins t the s ends, i.e u =U -U /L, the solution otined coincides with the clssicl one no mtte the vlue of the c pmete. The lst conclusion is vlidted when studying the s esponse to n pplied stin when oth its ends e fixed, thus the BCs eing u, u, u' nd u'. The following figues descie this ehvio fo vious L/g tios nd vious λ pmete vlues in the cse tht c=4, since, when the λ pmete is inseted to the solution, too mny pmete e pesent nd it is impossile to plot one single nomlized solution fo evey c nd λ vlue. Fig. 74 Nomlized xil displcement u / L vesus nomlized xil distnce x / L fo Fixed-Fixed. The clssicl BCs e u, u nd the non clssicl ones u', u' 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

92 Fig. 75 Nomlized xil stin u' / vesus nomlized xil distnce x / L of Fixed-Fixed. The clssicl BCs e u, u, nd the non clssicl ones u', u' Fig. 76 Nomlized xil doule stesses / EL vesus nomlized xil distnce x / L xxx Fixed-Fixed. The clssicl BCs e u, u, nd the non clssicl ones u', u' 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

93 Fig. 77 Nomlized xil doule stesses / EL vesus nomlized xil distnce x / L, xxx fo c=4 nd vious λ vlues. The clssicl BCs e u, u, nd the non clssicl ones u', u' Pesciing the stin t one end of the, while oth ends displcements e estined esults to non unifom, non symmeticl displcement, stin nd doule foces fields. Fo get c vlues these fields tend to get zeoed in the ette pt of the, nd diffe only t its ends, so the effect of the non clssicl BC to the is educed. In ode to pply these BCs, when the sufce elstic stin enegy is significnt λ gete doule foces need to e pplied, ut lwys thei mximum vlues ppe t the s ends. 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

94 Fig. 66 Fixed-Fixed. Sujected to xil displcments u U xˆ nd u U xˆ while the doule foces t its ends e pescied R R nd R R Next, is investigted the ehvio of fixed-fixed, whose ends stin is not diectly pescied, ut type R BCs e pplied i.e. the doule foces t the ends of the e pescied. The BCs e u U nd u U nd R R nd R R. This, too, is symmeticl polem. The displcement field tkes the following fom in the cse whee the l pmete is zeo R sinhc R sinhc* U U* * EA sinhc EA sinhc u The fist pt of this solution is the sme s the clssicl one. This mens tht the s ehvio diffes to its clssicl ehvio only if doule foces R e pplied t its ends. In othe cse, its esponse is clssicl egdless of how smll the o how significnt how get the c=l/g tio the micostuctue my e. Hence, fo non-clssicl ehvio fom the, eithe doule foces R e pplied t its ends, o thei defomtion is pescied nd diffeent to the stin of the espective clssicl solution. Pescied end stin, of couse, mens tht doule foces eing pplied, whose vlue depend on the pescied end displcements nd stins, s well s on mteil pmetes s will e discussed. Polizzoto Polizzotto, mde tht osevtion too, nd concluded tht the wy to otin clssicl ehvio fom gdient elstic the non-clssicl BCs need to e u ξ==u ξ==. This is not exctly ccute since it ppes to e n ity BC, moe of mthemticl tick in ode to otin line solution. Howeve, posteioi cn e sid tht this BC descies the condition of zeoing the doule foces t the ends of the, in cse of mteil of no sufce stin enegy λ=, which is the model Polizzoto used. Since the fist pt of the solution is the sme s the clssicl one, following, only the effect of doule foces t the s ends will e pesented. 94 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

95 Fig. 79 Nomlized xil displcement u /R / EA vesus nomlized xil distnce x / L of Fixed-Fixed.. The clssicl BCs e u, u, nd the non clssicl ones R R, R Fig. 8 Nomlized xil stin u' /R / LEA vesus nomlized xil distnce x / L of Fixed- Fixed. The clssicl BCs e u, u, nd the non clssicl ones R R, R 95 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

96 Fig. 8 Nomlized xil doule stesses /R / A vesus nomlized xil distnce x / L of xxx Fixed-Fixed. The clssicl BCs e u, u, nd the non clssicl ones R R, R In the gphs ove the displcements do not ppoch the clssicl solution s the c=l/g pmete inceses. This ehvio is ttiuted the estiction the s ends displcement. As will e shown futhe on, the ppliction of doule foces t fee s end ffects its length. The elongtes when the doule foces diection is wy fom the s cente, nd it shotens in the opposite cse. The R doule foces diection is towds the cente of the so the tends to get shote. By estining the s ends movement, pcticlly n opposite displcement to the one tht esulted fom the ppliction of the doule foces, is pplied, which is hidden clssicl kind of lod, esponsile fo this seemingly odd, non clssicl ehvio. In the cse of R whose diection is outwds of the, the doule foces effect elongtes the, nd the confined displcement of its ends imposes hidden clssicl set of BCs, too. As esult, fo get c pmete vlues, the displcement field is not educed to zeo field, which ws oiginlly thought to e the espective clssicl polem s solution. 96 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

97 When the λ pmete is significnt the ehvio of the devites dsticlly fom the clssicl one, even in the cse whee only clssicl BCs e pplied. The displcement field tkes the following fom: u u u u, u eing the displcement field in the cse tht λ=, descied ove,, c c tnhc / nd c c tnhc / u u cosh c / U coshc / U R R EA EA coshc R coshc* R c tnhc / c tnhc / * coshc EA coshc EA sinh tnhc / sinhc / sinhc sinhc c / sinh * R EA U tnhc / * c tnhc / c tnhc / c/ sinhc / sinhc* R * sinhc EA U Fig. 8 Nomlized xil displsement u /U cont, u /U dshed, L / g 4 distnce x / L of. Fixed-Fixed, fo vious λ vlues when c=l/g=4. vesus nomlized xil 97 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

98 Fig. 8 Nomlized xil stin u ' /U / L continuous, u ' /U / L dshed vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues Fig. 867 Nomlized xil doule stesses / U E continuous, / U E xxx xxx dshed vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues 98 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

99 Fig. 85 Nomlized xil displcement u /R / EA continuous, u /R / EA dshed vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues Fig. 86 Nomlized xil stin u ' /R / EAL continuous, u ' /R / EAL dshed vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues 99 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

100 Fig. 87 Nomlized xil doule stesses /R / EAL continuous, /R / EAL xxx xxx dshed vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues The effect of the λ pmete on the s ehvio is displyed in the gphs ove, fo of c vlue of fou c=4. Since, the λ pmete elimintes ny symmety fom the solution, the effect of BC is diffeent on ech end of the. In ech of the gphs ove, two cses e ddessed. In the fist thee, the effect of n imposed displcement to ech s end pesented. In the lst thee, the effect of pplied doule foces t ech s end while no doule foces e pplied to the othe is pesented. The λ pmete, i.e. the pesence of sufce enegy ffects stongly the displcement nd defomtion fields of. When only clssicl BCs e pplied, the sufce enegy effect on these fields hs level of symmety. Howeve, when non clssicl BCs e pplied, thei effect is vey stong nd ll symmety is eliminted fom those fields. The pesence of sufce enegy in this fom, though, is ely tken into ccount since it ely is oseved, nd, lso, following the odolemis odolemis, et l., nlogy, seems to e linked with nti symmetic popeties nd ehvios tht until ecently wee not commonly used. In this cse, too, in which sufce enegy is pesent nd the s ehvio gets vey complicted, clssicl ehvio cn e nticipted when the doule foces pplied e R = R = EA l U -U / L. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

101 Fig. 88 Fixed-Fee. The BCs e u U xˆ, P P xˆ Fig. 89 Fixed-Fixed. The BCs e u U xˆ u U xˆ, Clssicl theoy In the cse of fixed-fee, one end of the is sujected to n xil displcement U while the othe one is fee nd sujected to tensile foce P, thus the BCs cn e P P. The solution to this polem expessed s u U nd tht follows solution is vey simple, well known nd esily otined. u U P Gdient theoy In the gdient cse, the is mde of mteil with micostuctue. The clssicl BCs e the sme s efoe. Two sets of non-clssicl BCs e studied next. In the fist su cse, the stin of the oth ends of the is pescied, ε nd ε espectively. The BCs e u U nd P P nd u' nd u'. This is not symmeticl polem in espect to the BCs, so the displcement field is lso not expected to e symmeticl. The displcement field tkes the following fom: c sinh P tnhc / Uo EA c sinhc / u coshc coshc coshc* L csinhc csinhc L The field indeed is not symmetic one, so the effect of ech BC will e pesented septely. Fist, unde tension with estined ends stin is consideed. As shown in the following figues, the ehvio of the is stiffe thn the clssicl ehvio. Howeve, fo get c vlues, the solution ppoches the clssicl one, nd only t the ends of the the fields devite fom thei clssicl fom. The doule stesses e loclized only ne the ends nd thei vlues e smlle, fo gete c vlues. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

102 Fig. 9 Nomlized xil displcement u / PL / EA vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, P P, nd the non clssicl ones u', u' Fig. 9 Nomlized xil stin u' / P EA vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, P P, nd the non clssicl ones u', u' Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

103 Fig. 9 Nomlized xil doule stesses /P L / A vesus nomlized xil distnce x / L of Fixed-Fee xxx. The clssicl BCs e u, P P, nd the non clssicl ones u', u' Fig. 9 Nomlized doule stesses /P L / A vesus nomlized xil distnce x / L of Fixed-Fee, xxx fo c=4 nd vius λ vlues. The clssicl BCs e u, P P, nd the non clssicl ones u', u' Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

104 Fig. 94 Nomlized xil displcement u / L vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' Fig. 95 Nomlized xil displcement u' / vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

105 Fig. 96 Nomlized xil doule stesses / EL xxx vesus nomlized xil distnce L x / of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' Fig. 97 Nomlized xil doule stesses / EL xxx vesus nomlized xil distnce L x / of Fixed-Fee, fo c=l/g=4 nd vious λ vlues 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

106 Fig. 98 Nomlized xil displcement u / L vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' Fig. 99 Nomlized xil stin u' / vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

107 Fig. Nomlized xil doule stesses / EL xxx vesus nomlized xil distnce x / L of Fixed-Fee. The clssicl BCs e u, u, nd the non clssicl ones u', u' Fig. Nomlized xil doule stesses / EL xxx vesus nomlized xil distnce x / L of Fixed-Fee,fo c=l/g=4 nd vious λ vlues 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

108 As shown in the figues ove, in ode to estin the stins t the s ends doule stesses, diectly elted to the pplied foce P, e pplied t the s ends nd thei vlue is independent of the λ pmete, which though, modifies the doule stess distiution long the. In the cse whee the stin t the ends is detemined s P/EA -the vlue of the s ends in the clssicl polem- the solution is educed to the clssicl one, independent of the c tio. Thus, the s ehvio isn t lwys stiffe thn the one pedicted in clssicl theoy, it cn e the sme s the clssicl one, o eve less stiff povided tht the ight BCs e pplied. Widely in the litetue the ssumption of estined end stins is used s non clssicl BC. Tht indeed dicttes stiffe ehvio fo the s it is shown in the gphs ove. This isn t the cse thought if u >P/EA, in which the s displcement is gete thn the one of the clssicl cse. 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

109 Anothe fixed-fee is studied. It is the cse tht one end of the is sujected to n xil displcement U o, while tensile foce P is pplied to its othe end nd the doule foces R nd R t oth ends of the e known, i.e. the BCs e u U nd P P nd R R nd R. Fist, the cse whee the sufce stin enegy is R insignificnt/pmete λ=, will e discussed nd lte the wy the λ pmete ffects the displcement field will e pesented. The displcement field tkes the following fom, when λ=: P L R sinhc* R sinhc u U EA EA sinhc EA sinhc The function ove shows tht when no doule foces e pplied t the ends of the, the s ehvio is independent fo the g pmete, i.e. no scle effects e pesent. Next, is displyed the effect of the doule foces to the s ehvio, division of the solution tht cn e done due to the supeposition pinciple. It should e noted tht the doule foces on ech end of the ffect in diffeent wy its ehvio, i.e. they e not symmetic. Fig. Nomlized xil displcement u /R / EA vesus nomlized xil distnce x / L of Fixed-fee. The clssicl BCs e u, P, nd the non clssicl ones R R, R 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

110 Fig. Nomlized xil stin u' /R / LEA vesus nomlized xil distnce x / L of Fixed-fee. The clssicl BCs e u, P, nd the non clssicl ones R R, R Fig. 4 Nomlized xil doule stesses /R / A vesus nomlized xil distnce x / L of Fixedfee. The clssicl BCs e u, P, nd the non clssicl ones R R, R xxx Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

111 As in the cse of confined ends displcements, in this cse too when doule foces e pplied to n end of the its displcement field cnnot e educed to the clssicl one when the L/g tion inceses. Thus if le to pply such doule foces to the fixed s end, its length chnges, even when no xil foce is pplied. Note, though, tht only the pt of the which is ne the end is ffected y the doule foces. The est of the my e displced, ut no significnt stin is nticipted. Hence, the vition of the s length is ttiuted only to the defomtion of the ffected y the doule foces pt of the. Next, the doule foces effect on the fee end of the is pesented. The min diffeence of this cse to the one ove is tht the displcement nd stin fields e educed to the clssicl ones fo get c vlues. The ehvio of the, though, is vey simil in the two cses. They diffe due to hidden imposed displcement in the fist cse. Fig. 5 Nomlized xil displcement u /R / EA vesus nomlized xil distnce x / L of Fixed-fee. The clssicl BCs e u, P, nd the non clssicl ones R, R R Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

112 Fig. 6 Nomlized xil stin u' /R / LEA vesus nomlized xil distnce x / L of Fixed-fee. The clssicl BCs e u, P, nd the non clssicl ones R, R R Fig. 7 Nomlized xil doule stesses /R / A vesus nomlized xil distnce x / L of Fixed-fee xxx. The clssicl BCs e u, P, nd the non clssicl ones R, R R Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

113 In the cse of non zeo sufce stin enegy λ, the displcement field is u u u, whee u λ= is the displcement field of with λ= descied ove nd the u λ field is given elow. P L R sinhc* R sinhc u U EA EA sinhc EA sinhc P L cothc coshc* coshc u EA c coshc R EA R EA sinhc* sinhc sinhc sinhc c cosh c coshc / coshc* cothc coshc sinhc c tnh sinhc Due to the mny pmetes in this polem the effect of the l pmete is displyed in the cse whee c=l/g=4 Fig. 8 Nomlized xil displcement u /P L / EA vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P P, nd the non clssicl ones R, R Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

114 Fig. 9 Nomlized xil stin u' /P / EA vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P P, nd the non clssicl ones R, R Fig. Nomlized xil doule stesses /P L / A vesus nomlized xil distnce x / L of Fixedxxx Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P P, nd the non clssicl ones R, R 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

115 As in the cse of the fixed fixed with doule foces pplied to its ends, the sufce enegy esults to non unifom / non clssicl ehvio fom the even when only clssicl BCs e pplied. Notice tht gete displcements nd stins ppe thoughout the nd lso doule stesses ppe, even though doule foces e not pplied t the s ends. In the following figues the effect of doule foces t s ends is pesented, when λ. When pplying doule foces to the fixed end of the, the is shotened, nd the gete the λ pmete, the moe the is shotened. When pplying doule foces to the fee end of the, it elongtes. In contst with the fixed end, fo gete λ vlues, the elongtes less. Thus the λ pmete in ny cse contiutes in wy tht educes the s length. It is noted tht doule foces on eithe end, cete unique doule stess field fo ll λ vlues, no mtte thei diffeent effect on the displcement nd stin fields. The next figues pesent the effect of doule stesses to the fixed fee s ehvio. In is shown tht the ppliction of doule foces tigges the l length, in wy tht it ttempts to minimize the s length. And fo gete λ vlues the moe the is shotened y the ppliction of the doule foces. Also, it is noted tht the doule stess field in this polem is independent of the l pmete. So only the displcement nd stin fields e ffected 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

116 Fig. Nomlized xil displcement u /R / EA vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R R, R Fig. Nomlized xil stin u' /R / EAL vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R R, R 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

117 Fig. Nomlized xil doule stesses /R / A vesus nomlized xil distnce x / L of Fixedxxx Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R R, R Fig. 4 Nomlized xil displcement u /R / EA vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R, R R 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

118 Fig. 5 Nomlized xil stin u' /R / EAL vesus nomlized xil distnce x / L of Fixed-Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R, R R Fig. 6 Nomlized xil doule stesses /R / A vesus nomlized xil distnce x / L of Fixedxxx Fixed, fo c=l/g=4 nd vious λ vlues. The clssicl BCs e u, P nd the non clssicl ones R, R R Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

119 4.II..iii. Conclusions The following genel conclusions cn e dwn out the s ehvio. Pesciing the s degees of feedom genelly esults to non clssicl non unifom ehvios fom the. Restining the s ends stins does incese the s stiffness, especilly fo smll L/g tios The s ehvio when insted of stin BCs, zeo doule foces BCs e pplied is the sme s the espective clssicl one, no mtte L/g tio. The l pmete ffects the in wy tht minimizes its length, y educing its elongtion, when in tension, nd y dditionlly shotening the, when in compession. Any BC polem solution cn e educed to clssicl one, when ppopite BC comintions e pplied. The s ehvio, unde most loding cses, fo get c=l/g vlues is educed to the espective clssicl ehvio 9 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

120 4.II. -D composite stuctues 4.II..i. Node function in composite D stuctues Befoe ddessing ny specific polem, independent of the loding pplied, in composite stuctue fist one must investigte how tuss node would function in the fmewok of the gdient elsticity theoy. The clssicl elsticity node is consideed to e igid point ody tht mintins the continuity of the s ends displcement, while it llows them to otte independently. The node s ody only hs displcement degees of feedom nd no ottionl ones nd lso needs to stisfy the Newton s fist lw of motion ΣF=. In the cse of two colline s joined with node, the conditions ove cn e tnslted to extenl u u nd P P P In the clssicl elsticity fmewok one BC needs to e detemined t ech end of. The node joins two s thus the ove BCs t the node e sufficient fo well posed polem. This is not the cse in the gdient elsticity fmewok, whee two BCs need to e detemined t ech s end, thus t the gdient node joining two colline s fou BCs need to e detemined, fo well posed polem. The clssicl conditions esily cn e chosen to e the two BCs. The choice of the two ext non clssicl conditions is not s ovious nd needs to e discussed, since thei effect to ech s ehvio is cucil. Polizzoto Polizzotto, consideed the node s n intefce etween the s. This is fily esonle ssumption nd will e followed in the getest pt of the pesent wok. Polizzoto demnded tht esides the clssicl BCs, the stin nd the cuvtue of the s displcement to e equl t the intefce, i.e. u ' u' nd u " u" Deling with gdient elsticity nd highe ode diffeentil equtions, it is sue to use highe ode BCs t nodes-intefces s the ones discussed. A esonle choice thought needs to e mde so tht they epesent the el loding of the stuctue nd povide the el s ehvio. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

121 Demnding tht the stin of oth s t the intefce is equl is vey stict condition itself. Howeve stin mnipultion is one of the ilities given y the vition pinciple nd it is n cceptle BC, lthough questions ise egding the ility of the node s ody/intefce s to impose such ehvio. On the othe hnd, demnding tht oth s hve the sme cuvtue t the intefce s BC is not s esonle. It does not hve ny physicl suppot, nd seems even moe exteme thn the one ove. In the cse of dividing in multiple elements nd pplying ny BCs, using these non clssicl BCs t the ceted intefces, continuous displcement field is otined tht coincides with the displcement field of the oiginl. In othe cses though whee not ll the s e of the sme mteil, the sme Young s modulus nd the sme coss sectionl e, it is expected tht discontinuities will ppe in the composite odies displcement deivtives, which isn t the cse with these intefce conditions. Thus, demnding equl cuvtues is not consideed to e functionl BC nd othe intefce conditions e seeked fte. Insted of the cuvtue condition, demnding tht the doule foces R t the intefce e equl is consideed moe esonle s the fouth BC. This is not n ity condition since it is otined though the vition pinciple. Howeve, the doule foces μ xxx do not contiute neithe to the foce no the moment equiliium of ody, nd it should not e necessy to mke such demnd, sed on clssicl elsticity concepts. Howeve, n ssumption of gdient vesion of Newton s d lw, togethe with the ssumption of n elstic intefce etween the s, not ody-node, justifies this BC in the sense tht the two mteils lone intect with thei oundies, nd the doule foces e pplied fom one nothe. This is the BC poposed in othe intefce elsticity polems, s wve tnsmition y Yueqiu Li Li, et l., 5. Following the Κodolemis odolemis, et l., nlog of the petwisted em, it is oseved tht t n intefce tht no ext xil stess is pplied, oth the xil foce nd the imoment should e equl, thus the espective R tem should e equl too. This is the cse of n intefce of two simil in geomety nd popeties stuctul pts. Howeve, esides the loding, the R tem pesented depends on sevel Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

122 mteil nd coss sectionl pmetes of the ody thus the ssumption of the R equlity should not e genelized. It should e noted, though, tht the nlogy is not pefect since in this cse thee e exctly fou degees of feedom, while, the petwisted em hs six, nd ssumptions e mde fo ech of them. A diect nswe to this sech is otined y Weitzmn in the field of couple-stess theoies. He notes tht ssuming n elstic intefce, the enegy due to the couple-stesses E=μ ω θ in his cse should e fully tnsmitted coss the two odies intefce nd thus the BCs chosen e μ + =μ - nd ω θ- =ω θ+ in his study. This note is tnslted in the pesent study, in gdient elsticity, s μ xxx- = μ xxx+ nd u - =u + R - =R + nd u - =u +, in the cse of identicl coss sections tht e eing joined. This, though, is consideed to e the BC nd in the cse of n intefce etween two diffeent coss sections. As noted efoe, this does not come fully in geement with the conclusion otined though the petwisted em nlogy. But tht cn e ttiuted to the impefection of the nlogy. It is lso noted y Weitzmn tht it is eqully esonle to ssume tht no gdient stin enegy is sumitted though the intefce, thus the BCs cn e ny of the following comintions u' nd u' R nd u' u' nd R R nd R The lst one, Weitzmn notes, is the physiclly most possile nd, s shown elie, is cse tht the s ehves clssiclly independent of the micostuctue. Since the ehvio of tusses with such nodes is the clssicl, which is known nd no size effect is pedicted, this cse will not e studied. Thee seems to e no eson tht the node s - intefce s BCs e not symmeticl thus the second nd thid BC comintions seem not to hve some pcticl ppliction nd they will not e used. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

123 The fist BC comintion is the one most commonly met in the litetue, nd s noted efoe is consideed to e vey intusive BC, tht induces vey non-unifom stiffe thn clssicl ehvio of the. It so lso noted tht this BC implies tht the doule stesses t the two diffeent djoining odies tke diffeent vlues, which, howeve, e line functions of the xil foces pplied o the displcements pplies t ech s ends, R tnhc/ PL/c g tnhl/ g P o R tnhc / R R EA u u, in the cse l=. c tnhc / The node tht estins the stins of the connected s is heeupon efeed igid node. Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

124 4.II..ii. -D Stiffness Mtix Next step to solving multiple polems is otining the element stiffness nd flexiility mtices of gdient elstic element. Fist, the stiffness mtix will e cquied. The stiffness mtix clcultes the foces pplied to the in ode to ccomplish cetin displcements in the fmewok of the clssicl elsticity theoy. Thee e sevel wys to constuct the stiffness mtix. Hee the nlyticl solutions comined with the foce equiliium e used. The displcement field in the cse of pescied nodl displcements dxi nd dxj is u= dxi+ dxj- dxi ξ while fxj=pj=eau =EA/L dxj- dxi=k dxj- dxi nd the foce equiliium demnds tht fxi+ fxj=-> fx-=- fxj=- k dxjdxi thus the locl coodinte system stiffness mtix is f f xi xj k k k d k d xi xj In the gdient elsticity fmewok, thee e two moe degees of feedom the stins t the s ends - the espective wok conjugtes R nd R 4 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

125 The displcement field of with pescied nodl displcements nd stins is know, thus the f xk ns xk genelized foces cn e found. Once gin the stiffness mtix stisfies the eqution f d f d, in which cse i ij j f fxi xi fxj xj T P R P R T d dxi xi dxj xj T dxi qxi dxj qxj T The choice ove fo the f nd d vecto components is mde sed on the ssumption tht the D D gdient theoy y Polyzos, et l., cn e simplified nd used to -D polems. Thus, the s non clssicl degees of feedom e q u nˆ t the ends nd thei wok conjugtes R nˆ μ nˆ, u eing the displcement vecto, nˆ the unit noml vecto on the odies sufce, i.e. the s ends, nd μ eing the doule stess tenso. In D polems they e simplified to: u nˆ u' nˆ u' xˆ xˆ nd R R xˆ u nˆ u' nˆ u' xˆ xˆ nd R R xˆ The eltions ove show tht no mtte which end of the is consideed, the doule foces R positive diection is the sme s the longitudinl xis diection, while, the stin degee of feedom, depending on the coss section hs diffeent positive diection. Afte some simplifictions the locl coodinte system stiffness mtix is found to e the following 5 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

126 symm g tnhl / g g Ll Lg / tnhl / g g tnhl / g EA lg tnhl / g L g tnhl / g symm c c tnhc / L L c tnhc L g tnhl / g tnhc / EA c c tnhc / c L symm g tnhl / g g Lg / sinhl / g g tnhl / g g Ll Lg / tnhl / g lg tnhl / g tnhc / L L c sinhc tnhc / c tnhc L tnhc / c whee symm efes to the symmetic ntue of the stiffness mtix. In the cse tht λ= the stiffness mtix is simplified: EA L g tnhl / g symm g g tnhl / g Lg / tnhl / g g tnhl / g g g g tnhl / g Lg / g tnhl / g l sinhl / g Lg / tnhl / g EA c tnhc / c / L symm tnhc / L / c L / tnhc c / L tnhc / c / L L / c tnhc / L / sinhc tnhc / L / c L / tnhc The stiffness mtices of the pesent wok e exct, so one finite element pe physicl meme cn e ssigned nd the exct solution still e otined. Of couse, the expessions of the stiffness coefficients hee e moe complicted thn the ones in clssicl elsticity ut they e in closed fom nd hence the dditionl computtionl effot is eltively smll. 6 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

127 4.II..iii. -D Applictions Sevel exmples e pesented tht indicte how the elements cn e employed fo solving polems of moe complex loding thn the simple with end foces nd doule foces, without using complex distiuted foce functions nd with pefect geement with the ehvio pedicted y the theoy. The mjoity of the numeicl vlues used fo the diffeent pmetes of the polem in the following exmples e otined y hoiyn, et l., while some geometicl pmetes wee chosen popely. Fist, the elements esponse to fundmentl ledy known polem is tested. Assume of mde of gdient elstic mteil with intenl length pmete g=. μm, l=μm, Young s modulus E=.44 GP, coss section e = 78.54μm, fixed t one end t x=, while n xil foce P= 4 μn is pplied t its fee end nd the stins t oth ends e estined nd equl to zeo, i.e. u, u', u', P 4 N Fig. 7 FEM nd nlyticl solutions of the xil displcement u of in tension with estined end stins vesus nomlized xil distnce x / L. The clssicl BCs e u, P P, nd the non clssicl ones u', u' 7 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST

128 Fig. 8 FEM nd nlyticl solutions of the xil stin u ' of in tension with estined end stins vesus nomlized xil distnce x / L. The clssicl BCs e u, P P, nd the non clssicl ones u', u' InIn the figues ove, the s displcement nd stin fields e pesented s otin y dividing it into, 4, 8, 6 pts nd using the sme nume of elements connected y elstic nodes to model it. The FEM solutions descie pefectly the s displcement nd stin fields, no mtte how ig o smll ech element is chosen the discetiztion. This is vey sic gdient elstic loding cse, nd the esults e extemely stisfctoy. Next, less simple loding cse puts these elements to the test. Fig. 9 Schemtic epesenttion of the second polem, nd the fou pt discetiztion used. In this, second, polem the sme is studied, sujected to diffeent lod. This time the concentted foce P is pplied t the s middle insted of its fee end while no doule foces e pplied t its ends. The nlyticl solution ws otined using the distiuted lod function qx x L/ P, δ eing the dic function nd P the concentted lod, nd 8 Institutionl Repositoy - Liy & Infomtion Cente - Univesity of Thessly /7/8 :5: EEST