Chapter 2. Stress, Principal Stresses, Strain Energy

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1 Chapter Stress, Principal Stresses, Strain nergy Traction vector, stress tensor z z σz τ zy ΔA ΔF A ΔA ΔF x ΔF z ΔF y y τ zx τ xz τxy σx τ yx τ yz σy y A x x F i j k is the traction force acting on the area Δ A ΔAk Δ Δ Fx + Δ Fy + ΔFz ΔF lim z df σ z z ΔA ΔA 0, ΔF lim x df τ x zx ΔA ΔA 0, ΔFy τ zy lim ΔA ΔA 0 dfy τ xy τ yx, τ yz τ zy, τ zx τ xz Principal stresses At any point in a general state of stress, there are three mutually perpendicular principal planes which are free of shear stress. The normal stresses acting on the principal planes are called principal stresses and their directions are called principal directions. The highest and lowest principal stresses represent the maximum and minimum of all normal stresses on planes of any orientation at that point.

2 -Dimensional Case σ x τ xy τ yx σ y τ yx dy τ xy σ x d R a y cosθ m sinθ j m i d R n θ x y d R y d R dr σ d R n p d R x θp x σ y d R b n i + m j cosθ i + sin θj dr i j a σx τxy drb τxy mi σy mj dr drx i + dry j equilibrium requires that dr + dr + dr 0 a b dr σ x x xy dr y τxy σy m if dr is parallel to n then dr σ n 0 σx σ τxy 0 τxy σy σ m 0 τ nontrivial solution exists if σ σ x τ xy τ y xy σ σ 0 eigenvalues (principal stresses) σ x + σy σx σy σ ± ( ) +τ xy eigenvectors (principal directions) θ p 1 tan -1 τxy σx σy

3 3-Dimensional Case n i + mj + nk ( + m + n 1 ) dr + dr + dr + dr 0 a b c drx σx τxy τxz dry τxy σy τyz m dr n z τxz τyz σz dr σ n 0 if σx σ τxy τxz 0 τxy σy σ τ yz m 0 n 0 τxz τyz σz σ 3 I1 I I3 0 σ σ + σ I 1 σ x + σ y + σ z I σxσ y +σyσ z +σzσx τxy τyz τ zx I 3 σx σy σ z + τxy τyz τzx σxτyz σyτzx σz τ xy Because the stress tensor is symmetric, all three eigenvalues 1 3 σ σ σ are real and the eigenvectors are mutually orthogonal.

4 Normal and Shear Stress in Principal Directions e 3 e e 1 n e + me + ne ( + m + n 1 ) 1 3 dr [ σ] n or drx σ1 0 0 dry 0 0 σ m dr 0 0 σ z 3 n T dr σ n T σ e + σ me + σ n e T σ + σ m + σ n 1 3 normal stress drn drn σ n Tn 1 3 σ n σ + σ m + σ n shear stress τ s T σ n τ s ( σ 1 + σ m + σ3n ) ( σ 1 + σ m + σ3n ) τ s ( σ σ ) m + ( σ σ ) m n + ( σ σ ) n

5 Octahedral and Maximum Shear Stress e 3 e e 1 for the octahedral plane octahedral normal stress n e + e + e σ oct ( σ 1 + σ + σ 3 ) 3 1 σ oct ( σ x + σ y + σ z ) 3 σ oct I1 3 octahedral shear stress 1/ τ 1 oct ( σ1 σ ) + ( σ σ 3 ) + ( σ3 σ1 ) 3 1 1/ τ oct ( σx σ y ) + ( σy σ z ) + ( σz σ x ) + 6( τ xy + τ yz + τzx ) 3 1 1/ τ oct I1 6 I 3 maximum shear stress τ max σ1 σ3

6 Stress-Strain-Temperature Relations Isotropic material ε x σx ν( σ y + σz) + αδ T ε y σy ν( σ x + σz) + αδ T ε z σz ν( σ x + σy) + αδ T γ xy γ yz γ xz τxy G τ yz G τxz G (1 +ν ) G Plane state of stress ε x σx νσy + αδ T ε y σy νσx + αδ T σ x 1 ( x y ) αδt ε + νε ν 1 ν σ y 1 ( y x ) α ΔT ε + νε ν 1 ν τxy γ xy τ xy G γ xy G General state of stress σ x [(1 ) ] α Δ ν ε x + νε T y + νεz (1 +ν) (1 ν) 1 ν σ y α ΔT [(1 ν) ε y + νε z + νεx] (1 +ν) (1 ν) 1 ν σ z [(1 ) ] α Δ ν ε z + νε T x + νεy (1 +ν) (1 ν) 1 ν

7 Strain nergy Density D D kd F DF f du k u du 0 0 k dv V o 1 1 o ( εxσ x + εyσ y + εzσ z) + ( γxyτ xy + γyzτ yz + γxzτ xz) two-dimensional 1 1 G o ( σ x + σ y νσ x σ y ) + τ xy (1 ) ( ) G ε + ε + νε ε + γ ν o x y x y xy three-dimensional 1 o [ σ x +σ y +σz ν ( σxσ y +σyσ z +σxσz )] 1 ( + τ xy +τ yz +τxz ) G [(1 )( ) ( )] (1 +ν)(1 ν) o ν ε x +ε y +ε z + ν εxε y +εyε z +εxεz G + ( γ xy +γ yz +γxz)

8 Strain nergy of Beams I dv V o 1 1 [ ( )] ( ) G o σ x +σ y +σ z ν σ x σ y +σ y σ z +σ x σ z + τ xy +τ yz +τ xz extension o σ σ P A P 0 A A 0 P A torsion o τ G τ T r J T r 0 GJ A 0 T GJ

9 Strain nergy of Beams II bending o σ σ M y I M y 0 I A M 0 I shear-bending o τ G τ VQ I t V Q 0 GI A t 1 A Q k I A t 0 V kag

10 Strain nergy of Beams, Summary extension 0 P A torsion 0 T JG bending M 0 I shear-bending V k, AG 0 k A Q I t dydz Castigliano's first theorem Δ k P M k, k θk Castigliano's second theorem * Pk Δk, * M k θk

11 Strain nergy of Distortion Average normal stress 1 1 σ a ( σ x + σ y + σ z) ( σ 1 + σ + σ 3) 3 3 Deviatoric stresses s x σx σ a, s y σy σ a, s z σz σ a s τ s xy τ xy, s yz τ yz, zx zx Total strain energy density Distortion strain energy density 1 o [ σ x +σ y +σz ν ( σxσ y +σyσ z +σxσz )] 1 ( + τ xy +τ yz +τxz ) G 1 od [ sx + sy + sz ν ( sxsy + sysz + sxsz )] 1 ( + sxy + syz + sxz ) G 1 [( ) ( ) ( ) 6( od σ x σ y + σ y σ z + σ z σ x + τ xy + τ yz + τ xz)] 1G od 1 [( 1 ) ( 3 ) ( 3 1 ) ] 1G σ σ + σ σ + σ σ Octahedral stress 1 1/ τ oct ( σx σ y ) + ( σy σ z ) + ( σz σ x ) + 6( τ xy + τ yz + τzx 3 1/ τ 1 oct ( σ1 σ ) + ( σ σ 3 ) + ( σ3 σ1 ) 3

12 3 4G od τ oct ffective stress (also equivalent tensile stress or von Mises stress) σ τ e oct 3 1 [( ) ( ) ( ) 6( )] 1/ σ e σx σ y + σy σ z + σz σ x + τ xy + τ yz + τ xz 1 σ e ( σ σ ) + ( σ σ ) + ( σ σ ) in uniaxial tension σ e σ x in hydrostatic stress σ e 0 1 od σ e 6 G 1/

13 Stress Concentration σ max d σ 0 P Dt D t σ nom P ( D d) t σ K σ max t nom 3 d d d Kt D D D b σ max σ max a a max (1 ) 0 b σ + σ σ max σ 0

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