= l. = l. (Hooke s Law) Tensile: Poisson s ratio. σ = Εε. τ = G γ. Relationships between Stress and Strain

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1 Relationships between tress and train (Hooke s Law) When strains are small, most of materials are linear elastic. Tensile: Ε hear: Poisson s ratio Δl l Δl l Nominal lateral strain (transverse strain) Poisson s ratio: lateral tensile strain strain

2 Relationships between tress and train An isotropic material has a stress-strain relationships that are independent of the orientation of the coordinate sstem at a point. A material is said to be homogenous if the material properties are the same at all points in the bod For isotropic materials lastic tress-train Relationships Uniaial Principal tresses 3 Principal trains 3

3 [ ] [ ] [ ] [] υ υ υ υ υ υ Uniaial tresses

4 ( ) ( ) 3 Principal tresses Principal trains 3 Biaial ( ) ( ) ( ) ( ) ( ) ( ) Principal tresses Principal trains Triaial

5 [ ] [ ] [] υ υ υ υ υ υ [ ] Triaial tresses

6 For an isotropic material, the principal aes for stress and the principal aes for strain coincide. tan θ XY X Y ( )( ) XY tan θ tan θ X Y ( )( ) ( ) ( )

7 Plane tress ( ) ( ) ( ) Plane train ( )( ) ( ) [ ] ( )( ) ( ) [ ] ( )( ) ( )

8 Isotropic Materials

9 [ ] [ ][ ] is the compliance matri

10 Isotropic Materials An isotropic material has stress-strain relationships that are independent of the orientation of the coordinate sstem at a point. The isotropic material requires onl two independent material constants, namel the lastic Modulus and the Poisson s Ratio.

11 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) [ ] [ ][ ] is the elastic or stiffness matri The isotropic material requires onl two independent material constants, namel the lastic Modulus and the Poisson s Ratio.

12 Isotropic Materials The isotropic material requires onl two independent material constants, namel the lastic Modulus and the Poisson s Ratio. ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )

13 Anisotropic Materials Up to this point we have limited the stud of the properties of materials to isotropic materials. For the most general linearl elastic anisotropic materials, a particular component of stress is assumed to depend of all si components of strain Where ij are constants if the material is homogeneous

14 Taking energ considerations the coefficients of this matri are smmetric. Hence, instead of 36 independent constant, we have independent constants is referred to as the elastic matri or stiffness matri. [ ]

15 Hence, we can also write [] [][ ] The matri is referred to as the compliance matri and the elements of are the compliances.

16 elastic constants are required to describe the most general anisotropic material (full anisotropic). This is in contrast to an isotropic material for which there are onl two independent elastic constants (tpicall the Young Modulus and the Poisson s ratio). ( ) ( )( ) ( )( ) ( )( ) ( ) [ ] ) ) ( (

17 Man materials of practical interest contain certain material smmetries with respect to their elastic properties (elastic smmetries). Other tpe of smmetries are possible optical, electrical and thermal properties. Neumann s Principle This is the most important concept in crstal phsics. It states;... the smmetr of an phsical propert of a crstal must include the smmetr elements of the point group of the crstal. This means that measurements made in smmetr-related directions will give the same propert coefficients. Let us determine the structure of the elastic matri for a material with a single plane of elastic smmetr. rstals whose crstalline structure is monoclinic as eamples of materials possessing a single plane of elastic smmetr. ample Iron aluminide, gpsum, talc, ice, selenium Materials with one plane of smmetr are referred to as Monoclinic materials.

18 rstal stems rstallographers have shown that onl seven different tpes of unit cells are necessar to create all point lattice ubic a b c ; α β 9 Tetragonal a b c ; α β 9 Rhombohedral a b c ; α β 9 Heagonal a b c ; α β 9, Orthorhombica b c ; α β 9 Monoclinic a b c ; α 9 β Triclinic a b c ; α β 9

19 Monoclinic Materials Let us assume that the -plane is the plane of elastic smmetr. For such a material the elastic coefficients in the stress-strain law must remain unchanged when subjected to a transformation that represents a reflection in the smmetr plane. For monoclinic materials (due to one plane of elastic smmetr) the number of independent elastic constants is reduced from to

20 KY TO NOTATION TRILINI () MONOLINI (3)

21 ORTHORHOMBI (9) UBI (3) (7) TTRAONAL (6)

22 HXAONAL (5) IOTROPI () (7) TRIONAL (6)

23 Orthotropic Materials Let us consider a material with a second plane of elastic smmetr. The -plane and the -plane are the planes of elastic smmetr and are perpendicular to each other. Again, for such a material the elastic coefficients in the stress-strain law must remain unchanged when subjected to a transformation that represents a reflection in the smmetr plane. For orthotropic materials (due to the two planes of elastic smmetr) the number of independent elastic constants is reduced from to

24 Materials possessing two perpendicular planes of elastic smmetr must also possess a third mutuall perpendicular plane of elastic smmetr. Materials having three mutuall perpendicular planes of elastic smmetr are referred to as orthotropic (orthogonall anisotropic) materials. Long Fiber omposite

25 Transversel Isotropic Materials Materials that are isotropic in a plane. Transversel isotropic materials require five independent material constants. ( )

and p, the Young's modulus and poisson ratio in the -direction, p and p, and

26 B convention, the 5 elastic constants in transverse isotropic constitutive equations are the Young's modulus and poisson ratio in the - smmetr plane, p and p, the Young's modulus and poisson ratio in the -direction, p and p, and the shear modulus in the -direction p. The compliance matri takes the form, where.

27 The stiffness matri for transverse isotropic materials, found from the inverse of the compliance matri, is given b, where,

28 ngineering Material onstants for Orthotropic Materials The quantities appearing in the coefficient matri can be written in terms of well understood engineering constants such as the Young Modulus and the Poisson s ratio. For the, and coordinate aes we can write: Where the Young Modulus in the -, - and - directions are not necessaril equal. An etension in the -ais is accompanied b a contraction in the - and - ais. However, this quantities are not necessaril equal in orthotropic materials. Where is the contraction in the -direction due to the stress in the -direction

29 If all three stresses are applied simultaneousl, then: [ ] [ ][ ] omparing with the compliance matri for orthotropic materials: Where is the contraction in the -direction due to the stress in the -direction

30 Whereas with isotropic materials the relationship between shear stress and shear strain is the same in an coordinate planes, for orthotropic materials these relationships are not the same

31 Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ

32 33 In -D

33 ample: At a point on a free surface of aluminum (, ksi and 4, ksi) the strains recorded b the three strain gages shown in the figure below are as given. Determine the stresses,, and.

34 olution: ) ( ) ( ) ( 6 4 (35) (35) (35) (35) ) 4 3 ( ) 4 3 ( ) 4 ( 6 5 (6) (6) (6) (6) 6 c c b b a os in in os os in in os μ μ μ μ μ

35 μ μ μ ,, , ,,666.7,666.7,666.7, , ,.5,.5,5.5,5.5, psi,35.4 9,76. 3,568.5

36 ample An orthotropic material has the following properties 7,5ksi,,5ksi,,5ksi and.5. Determine the principal stresses and strains at a point on a free surface where the following strains were measured: -4μ ; 6μ ; -5μ. onsider plane stress conditions olution:

37 psi psi psi psi psi psi Ma μ μ μ Ma

38 tan θ tan θ XY X ( ) Y ( 65) Different angles to obtain the principal stresses and the principal strains.

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