MACROMECHANICS

1. MACROMECHANICS Ahmet Erkliğ

2. Objectives • Review definitions of stress, strain, elastic moduli, and strain energy. • Develop stress–strain relationships for different types of materials. • Develop stress–strain relationships for a unidirectional/bidirectional lamina. • Find the engineering constants of a unidirectional/bidirectional lamina in terms of the stiffness and compliance parameters of the lamina. • Develop stress–strain relationships, elastic moduli, strengths, andthermal and moisture expansion coefficients of an angle ply basedon those of a unidirectional/bidirectional lamina and the angle ofthe ply.

3. LaminaandLaminate A lamina is a thin layer of a composite material. A laminate is constructed by stackinga number of such lamina in the direction of the lamina thickness

4. Review of Definitions Stress There are thus six independent stresses. The stresses σx ,σy, and σzare normal to the surfaces of the cuboid and the stresses τyz, τzx ,and τxyare along the surfaces of the cuboid.

5. Review of Definitions Strain Under loads, the lengths of the sides of the infinitesimal cuboidchange. The faces of the cube also get distorted. The change in length corresponds to a normal strain and the distortion corresponds to the shearingstrain.

6. Review of Definitions ElasticModuliforisotropichomogeneousmaterial; Thecompliancematrix [S]

7. Review of Definitions ElasticModuliforisotropichomogeneousmaterial; Thestiffnessmatrix [C]

8. Review of Definitions StrainEnergy; Energy is defined as the capacity to do work. In solid, deformable, elasticbodies under loads, the work done by external loads is stored as recoverablestrain energy.

9. Hooke’s Law for Different Types of Materials where the 6 × 6 [C] matrix is called the stiffness matrix. The stiffness matrixhas 36 constants. Dueto the symmetry of the stiffness matrix [C],36 constantsarereducedto 21 independentconstants

10. compliance matrix [S]

11. Anisotropic Material The material that has 21 independent elastic constants at a point is called ananisotropic material.

12. Monoclinic Material If, in one plane of material symmetry, for example, direction3 is normal to the plane of material symmetry, then the stiffness matrixreduces to 13 independentconstants.

13. Orthotropic Material (Orthogonally Anisotropic)/SpeciallyOrthotropic If a material has three mutually perpendicular planes of material symmetry,then the stiffness matrix has 9 independent elastic constants (E1, E2, E3, G12, G13, G23, ν12, ν13, ν23)

14. OrthotropicMaterial

15. OrthotropicMaterial

16. Transversely Isotropic Material Consider a plane of material isotropy in one of the planes of an orthotropicbody. If direction 1 is normal to that plane (2–3) of isotropy, then the stiffnessmatrix is 4 independentconstants

17. Isotropic Material If all planes in an orthotropic body are identical, it is an isotropic material;then, the stiffness matrix is 2 independentconstants (E, ν)

18. Summary • Anisotropic: 21 • Monoclinic: 13 • Orthotropic: 9 • Transversely isotropic: 5 • Isotropic: 2 independent elastic constants

19. Plane Stress Assumption A thin plate is a prismatic member having a small thicknessand there are no out-of-plane loads,it can be considered to be under plane stress. If the upper andlower surfaces of the plate are free from external loads, then σ3 = 0, τ31 = 0,and τ23 = 0.

21. E1= longitudinal Young’s modulus (in direction 1) E2= transverse Young’s modulus (in direction 2) ν12= major Poisson’s ratio, where the general Poisson’s ratio, νij is definedas the ratio of the negative of the normal strain in directionj to the normal strain in direction i, when the only normal load isapplied in direction i G 12 = in-plane shear modulus (in plane 1–2)

22. Coefficients of ComplianceandStiffnessMatrices

23. Hooke’s Law for a 2D Angle Lamina Localand global axes of an anglelamina Transversedirection Longitudinaldirection Global axes, x, y

24. Global andLocalStresses [T] is called the transformation matrix

25. Global andLocalStresses [R] is the Reuter matrix

26. Global andLocalStresses

27. ReducedStiffnessMatrix

28. Global andLocalStresses Inverting stress equation Transformedreduced compliance matrix

29. Global andLocalStresses

30. Example Find the following for a 60° angle lamina of graphite/epoxy.Use the properties of unidirectional graphite/epoxy lamina from Table 2.1. Transformed compliance matrix Transformed reduced stiffness matrix Global strains Local strains Local stresses Principal stresses Maximum shear stress Principal strains Maximum shear strain

31. Solution Thecompliance matrix elements are

32. Transformedcompliance matrix elements are

33. Transformedreducedstiffnessmatrix elements are

34. 3. The global strains in the x–y plane are given by Equation (2.105)

38. The value of the angle at which the maximum normal stresses occur is

43. 9. The maximum shearing strain is given by

45. Engineering Constants of an Angle Lamina The engineering constants for a unidirectional lamina were related to thecompliance and stiffness matrices 1. For finding the engineering elastic moduli in direction x (Figure2.23a), apply The elastic moduli in direction x is defined as Then

46. In an angle lamina, unlike in a unidirectional lamina, interaction alsooccurs between the shear strain and the normal stresses. This iscalled shear coupling. The shear coupling term that relates the normal stress in the x-direction to the shear strain is denoted by mx andis defined as Note that mxis a nondimensional parameter like the Poisson’s ratio.

47. 2. Similarly, by applying stresses

48. 3. Also, by applying the stresses

49. Strain-Stress Equation of an Angle Lamina

50. Engineering Constants of an Angle Lamina