Macromechanics Study of stress-strain behavior of composites using effective properties of an equivalent homogeneous material. Only the globally averaged stresses and strains are considered, not the local fiber and matrix values.
Stress-Strain Relationships for Anisotropic Materials First, we discuss the form of the stress-strain relationships at a point within the material, then discuss the concept of effective moduli for heterogeneous materials where properties may vary from point-to-point.
General Form of Elastic - Relationships for Constant Environmental Conditions Each component of stress, ij, is related to each of nine strain components, ij (Note: These relationships may be nonlinear)
Expanding F ij in a Taylor’s series and Retaining only the first order terms, for a linear elastic material
3D state of stress
Generalized Hooke’s Law for Anisotropic Material = (2.2)
Symmetry Simplifies the Generalized Hooke’s Law 1.Symmetry of shear stresses and strains: Same condition for shear strains, 2.Material property symmetry – several types will be discussed. O 1 2
Symmetry of shear stresses and shear strains: Thus, only 6 components of ij are independent, and likewise for ij. This leads to a contracted notation.
Geometry of Shear Strain xy = Engineering Strain xy = Tensor Strain Total change in original angle = xy Amount each edge rotates = xy /2 = xy
Using contracted notation or in matrix form where and are column vectors and [C] is a 6x6 matrix (the stiffness matrix) (2.3) (2.4)
(2.5) (2.6) Alternatively, or where [S] = compliance matrix and
Up to now, we only considered the stresses and strains at a point within the material, and the corresponding elastic constants at a point. What do we do in the case of a composite material, where the properties may vary from point to point? Use the concept of effective moduli of an equivalent homogeneous material.
StressStrain Heterogeneous composite under varying stresses and strains Equivalent homogeneous material under average stresses and strains Concept of an Effective Modulus of an Equivalent Homogeneous Material Stress,Strain,
Effective moduli, C ij (2.9) where, (2.7) (2.8)
3-D Case General Anisotropic Material [C] and [S] each have 36 coefficients, but only 21 are independent due to symmetry. Symmetry shown by consideration of strain energy. Proof of symmetry: Define strain energy density
(2.12) Now, differentiate: butif i=j if i j
(show) (2.11) (2.13) But if the order of differentiation is reversed, (2.14)
Since order of differentiation is immaterial, (Symmetry) Similarly, and Only 21 of 36 coefficients are independent for anisotropic material.
(2.15) Stiffness matrix for linear elastic anisotropic material with no material property symmetry
3-D Case, Specially Orthotropic Fiber direction , 2, 3 principal material coordinates
(a)(b)(c) Simple states of stress used to define lamina engineering constants for specially orthotropic lamina.
Consider normal stress 1 alone: Resulting strains, (2.19)
Typical stress-strain curves from ASTM D3039 tensile tests Stress-strain data from longitudinal tensile test of carbon/epoxy composite. Reprinted from ref.  with permission from CRC Press.
where E 1 = longitudinal modulus ij = Poisson’s ratio for strain along j direction due to loading along i direction Similarly,
1 2 3 Now consider normal stress 2 alone: Strains: (2.20) Where E 2 = transverse modulus Similar result for 3 alone
Observation: All shear strains are zero under pure normal stress (no shear coupling). For alone
Now, consider shear stress alone, Strain Where G 12 = Shear modulus in 1-2 plane (No shear coupling) (2.21)
Similarly, for alone and for alone Now add strains due to all stresses using superposition
Specially Orthotropic 3D Case (2.22) 12 coefficients, but only are 9 independent
Symmetry: Only 9 independent coefficients. Generally orthotropic 3-D case – similar to anisotropic with 36 nonzero coefficients, but 9 are independent as with specially orthotropic case
Specially Orthotropic – Transversely Isotropic Reinforcement Fibers randomly packed in 2-3 plane, so properties are invariant to rotation about 1-axis (2 same as 3)
Specially orthotropic, transversely isotropic (2 and 3 interchangeable) (2.23) Now, only 5 coefficients are independent.
Isotropic 2 independent coefficients Usually measure E, υ – calculate G
Isotropic – 3D case Same form for any set of coordinate axes
3-D Isotropic – stresses in terms of strains
3-D Case, Generally Orthotropic Reinforcement direction x y z Material is still orthotropic, but stress-strain relations are expressed in terms of non-principal xyz axes
Generally Orthotropic Same form as anisotropic, with 36 coefficients, but 9 are independent as with specially orthotropic case
Elastic coefficients in the stress-strain relationship for different materials and coordinate systems Material and coordinate systemNumber of nonzero coefficients Number of independent coefficients Three – dimensional case Anisotropic3621 Generally Orthotropic (nonprincipal coordinates) 369 Specially Orthotropic (Principal coordinates)129 Specially Orthotropic, transversely isotropic125 Isotropic122 Two – dimensional case (lamina) Anisotropic96 Generally Orthotropic (nonprincipal coordinates) 94 Specially Orthotropic (Principal coordinates)54 Balanced orthotropic, or square symmetric (principal coordinates) 53 Isotropic52
2-D Cases Use 3-D equations with, Plane stress, Or
Experimental Characterization of Orthotropic Lamina Need to measure 4 independent elastic constants Usually measure E 1, E 2, υ 12, G 12 (see ASTM test standards later in Chap. 10)
Stresses in terms of tensor strains, (2.26)
Off – Axis Compliances: Off – Axis Stiffnesses: Where f ij and f ij ’ are found from transformations of stress and strain components from 1,2 axes to x, y axes
Sign convention for lamina orientation x y 2 1 x y 2 1
Stress Transformation: 1 2 Y X dA and
(2.29) Equations used to generate Mohr’s circle.
Resulting stress Transformation: Where (2.30) or (2.31)
Where (2.32) Strain Transformation: (2.33)
Recall: Tensor shear strain Where xy = engineering shear strain or
Substituting (2.33) into (2.26), then substituting the resulting equations into (2.30) Carrying out matrix multiplications and converting back to engineering strains, (2.34) (2.35)
Where (2.36) Alternatively (2.37) Where
Generally Orthotropic Lamina (Off Axis) 1 2 Or (2.37) 9 Coefficients - 6 independent y x
(2.38) In expanded form:
Off-axis lamina engineering constants Young’s modulus, E x 2 1 y x When or (2.39) (2.40)
Complete set of transformation equations for lamina engineering constants (2.40)
Variations of off-axis engineering constants with lamina orientation for unidirectional carbon/epoxy, boron/aluminum and glass/epoxy composites. (From Sun, C.T Mechanics of Aircraft Structures. John Wiley & Sons, New York. With permission.)
Shear Coupling Ratios, or Mutual Influence Coefficients Quantitative measures of interaction between normal and shear response. Example: when Shear Coupling Ratio Analogous to Poisson’s Ratio (2.41)
Example of off-axis strain in terms of off-axis engineering constants (2.43)
Compliance matrix is still symmetric for off-axis case, so that, for example and