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Micromechanics Macromechanics Fibers Lamina Matrix Laminate Structure

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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.

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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.

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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)

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Expanding F ij in a Taylor’s series and Retaining only the first order terms, for a linear elastic material

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3D state of stress

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Generalized Hooke’s Law for Anisotropic Material = (2.2)

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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

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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.

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Stresses Tensor NotationContracted Notation 11 11 22 22 33 33 23 = 32 44 13 = 31 55 12 = 21 66

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Strains Tensor NotationContracted Notation 11 1 22 2 33 3 2 23 = 2 32 = 23 = 32 4 2 13 = 2 31 = 13 = 31 5 2 12 = 2 21 = 12 = 21 6

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Geometry of Shear Strain xy = Engineering Strain xy = Tensor Strain Total change in original angle = xy Amount each edge rotates = xy /2 = xy

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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)

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(2.5) (2.6) Alternatively, or where [S] = compliance matrix and

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Expanding:

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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.

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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,

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Effective moduli, C ij (2.9) where, (2.7) (2.8)

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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

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(2.12) Now, differentiate: butif i=j if i j

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(show) (2.11) (2.13) But if the order of differentiation is reversed, (2.14)

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Since order of differentiation is immaterial, (Symmetry) Similarly, and Only 21 of 36 coefficients are independent for anisotropic material.

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(2.15) Stiffness matrix for linear elastic anisotropic material with no material property symmetry

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3-D Case, Specially Orthotropic Fiber direction 1 2 3 1, 2, 3 principal material coordinates

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(a)(b)(c) Simple states of stress used to define lamina engineering constants for specially orthotropic lamina.

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Consider normal stress 1 alone: 1 2 3 Resulting strains, (2.19)

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Typical stress-strain curves from ASTM D3039 tensile tests Stress-strain data from longitudinal tensile test of carbon/epoxy composite. Reprinted from ref. [8] with permission from CRC Press.

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where E 1 = longitudinal modulus ij = Poisson’s ratio for strain along j direction due to loading along i direction Similarly,

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1 2 3 Now consider normal stress 2 alone: Strains: (2.20) Where E 2 = transverse modulus Similar result for 3 alone

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Observation: All shear strains are zero under pure normal stress (no shear coupling). For alone

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Now, consider shear stress alone, 1 2 3 Strain Where G 12 = Shear modulus in 1-2 plane (No shear coupling) (2.21)

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Similarly, for alone and for alone Now add strains due to all stresses using superposition

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Specially Orthotropic 3D Case (2.22) 12 coefficients, but only are 9 independent

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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

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Specially Orthotropic – Transversely Isotropic Reinforcement 1 2 3 Fibers randomly packed in 2-3 plane, so properties are invariant to rotation about 1-axis (2 same as 3)

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Specially orthotropic, transversely isotropic (2 and 3 interchangeable) (2.23) Now, only 5 coefficients are independent.

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Isotropic 2 independent coefficients Usually measure E, υ – calculate G

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Isotropic – 3D case Same form for any set of coordinate axes

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3-D Isotropic – stresses in terms of strains

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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

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Generally Orthotropic Same form as anisotropic, with 36 coefficients, but 9 are independent as with specially orthotropic case

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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

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2-D Cases Use 3-D equations with, Plane stress, Or

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Specially Orthotropic Lamina 1 2 Or (2.24) (2.26) 5 Coefficients - 4 independent

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Specially Orthotropic Lamina in Plane Stress 5 nonzero coefficients 4 independent coefficients (2.24)

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Or in terms of ‘engineering constants’ (2.25)

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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)

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Stresses in terms of tensor strains, (2.26)

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Inverting [S]:

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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

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Sign convention for lamina orientation x y 2 1 x y 2 1

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Stress Transformation: 1 2 Y X dA and

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(2.29) Equations used to generate Mohr’s circle.

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Resulting stress Transformation: Where (2.30) or (2.31)

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Where (2.32) Strain Transformation: (2.33)

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Recall: Tensor shear strain Where xy = engineering shear strain or

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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)

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Where (2.36) Alternatively (2.37) Where

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Generally Orthotropic Lamina (Off Axis) 1 2 Or (2.37) 9 Coefficients - 6 independent y x

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(2.38) In expanded form:

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Off-axis lamina engineering constants Young’s modulus, E x 2 1 y x When or (2.39) (2.40)

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Complete set of transformation equations for lamina engineering constants (2.40)

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Variations of off-axis engineering constants with lamina orientation for unidirectional carbon/epoxy, boron/aluminum and glass/epoxy composites. (From Sun, C.T. 1998. Mechanics of Aircraft Structures. John Wiley & Sons, New York. With permission.)

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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)

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Example of off-axis strain in terms of off-axis engineering constants (2.43)

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Compliance matrix is still symmetric for off-axis case, so that, for example and

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Balanced Orthotropic Lamina (Ex: Woven cloth, cross-ply) 1 2 Only 3 independent coefficients

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Lamina Stiffness Transformations

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Use of Invariants The lamina stiffness transformations can be written as: (2.44)

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Where the invariants are (2.45)

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Alternatively, the off-axis compliances can be expressed as (2.46)

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where the invariants are (2.47)

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Example: decomposition of using invariants References: Mechanics of Composite Materials, Jones Introduction to Composite Materials, Tsai & Hahn

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Invariants in transformation of stresses: Mohr’s circle Where Invariant (2.48) I R

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Similar graphical interpretation of stiffness transformations Ex: (2.49) Isotropic Part Orthotropic Part

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U 1 and U 4 : First order invariants U 2 and U 3 : Second order invariants Radii of circles indicates degree of orthotropy. (i.e., if U 2 =U 3 =0, we have isotropic material)

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