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Micromechanics Macromechanics Fibers Lamina Laminate Structure Matrix

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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 Fij 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**

Symmetry of shear stresses and strains: Same condition for shear strains, Material property symmetry – several types will be discussed. 2 O 1

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

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Strains Tensor Notation Contracted 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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**where [S] = compliance matrix and**

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

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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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**Concept of an Effective Modulus of an Equivalent Homogeneous Material**

Heterogeneous composite under varying stresses and strains Stress, Strain, Equivalent homogeneous material under average stresses and strains Stress Strain

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Effective moduli, Cij (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: but if i=j if i j

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**But if the order of differentiation is reversed,**

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

(2.15)

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**3-D Case, Specially Orthotropic**

Fiber direction 2 1, 2 , 3 principal material coordinates 1

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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:**

3 1 2 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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Similarly, where E1 = longitudinal modulus ij = Poisson’s ratio for strain along j direction due to loading along i direction

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**Now consider normal stress 2 alone:**

Strains: 3 1 2 (2.20) Where E2 = 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,**

3 1 2 Strain Where G12 = Shear modulus in 1-2 plane (2.21) (No shear coupling)

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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**

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

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**Now, only 5 coefficients are independent.**

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**

z Reinforcement direction y Material is still orthotropic, but stress-strain relations are expressed in terms of non-principal xyz axes x

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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 system Number of nonzero coefficients Number of independent coefficients Three – dimensional case Anisotropic 36 21 Generally Orthotropic (nonprincipal coordinates) 9 Specially Orthotropic (Principal coordinates) 12 Specially Orthotropic, transversely isotropic 5 Isotropic 2 Two – dimensional case (lamina) 6 (nonprincipal coordinates) 4 Balanced orthotropic, or square symmetric (principal coordinates) 3

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

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**Specially Orthotropic Lamina**

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

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**5 nonzero coefficients 4 independent coefficients**

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

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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 E1, E2, υ12, G12 (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 fij and fij’ 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**

y y 1 2 2 x x 1

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**Stress Transformation:**

Y 2 1 X dA and

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**Equations used to generate Mohr’s circle.**

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

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**Resulting stress Transformation:**

(2.30) Where or (2.31)

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**Strain Transformation:**

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)**

(2.34) Carrying out matrix multiplications and converting back to engineering strains, (2.35)

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Where (2.36) Alternatively (2.37) Where

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**Generally Orthotropic Lamina (Off Axis)**

(2.37) Or x 2 y 1 9 Coefficients - 6 independent

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In expanded form: (2.38)

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**Off-axis lamina engineering constants**

Young’s modulus, Ex 2 When y (2.39) or 1 x (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 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)**

2 1 Only 3 independent coefficients

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**Lamina Stiffness Transformations**

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**The lamina stiffness transformations can be written as:**

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 (2.48) Invariant I Invariant

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**Similar graphical interpretation of stiffness transformations**

Ex: (2.49) Orthotropic Part Isotropic Part

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**U1 and U4: First order invariants**

U2 and U3: Second order invariants Radii of circles indicates degree of orthotropy. (i.e., if U2=U3=0, we have isotropic material)

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