2MicromechanicsThe analysis of relationships between effective composite properties (i.e., stiffness, strength) and the material properties, relative volume contents, and geometric arrangement of the constituent materials.
3Micromechanics - Stiffness Mechanics of materials models –Simplifying assumptions make it unnecessary to specify details of stress and strain distribution – fiber packing geometry is arbitrary. Use average stresses and strains.
4Micromechanics - Stiffness Theory of elasticity models -“Actual” stress and strain distributions are used – fiber packing geometry taken into account.Closed form solutionsNumerical solutions such as finite elementVariational methods (bounds)
14Fiber volume fraction – packing geometry relationships Real composites:Random fiber packing arrayUnidirectional:Chopped:Filament wound: close to theoretical
15Photomicrograph of carbon/epoxy composite showing actual fiber packing geometry at 400X magnification
16Voronoi cell and its approximation. (From Yang, H. and Colton, J.S Polymer Composites, 51, 34–41. With permission.)Random nature of fiber packing geometry in real composites can bequantified by the use of the Voronoi cell. Each point within the spaceof a Voronoi cell for a particular fiber is closer to the center of that fiberthan it is to the center of any other fibersVoronoi cellsEquivalent squarecells, with Voronoicell size, s
17Typical histogram of Voronoi distances and corresponding Wiebull distribution for a thermoplastic matrix composite. (From Yang, H. and Colton, J.S PolymerComposites, 51, 34–41. With permission.)
18Elementary Mechanics of Materials Models for Effective Moduli Fiber packing array not specified – RVE consists of fiber and matrix blocks.Improved mechanics of materials models and elasticity models do take into account fiber packing arrays.
19Assumptions:Area fractions = volume fractionsPerfect bonding at fiber/matrix interface – no slipMatrix is isotropic, fiber can be orthotropicFiber and matrix linear elasticLamina is macroscopically homogeneous, linear elastic and orthotropic
20Concept of an Effective Modulus of an Equivalent Homogeneous Material. Heterogeneous composite under varying stresses and strainsStress,Strain,Equivalent homogeneous material under average stresses and strainsStressStrain
21Representative volume element and simple stress states used in elementary mechanics of materials models
22Representative volume element and simple stress states used in elementary mechanics of materials modelsLongitudinal normal stressTransverse normal stressIn-plane shear stress
23Average stress over RVE: (3.19)Average strain over RVE:(3.20)Average displacement over RVE:(3.21)
24Longitudinal ModulusRVE under average stress governed by longitudinal modulus E1.Equilibrium:Note: fibers are often orthotropic.Rearranging, we get “Rule of Mixtures” for longitudinal stress(3.22)Static Equilibrium(3.23)
25Hooke’s law for composite, fiber and matrix Stress – strain Relations(3.24)So that:(3.25)
26Assumption about average strains: Geometric Compatibility(3.26)Which means that,(3.27)“Rule of Mixtures” – generally quite accurate – useful for design calculations
27Variation of composite moduli with fiber volume fraction Eq. 3.27Eq. 3.40Predicted E1 and E2 from elementary mechanics of materials models
28Variation of composite moduli with fiber volume fraction Comparison of predicted and measured E1 for E-glass/polyester. (From Adams, R.D., Engineered Materials Handbook, Vol. 1, Composites, 206–217.)
29Strain Energy Approach (3.28)Where strain energy in composite, fiber and matrix are given by,(3.29a)(3.29b)(3.29c)
30Subst. in “Rule of Mixtures” for longitudinal stress: Strain energy due to Poisson strain mismatch at fiber/matrix interface is neglected.Let the stresses in fiber and matrix be defined in terms of the composite stress as:(3.30)Subst. in “Rule of Mixtures” for longitudinal stress:(3.23)
31Combining (3.30), (3.24) & (3.29) in (3.28), Or(3.31)Combining (3.30), (3.24) & (3.29) in (3.28),(3.32)Solving (3.31) and (3.32) simultaneously for E-glass/epoxy with known properties:Find a1 and b1, then
33Transverse Modulus RVE under average stress Response governed by transverse modulus E2Geometric compatibility:From definition of normal strain,(3.34)(3.35)
341-D Hooke’s laws for transverse loading: Thus, Eq.(3.34) becomes(3.36)Or(3.37)Where1-D Hooke’s laws for transverse loading:(3.38)
35Where Poisson strains have been neglected. Combining (3. 37) and (3 (3.39)Assuming thatWe get(3.40)
36- “Inverse Rule of Mixtures” – Not very accurate - Strain energy approach for transverse loading, Assume,(3.41)Substituting in the compatibility equation (Rule of mixture for transverse strain), we get(3.42)
37Then substituting these expressions for and in (3.28)We get(3.43)Solving (3.42) and (3.43) simultaneously for a2 and b2, we get for E-glass/epoxy,
39In-Plane Shear Modulus, G12 Using compatibility of shear displacement and assuming equal stresses in fiber and matrix:(Not very accurate)(3.47)Major Poisson’s Ratio, υ12Using compatibility in 1 and 2 directions:(Good enough for design use)(3.45)
40Design EquationsElementary mechanics of materials Equations derived for G12 and E2 are not very useful – need to develop improved models for G12 and E2.
41Improved Mechanics of Materials Models for E2 and G12 Mechanics of materials models refined by assuming a specific fiber packing array.Example: Hopkins – Chamis method of sub-regionsRVE
42Convert RVE with circular fiber to equivalent RVE having square fiber whose area is the same as the circular fiber.RVESub Region AAsfASub Region BdsBsfBASub Region AADivision of representative volume element into sub regions based on square fiber having equivalent fiber volume fraction.
43Equivalent Square Fiber: (from )(3.48)Size of RVE:(3.49)For Sub Region B:ssf
44Following the procedure for the elementary mechanics of materials analysis of transverse modulus: (3.50)but(3.51)So that(3.52)
45For sub regions A and B in parallel, (3.53)Or finally(3.54)Similarly,
46Simplified Micromechanics Equations (Chamis) Only used part of the analysis for sub region B in Eq. (3.52):(3.52)Fiber properties Ef2 and Gf12 in tables inferred from these equations.
49Semi empirical ModelsUse empirical equations which have a theoretical basis in mechanicsHalpin-Tsai Equations(3.63)Where(3.64)
50And curve-fitting parameter 2 for E2 of square array of circular fibers1 for G12As Rule of MixturesAs Inverse Rule of Mixtures
51Tsai-Hahn Stress Partitioning Parameters let(3.65)Get(3.66)Where stress partitioning parameter(when get inverse Rule of Mixtures)
52Transverse modulus for glass/epoxy according to Tsai-Hahn equation (Eq Transverse modulus for glass/epoxy according to Tsai-Hahn equation (Eq. 3.66). (From Tsai, S.W. and Hahn, H.T Introduction to Composite Materials. Technomic Publishing Co., Lancaster, PA. With permission from Technomic Publishing Co.)Eq. 3.66
53Micromechanical Analysis of Composite Materials Using Elasticity Theory Micromechanical analysis of composite materials involve the development of analytical models for predicting macroscopic composite properties in terms of constituent material properties and information on geometry and loading. Analysis begins with the selection of a representative volume element, or RVE, which depends on the assumed fiber packing array in the composite.
54Example: Square packing array RVEMatrixFiberDue to double symmetry, we only need to consider one quadrant of RVEMatrixFiber
55The RVE is then subjected to uniform stress or displacement along the boundary. The resulting boundary value problem is solved by either stress functions, finite differences or finite elements.Later in this course we will discuss specific examples of finite difference solutions and finite element solutions for micromechanics problems.