Micromechanics The 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.
1.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. Micromechanics - Stiffness
2.Theory of elasticity models - “Actual” stress and strain distributions are used – fiber packing geometry taken into account. a)Closed form solutions b)Numerical solutions such as finite element c)Variational methods (bounds) Micromechanics - Stiffness
Real composites: Random fiber packing array Unidirectional: Chopped: Filament wound: close to theoretical Fiber volume fraction – packing geometry relationships
Photomicrograph of carbon/epoxy composite showing actual fiber packing geometry at 400X magnification
Voronoi 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 be quantified by the use of the Voronoi cell. Each point within the space of a Voronoi cell for a particular fiber is closer to the center of that fiber than it is to the center of any other fiber Voronoi cells Equivalent square cells, with Voronoi cell size, s s
Typical histogram of Voronoi distances and corresponding Wiebull distribution for a thermoplastic matrix composite. (From Yang, H. and Colton, J.S Polymer Composites, 51, 34–41. With permission.)
Elementary 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.
Assumptions: 1.Area fractions = volume fractions 2.Perfect bonding at fiber/matrix interface – no slip 3.Matrix is isotropic, fiber can be orthotropic 4.Fiber and matrix linear elastic 5.Lamina is macroscopically homogeneous, linear elastic and orthotropic
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,
Representative volume element and simple stress states used in elementary mechanics of materials models
Longitudinal normal stress In-plane shear stress Transverse normal stress Representative volume element and simple stress states used in elementary mechanics of materials models
Average stress over RVE: (3.19) Average strain over RVE: (3.20) Average displacement over RVE: (3.21)
Longitudinal Modulus RVE under average stress governed by longitudinal modulus E 1. Equilibrium: Note: fibers are often orthotropic. Rearranging, we get “Rule of Mixtures” for longitudinal stress (3.22) (3.23) Static Equilibrium
Hooke’s law for composite, fiber and matrix Stress – strain Relations (3.24) So that: (3.25)
Which means that, Assumption about average strains: (3.26) Geometric Compatibility (3.27) “Rule of Mixtures” – generally quite accurate – useful for design calculations
Variation of composite moduli with fiber volume fraction Predicted E 1 and E 2 from elementary mechanics of materials models Eq Eq. 3.40
Variation of composite moduli with fiber volume fraction Comparison of predicted and measured E 1 for E-glass/polyester. (From Adams, R.D., Engineered Materials Handbook, Vol. 1, Composites, 206–217.)
Strain Energy Approach (3.28) Where strain energy in composite, fiber and matrix are given by, (3.29a) (3.29b) (3.29c)
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)
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 a 1 and b 1, then
Transverse Modulus RVE under average stress Response governed by transverse modulus E 2 Geometric compatibility: From definition of normal strain, (3.34) (3.35)
Thus, Eq.(3.34) becomes (3.36) Or (3.37) Where (3.38) 1-D Hooke’s laws for transverse loading:
Where Poisson strains have been neglected. Combining (3.37) and (3.38), (3.39) Assuming that We get (3.40)
- “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)
Then substituting these expressions for and in (3.28) We get (3.43) Solving (3.42) and (3.43) simultaneously for a 2 and b 2, we get for E-glass/epoxy,
In-Plane Shear Modulus, G 12 Using compatibility of shear displacement and assuming equal stresses in fiber and matrix: (Not very accurate) (3.47) Major Poisson’s Ratio, υ 12 Using compatibility in 1 and 2 directions: (Good enough for design use) (3.45)
Design Equations Elementary mechanics of materials Equations derived for G 12 and E 2 are not very useful – need to develop improved models for G 12 and E 2.
Improved Mechanics of Materials Models for E 2 and G 12 Mechanics of materials models refined by assuming a specific fiber packing array. Example: Hopkins – Chamis method of sub-regions RVE
Convert RVE with circular fiber to equivalent RVE having square fiber whose area is the same as the circular fiber. Division of representative volume element into sub regions based on square fiber having equivalent fiber volume fraction. d RVE s A B A sfsf sfsf A B A Sub Region A Sub Region B Sub Region A
Equivalent Square Fiber: (from ) (3.48) Size of RVE: (3.49) For Sub Region B: s sfsf sfsf
Following the procedure for the elementary mechanics of materials analysis of transverse modulus: (3.50) but (3.51) So that (3.52)
For sub regions A and B in parallel, (3.53) Or finally (3.54) Similarly,
Simplified Micromechanics Equations (Chamis) Only used part of the analysis for sub region B in Eq. (3.52): (3.52) Fiber properties E f2 and G f12 in tables inferred from these equations.
Semi empirical Models Use empirical equations which have a theoretical basis in mechanics Halpin-Tsai Equations (3.63) Where (3.64)
And curve-fitting parameter 2 for E 2 of square array of circular fibers 1 for G 12 As Rule of Mixtures As Inverse Rule of Mixtures
Tsai-Hahn Stress Partitioning Parameters let Get (3.66) Where stress partitioning parameter (when get inverse Rule of Mixtures) (3.65)
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
Micromechanical 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.
RVE Example: Square packing array Matrix Fiber Due to double symmetry, we only need to consider one quadrant of RVE Matrix Fiber
The 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.