Stress and Strain Concentrations Inside Random Two-Phase Elastic Composites

Failure initiation in composite materials is a multiscale phenomena. Central to the analysis is the assessment of the load transfer between length scales. It is common knowledge that the load transfer can result in local stress and strain fields that are significantly greater than the applied macroscopic forces. Quantities useful for the study of load transfer include higher order moments of the stress and strain fields inside the composite. The higher moments are sensitive to local field concentrations generated by the interaction between the microstructure and the macroscopic load. These quantities have seen extensive application in the theoretical analysis of material failure, see [3].

In this project, composites made from two linear isotropic elastic materials are considered. It is assumed that only the volume (area) fraction and elastic properties of each elastic material are known. Optimal lower bounds on the local stress and strain fields are established for several loading conditions. These bounds provide the minimum amount of local field amplification that can be expected from this class of composites. The cases covered by this analysis include:

• Optimal lower bounds on the higher order moments and the L^∞ norm of the local stress and strain fields when the applied macroscopic loading is hydrostatic.
• Optimal lower bounds on the higher order moments and the L^∞ norm of the local stress and strain fields when the applied macroscopic loading is deviatoric.
• Optimal lower bounds on the higher order moments of the hydrostatic component of the local stress and strain fields for general applied macroscopic loading when the bulk moduli of the two materials are the same.
• Optimal lower bounds on the higher order moments and the L^∞ norm of the Von Mises equivalent stress and the deviatoric component of the strain for general applied macroscopic loading when the shear moduli of the two materials are the same.
• Optimal lower bounds on the higher order moments of the local stress and strain fields for a subspace of mixed mode loading characterized by a special dimensionless group of material parameters when the shear moduli of the two materials are the same.

The microgeometries that attain these bounds depend upon the macroscopic loading and material properties. Several distinguished parameter regimes are identified where the optimal configurations are given by layered materials, Hashin and Shtrikman coated sphere (cylinder) assemblages [2], or coated confocal ellipsoid (ellipse) assemblages [4] and [5]. It is well-known that these microgeometries give extreme effective properties, see for example [1]. In this analysis, it is shown that these microgeometries give extreme field properties.

To give an idea of the results, we present optimal lower bounds on the local stress field when the sample is subjected to a hydrostatic loading. These bounds hold for all configurations of the two elastic materials subject to prescribed constraints on θ₁ and θ₂ (proportions of each material). In what follows, χ_i denotes the indicator function of the set occupied by the i^th material and d gives the dimensionality of the elastic problem under consideration. The volume or area average of a quantity q over a period cell Q is denoted by <q>.

Optimal lower bounds on the moments of the local stress.

The stress field inside material-1 satisfies

Moreover, for d=2(3), the local stress inside material-1 for the coated cylinder (sphere) assemblage with core of material-1 and coating of material-2 attains the lower bound (1) for every r in [2,∞].

The stress field inside material-2 satisfies

Moreover, for d=2(3), the local stress inside material-2 for the coated cylinder (sphere) assemblage with core of material-2 and coating of material-1 attains the lower bound (2) for every r in [2,∞].

Optimal lower bound on the L^∞ norm of the local stress.

The stress field inside the composite satisfies

Moreover, for d=2(3), the stress field inside the coated cylinder (sphere) assemblage with core of material-1 and coating of material-2 attains the lower bound (3).

References

[1]

Allaire, G. and R.V. Kohn (1993). “Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions.”Quart. Appl. Math. 51, pp. 675–699.

[2]

Hashin, Z. and Shtrikman, S., (1963). “A variational approach to the theory of the elastic behavior of multiphase materials.” J. Mech. Phys. Solids. 11, pp. 127–140.

[3]

Kelly, A. and Macmillan, N.H., (1986). Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford.

[4]

Milton, G.W., (1980). “Bounds on the complex dielectric constant of a composite material.” Appl. Phys. Lett., 37, pp. 300–302. er-Verlag, New York.

[5]

Tartar, L., (1985). “Estimation Fine des Coefficients Homogénéisés.” in P. Kree (ed.), E. De Giorgi colloquium (Paris, 1983), Pitman Publishing Ltd., London, pp. 168–187.

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