Generalized Eshelby Integral Formula for Multiple Inclusion Composite Materials

Abstract

Eshelby integral formula, which was initially derived for a single inclusion embedded in all matrix system, plays a fundamental role in the micromechanics of composite or heterogeneous materials. The integral formula is remarked to be readily generalized to the case of multiple inclusions. However, the proof for such case has never been established. Herein, the integral formula is reformulated within the frameworks of generalized self-consistent method for the case of multiple inclusion composite system and thus, establishing the proof of its generalized version.

Appendix

Herein, the mathematical treatments based on the continuum tensor mechanics in multiscale theory of solids [13,14,15,16,17], converting the energy expression from the volume integral shown in Eq. (5) to a surface integral form given in Eq. (6) of the manuscript, are illustrated and explained for the sake of clarity in understanding the conversion procedures. Focusing on the first term in the integral expression of Eq. (5) without specifically referring to any of the mediums, we have

$$ \frac{1}{2V}\int \limits_{V} \sigma_{ij} \varepsilon_{ij} dV $$

(32)

where σ_ij and ε_ij are the respective stress and strain tensor fields of a microscale medium under uniform boundary conditions. Next, we wish to get a different expression for the strain tensor field shown in Eq. (32). Under the assumption of where the displacement gradient fields in the RVE are small compared to the unity, the displacement gradients u_i,j can be represented as

$$ \left\langle {u_{i,j} } \right\rangle = \left\langle {\varepsilon_{ij} } \right\rangle + \left\langle {\omega_{ij} } \right\rangle $$

(33)

where comma denotes derivative, ε_ij is the infinitesimal strain tensor and ω_ij is the infinitesimal rigid body rotation tensor. In micromechanics analysis, the small rigid body rotation tensor in microscale is typically taken to be zero, i.e. ω_ij = 0. As such, with the help of Eqs. (33), (32) can now be rewritten as

$$ \frac{1}{2V}\int \limits_{V} \sigma_{ij} u_{i,j} dV. $$

(34)

Next, we wish to have a different form of Eq. (34) so that it will be convenient for us to apply divergence theorem to that equation. Recalling the full form of equilibrium equations, we have

$$ \left\langle {\sigma_{ij,j} } \right\rangle + \left\langle {b_{i} } \right\rangle = \left\langle {a_{i} } \right\rangle $$

(35)

where σ_ij are components of Cauchy stress, b_i is a body force, ρ is mass density and a_i is an acceleration. Under static condition where a_i = 0, the microscale inertia force shown on the right-hand side of Eq. (35) vanishes. Furthermore, by considering no microscale body forces exist in the RVE, we have b_i = 0. Even if exist such as in the case of Fig. 3c, these body forces are considered ‘invisible’ in the RVE equilibrium equations since they are non-fluctuated body forces [17]. Moreover, one can treat the problems of the body forces negligible in comparison to the boundary conditions especially in the context of the GSC method where its RVE has an infinite radius or size. Hence, the equilibrium equations of Eq. (35) become

$$ \left\langle {\sigma_{ij,j} } \right\rangle = 0. $$

(36)

With Eqs. (36), (34) is implied to have the following form

$$ \frac{1}{2V}\int \limits_{V} \left( {\sigma_{ij,j} u_{i} + \sigma_{ij} u_{i,j} } \right)dV. $$

(37)

Meaningfully, Eq. (37) can be rewritten as

$$ \frac{1}{2V}\int \limits_{V} \left( {\sigma_{ij} u_{i} } \right)_{,j} dV. $$

(38)

Equation (38) is now ready for the application of divergence theorem, which is also known as the Gauss’s or Ostrogradsky’s theorem, or sometimes referred to as the Gauss-Ostrogradsky’s theorem. In general, the theorem mathematically states that

$$ \int \limits_{V} \left( {t_{ij \ldots k} } \right)_{,q} dV = \int \limits_{S} t_{ij \ldots k} n_{q} dS $$

(39)

where t_ij…k is an arbitrarily order of continuous differentiable tensor field, n_q is a unit vector normal to surface S, which bounds the volume V. Basically, it is a relation, which relates an integral over a closed volume to an integral over its bounding surface. Thus, by using divergence theorem exemplified in Eq. (39), the volume integral of Eq. (38) can now be desirably transformed to its equivalent surface integral form, which gives us

$$ \frac{1}{2V}\int \limits_{S} \sigma_{ij} u_{i} n_{j} dS. $$

(40)

As we can see, Eq. (40) exhibits the typical average energy expression in surface integral form whenever the divergence theorem along with the equations of equilibrium are invoked and applied to its equivalent volume integral form given in Eq. (32) earlier.