Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain

This paper presents the Eshelby’s tensor of a polygonal inclusion with a polynomial eigenstrain, which can provide an elastic solution to an arbitrary, convex inclusion with a continuously distributed eigenstrain by the Taylor series approximation. The Eshelby’s tensor for plane strain problem is derived from the fundamental solution of isotropic Green’s function with the Hadmard regularization, which is composed of the integrals of the derivatives of the harmonic and biharmonic potentials over the source domain. Using the Green’s theorem, they are converted to two line (contour) integrals over the polygonal cross section. This paper evaluates them by direct analytical integrals. Following Mura’s work, this paper formulates the method to derive linear, quadratic, and higher order of the Eshelby’s tensor in the polynomial form for arbitrary, convex polygonal shapes of inclusions. Numerical case studies were performed to verify the analytic results with the original Eshelby’s solution for a uniform eigenstrain in an ellipsoidal domain. It is of significance to consider higher order terms of eigenstrain for the polygon-shape inclusion problem because the eigenstrain distribution is generally non-uniform when Eshelby’s equivalent inclusion method is used. The stress disturbance due to a triangle particle in an infinite domain is demonstrated by comparison with the results of the finite element method (FEM). The present solution paves the way to accurately simulate the particle-particle, partial-boundary interactions of polygon-shape particles.

Micromechanics-based approach for the effective estimation of the elastic properties of fiber-reinforced polymer matrix composite

In this paper, we proposed a revised Mori–Tanaka model for the effective estimation of the elastic properties at lower fiber volume fraction. A review of some notable micromechanics-based models with the theories proposed by Voigt and Reuss, Hashin–Shtrikman model, Mori–Tanaka method and dilute dispersion scheme is carried out, and a critique is presented focusing on the limitations of these models. Finite Element (FE) simulations are performed using Representative Volume Element (RVE) technique to rationalize the analytical results. Our results revealed that revised Mori–Tanaka estimates and FE predictions are in agreement. Elastic properties of the test material are dependent on size of RVE suggesting the effective elastic modulus evaluated using RVE forms the lower bounds of true effective values. However, we still believe that there is room for the debate for evaluating the elastic properties of these composites at larger volume fractions with the inclusion of Eshelby’s tensor in Mori–Tanaka scheme. Thus the efficacy of micromechanics-based models for the effective estimation of elastic properties of polymer matrix composites is highlighted. Our findings may provide new significant insights of the effective estimation of elastic properties of PMC using micromechanics-based approach.

Green’s Function and Eshelby’s Tensor Based on Mindlin’s 2nd Gradient Model: An Explicit Study of Cylindrical Inclusion Case

Based on Mindlin’s 2nd gradient model that involves two length-scale parameters, Green’s function, Eshelby tensor and Eshelby-like tensor for an inclusion of arbitrary shape are derived. It is proved that the Eshelby tensor consists of two parts: the classical Eshelby tensor and a gradient part including the length-scale parameters, which enable the interpretation of the size effect. When the strain gradient is not taken into account, the obtained Green’s function and Eshelby tensor reduce to its analogue based on the classical elasticity. For the cylindrical inclusion case, the Eshelby tensor in and outside the inclusion, the volume average of the gradient part and the Eshelby-like tensor are explicitly obtained. Unlike the classical Eshelby tensor, the results show that the components of the new Eshelby tensor vary with the position and the inclusion dimensions. It is demonstrated that the contribution of the gradient part should not be neglected.

Eshelby's tensor for embedded inclusions and the Elasto-Capillary phenomenon

The elastic state of an embedded inclusion undergoing a stress-free transformation strain was the subject of John Douglas Eshelby's now classical paper in 1957. This paper, the subject of which is now widely known as “Eshelby's inclusion problem”, is arguably one of the most cited papers in solid mechanics and several other branches of physical sciences. Applications have ranged from geophysics, quantum dots to composites. Over the past two decades, due to an interest in all things “small”, attempts have been made to extend Eshelby's elastic analysis to the nanoscale by incorporating capillary or surface energy effects. In this note, we revisit a particular formulation that derives a very general expression for the elasto-capillary state of an embedded inclusion. This approach, that closely mimics that of Eshelby's original paper, appears to have the advantage that it can be readily used for inclusions of arbitrary shape (for numerical calculations) and provides a facile route for approximate solutions when closed-form expressions are not possible. Specifically, in the case of inclusions of constant curvature (sphere, cylinder) subject to some simplifications, closed-form expressions are obtained.