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

   https://doi.org/10.1115/1.4050279  

   Wu, Chunlin; Yin, Huiming (June 2021, Journal of Applied Mechanics)

   null (Ed.)

   Abstract 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. 

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2. The effects of boundary and inhomogeneities on the delamination of a bi-layered material system

   https://doi.org/10.1177/10567895231216008  

   Wu, Chunlin; Yin, Huiming (November 2023, International Journal of Damage Mechanics)

   The inclusion-based boundary element method (iBEM) is developed to calculate the elastic fields of a bi-layered composite with inhomogeneities in one layer. The bi-material Green’s function has been applied to obtain the elastic field caused by the domain integral of the source fields on inclusions and the boundary integral of the applied loads on the surface. Using Eshelby’s equivalent inclusion method (EIM), the material mismatch between the particle and matrix phases is simulated with a continuously distributed source field, namely eigenstrain, on inhomogeneities so that the iBEM can calculate the local field. The stress singularity along the interface leads to the delamination of the bimaterials under a certain load. The crack’s energy release rate (  J) is obtained through the J-integral, which predicts the stability of the delamination. When the stiffness of one layer increases, the J-integral increases with a higher gradient, leading to lower stability. Particularly, the effect of the boundary and inhomogeneity on the J-integral is illustrated by changing the crack length and inhomogeneity configuration, which shows the crack is stable at the beginning stage and becomes unstable when the crack tip approaches the boundary; a stiffer inhomogeneity in the neighborhood of a crack tip decreases J and improves the fracture resistance. For the stable cracking phase, the J-integral increases with the volume fraction of inhomogeneity are evaluated. The model is applied to a dual-glass solar module with air bubbles in the encapsulant layer. The stress distribution is evaluated with the iBEM, and the J-integral is evaluated to predict the delamination process with the energy release rate, which shows that the bubbles significantly increase the J-integral. The effect of the bubble size, location, and number on the J-integral is also investigated. The present method provides a powerful tool for the design and analysis of layered materials and structures. 

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3. Effect of defects and reinforcements on the mechanical behaviour of a bi-layered panel

   https://doi.org/10.1098/rspa.2022.0754  

   Wu, C. L.; Zhang, L. L.; Yin, H. M. (March 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This paper investigates the mechanical behaviour of a bi-layered panel containing many particles in one layer and demonstrates the size effect of particles on the deflection. The inclusion-based boundary element method (iBEM) considers a fully bounded bi-material system. The fundamental solution for two-jointed half spaces has been used to acquire elastic fields resulting from source fields over inclusions and boundary-avoiding multi-domain integral along the interface. Eshelby’s equivalent inclusion method is used to simulate the material mismatch with a continuously distributed eigenstrain field over the equivalent inclusion. The eigenstrain is expanded at the centre of the inclusion, which provides tailorable accuracy based on the order of the polynomial of the eigenstrain. As a single-domain approach, the iBEM algorithm is particularly suitable for conducting virtual experiments of bi-layered composites with many defects or reinforcements for both local analysis and homogenization purposes. The maximum deflection of solar panel coupons is studied under uniform vertical loading merged with inhomogeneities of different material properties, dimensions and volume fractions. The size of defects or reinforcements plays a significant role in the deflection of the panel, even with the same volume fraction, as the substrate is relatively thin. 

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4. Finite elements for div div conforming symmetric tensors in three dimensions

   https://doi.org/https://doi.org/10.1090/mcom/3700  

   Chen, Long; Huang, Xuehai (December 2021, Mathematics of computation)

   Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. Besides the normal-normal component, another trace involving combination of first order derivatives of stress should be continuous across the face. Due to the rigid of polynomials, the symmetric stress tensor element is continuous at vertices, and on the plane orthogonal to each edge. Hilbert complex and polynomial complexes are presented and several decomposition of polynomial vector and tensors spaces are revealed from the complexes. The constructed divdiv conforming elements are exploited to discretize the mixed formulation of the biharmonic equation. Optimal order and superconvergence error analysis is provided. Hybridization is given for the ease of implementation. 

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5. Chance-constrained optimal design of porous thermal insulation systems under spatially correlated uncertainty

   https://doi.org/10.1007/s00158-025-04125-5  

   Singh, Pratyush Kumar; Faghihi, Danial (September 2025, Structural and Multidisciplinary Optimization)

   This paper introduces a computationally efficient framework for the optimal design of engineering systems governed by multiphysics, nonlinear partial differential equations (PDEs) and subject to high-dimensional spatial uncertainty. The focus is on 3D printed silica aerogel-based thermal break components in building envelopes, where the objective is to maximize thermal insulation performance while ensuring mechanical reliability by mitigating stress concentrations. Material porosity is modeled as a spatially correlated Gaussian random field, yielding a high-dimensional stochastic design space whose dimensionality corresponds to the mesh resolution after finite element discretization. A robust design objective is employed, incorporating statistical moments of the thermal performance metric and in conjunction with a probabilistic (chance) constraint that restricts the p-norm of the von Mises stress field below a critical threshold, effectively controlling stress concentrations across the domain. To alleviate the substantial computational burden associated with Monte Carlo estimation of statistical moments, a second-order Taylor series approximation is introduced as a control variate, significantly accelerating convergence. Furthermore, a continuation-based strategy is developed to regularize the non-differentiable chance constraints, enabling the use of an efficient gradient-based Newton–Conjugate Gradient optimization algorithm. The proposed framework achieves computational scalability that is effectively independent of the stochastic design space dimensionality. Numerical experiments on two- and three-dimensional thermal breaks in building insulation demonstrate the method’s efficacy in solving large-scale, PDE-constrained, chance-constrained optimization problems with uncertain parameter spaces reaching dimensions in the hundreds of thousands. 

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