skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization
Abstract 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 đ¶0 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.  more » « less
Award ID(s):
1738802 1935773 1941244 1762891
PAR ID:
10282844
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of Applied Mechanics
ISSN:
0021-8936
Page Range / eLocation ID:
1 to 35
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, 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. 
    more » « less
  2. ; (, 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 decreasesJand 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. 
    more » « less
  3. ; ; (, 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. 
    more » « less
  4. ; (, 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. 
    more » « less
  5. ; ; (, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)
    Eshelby’s equivalent inclusion method (EIM) has been formulated to solve harmonic heat transfer problems of an infinite or semi-infinite domain containing an inclusion or inhomogeneity. For the inclusion problem, the heat equation is reduced to a modified Helmholtz’s equation in the frequency domain through the Fourier transform, and the harmonic Eshelby’s tensor is derived from the domain integrals of the corresponding Green’s function in the form of Helmholtz’s potential. Using the convolution property of the Fourier space, Helmholtz’s potential with polynomial-form source densities is integrated over an ellipsoidal inclusion, which is reduced to a one-dimensional integral for spheroids and an explicit, exact expression for spheres. The material mismatch in the inhomogeneity problem is simulated by continuously distributed eigen-fields, namely, the eigen-temperature-gradient (ETG) and eigen-heat-source (EHS) for thermal conductivity and specific heat, respectively. The proposed EIM formulation is verified by the conventional boundary integral method with the harmonic Green’s function and multi-domain interfacial continuity, and the accuracy and efficacy of the solution are discussed under different material and load settings. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email