Theses Doctoral

Virtual experiments and designs of composites with the inclusion-based boundary element method (iBEM)

Wu, Chunlin

This dissertation develops and implements an effective numerical scheme, the inclusion-based boundary element method (iBEM), to investigate the mechanical and multi-physical properties of the composites containing arbitrarily shaped particles. Besides the linear elasticity and transient heat conduction problems shown in the dissertation, it can be extended to other problems, such as potential flows and Stokes flows. Through the combination of conventional boundary element method (BEM) and the Eshelby's equivalent inclusion method (EIM), the local field is obtained through superposition of the domain integral of eigen-fields and boundary integral equations.

Firstly, the boundary value problems of a composite containing various fully bonding phases of subdomains is introduced. Due to the continuity of displacement (potential) and traction (flux) at the interfaces between different material phases, the interfacial continuity equations are established, which can be solved with the multi-region BEM conventionally. Thanks to Eshelby's celebrated contribution, the material difference in inhomogeneity problems is simulated by an eigenstrain on the inclusion domain but with the same material properties as the matrix. Therefore, the boundary value problems with inhomogeneities can be transformed as domain integral of Green's function with the eigenstrain over the inclusion, where can be determined by the equivalent stress conditions in EIM. Hence, the algorithm of iBEM is formulated and established on the basis of boundary conditions and equivalent stress equations instead of various continuity constraint equations, which saves efforts in computational resources and pre/post-process.

The domain integral of Green's function is the key to the algorithm of iBEM, as it bridges the inhomogeneities and the boundary. The closed-form expression of domain integrals for ellipsoidal / elliptical inclusions with polynomial eigenstrain, polygonal and polyhedral inclusions with constant eigenstrain have already existed in the literature. However, it is not applicable to arbitrary particles with varying eigenstrain. This dissertation derives the closed-form domain integrals for polygon and polyhedral inclusions with polynomial eigenstrain source terms, which creates feasibility to solve the local field and effective material properties for composites with arbitrary particles.

Although the EIM with polynomial-form eigenstrain has been applied to simulate the material mismatch for ellipsoidal / elliptical inhomogeneities by using the Taylor's of eigenstrain field at the particle center, when it is extended to angular particles, the inaccuracy is significantly reduced due to the rapid and complicated eigenstrain variation in the neighborhood of vertices with the strong singular effects. Therefore, the domain discretization of an angular particle is proposed to tackle the complicated distribution of elastic fields, which keeps the features of exactness (no approximation of interior field) and 𝐂⁰ continuity of eigenstrain. Hereby, the iBEM is proposed to serve as an effective and powerful tool, which takes the advantages of both BEM and EIM. The interaction of inhomogeneities is considered in the process of constructing EIM equations, and boundary effects are taken into account as the contribution to displacement of the eigen-field over inhomogeneities, hence, a complete linear equation system can be established.

For the inclusion problems with a prescribed eigenstrain, no domain discretization is required because the exact elastic solution is obtained given the specific dimension of the geometry. Regarding to inhomogeneity problems, 1) the ellipsoidal / elliptical shape is versatile, which could be switched to various of shapes by adjusting the aspect ratio and orientations; 2) though the angular subdomain requires discretization, this method is rapidly convergent and no mesh is needed for the matrix. Therefore, this method enables the simulation of thousands 3𝐷 and 2𝐷 arbitrary shaped particles in a desk-top computer and the effective moduli can be obtained through virtual experiments (i.e, uni-axial loading) or periodic boundary conditions. This method can be easily extended to multi-physical problems, such as transient hear transfer, steady state heat, through changing the fundamental solutions accordingly. Three major packages have been added to the iBEM software, as transient heat transfer, closed-form 2D/3D domain integrals, and domain discretization method. Some case studies demonstrate the capability and applications of this method and software. This main contributions of the PhD studies are as follows:

1) The closed-form domain integrals for polygonal and polyhedral inhomogeneities have been derived based on the gravitational potential theory and transformed coordinates. The solutions are verified with the classic solution of circular and spherical potentials with polynomial source terms (i.e, linear and quadratic) by using many triangular and tetrahedral elements. It enables to solve the inhomogeneity problems with arbitrary particles.

2) Due to the discontinuity on the surfaces and edges of the subdomains and strong singular effects on the vertices, the variation of eigenstrain field is complicated in the neighborhood of edges and vertices. The domain discretization approach is proposed to provide a rapid convergent and effective solution in the infinite space. Different from the Taylor's expansion, the eigenstrain is assigned exactly at the nodes with shape functions instead of at the centroid of the elements, therefore, a 𝐂⁰ continuity is enforced. Here 3-node, 6-node triangular elements and 4-node, 10-node tetrahedral elements are implemented in the code of iBEM, which agree well with FEM but with much fewer of elements. Other types of element are also implementable in the same fashion.

3) The discretization method is applied to investigate the stress singularities of a vertex on an isosceles triangle embedded in an unbounded matrix. Two types of stress singularities are investigated: when the load is applied to the triangular inclusion with the same stiffness as the matrix, the singularity is caused by the irregular load distribution, namely load singularity, and can be exactly evaluated by integral of the potentials on the source with Eshelby's tensor. The second singularity, namely material singularity, is caused by the stiffness mismatch between the triangular inhomogeneity and the matrix under a uniform far field stress, in which the material mismatch is simulated by an eigenstrain. The relationship between the load singularity and material singularity is investigated, and the linkages of these singularities with line distributed force, cracking, and point force are discussed.

4) A parametric study of accuracy on stress field for uniform, linear and quadratic eigenstrain fields was performed and case studies have been presented to demonstrate the capability of iBEM for virtual experiments of ellipsoidal / elliptical inhomogeneities. Subsequently, combining the domain discretization method, iBEM is also applied to study the local elastic fields of the angular inhomogeneities. The effective material behavior is obtained with either large number of particles or periodic boundary condition (PBC) and some interesting discoveries of microstructure-dependent material behavior are reported with the aid of virtual experiments.

5) The iBEM is extended to multiphysical problems. The temperature and hear flux fields of composite materials containing phase change materials (PCM) for energy efficient buildings is demonstrated. Different from the static EIM, the thermal property mismatch between PCM particle and matrix phase is simulated with a uniformly distributed eigen-temperature gradient field and a fictitious heat source on the particle. With the equivalent heat flux conditions and the specific heat-temperature relationship, the eigen-temperature gradient and fictitious heat source can be solved and temperature field of the bounded domain can be calculated. Verified with FEM and laboratory measurements of the transient heat transfer within a building block containing a PCM capsule. Parametric studies have also been conducted to study the influences of the PCM location and volume fraction on the temperature fields of composites with multiple particles. The virtual experiments demonstrate the energy saving and phase delay by using the PCM-concrete wall panel.

In summary, the proposed iBEM algorithm bridges the gap between conventional EIM and BEM for virtual experiments of composites samples. The combination of shape functions and domain integrals of polygonal / polyhedral subdomain enables its application to arbitrary shaped particles. It serves as a powerful tool to conduct virtual experiments for composite materials with various geometry and investigate the effective moduli under uni-axial load of samples with large number of particles or under the periodic boundary condition. In the future, the iBEM will be implemented for time independent and dependent nonlinear behavior of composites, such as elastoplastic, viscoelastic, and dynamic elastic problems. In addition to the current parallel computing scheme, GPU can be employed to speed up particle - particle interactions.


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More About This Work

Academic Units
Civil Engineering and Engineering Mechanics
Thesis Advisors
Yin, Huiming
Ph.D., Columbia University
Published Here
September 8, 2021