A 3D multi-physics boundary element computational framework for polycrystalline materials micro-mechanics

A recently developed novel three-dimensional (3D) computational framework for the analysis of polycrystalline materials at the grain scale is described in this lecture. The framework is based on the employment of: i) 3D Laguerre-Voronoi tessellations for the representation of the micro-morphology of polycrystalline materials; ii) boundary integral equations for the representation of the mechanics of the individual grains; iii) suitable cohesive traction-separation laws for the representation of the multi-physics behavior of the interfaces (either inter-granular or trans-granular) within the aggregate, which are the seat of damage initiation and evolution processes, up to complete decohesion and failure. The lecture will describe the main features of the proposed framework, its main advantages, current issues and direction of potential further development. Several applications to the computational analysis of damage initiation and micro-cracking of common and piezoelectric aggregates under different loading conditions will be discussed. The framework could find profitable application in the multiscale analysis of polycrystalline components and in the design of micro-electromechanical devices (MEMS)

two-scale three-dimensional boundary element framework for degradation and failure in polycrystalline materials

A fully three-dimensional two-scale boundary element approach to degradation and failure in polycrystalline materials is proposed. The formulation involves the engineering component level (macroscale) and the material grain scale (micro-scale). The damage-induced local softening at the macroscale is modelled employing an initial stress approach. The microscopic degradation processes are explicitly modelled by associating Representative Volume Elements (RVEs) to relevant points of the macro continuum and employing a three-dimensional grain-boundary formulation to simulate intergranular degradation and failure in the microstructural Voronoi-type morphology through cohesive-frictional contact laws. The scales coupling is achieved downscaling macro-strains as periodic boundary conditions for the RVE, while overall macro-stresses are obtained via volume averages of the micro-stress field. The comparison between effective macro-stresses for the damaged and undamaged RVE allows to define a macroscopic measure of material degradation. Some attention is devoted to avoiding pathological damage localization at the macro-scale. The multiscale processing algorithm is described and some preliminary results are illustrated

A cohesive boundary element approach to material degradation in three-dimensional polycrystalline aggregates

A new three-dimensional grain-level formulation for intergranular degradation and failure in polycrystalline materials is presented. The polycrystalline microstructure is represented as a Voronoi tessellation and the boundary element method is used to express the elastic problem for each crystal of the aggregate. The continuity of the aggregate is enforced through suitable conditions at the intergranular interfaces. The grain-boundary model takes into account the onset and evolution of damage by means of an irreversible linear cohesive law, able to address mixed-mode failure conditions. Upon interface failure, a non-linear frictional contact analysis is introduced for addressing the contact between micro-crack surfaces. An incremental-iterative algorithm is used for tracking the micro-degradation and cracking evolution. The behavior of a polycrystalline specimen under tensile load has been performed, to show the capability of the formulation

Brittle failure in polycrystalline RVEs by a grain-scale cohesive boundary element formulation

Polycrystalline materials are commonly employed in engineering structures. For modern applica- tions a deep understanding of materials degradation is of crucial relevance. It is nowadays widely recognized that the macroscopic material properties depend on the microstructure. The polycrystalline microstructure is characterized by the features of the grains and by the phys- ical and chemical properties of the intergranular interfaces, that have a direct influence on the evolution of the microstructural damage. The experimental investigation of failure mechanisms in 3D polycrystals still remains a challenging task. A viable alternative, or complement, to the experiments is Computational Micromechanics. The present-day availability of cheaper computational power is favoring the advancement of the sub- ject. A popular approach for polycrystalline fracture problems consists in the use of cohesive sur- faces embedded in a Finite Element (FE) representation of the microstructure, so that the evolution of microcracks stems as an outcome of the simulation, without any assumptions, see e.g. [4]. An alternative to the FEM is the Boundary Element Method (BEM). A 2D cohesive BE formula- tion for intergranular failure and a 3D BE formulation for polycrystalline materials homogeniza- tion have been recently proposed [1–3]. In this work, a novel 3D grain-level model for the study of polycrystalline intergranular degra- dation and failure is presented. The microstructures are generated as Voronoi tessellations, that mimic the main statistics of polycrystals. The formulation is based on a grain-boundary integral representation of the elastic problem for the crystals, seen as anisotropic domains with random crystallographic orientation in space. The integrity of the aggregate is restored by enforcing suit- able intergranular conditions. The evolution of intergranular damage is modeled using an extrinsic irreversible mixed-mode cohesive linear law. Upon interface failure, non-linear frictional con- tact analysis is used, to address separation, sliding or sticking between micro-crack surfaces. An incremental-iterative algorithm is used for tracking the micro-cracking evolution. Several numeri- cal tests have been performed and they demonstrated the capability of the formulation to track 3D micro-cracking, under either tensile or compressive loads

A multiscale approach to polycrystalline materials damage and failure

A two-scale three-dimensional approach for degradation and failure in polycrystalline materials is presented. The method involves the component level and the grain scale. The damageinduced softening at the macroscale is modelled employing an initial stress boundary element approach. The microscopic degradation is explicitly modelled associating Representative Volume Elements (RVEs) to relevant points of the macro continuum and employing a cohesive-frictional 3D grain-boundary formulation to simulate intergranular degradation and failure in the Voronoi morphology. Macro-strains are downscaled as RVEs' periodic boundary conditions, while overall macro-stresses are obtained upscaling the micro-stress field via volume averages. The comparison between effective macro-stresses for the damaged and undamaged RVEs allows to define a macroscopic measure of local material degradation. Some attention is devoted to avoiding pathological damage localization at the macro-scale. The multiscale processing algorithm is described and some preliminary results are illustrated

Fast Solution of 3D Elastodynamic Boundary Element Problems by Hierarchical Matrices

In this paper a fast solver for three-dimensional elastodynamic BEM problems formulated in the Laplace transform domain is presented, implemented and tested. The technique is based on the use of hierarchical matrices for the representation of the collocation matrix for each value of the Laplace parameter of interest and uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy for speeding up the overall analysis is presented and tested. The reported numerical results demonstrate the effectiveness of the technique

A computationally effective 3D Boundary Element Method for polycrystalline micromechanics

An effective computational framework for homogenization and microcracking analysis of polycrystalline RVEs is presented. The method is based on a recently developed grain-boundary formulation for polycrystalline materials and several enhancements over the original technique are introduced to reduce the computational effort needed to model three-dimensional polycrystalline aggregates, which is highly desirable, especially in a multiscale perspective. First, a regularization scheme is used to remove pathological entities, usually responsible for unduly large mesh refinements, from Voronoi polycrystalline morphologies. Second, an improved meshing strategy is used, with an aim towards meshing robustness, a requirement often challenged by the inherent high statistical variability of Voronoi tessellations. Additionally, for homogenization purposes, the use of periodic non-prismatic polycrystalline RVEs is proposed as an alternative to the classical prismatic RVEs, generally employed in the literature. The proposed overall scheme promotes a remarkable reduction in the number of DoFs of the problem in hand, and thus outstanding savings in terms of computational time and memory storage. Furthermore, the smoother meshing strategy, combined with a Newton-Raphson method, enhances the convergence of the microcracking algorithm

A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials

In this study, a novel three-dimensional micro-mechanical crystal-level model for the analysis of intergranular degradation and failure in polycrystalline materials is presented. The polycrystalline microstructures are generated as Voronoi tessellations, that are able to retain the main statistical features of polycrystalline aggregates. The formulation is based on a grain-boundary integral representation of the elastic problem for the aggregate crystals, that are modeled as three-dimensional anisotropic elastic domains with random orientation in the three-dimensional space. The boundary integral representation involves only intergranular variables, namely interface displacement discontinuities and interface tractions, that play an important role in the micromechanics of polycrystals. The integrity of the aggregate is restored by enforcing suitable interface conditions, at the interface between adjacent grains. The onset and evolution of damage at the grain boundaries is modeled using an extrinsic non-potential irreversible cohesive linear law, able to address mixed-mode failure conditions. The derivation of the tractionseparation law and its relation with potential-based laws is discussed. Upon interface failure, a non-linear frictional contact analysis is used, to address separation, sliding or sticking between micro-crack surfaces. To avoid a sudden transition between cohesive and contact laws, when interface failure happens under compressive loading conditions, the concept of cohesive-frictional law is introduced, to model the smooth onset of friction during the mode II decohesion process. The incremental-iterative algorithm for tracking the degradation and micro-cracking evolution is presented and discussed. Several numerical tests on pseudo- and fully three-dimensional polycrystalline microstructures have been performed. The influence of several intergranular parameters, such as cohesive strength, fracture toughness and friction, on the microcracking patterns and on the aggregate response of the polycrystals has been analyzed. The tests have demonstrated the capability of the formulation to track the nucleation, evolution and coalescence of multiple damage and cracks, under either tensile or compressive loads

Modelling Polycrystalline Materials: An Overview of Three-Dimensional Grain-Scale Mechanical Models

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Inter-Element Crack Propagation with High-Order Stress Equilibrium Element

The present contribution proposes a formulation based on the use of hybrid equilibrium elements (HEEs), for the analysis of inter-element delamination and fracture propagation problems. HEEs are defined in terms of quadratic stress fields, which strongly verify both the homogeneous and inter-element equilibrium equations and they are employed with interfaces, initially exhibiting rigid behavior, embedded at the elements’ sides. The interface model is formulated in terms of the same degrees of freedom of the HEE, without any additional burden. The cohesive zone model (CZM) of the extrinsic interface is rigorously developed in the damage mechanics framework, with perfect adhesion at the pre-failure condition and with linear softening at the post-failure regime. After a brief review, the formulation is computationally tested by simulating the behavior of a double-cantilever-beam with diagonal loads; the obtained numerical results confirm the accuracy and potential of the method