Mathematical Problems in Engineering

Volume 2016, Article ID 4715696, 14 pages

http://dx.doi.org/10.1155/2016/4715696

## Modeling Crack Propagation in Polycrystalline Microstructure Using Variational Multiscale Method

Department of Aerospace Engineering, University of Michigan, 3025 FXB Building, 1320 Beal Avenue, Ann Arbor, MI 48109, USA

Received 3 February 2016; Accepted 24 April 2016

Academic Editor: Zhiqiang Hu

Copyright © 2016 S. Sun and V. Sundararaghavan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Crack propagation in a polycrystalline microstructure is analyzed using a novel multiscale model. The model includes an explicit microstructural representation at critical regions (stress concentrators such as notches and cracks) and a reduced order model that statistically captures the microstructure at regions far away from stress concentrations. Crack propagation is modeled in these critical regions using the variational multiscale method. In this approach, a discontinuous displacement field is added to elements that exceed the critical values of normal or tangential tractions during loading. Compared to traditional cohesive zone modeling approaches, the method does not require the use of any special interface elements in the microstructure and thus can model arbitrary crack paths. The capability of the method in predicting both intergranular and transgranular failure modes in an elastoplastic polycrystal is demonstrated under tensile and three-point bending loads.

#### 1. Introduction

Efficient microscale modeling tools are needed to compute microstructure-dependent properties of advanced structural alloys used in aerospace, naval, and automotive applications [1]. Microstructural effects become an important consideration in regions of stress concentrations such as notches, cracks, and contact surfaces. While crystal plasticity finite element (CPFE) method [2, 3] has emerged as an effective tool for simulating the mechanical response of the microscale containing aggregates of few hundred metallic crystals, simulation of “macroscale” components that contain millions of grains is a challenging task even when using current state-of-the-art supercomputers. Multiscale methods developed to address this problem can be broadly classified into two categories: concurrent and hierarchical methods. In hierarchical methods, microscale models that contain features significantly smaller than the coarse scale mesh resolution are simulated to recover the material properties at the macroscale. An example is the method of* computational homogenization* in which stresses in a polycrystal are volume averaged to compute the macroscopic stress state [4]. Homogenization approaches are inadequate in problems where there is no clear separation of length scales, for example, when simulating localization or failure phenomena [5].

In concurrent methods, both micro- and macroscale problems are solved together in a single domain using multiresolution meshes. Most popular among these methods are quasicontinuum methods, where resolution of meshes from atomistic to continuum scales has been considered in the past to study crack propagation [6]. In concurrent methods, treatment of interfaces between fine and coarse scale regions becomes problematic when multiphysical domains (e.g., crystal plasticity at fine scale and continuum plasticity at coarse scale) are considered. For example, Dunne et al. [7] employed a combined continuum crystal plasticity finite element approach for modeling a three-point bend test. One of the potential difficulties identified in the model was the inconsistency of yield stress at the interface between the domains. To minimize the effect, the choice of the critical resolved shear stress in the crystal plasticity model was made such that polycrystal yielding occurred at a value close to the continuum yield stress. In this paper, the interface problem is solved through the use of a single physical model at both the fine and coarse scales. This is achieved (i) by explicitly resolving single crystal deformation at the* microscale* (crack tip) using a crystal plasticity model and (ii) by employing a statistical model at the* macroscale* (far field), where polycrystal aggregates are represented using an orientation distribution function (ODF). Under an applied deformation, microstructure evolution in the far field is simulated by numerically evolving the ODF (rather than the microstructure itself) using conservation laws [8]. Such statistical approaches are significantly (several orders of magnitude) faster than full order (CPFE) methods [9, 10] and balance the trade-off between accuracy and computational efficiency in far-field regions.

The paper presents the first use of this efficient multiscaling framework for prediction of crack trajectories in a polycrystalline microstructure. Towards this end, a cohesive zone model (CZM) [11, 12] is introduced at the microscale for modeling the crack kinematics. In this approach, the crack is represented using special interface elements that obey a relationship between the interface traction and the crack separation. At the microscale, these relationships have been motivated from the results of atomistic simulations of grain boundaries [13–15]. Cohesive element-based approaches have been previously used for modeling intergranular crack paths [16–18]. However, in order to simulate both transgranular and intergranular crack paths in an elastoplastic material, these interface elements are needed practically at every element-to-element boundary. This both makes the problem computationally expensive and also results in unsatisfactory mesh convergence properties [19]. Other computational methods have emerged in the past two decades that incorporate displacement discontinuities using enrichment functions that are classified into element enrichment methods (e.g., variational multiscale method (VMM) [20, 21]) and nodal enrichment methods (e.g., Extended Finite Element Method (XFEM) [22, 23]). Sukumar et al. [24] employed XFEM to simulate intergranular cracks in elastic polycrystals. Rudraraju [25] and Rudraraju et al. [26] recently explored the use of VMM for prediction of crack propagation in elastic composites and show good comparison with experiments. However, the use of these methods has not yet been demonstrated in elastoplastic regimes, in particular, for cases involving intergranular to transgranular failure mode transitions at the microstructural level. In this work, a numerical framework based on VMM is proposed for addressing such a case. The paper is organized as follows. In Section 2, VMM scheme is briefly introduced. In Section 3, the polycrystalline constitutive model will be briefly introduced followed by the statistical (ODF) modeling approach in the far field. In Section 4, examples illustrating crack propagation at the microstructural level are presented, while concluding remarks are presented in Section 5.

#### 2. Introduction of Variational Multiscale Method

The microstructural simulations (in 2D) described here employ an updated Lagrangian framework wherein the configuration at the previous time step () serves as a reference configuration for the next time step (). Let refer to the total deformation gradient in the reference configuration () at time with respect to the initial undeformed configuration () at time . Similarly, the total deformation gradient in the current configuration () at time is written as . Then denotes the relative deformation gradient between the two configurations; that is,

A crack in the reference configuration is represented as a surface . The displacement jump in the crack is denoted by . The crack surface in this configuration has a normal direction** n** and tangential direction** m** as shown in Figure 1.