An adaptive phase-field model based on bilinear elements for tensile-compressive-shear fracture

Highlights

• A new phase-field model is proposed for tensile-compressive-shear fracture.

• A mesh adaptive scheme with a robust adaptive criterion is developed.

• The bilinear multi-node element is embedded in the adaptive scheme.

• A new degenerative triangular multi-node element is proposed.

Abstract

An efficient adaptive phase-field method based on bilinear elements for tensile-compressive-shear fracture is developed. On the one hand, refined meshes are needed near the crack path to obtain accurate results, and the computational efficiency is very low if a prior mesh refinement is applied, especially when the crack paths are unknown. On the other hand, conventional phase-field models are not applicable to tensile-compressive-shear fracture under complex stress states at present. In this paper, an adaptive scheme is proposed to improve the computational efficiency, in which a bilinear multi-node element is adopted to avoid using high order quadrature and shape functions when new nodes are inserted in elements, and a new multi-node triangular element is outlined to expand the scope of application of the proposed adaptive method. Another important aspect of the contribution is the development of the phase-field model for tensile-compressive-shear fracture under complex stress states, in which a universal fracture criterion is embedded. To determine the optimal parameters in the adaptive scheme and demonstrate the advantages of the proposed bilinear adaptive method, a series of numerical examples are performed for sensitivity analysis. Comparison with experimental results is also conducted to validate the proposed phase-field model for tensile-compressive-shear fracture under complex stress states.

Introduction

Prediction of crack evolution and fracture of materials is still challenging in engineering. Numerical methods are important tools in fracture analysis due to their capacity in dealing with complex fracture strengths, patterns, and processes. In 1920, Griffith proposed the fracture criterion and established linear elastic fracture mechanics (LEFM). From then on, fracture mechanics has become an important branch of solid mechanics. Although LEFM quantified the relationship between crack length and material inherent resistance, it could not predict the process of crack nucleation, propagation, branching, and coalescence. To resolve the problem, a series of methods to obtain the crack evolution process [1] was proposed gradually, such as eXtended Finite Element Method (XFEM) [2], [3], [4], [5], Boundary Element Method (BEM) [6], [7], [8], Mesh-less Method (MM) [9], [10], [11], [12], [13], PeriDynamics [14], [15], [16], Discrete element method (DEM) [17] and Lattice Discrete Element Model (LDEM) [18]. However, some of these approaches need additional criteria to track the crack path explicitly due to the displacement field discontinuity on the crack [19], and there is a strong singularity at the crack tip in solving the constitutive equation [20], [21].

Confronting with the above challenge, Francfort and Marigo [22] proposed a variational formulation for fracture which substituted the stationary condition in Griffith's energy theory with a global minimization of the total energy in a fracture solid. Later, the regularization equation based on Γ-convergence [23] for the phase-field method was established by Bourdin et al. [24], [25]. Then Miehe et al. [26] outlined a thermodynamically consistent framework of the phase-field model and extended it to dynamic failure problems [27]. Other early contributions can be found in the works of Amor et al. [28], Kuhn and Müller [29], and Hakim and Karma [30]. By means of diffusing the crack in a finite width, the phase-field model successfully overcomes the limitation of Griffith's theory for which a pre-existing crack and a well-defined crack path are needed.

Just name a few theories that have been developed based on the phase-field model (PFM) in recent years. Ambati et al. [31] applied phase-field theory in ductile fracture of elastoplastic solids within the framework of the quasi-static kinematically linear regime. Further and similar work can be found in [32], [33], [34], [35]. Clayton and Knap [36] simulated the polycrystals with individual crystals having anisotropic elastic coefficients and anisotropic fracture properties. On this basis, Nguyen et al. [37] represented an improved model giving phase-field theory the ability to analyze the interactions and the competition between intergranular and transgranular fractures of polycrystal materials by combining cohesive zone model. Moreover, a general modular phase-field framework was established for anisotropic brittle failure by Teichtmeister et al. [38]. Some researchers [39], [40], [41], [42], [43], [44] took the phase-field theory into the hydraulic fracturing problem of porous materials. Zhou et al. [45] regarded phase-field as an interpolation function and outlined a fracture model driven by the elastic energy for poroelastic media. In addition, a large volume of research work on the phase-field model in other fields [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56] has also been published in recent years.

A parameter called the internal length scale is introduced as an additional parameter in the phase-field model, and it enables establishing a connection with the material strength, which itself allows capturing crack size effects that are not captured by Griffith's theory, i.e., the presence of a length scale enriches the model beyond Griffith. However, for the conventional phase-field model, the global response of the simulation result is largely dependent upon the internal length scale [57], [58], which is difficult to calibrate and causes some problems. The stress and displacement response such as the maximum critical stress can be considerably influenced by the change of the internal length scale. Although the rational value has been discussed [57], it is insufficient to reduce the sensitivity caused by it. Wu [59] proposed a unified phase-field model, within the framework of thermodynamics [26], in which the influence of the internal length scale can be eliminated in the overall global responses in some examples, and more details can be found in [60], [61], [62], [63], [64], [65]. On the other hand, in the conventional phase-field model, the cracks are mainly caused by tensile stress, and only tensile strain energy is taken into account for fracture. As a result, it is only applicable to some specific situations where shear and compressive stress are not important for fracture. Zhang et al. [66] considered the difference of energy release rates for shear crack and tensile crack, and proposed a modified model for Mode-I and Mode-II fracture. Recently, Wang et al. [67] embedded the unified tensile fracture criterion [68] in the unified phase-field model [59] and proposed a new model for mixed-mode fracture. Meanwhile, a model for compressive-shear fractures for rock-like materials has also been developed by Zhou et al. [69]. Some other relevant work has been reported in [70], [71] for different fracture mode of geologic materials. Although these models can simulate tensile-shear or compressive-shear fractures very well, there is few phase-field models that can simulate both tensile-shear and compressive-shear fractures at the same time under complex stress states. Fei et al. [72] proposed a relative model with two phase-fields to show tensile and shear cracks, but the theory was quite complex.

The application of the phase-field theory is limited due to its high computational costs. In particular, the process of crack prediction and the need for a prior refined mesh on the crack path are contradictory. Apparently, the adaptive method is one of the best ways to resolve the above problem. Patil et al. [73] proposed an adaptive method for the phase-field model where a refined mesh can be coarsened if necessary. Hirshikesh et al. [74] outlined another adaptive model for brittle fracture based on the scaled boundary finite element method. Some other researchers [75], [76], [77], [78], [79], [80], [81], [82] also applied adaptive methods in the phase-field model for fracture.

A novel bilinear multi-node element and the corresponding adaptive process were developed by Lim et al. [83]. The proposed element is different from the conventional multi-node element. It can insert nodes on the element boundary arbitrarily and does not raise the orders of quadrature and shape functions. As a result, the number of sub-elements that one element can be divided into is unlimited. Furthermore, it is proved to have advantages in both computational time and accuracy.

This paper aims to establish an adaptive phase-field method with high efficiency, flexibility, and accuracy for complex failure modes. On the one hand, a new phase-field model for tensile-compressive-shear fracture is proposed. More complex fractures under tensile-shear and compressive-shear conditions can be predicted using the proposed phase-field model.

On the other hand, to improve the computing efficiency as much as possible and overcome the limitation on the construction of shape functions for conventional adaptive elements, the bilinear multi-node element is adopted for adaptation. Moreover, a robust and specific adaptive criterion for the phase-field model is presented in the paper, which can obtain the most reasonable mesh distribution and global response. The parameters in the adaptive criterion are determined by sensitivity analysis. Besides, the detailed formulas for the bilinear quadrilateral multi-node element are given, and a new degenerative bilinear triangular multi-node element including its integral form is proposed. Although some adaptive methods for the phase-field model have been proposed [74], [75], [80], [84], they did not take the initial size of elements into account and the models may cause unnecessary mesh refinement for the smaller elements when the mesh is nonuniform. Therefore, a specific linear adaptive criterion is outlined in this paper. By combining the above contributions, an accurate and efficient adaptive method is developed.

This paper is organized as follows. The theory of a new phase-field model for tensile-compressive-shear fracture is proposed in Section 2. The bilinear multi-node quadrilateral and triangular elements are addressed in Section 3. The procedure for numerical implementation is briefly summarized, and the adaptive scheme based on the phase-field variable is proposed in Section 4. Section 5 provides some numerical examples to determine the optimal adaptive parameters and demonstrate the effectiveness of the proposed adaptive method for complex fracture mode in comparison with experiments.