Anisotropic thermal-conductivity degradation in the phase-field method accounting for crack directionality

Highlights

• Thermal-conductivity degradation is used to model heat-transfer physics across cracks.

• A new anisotropic approach that depends on the phase-field gradient is proposed.

• The temperature and heat flux errors are reduced compared with isotropic approaches.

• The solution to heat transfer problems with fracture becomes length scale insensitive.

• Inaccurate modeling of heat transfer may lead to poor capture of failure modes.

Abstract

Dynamic loads applied to metals may lead to brittle or ductile fracture depending on the imposed strain rates, material properties, and specimen geometry. With an increase in the applied velocity, brittle to ductile failure mode transition may also be observed. Furthermore, at high strain rates, ductile fracture may be preceded by shear bands, which are narrow bands of intense plastic deformation, typically accompanied by a significant rise in temperature. Reliable models are needed to predict the response of metals subject to dynamic loads. In particular, capturing the interplay between heat conduction and crack propagation using the phase-field fracture method is still an open research field. To accurately capture the heat transfer physics across crack surfaces, damage models degrading thermal-conductivity are necessary. While isotropic thermal-conductivity degradation models were proposed, they may lead to errors as they are indifferent to crack directionality. In our current work, a new anisotropic approach is proposed in which thermal-conductivity, which depends on the phase-field gradient, is degraded solely across the crack. It is shown that this approach improves the near-field approximation of temperature and heat flux compared with isotropic degradation, when taking the discontinuous crack solutions as reference. Additionally, the anisotropic degradation approach is implemented in a unified dynamic fracture model and a couple of examples demonstrate the viability of this approach.

Introduction

Dynamic loads may result in brittle or ductile fracture of metallic structures, depending on the imposed strain rates, material type, and geometry. The occurrence of brittle fracture is characterized by fast crack propagation in materials with low ductility accompanied by minimal plastic deformation and heat dissipation. On the other hand, ductile fracture is the failure mode in materials which can accommodate large plastic deformation. Additionally, ductile fracture developed under high strain rate loading may be preceded by shear bands, which are intense plastic deformation zones, typically accompanied by significant temperature increase [1], [2], [3], [4], [5]. Presumably, shear bands result from a thermal softening mechanism due to plastic heating although other softening mechanisms, such as dynamic recrystallization, are believed to play a prominent role [6], [7], [8], [9], [10], [11], [12], [13], [14]. The microstructural and textural evolution of shear bands in metals are studied in [15] along with the estimation of their width. Brittle/ductile fracture and shear banding are all failure modes that occur at distinct spatial and temporal scales. Hence, for predictive numerical simulations, it is crucial to account for all these failure modes (shear bands and cracks) explicitly.

The dynamic fracture of metals under high strain rate loading is a coupled thermo-mechanical problem, in which a fraction of the inelastic mechanical work is converted to heat during the deformation, through the so-called Taylor–Quinney parameter. While this fraction is nearly universally assumed to be a constant close to one [16], experimental data and theoretical arguments show that in general the Taylor–Quinney parameter is variable, and depends on the strain, strain rate, and temperature [17]. In addition, the formation of shear bands under dynamic loading conditions is a rapid process, and as a result, an adiabatic assumption in which thermal-conductivity is neglected has been extensively used in the literature; see e.g. [18]. Although this assumption simplifies the model, it omits the steep temperature gradients, which counterbalance heat production. Hence, neglecting thermal diffusion eliminates an important physical length scale in the system needed to regularize the equations, which is required for mesh insensitive numerical formulations [19], [20], [21], [22]. Another approach to derive a mesh-insensitive formulation is to introduce an additional plastic strain measure as an internal variable (apart form the standard equivalent plastic strain) and use a large deformation gradient-plasticity theory as presented in [23].

Modeling the interplay between heat transfer and crack propagation is also important in engineering science as shown, for example, by the experiments in [24], which analyzed the temperature distribution around propagating crack tips in steel components. Specifically, it is speculated that the nucleation growth and coalescence of voids in front of the crack tip lead to a dramatic reduction of yield stress inside the process zone, which enhances the mechanism of shear band formation between coalescing voids and may reduce the dynamic fracture toughness. The yield stress reduction can be described by a GTN model [25], [26], [27], and the void coalescence and growth play important role to ductile fracture because they affect both the heat conduction and plastic dissipation in a ductile material. To model ductile fracture for a range of stress triaxialities, a void damage model is combined with the Mohr–Coulomb model at the slip system scale in [28]. Additional important experimental work investigated the impact of crack tip velocity on the plastic zone and temperature distribution around crack tips [29]. Experimental results, such as thermal images in [30], provide a way to test predictions of models which estimate temperature distributions and plastic zone sizes at stationary cracks. In [31], it is reported that (i) crack initiation can be affected by the slip patterns and plastic zone size and shape, and (ii) kink shear bands can be observed perpendicular to the slip direction.

In this work, cracks are modeled as continuous diffusive entities using the phase-field method. The extent of material damage is characterized by a scalar internal field variable $c$ (called phase-field) that ranges from 0 (undamaged material) to 1 (fully fractured material) [32], [33], [34], [35]. For elastic materials, the phase-field fracture method is derived from minimization of the stored energy using the Griffith criterion for crack growth [36], which is expressed as the summation of its elastic and crack surface energies. The elastic energy is degraded using a stress degradation function to account for the energy which is released due to crack formation, and the surface energy is defined in terms of the phase-field, its gradient, and a length scale parameter ${\ell }_0$ [37], [38]. In the case of brittle fracture, ${\ell }_0$ should be assigned the smallest value possible, taking into consideration available computational resources, to obtain a sharp representation of the crack discontinuity. However, the choices of a stress degradation function and length scale parameter ${\ell }_0$ affect the critical values of stress ${\sigma }_c$ (maximum stress) and the corresponding strain ${\epsilon }_c$ as shown in [39]. In the case of cohesive fracture, a phase-field formulation is proposed in [40] where a new degradation function is derived based on a local stress–strain response in the cohesive zone. Also, a length scale insensitive phase-field model of cohesive fracture is presented in [41]. Finally, in the case of ductile fracture, ${\ell }_0$ can be considered a material parameter for engineering applications and may define a non-negligible crack process zone since the critical stress value should be finite. In phase-field models based on the Griffith criterion, the surface stresses which can be applied on material interfaces are neglected. To address this issue, a thermodynamically consistent phase-field approach is proposed in [42] where the free energy includes coupled terms required for reproducing surface stresses. It is noteworthy in this approach that the normalized phase-field gradient is used to express the elastic and surface stresses.

To correctly capture the heat transfer across cracks in the context of the phase-field method, damage models degrading the effective thermal-conductivity are necessary in order to impose correct Neumann boundary conditions on the crack surfaces. Analytical expressions of the effective thermal conductivity in heterogeneous materials were derived by Maxwell [43] and Rayleigh [44], introducing assumptions which simplify the study of such problems. In Maxwell’s model, it is assumed that the dispersion of inclusions embedded in matrix is dilute, and therefore the thermal interactions between these filler particles are ignored. Later, less restrictive assumptions were made to calculate the effective thermal conductivity analytically [45], [46], [47], [48], [49]. Specifically, Eucken [45] modified Maxwell’s model (the so-called Maxwell–Eucken model) to consider multiple different phases of filler particles as one continuous matrix phase. Moreover, the effective thermal conductivity can be calculated by introducing infinitesimal changes to an existing heterogeneous material and constructing a composite material incrementally as shown in [47], [48]. Levy [46] proposed a modified Maxwell–Eucken model to account for phenomena related to the ratio of thermal conductivities of components. A detailed review of these models can been found in [50].

While isotropic quadratic thermal-conductivity degradation functions were employed in [37], [51], they lack any physical basis. Inspired by the analytical work of Maxwell [43], the derivation of thermal-conductivity degradation functions is presented in [52], based on a micro-mechanics void extension model of Laplace’s equation. Specifically, a material point is considered as a sphere with a concentric expanding void, and its effective conductivity is estimated. Using this thermal-conductivity degradation, we obtain a heat flux vector field and temperature distribution which clearly reflect the crack topology, however, effects of crack directionality on thermal-conductivity are neglected in models of isotropic degradation. In particular, thermal-conductivity is degraded both across and along cracks in the crack process zone, which introduces some error as only normal directions should be degraded according to Neumann boundary conditions.

In this paper, a new anisotropic approach is proposed in which thermal-conductivity is degraded solely across the crack, and depends on the phase-field gradient. The key idea is that the phase-field gradient and the normal vector to a crack are approximately collinear, taking advantage of the phase-field method to track the crack path. It is noteworthy that the anisotropy of the thermal conductivity tensor is not due to anisotropic material properties, but rather serves as a means of accounting for the crack directionality, and consequently approximating the temperature and heat flux solution more accurately. Comparing the results of the isotropic and anisotropic formulations to discontinuous crack solutions, it is shown that the error in the near-field approximation of temperature and heat flux in the case of the anisotropic approach is reduced.

We investigate the performance of the new approach on a set of dynamic fracture benchmark problems using a unified dynamic fracture model, which accounts for fracture and shear bands simultaneously. The unified dynamic fracture model, developed by McAuliffe and Waisman [53], [54], captures these failure modes with a single set of governing equations. Shear bands are modeled as an elastic–viscoplastic material with thermal softening, while cracks are modeled with the phase field method and driven by both elastic and plastic energies. Although shear bands and fracture have been successfully modeled using embedded discontinuities [55], [56], [57], in our approach the standard finite element method is used. Moreover, the interplay between shear banding and fracture is achieved by incorporating an additional term in the phase-field model to account for the contribution of inelastic energy to fracture. It is noteworthy that the unified model [53], [54] can capture the brittle–ductile failure transition, a phenomenon observed in notched plate impact experiments of Kalthoff and Winkler [58]. In our current work, the new anisotropic thermal-conductivity, which depends on the phase-field gradient, is adopted in conjunction with the physics-based degradation functions [52]. This enables the unified model to capture the heat transfer process more accurately in the post-failure regime.

The structure of the paper has the following parts. In Section 2, motivation for an anisotropic thermal-conductivity degradation is presented. In Section 3, the model for capturing shear bands and fracture is presented briefly, incorporating the proposed conductivity enhancement. In Section 4, the effects of heat transfer on the crack are studied on two benchmark problems. To demonstrate the viability of the novel anisotropic thermal-conductivity degradation, the proposed technique is compared with the standard isotropic quadratic thermal-conductivity degradation, commonly used in the literature. Finally, conclusions of the current study are presented in Section 5.