Towards Phase Field Modeling of Ductile Fracture in Gradient-Extended Elastic-Plastic Solids

The modeling of failure in ductile metals must account for complex phenomena at a micro-scale as well as the final rupture at the macro-scale. Within a top-down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic-plastic model with a concept for the modeling of macroscopic crack discontinuities. In this context, it is important to account for material length scales and thermo-mechanical coupling effects due to dissipative heating. This can be achieved by the construction of non-standard, gradient-enhanced models of plasticity with a full embedding into continuum thermodynamics [1,2]. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture based on regularized crack discontinuities. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns within a pure continuum formulation. Moreover, the phase field modeling of fracture is related to gradient theories of continuum damage mechanics, and fits nicely the structure of constitutive models for gradient plasticity. The main focus of this work is the extensions to gradient thermoplasticity and phase field formulation of ductile fracture, conceptually in line with the work [3]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

饱和多孔介质耦合系统的变分原理

本文采用变积方法,建立了等温准静态下饱和多孔介质的六类变量的广义变分原理.在此基础上,通过引入约束条件得到各级变分原理,其中包括五类变量,四类变量,三类变量和二类变量的变分原理.除得到文献中已有的变分原理外,本文给出了许多新的变分原理,为建立饱和多孔介质的有限元模型提供了基础.     

Phase Field Modeling of Fracture in Plates and Shells

The numerical modeling of failure mechanisms in plates and shells due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. In this paper, we extend ideas recently outlined in [1, 2] towards the phase field modeling of fracture in dimension-reduced continua with application to Kirchhoff plates and shells. The introduction of history fields, containing the maximum reference energy obtained in history, provides a very transparent representation of the coupled balance equations and allows the construction of an extremely robust operator split technique. The performance of the proposed models is demonstrated by means of representative numerical examples. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Approximation of Dynamic Brittle Fracture

The numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations - the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Approach for Dynamic Fracture

In phase field fracture models cracks are indicated by the value of a scalar field variable which interpolates smoothly between broken and undamaged material. The evolution equation for this crack field is coupled to the mechanical field equations in order to model the mutual interaction between the crack evolution and mechanical quantities. In finite element simulations of crack growth at comparatively slow loading velocities, a quasi-static phase field model yields reasonable results. However, the simulation of fast loading or the nucleation of new cracks challenges the limits of such a formulation. Here, the quasi-static phase field model predicts brutal crack extension with an artificially high crack speed. In this work, we analyze to which extend a dynamic formulation of the mechanical part of the phase field model can overcome this paradox created by the quasi-static formulation. In finite element simulations, the impact of the dynamic effects is studied, and differences between the crack propagation behavior of the quasi-static model and the dynamic formulation are highlighted. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Geometrically Exact Incremental Variational Formulation for Phase Field Models in Micromagnetics

Magnetic materials have been finding increasingly wide areas of application. We focus here on the continuum modeling of such materials and present an incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetically as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. Furthermore, the algorithmic preservation of the geometrical nature of the variables is an important challenge from the numerical perspective and to this end we present a novel FE discretization whereby the geometric property of the magnetization director is pointwise exactly preserved by nonlinear rotational updates at the nodes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of a Phase Field Model for Elasto-plastic Fracture

Phase field modelling of brittle fracture is very well understood today. However, the attempts of investigation of elasto-plastic fracture by the phase field approach are limited. This contribution deals with the investigation of a phase field model for elasto-plastic fracture. Based on a free energy density comprising elastic, fracture and plastic contributions, the model describes an extension of the linear elastic model towards von Mises plasticity. In this work it is analyzed numerically to which extend analytical findings concerning the interpretation of the model parameters in 1D are transferable to 2D scenarios. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Approach to Multiphase Flow Modeling

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals?? theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick??s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.     

Exponential Finite Elements for a Phase Field Fracture Model

Sharp interface material models can be related to phase field models by introducing an order parameter, whose value is assigned to the different phases of a material. The elastic material law is coupled to the evolution equation of the order parameter and cracking is addressed as a phase transition problem instead of a moving boundary value problem. A regularization parameter ϵ controls the width of the diffuse cracks represented by the order parameter and the underlying sharp interface model can be recovered from the phase field model by the limit ϵ → 0. However, in numerical simulations using standard finite elements with linear shape functions, the minimum value of ϵ is restricted by the grid size and therefore the discretization of the crack field requires extensive mesh refinement for small values of ϵ. In this work, we construct special 2d shape functions which take into account the exponential character of the crack field and its dependence on the parameter ϵ. Especially in simulations with small values of ϵ and a rather coarse mesh, the elements with exponential shape functions perform significantly better than standard linear elements. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an Energetic Interpretation of a Phase Field Model for Fracture

In the pioneering work by Griffith, it is assumed that a crack propagates, if this is energetically favorable. However, this original formulation requires a pre-existing initial crack. In order to bypass this deficiency of classical Griffith theory, Francfort and Marigo advocate a global variational criterion, where the total energy is minimized with respect to any admissible displacement field and crack set. Bourdin's regularized approximation of this variational formulation makes use of a continuous scalar field to indicate cracks. Based on this regularization a phase field fracture model is formulated. The crack field is assumed to follow a Ginzburg-Landau type evolution equation, and cracking is addressed as a phase transition problem. The coupled problem of mechanical balance equations and the evolution equation is solved using the finite element method combined with an implicit time integration scheme. The numerical solution naturally yields the crack evolution including crack propagation, kinking, branching and initiation without any additional criteria. In this work we study the driving mechanisms behind the crack evolution in the phase field fracture model and compare to the purely energetic considerations of the underlying variational formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非牛顿流体力学变分原理

本文将钱伟长教授在文献[1]中提出的不可压缩粘性流的最大功率消耗原理进一步推广到本构方程为ε_ij=?τ/?σ'_ij的非牛顿流体流动问题,并采用识别的拉氏乘子法解除变分约束条件,导出其广义变分原理。     

一类非牛顿流体流动问题的变分原理和广义变分原理

本文将钱伟长教授[1]的不可压缩粘性流的最大功率消耗原理推广到一类特殊的非牛顿流体─-广义牛顿流体的流动问题,并采用识别的拉氏乘子法来解除变分约束条件,导出其广义变分原理。     

滑移和感应磁场对Johnson-Segalman流体蠕动传输的影响

存在感应磁场和滑移条件下,研究Johnson-Segalman(J-S)流体在平面通道中的蠕动流.通道中的流动认为是对称的,并在剪切应力项中考虑了速度的滑移条件.首先给出问题的数学公式,然后在长波长和低Reynolds数近似下,求解该方程组得到摄动解,确定沿管道截面的压力增量、轴向速度、微转动分量、流函数、磁力函数、轴向感应磁场和电流密度分布公式.导出了小数值Weis-senberg数时解的表达式,分析并勾画出诸流动物理量的有趣变化.     

A Variational Homogenization Approach to Electromechanical Hystereses Caused by Domain Wall Movement

In recent years, increasing interest in so-called smart materials such as ferroelectric polymers and ceramics has been shown. Those materials are used in various actuators, sensors, and also in medical devices. In this paper, we outline a micro-macro approach to the modeling of macroscopic hystereses which directly takes into account the microstructural evolution of electrically poled domains. To this end, an incremental variational formulation for a gradient-type phase field model is developed and exploited for the simulation of electromechanically coupled problems. The formulation determines the hysteretic response of the material in terms of an energy-enthalpy and a dissipation function which both depend on the microscopic remanent polarization treated as an order parameter. The gradient-type balance law for the phase field can be considered as a generalization of Biot's equation for standard dissipative materials and may be related to the classical Ginzburg-Landau equation. Furthermore, the variational formulation serves as natural starting point for a compact and symmetric finite element implementation of the coupled micromechanical problem covering the displacement, the electric potential, and the microscopic polarization vector. For this three-field scenario we develop a variational-based homogenization method which determines the overall macroscopic hysteretic properties of a polycrystalline aggregate. The proposed computational method can be used as a numerical laboratory for the improvement of microstructural properties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)