A continuum phase field model for fracture

A phase field model based on a regularized version of the variational formulation of brittle fracture is introduced. The influences of the regularization parameter that controls the interface width between broken and undamaged material and of the mobility constant of the evolution equation are studied in finite element simulations. A generalized Eshelby tensor is derived and analyzed for mode I loading in order to evaluate the energy release rate of the diffuse phase field cracks. The numerical implementation is performed with finite elements and an implicit time integration scheme. The configurational forces are computed in a postprocessing step after the coupled problem of mechanical balance equations and the evolution equation is solved. Some of the numerical results are compared to analytical results from classical Griffith theory.     

Application of the material force method to isotropic continuum damage

 The objective of this work is the exploitation of the notion of material forces in computational continuum damage mechanics. To this end we consider the framework of isotropic geometrically non–linear continuum damage and investigate the spatial and material settings that lead to either spatial or material forces, respectively. Thereby material forces essentially represent the tendency of material defects to move relative to the ambient material. In this work we combine an internal variable approach towards damage mechanics with the material force method. Thus the appearance of distributed material volume forces that are conjugated to the damage field necessitates the discretization of the damage variable as an independent field in addition to the deformation field. Consequently we propose a monolithic solution strategy for the corresponding coupled problem. The underlying kinematics, strong and weak forms of the coupled problem will be presented and implemented within a standard Galerkin finite element procedure. As a result in particular global discrete nodal quantities, the so–called material node point (surface) forces, are obtained and are studied for a number of computational examples. Received: 19 August 2002 / Accepted: 16 October 2002     

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.     

A damage-based temperature-dependent model for ductile fracture with finite strains and configurational forces

In this paper we assess a crack propagation criterion based on the notion of configurational force in the spirit of Gurtin (Configurational forces as basic concepts of continuum physics. Applied mathematical sciences. Springer, Berlin, 2000). We extend the theory of Gurtin to finite strain elasto-plastic fracture and in addition take thermal effects into account. The global model is a system of nonlinear and non-smooth equations which are solved directly with a finite element discretization. Comparison with laboratory experiments is provided, thereby showing that the concept of configurational force can be successfully used for computational damage-based fracture tests on ductile materials.     

On the interaction between an edge dislocation and a coherent precipitate in anisotropic materials

A numerical method is presented to investigate the interaction between an edge dislocation and a misfitting precipitate in an anisotropic material. The technique is based on the concept of configurational (material) forces in linear elasticity. After briefly discussing the theory of configurational forces, a generalized balance equation is obtained to calculate the material forces on the defects under consideration. Utilizing a semi-analytical method via a finite element formulation, the generalized balance equation is solved and results are presented for the diffusion process and the dislocation movement in a cubic material. Furthermore, the microstructural change of the inclusion morphology associated with this interaction is discussed. To simplify matters, we consider a purely dilatational misfit of the inclusion and no far field stresses are taken into account.     

On configurational forces in the context of the finite element method

The theory of configurational forces is briefly recast, together with the underlying balance laws. It is shown, that in the case of a homogeneous body without body forces the additional balance laws are identically satisfied if the standard force balance holds. In approximate solutions, for example obtained by finite elements, the equilibrium is not satisfied exactly, thus configurational forces occur on discretization nodes. An implementation of the configurational force balance into the finite element scheme is presented. The use of configurational forces is discussed with three main aspects. It is demonstrated how configurational forces can be used to check and to improve the finite element solution. Examples from fracture mechanics and problems with material inhomogeneities are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Crack Tip Shielding or Anti-shielding due to Smooth and Discontinuous Material Inhomogeneities

This paper describes a theoretical model and related computational methods for examining the influence of inhomogeneous material properties on the crack driving force in elastic and elastic-plastic materials. Following the configurational forces approach, the crack tip shielding or anti-shielding due to smooth (e.g. graded layer) and discontinuous (e.g. bimaterial interface) distributions in material properties are derived. Computational post-processing methods are described to evaluate these inhomogeneity effects. The utility of the theoretical model and computational methods is demonstrated by examining a bimaterial interface perpendicular to a crack in elastic and elastic-plastic compact tension specimens.     

Use of material forces in adaptive finite element methods

The theory of material forces for a hyperelastic material is briefly presented using the translational invariance of the control volume. The theory is derived for the dynamical setting, while the numerical implementation is limited to the static case. The finite element (FE) method is used to solve the standard field equations. After obtaining the solution the material force balance is consistently discretized with FE. As a result of this post-processing discrete material forces are obtained. They are then used to set up an adaptive scheme, in which the magnitude of the material forces acts as an indicator for mesh refinement. Special consideration is given to the boundary, where two different refinement strategies are proposed. The two strategies are compared by studying the refinement process for three examples.     

On the static equilibria of branched elastic rods

Structures featuring rod-like bodies connected in a branched ensemble are ubiquitous. They appear in antennae, replicating DNA strands, and, most visibly, in the plant kingdom. In spite of this, modeling these branched structures using rod theories have received little attention in the literature. With the help of a general rod theory, which was originally developed by Green and Naghdi, balance laws for these structures are discussed. The governing equations established are then shown to be equivalent to variational principles for tree-like structures. These principles are adopted from works by Ivanov and Tuzhilin and may lead to the development of nonlinear stability criteria for these structures. It is also shown how the conditions at a branching point in these structures can be understood using recent interpretations of Eshelby’s work on material (configurational) forces in adhesion problems. The static configurations are illuminated for several structures using Euler’s theory of the elastica. Instances of multiple possible configurations and novel extensions to the classical problem of the tallest column are also discussed.     

On the convergence of 3D free discontinuity models in variational fracture

Free discontinuity problems arising in the variational theory for fracture mechanics are considered. A Γ -convergence proof for an r-adaptive 3D finite element discretization is given in the case of a brittle material. The optimal displacement field, crack pattern and mesh geometry are obtained through a variational procedure that encompasses both mechanical and configurational forces. Possible extensions to cohesive fracture and quasi-static evolutions are discussed.     

The wave propagation in a beam on a random elastic foundation

This paper presents a study of wave propagation in an infinite beam on a random Winkler foundation. The spatial variation of the foundation spring constant is modelled as a random field and the influence of the correlation length is studied. As it is impossible to determine the general stochastic Green’s function, the configurational average of the Green’s function and its correlation function are considered. These functions are found through the transformation of the stochastic equation of motion into the Dyson equation for the mean or coherent field and the Bethe–Salpeter equation for the field correlation, as used in the study of wave propagation in random media. The approximate solutions of the Dyson and the Bethe–Salpeter equations are validated by means of a Monte Carlo simulation and compared with the results of a classical Neumann expansion method. Although both methods only involve the second order statistics of the random field, the approximation of the Dyson and the Bethe–Salpeter equations gives better results than the Neumann expansion, at the expense of a larger computational effort. Furthermore, the results show that a small spatial variation of the spring constant has an influence on the response if the correlation length and the wavelength have a similar order of magnitude, while the waves in the beam cannot resolve the spatial variation in the case where the correlation length is much smaller than the wavelength.     

On configurational forces in hyperelastic materials under shock and impact

In the present work, we discuss configurational forces (also known as material forces) in the context of numerical simulations of short-time dynamical problems with an explicit finite element solver. In extension to the work presented in Kolling and Mueller (Comput Mech 35:392–399, 2005), the fully 3D-case for hyperelastic materials and large strains is shown. The intention of the present paper is the investigation of the influence of inertia effects, which become increasingly important at high strain rates. At first, the dynamical part of configurational forces will be discussed at moderate strain rates. Finally, we use configurational forces in the context of shockwave propagation.     

Electromagnetic forming by distributed forces in magnetic and nonmagnetic materials

In this paper, we discuss the electromechanical force densities associated with pulsed electromagnetic fields in inhomogeneous, linear media with conductive losses, in the context of a process of shaping metal objects. We show that the conductivity and the gradients in permittivity and in permeability lead to volume forces, while jump discontinuities in permittivity and permeability lead to surface forces. These electromagnetic forces are assumed to act as volume (body) source densities in the elastodynamic equations and as surface source densities in the corresponding boundary conditions that govern the elastic motion of deformable matter. As an example, we apply the theory to the calculation of the elastic field in a hollow cylindrical object made of a conducting magnetic or nonmagnetic material. We compare the numerical results with those for the classical theory of elasticity with concentrated forces on the boundaries of the material as the source of the elastodynamic field.     

Material forces due to crack-inclusion interaction

The interaction between a crack and an elastically misfitting inclusion is investigated numerically. The microstructural change of the inclusion's morphology associated with this interaction is discussed using linear elasticity. The misfit is considered purely dilatational and no far field stresses are taken into account. After briefly discussing the theory of configurational forces using Eshelby's energy momentum tensor, a generalized balance equation is obtained to calculate material forces on a crack tip and on the interface of an inclusion. Utilizing a finite element formulation, the generalized balance equation is solved and results are presented for cracks interacting with both hard and soft inclusions.     

A multiscale approach to damage configurational forces

A two-scale homogenization method is used to construct a damage model in the framework of configurational mechanics. The upscaling procedure allows for the identification of damage configurational forces as the result of the microscopic fracture analysis. The obtained damage equation incorporates stiffness degradation, material softening, unilaterality, induced anisotropy. The balance of configurational forces naturally captures a microscopic length, leading to size effects in the overall damage response. The new approach is illustrated in the case of brittle damage, for a three point bending test. Extended finite elements are used for the numerical modeling of macro-crack initiation and growth. The influence of the microscopic size on the failure initiation stress is analyzed and it is shown that this dependence follows a Hall–Petch type rule.     

Configurational forces and the application to dynamic fracture in electroelastic medium

Aimed at a fresh look onto the defect mechanics of electroelastic medium, the configurational force approach is developed for these materials. The notion of working is defined in a more general framework by introducing the conjugates of configurational force and the evolution velocity of migrating control volume. The balance of configurational forces is established through the invariance condition of working under the change of material observer. Eshelby relation is identified by using the invariance requirement of configurational working under reparameterization of the motion of the boundary of migrating control volume. Energy dissipation concentrated at crack tip is evaluated through the generalized mechanical version of the second law of thermodynamics applicable to migrating control volume. Theoretical investigation shows that the negative projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of energy release rate which depends on constitutive response of materials and is independent of the energy of free electric field.     

Modeling of heat and mass transfer in capillary-porous materials

A system of differential equations for heat and mass transfer in capillary-porous materials in a hydroscopic moisture-content field is derived with allowance for kinetics of moisture desorption, two-phase filtration, and liquid pressure determined by the action of surface forces. An analysis is made of the results of a computational experiment illustrating evolution of the moisture content, temperature, and vapor pressure fields against the material characteristics and kinetic coefficient. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 2, pp. 225–232, March–April, 1998.     

The concept of a balance law for a cracked elastic body and the configurational force and moment at the crack tip

In this work, the derivation of the configurational equations for a cracked elastic body by postulating balance laws, is studied. To this end, a proper kinematics is proposed, according to which the evolution of the crack in the material configuration and the physical motion (deformation) are separated. A rigorous localization procedure provides the local equations in material space holding in the smooth part of the body, the corresponding jump conditions along the crack, as well as the configurational force and moment at the crack tip. The results are discussed in connection with the corresponding ones of fracture mechanics and some new interpretations are proposed.     

Discretized peridynamics for linear elastic solids

Peridynamics is a theory of continuum mechanics employing a nonlocal model that can simulate fractures and discontinuities (Askari et?al. J Phys 125:012–078, 2008; Silling J Mech Phys Solids 48(1):175–209, 2000). It reformulates continuum mechanics in forms of integral equations rather than partial differential equations to calculate the force on a material point. A connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node. The peridynamic stress and the constitutive law in elasticity are used for the derivation of one- and three-dimensional numerical micromoduli. For three-dimensional discretized peridynamics, the numerical micromodulus is larger than the analytical micromodulus, and converges to the analytical value as the horizon to grid spacing ratio increases. A comparison of material responses in a three-dimensional discretized peridynamic model using numerical and analytical micromoduli, respectively, is performed for different horizons. As the horizon increases, the boundary effect is more conspicuous, and the errors increase in the back-calculated Young’s modulus and strains. For the simulation of materials of Poisson’s ratios other than 1/4, a pairwise compensation scheme for discretized peridynamics is proposed. Compared with classical (local) elasticity solutions, the computational results by applying the proposed scheme show good agreement in the strain, the resultant Young’s modulus and Poisson’s ratio.