Configurational Forces and the Dynamics of Planar Cracks in Three-Dimensional Bodies

This paper develops a three-dimensional framework for the evolution of planar cracks, concentrating on the derivation of balances and constitutive equations that describe the motion of the crack tip. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law. This revised version was published online in August 2006 with corrections to the Cover Date.     

The role of the configurational force balance in the nonequilibrium epitaxy of films

We discuss the epitaxial growth of an elastic film, allowing for stress and diffusion within the film surface as well as nonequilibrium interactions between the film and the vapor. Our approach, which relies on recent ideas concerning configurational forces, is based on: (i) standard (Newtonian) balance laws for forces and moments together with an independent balance law for configurational forces; (ii) atomic balances, one for each species of mobile atoms; (iii) a mechanical version of the second law that accounts for temporal changes in free energy, energy flows due to atomic transport, and power expended by both standard and configurational forces; (iv) thermodynamically consistent constitutive relations for the film surface and for the interaction between the surface and the vapor environment. The normal component of the configurational force balance at the surface represents a generalization, to a dynamical context involving dissipation, of a condition that would arise in equilibrium by considering variations of the total free energy with respect to the configuration of the film surface. Our final results consist of partial differential equations that govern the evolution of the film surface.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

Interface Evolution in Three Dimensions¶with Curvature-Dependent Energy¶and Surface Diffusion:¶Interface-Controlled Evolution,Phase Transitions,Epitaxial Growth of Elastic Films

When the interfacial energy is a nonconvex function of orientation, the anisotropic-curvature-flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions. The theory is based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances; a mechanical version of the second law that includes, via the configurational moments, work that accompanies changes in the curvature of the interface; a constitutive theory whose main ingredient is a positive-definite, isotropic, quadratic dependence of the interfacial energy on the curvature tensor. Two special cases are investigated: (i) the interface is a boundary between bulk phases or grains, and (ii) the interfaceseparates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface.     

Rolling resistance of a rigid sphere with viscoelastic coatings

We present a novel three-dimensional boundary-element formulation that fully characterizes the mechanical behavior of the external boundary of a multi-layered viscoelastic coating attached to a hard rotating spherical core. The proposed formulation incorporates both, the viscoelastic, and the inertial effects of the steady-state rolling motion of the sphere, including the Coriolis effect. The proposed formulation is based on Fourier-domain expressions of all mechanical governing equations. It relates two-dimensional Fourier series expansions of surface displacements and stresses, which results in the formation of a compliance matrix for the outer boundary of the deformable coating, discretized into nodes. The computational cost of building such a compliance matrix is optimized, based on configurational similarities and symmetry. The proposed formulation is applied, in combination with a rolling contact solving strategy, to evaluate the viscoelastic rolling friction of a coated sphere on a rigid plane. Steady-state results generated by the proposed model are verified by comparison to those obtained from running dynamic simulations on a three-dimensional finite element model, beyond the transient. A detailed application example includes a verification of convergence and illustrates the dependence of rolling resistance on the applied load, the thickness of the coating, and the rolling velocity.     

Investigation on thermomechanical fracture in the framework of configurational forces

A configurational force approach is developed for providing a fresh look onto classical aspects of thermomechanical fracture. The theoretical framework is based on the finite deformation and makes no restrictions on the material response. The integral form of configurational force balance at the crack tip is constructed, and the concentrated configurational body force is decomposed into the inertial and internal parts. The energy release rate is evaluated through the generalized second law of thermodynamics applicable to configurational force system. The theoretical investigation shows that the negative of the projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of the energy release rate and acts directly in response to crack propagation. This finding enables us to deal with the thermomechanical fracture problems in material space.     

Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing

The weight function theory for three-dimensional elastic crack analysis received great attention after the work of Rice (1985, 1989). Several applications have been considered since then, particularly in the context of configurational stability, crack path prediction, stress intensity factor expansions, perturbation approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order to formulate the problem in terms of Cauchy principal value. In the present note, such hypothesis is further investigated and consequences discussed. A variational statement given in Salvadori and Fantoni (2013a) is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to accurate approximation of crack front propagation.     

Configurational forces in discrete elastic systems

The concept of configurational forces is applied to a simple, one-dimensional problem that is solved by finite elements. Both the exact solution and its finite-element approximation are provided in closed form. The total energy according to the approximate solution depends on the choice of the nodes. Any virtual shift of a node results in a virtual change of energy, which can be interpreted as the virtual work done by a configurational force acting on that node. It is shown that, in the case of equidistant nodes, the configurational forces acting on the interior nodes vanish. Also, the relation between the nodal configurational forces and the Eshelby stress resultant along the rod is investigated.     

On energetic changes due to configurational motion of standard continua

The appropriate thermodynamic setting for the combined configurational and deformational motion of standard continua is discussed in this paper. A key ingredient is an absolute (fixed) reference configuration, relative to which configurational and deformational changes (rates) with respect to the material (undeformed) and spatial (deformed) configurations, respectively, can be described in a unified fashion. In particular, we are interested in formulating the local as well as global measures of the energy dissipation due to configurational changes of a given physical system. It is believed that the presentation in this paper provides the following advantages: a unified kinematic and thermodynamic setting of the configurational and deformational motions, in particular the generic balance law accounting for configurational flux, increases the general clarity; the separation of the total dissipation in terms of the intrinsic change in elastic energy and in terms of the material dissipation that is induced by configurational changes becomes transparent. All results are obtained without any restrictions on dynamics, thermomechanical couplings, etc.     

基于数字散斑相关实验测量的材料构型力的计算方法

材料构型力学主要研究材料中的缺陷(夹杂、空穴、位错、裂纹、塑性区等)的构型(形状、尺寸和位置)改变时,所引起的系统自由能的变化。本研究将基于数字散斑相关技术,实验测量材料试件的位移场分布,随后通过材料构型力的定义式,计算求得弹塑性材料中缺陷构型力的分布。其方法概括如下:位移场通过数字图像相关技术测得;应变及位移梯度场利用三次样条拟合获得;线弹性材料应力通过简单线弹性本构方程获取,而塑性材料的表面应力场通过Ramberg-Osgood本构方程计算求得;弹塑性应变能密度分布则由应力-应变曲线数值积分获得。该方法对普通弹性材料或者弹塑性材料均适用,可以用于各种不同的缺陷及缺陷群的材料构型力测量。     

On the role of material dissipation for the crack-driving force

The thermodynamic setting for the formulation of the “crack-driving force” for a singular crack in conjunction with rate-independent material response is discussed. One key ingredient is the introduction of a fixed (absolute) configuration, relative to which both physical and (virtual) configurational and spatial changes can be described. Only quasistatic and isothermal conditions are considered in this paper. A variational framework is established for the rate of global energy dissipation (integrated over the whole material domain) due to the combined action of a (discrete) crack extension and continuum inelasticity, whereby the material time derivative of internal variables and the rate of crack extension are coupled. The classical assumption (previously adopted in the literature) is that there is no coupling, i.e. the internal variables are considered as fixed (material) fields just like an inhomogeneous material property. The other (extreme) assumption is that the internal variables fields are convected with the configurational motion due to the virtual crack extension. Both cases are investigated in this paper for a simple 2D example of an edge crack in a plate in a setting of small strains and hardening plasticity. In particular, we consider convergence issues from mesh refinement.     

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Configurational forces – morphology evolution and finite elements

A brief introduction into the theory of configurational forces is presented and two possible applications of the theory are discussed. The first application is microstructure evolution in two-phase materials. In an inclusion problem the driving force on the interface is interpreted in the context of configurational forces. The knowledge of the driving force allows the treatment of microstructure evolution and equilibrium morphologies. The second application deals with configurational forces in the framework of the Finite Element Method. The calculation of configurational forces induced by the numerical method is discussed.     

三维编织CMC的弯曲断裂研究

从界面断裂的角度出发，对三维编织CMC的断裂作了理论研究和数值分析，对于三点弯曲试件，通过数值拟合修正了能量释放率G的理论表达式中的自由常数A，同时也研究了材料的各个参变量对于断裂韧性的影响，由此得出了一个基本完善的三点弯曲试件断裂韧性G的理论公式，该能量释放率方法可以应用于单试件的试验计算，与断裂韧性的柔度标定方法相比，该方法一方面可以减少试验件数量；另一方面，试验结果显示出在试件切口尺雨处于0．4≤a/W≤0.5时，可以获得比较稳定的断裂韧性值。     

Viscoplastic flow rule for dislocation-mediated plasticity from systematic coarse-graining

The plastic response of metals is determined by the collective, coarse-grained dynamics of dislocations, rather than by the dynamics of individual dislocations. The evolution equations at both levels are quite different, for example considering their dependence on the applied mechanical load. On the one hand, the relation between the configurational force and dislocation velocity for individual dislocations is linear. On the other hand, in phenomenological crystal plasticity models, the relation between load and plastic slip is highly non-linear and often taken of power-law form. In this work, it is shown that this difference is justified and a consequence of emergent effects. Previously, an expression for the macroscopic dislocation flux was derived by systematic coarse graining (Kooiman et al., 2015). This expression has been evaluated numerically in this paper. The resulting relation between dislocation flux and applied mechanical load is found to be of power-law form with an exponent 3.7, while the underlying Discrete Dislocation Dynamics has a linear flux–load relation.     

Boundary Element Analysis of Vibro-Acoustic Interaction between Vibrating Membrane and Thin Fluid Layer

This paper presents a variational formulation for the study of the acoustic propagation and radiation of a vibrating membrane coupled to a cavity filled with a visco-thermal fluid. This formulation is obtained by combining a variational formulation by integral equations of both internal and external fluids, which takes account of acoustic and entropic waves coupling, with a classical variational formulation of the membrane. Numerical results obtained by this new formulation are compared to those obtained when the viscous and thermal effects are not considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

A deformational and configurational framework for geometrically non-linear continuum thermomechanics coupled to diffusion

A general framework encompassing both the (conventional) deformational and configurational settings of continuum mechanics is presented. A systematic application of balance principles over a migrating control volume in the undeformed configuration of the continuum body yields the system of governing equations in the bulk, on the surface and on a coherent interface within the continuum. The equations governing the response of the bulk agree with those of the conventional deformational approach. The localised balance equations are expressed in the configurational setting using a pull-back operator and reformulated in terms of the Eshelby stress. The configurational expression of the dissipation elucidates the energy loss associated with configurational changes. The general framework is introduced by considering the problem of coupled deformation, heat conduction and species diffusion within a geometrically non-linear continuum body intersected by a coherent interface. The nature of the coupling is emphasised throughout the presentation and via an example.     

A Note on the Thermomechanics of Curvature Flows in IR3 and on Surfaces in IR3

Equations for the evolution of curves in IR³ and on surfaces in IR³ are derived from a configurational force balance, a mechanical version of the second law, and suitable constitutive assumptions. Both the isotropic and anisotropic cases are considered.Sommario.In questo lavoro si derivano le equazioni di evoluzione per curve in IR³ e su superfici di IR³, utilizzando un bilancio di forze configurazionali, una versione meccanica del secondo principio e opportune ipotesi costitutive. Sono trattati sia il caso isotropo che anisotropo.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

On the sensitivity of the rate of global energy dissipation due to configurational changes

The thermodynamic framework for combined configurational and deformational changes was recently discussed by [Runesson, K., Larsson, F., Steinmann, P., 2009. On energetic changes due to configurational motion of standard continua. Int. J. Solids Struct, 46, 1464–1475.]. One key ingredient in this setting is the (fixed) absolute configuration, relative to which both physical and virtual (variational) changes of the material and spatial configurations can be described. In the present paper we consider dissipative material response and emphasize the fact that it is possible to identify explicit energetic changes due to configurational changes for “frozen” spatial configuration (a classical view) and the configuration-induced material dissipation. The classical assumption (previously adopted in the literature) is to ignore this dissipation, i.e. the internal variables are considered as fixed fields in the material configuration. In this paper, however, we define configurational forces by considering the total variation of the total dissipation with respect to configurational changes. The key task is then to compute the sensitivity of the internal variable rates to such configurational changes, which results in a global tangent problem based on the balance equations (momentum and energy) for a given body. In this paper we restrict to quasistatic loading under isothermal conditions and for elastic-plastic response, and we apply the modeling to the case of a moving interface of dissimilar materials.