Material Motion Problem Research Articles

On the spatial and material motion problems in nonlinear electro-elastostatics with consideration of free space

In this work the formulation of spatial and material motion problems in nonlinear electro-elastostatics is considered using an energy approach, which takes into account the contribution of the free space surrounding a nonlinearly polarized body undergoing large deformation. The free space can have a huge impact on the electric field and on the deformation field inside a body made of materials with low electric permittivity such as the so-called electronic electroactive polymers (EEAPs). The contribution of the free space can be taken into account by using the electric flux and the Maxwell's traction acting on the boundary of the body. These two quantities can be expressed in terms of a stored energy density function. By using a stored energy density function for both the material body and the (finite or infinite) free space, the governing equations of both spatial and material motion problems are derived by considering the change of energy with respect to a change in the spatial or material configuration. In the spatial motion problem, well-known definitions for electric and mechanical quantities are derived. In the material motion problem, in addition to the derivation of configurational forces, this approach reveals the formulas for the part of energy that is released from the system material body, applied forces in response to a change in the material configuration, which are particularly useful in the study of defects such as crack propagation. The same approach can be used in the case of nonlinear electro-thermo-mechanical coupling and constitutes the direction for future works.

A comparison of different approaches in the multi-scale computation of configurational forces

In the present work the FE2 scheme is extended towards the homogenization of material quantities like the Eshelby stress and material node point forces. Therefore, in contrast to standard computational homogenization schemes volume forces on the micro-and on the macro-level have to be taken into account which emerge in the material motion problem due to inhomogeneities in the material. Different approaches in the determination of the material stresses are compared and it is shown that the direct calculation of the material stresses in terms of averaged material quantities requires an internal extra term to fulfill the energy consistency represented by a Hill-Mandel type condition. Furthermore, two approaches based on the average of the material two-point stress and the Eshelby stress are compared which require a further scale-transition which is performed within a postprocessing step. The influence of different micro-structures onto the macroscopic material quantities is studied within numerical examples.

Material force method: theoretical and numerical aspects in nonlinear electro‐elastostatics

AbstractFormulations of the spatial and material motion problem in nonlinear electroelastostatics are considered in this work by a virtual work approach. Based on these formulations, a finite element discretization is realized and a numerical example is presented to demonstrate possible uses of the material force method in studying the closing process of cracks. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fracture mechanical behaviour of visco‐elastic materials

AbstractThe material force approach is an efficient, elegant and accepted means to compute the J‐integral as a fracture mechanical parameter for elastic and inelastic materials. With the formulation of a multiplicative split of the deformation gradient at hand, rate‐dependent (visco‐elastic) materials based on the Bergström‐Boyce‐Model can be investigated. For these investigations, the so‐call material body forces have to be computed to separate the driving force acting on the inelastic zone around the crack tip from the driving force acting on the crack tip itself, representing the crack driving force or the J‐integral, respectively. For that purpose, the complete balance for the material motion problem for rate‐dependent, inelastic materials has to be known and the balance has to be fulfilled at every nodal point in a finite element discretization.In this contribution, the so‐called dwell effect should serve as an example for the application of the material force method for inelastic materials. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Theoretical and numerical aspects of the material and spatial settings in nonlinear electro-elastostatics

The formulation of the spatial and material motion problem in nonlinear electro-elastostatics is revisited in this work. A finite element discretization is realized and a numerical example is presented to demonstrate possible application of the formulation in studying the closing process of cracks.

Editorial

Elastomeric materials show a wide range of different elastic and inelastic properties. Additionally, this class of materials is subjected to large deformations. Considering all these effects, fracture mechanical investigations are very challenging tasks and cannot be performed with standard approaches. Effects of inhomogeneities and discontinuities such as cracks can be investigated with the so-called material force approach in an efficient and elegant way. For comprehensive investigations of inelastic materials, the complete balance of the material motion problem has to be formulated. In this case, the material volume forces depend on the internal history variables which are required for the inelastic constitutive model. This paper derives a general formulation for rate-dependent and rate-independent inelastic materials based on a multiplicative split of the deformation gradient to cover viscoelastic and elastoplastic materials at finite deformations.

Material forces for inelastic models at large strains: application to fracture mechanics

Elastomeric materials show a wide range of different elastic and inelastic properties. Additionally, this class of materials is subjected to large deformations. Considering all these effects, fracture mechanical investigations are very challenging tasks and cannot be performed with standard approaches. Effects of inhomogeneities and discontinuities such as cracks can be investigated with the so-called material force approach in an efficient and elegant way. For comprehensive investigations of inelastic materials, the complete balance of the material motion problem has to be formulated. In this case, the material volume forces depend on the internal history variables which are required for the inelastic constitutive model. This paper derives a general formulation for rate-dependent and rate-independent inelastic materials based on a multiplicative split of the deformation gradient to cover viscoelastic and elastoplastic materials at finite deformations.

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.

On configurational forces in short-time dynamics and their computation with an explicit solver

A theoretical description and a computational method to calculate configurational forces in the context of the finite element (FE) method is presented. With respect to problems in short-time dynamics, the fully 3D-case and large deformations in hyper-elastic materials are taken into account. The FE implementation and numerical analysis of different structures demonstrates the applicability of this field of mechanics. In the chosen derivation, the Lagrangian depends on the deformation gradient and on the position (in the reference configuration) explicitly, which accounts for inhomogeneous materials, e.g. materials with phase boundaries, voids or cracks. Analogous to the local balance of linear momentum, the so-called Eshelby stress satisfies a configurational force balance (balance of momentum for the material motion problem), where configurational (or material) forces appear as volume forces in the physical space. A consistent FE description is obtained by formulating the weak form of the configurational force balance. Thus, the configurational forces acting on the finite element nodes may be computed after the physical boundary value problem is solved. For the static case and small deformations, the configurational force is strongly related to the well known J-integral in fracture mechanics.

Material forces in open system mechanics

The basic concern of the present work is the exploitation of the notion of material forces in the theoretical and computational analysis of open systems with particular application to biomechanical problems. Based on a completely dual framework for the spatial and the material motion problem, we introduce the balance equations for open system thermodynamics. In combination with the appropriate constitutive equations, they constitute the basis of the finite element formulation derived thereafter. For the spatial motion problem, the solution of the governing equations, basically the balance of mass and momentum, renders the discrete nodal values of the density and the deformation as primary unknowns. For the material motion problem, the computational analysis of the balance of momentum yields the discrete material node point forces which can be interpreted as driving forces for the local rearrangement of material inhomogeneities. Once the spatial motion problem has been solved, the material force method is nothing but a mere post-processing step from an algorithmic point of view. As a convincing benefit of the proposed strategy, the computation of the material forces is extremely cheap and requires no additional finite element data structure. In the context of biomechanics, material forces give further insight into complex biomechanically induced processes, such as functional adaption, morphogenesis, healing or growth. For example, material forces on the boundary can be considered as a measure of shape sensitivity and thus indicate morphological changes while material forces in the interior indicate the tendency to create new mass locally or to equilibrate local density inhomogeneities.

Application of the material force method to thermo-hyperelasticity

The numerical analysis of material forces in the context of thermo-hyperelasticity constitutes the central topic of the present paper. In contrast to classical spatial forces in the sense of Newton, material forces in the sense of Eshelby indicate the tendency of material inhomogeneities to move relative to their surrounding material. Material forces are thus considered of particular importance in the context of thermo-elasticity where thermal effects can be understood as a potential source of inhomogeneity. The relevant balance equations of thermo-elasticity, i.e. the balance of momentum and energy, which essentially govern the evolution of the deformation and the temperature field are thus illustrated for both, the classical spatial and the material motion context. Guided by arguments of duality, the corresponding weak forms are derived. Next, we carry out the finite element discretization of both problems. While the numerical solution of the spatial motion problem renders the discrete spatial deformation map and the temperature as nodal degrees of freedom, the solution of the material motion problem provides the discrete material node point forces. The former typically relies on the solution of a global system of equations whereas the latter is introduced as a mere post-processing procedure. Since we apply a simultaneous solution of the mechanical and the thermal problem with the deformation and the temperature interpolated in a C 0 -continuous way, all the relevant information for the material force method is readily available once the spatial motion problem has been solved. Selected examples from the field of fracture mechanics illustrate the additional insight that is provided by the results of the material force method.

On spatial and material settings of thermo-hyperelastodynamics for open systems

The present treatise aims at deriving a general framework for the thermodynamics of open systems typically encountered in chemo- or biomechanical applications. Due to the fact that an open system is allowed to constantly gain or lose mass, the classical conservation law of mass has to be recast into a balance equation balancing the rate of change of the current mass with a possible in- or outflux of matter and an additional volume source term. The influence of the generalized mass balance on the balance of momentum, kinetic energy, total energy and entropy is highlighted. Thereby, special focus is dedicated to the strict distinction between a volume specific and a mass specific format of the balance equations which is of no particular relevance in classical thermodynamics of closed systems. The change in density furnishes a typical example of a local rearrangement of material inhomogeneities which can be characterized most elegantly in the material setting. The set of balance equations for open systems will thus be derived for both, the spatial and the material motion problem. Thereby, we focus on the one hand on highlighting the remarkable duality between both approaches. On the other hand, special emphasis is placed on deriving appropriate relations between the spatial and the material motion quantities.

On spatial and material settings of hyperelastodynamics

The relevant relations pertaining to hyperelastodynamics are stated in spatial and material setting. Thereby the spatial setting allows for the computation of the spatial motion problem, whereas the material setting allows for the consideration of the material motion problem. Thus the former serves to compute the dynamic response of a hyperelastic continuum due to given data, while the latter serves to consider any possible nonsmoothness in the solution to the former. It is the aim of this treatise to establish a formalistic framework that (i) highlights the intriguing duality of the spatial and material setting, and (ii) provides the necessary tools for an elegant transition between the spatial and material setting. To this end, the essential vehicle of the present approach is the variational approach which is firstly applied to hyperelastostatics and secondly to hyperelastodynamics. By adopting a tedious but strict formalism a number of new aspects are discovered.

On spatial and material settings of hyperelastostatic crystal defects

Based on the spatial and material settings of hyperelastostatics the appropriate volume force densities associated to either spatial or material dislocation densities are derived. Thereby spatial and material dislocation densities are defined corresponding to incompatibilities of either the spatial or the material configuration, respectively. All derivations are performed within the geometrically nonlinear framework of continuum mechanics in order to emphasize the intriguing duality of the spatial and the material motion problem. For the particular cases of a single dislocation with either spatial or material Burgers vector the corresponding single forces in material or spatial description, respectively, take the format and interpretation of the well-known Peach–Koehler force (Peach and Koehler, Phys. Rev. 80 (1950) 436). As a result, an attempt is made to clarify some possible irritations documented in the literature (Ericksen, Int. J. Plasticity 14 (1998) 9) of how to set up the Peach–Koehler force within the framework of geometrically nonlinear hyperelastostatics.