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This paper discusses the calculation of discrete material forces within the Finite Element Method. The possible use of these discrete material forces in - and -adaptive procedures is demonstrated by two short examples from non-linear elasticity.

The present contribution aims at deriving a generic hyperelastic Arbitrary Lagrangian Eulerian formulation embedded in a consistent variational framework. The governing equations follow straightforwardly from the Dirichlet principle for conservative mechanical systems. Thereby, the key idea is the reformulation of the total variation of the potential energy at fixed referential coordinates in terms of its variation at fixed material and at fixed spatial coordinates. The corresponding Euler-Lagrange equations define the spatial and the material motion version of the balance of linear momentum, i.e. the balance of spatial and material forces, in a consistent dual format. In the discretised setting, the governing equations are solved simultaneously rendering the spatial and the material configuration which minimise the overall potential energy of the system. The remeshing strategy of the ALE formulation is thus no longer user-defined but objective in the sense of energy minimisation. As the governing equations are derived from a potential, they are inherently symmetric, both in the continuous case and in the discrete case.

It is shown that the concept of material forces together with associated balance laws, besides the classical laws of linear and angular momentum in the physical space, provide a firm theoretical framework within which weakly nonlocal (gradient type) models of damage and plasticity can be formulated in a clear and rigorous manner. The appropriate set of balance laws for physical and material forces as well as the first and second law of thermodynamics are formulated in integral form. The corresponding local laws are next derived and the general structure of thermodynamically consistent constitutive equations is formulated.

In the proposed model, the plastic distortion and the plastic connection with a non-zero Cartan’s torsion and a non-metric structure, characterize the irreversible behaviour of the crystalline materials, via the appropriate evolution equations. The material behaves like an elastic material element with respect to a second order reference configuration, attached to plastic material structure.

We consider dislocations in the framework of Eringen’s nonlocal elasticity. The fundamental field equations of nonlocal elasticity are presented. Using these equations, the nonlocal force stresses of a straight screw and a straight edge dislocation are given. By the help of these nonlocal stresses, we are able to calculate the interaction forces between dislocations (Peach-Koehler forces). All classical singularities of the Peach-Koehler forces are eliminated. The extremum values of the forces are found near the dislocation line.

The notion of force arises in a natural way in materials with ‘defects’ or in inhomogeneous materials. This notion, which is intimately related with that of energy-momentum tensor, has been widely and successfully employed in crack-propagation problems, in phase-transition problems and others. In elasticity, there is general agreement on the form of the energy-momentum tensor, also known as the stress tensor. In dielectrics, slightly different forms are introduced in the literature for this tensor. One can also find in the literature different expressions for the energy-momentum tensor that is inherent to magnetic materials. Attention is focused here on these differences, which apparently stem from different standpoints and approaches. On the one hand, one can note that the discrepancies can be readily composed. On the other hand, each of the various forms of the aforementioned tensor seems to be useful, though in the appropriate context. Some of them need to be properly interpreted; others are apparently pertinent to specific boundary value problems.

The purpose of this work is the continuum thermodynamic and variational formulation of spatial and configurational models for a class of elastic, viscous continua with microstructure and material inhomogeneity. As in the standard case, the variational formulation of such models relies on the basic results of the direct continuum thermodynamic formulation obtained in the context of the total energy balance and dissipation principle. On this basis, spatial field relations in the bulk and across a singular surface, as well as the corresponding boundary conditions, are derived for the positional and microstructure degrees-of-freedom as the rate stationarity conditions of a corresponding rate-functional. Finally, variation of the incremental form of the rate functional with respect to the reference configuration yields the configurational balance and field relations for the materials in question. For simplicity, attention is restricted here to isothermal and quasi-static conditions.

In the abstract of his 1970 paper, Eshelby stated: “.” He then went on to demonstrate that the elastic energy-momentum tensor proves to be a useful tool in calculating such forces. The ‘forces’ turn out to be the appropriate traction vectors associated with the energy-momentum (stress) tensor. It is therefore natural and perhaps even fundamental to look for the ’strain tensor’ that can be paired with the ’stress tensor’ to form work. The ’strain rate’ would then be that some kind of departure from uniformity within a physical system. In this paper, we examine the configurational changes brought about by atomic diffusion in a nonuniform alloy crystal. The transformation from a reference, single-parameter simple cubic cell to a six-parameter alloy crystal cell, called the eigentransformation, is identified as the needed kinematic tensor.

Some elements of fracture mechanics, such as energy-release rates and stress-intensity factors, might be examined not on the basis of continuum theories, but on the basis of the much simpler theories of strength-of-materials. Thus it becomes possible to teach fracture mechanics in undergraduate courses for engineers.

We present a new formulation of the configurational force balance in a constitutive-independent framework of thermomechanics. To this end we use invariance requirements for the configurational working - here defined following the ideas of Green and Naghdi on the basic postulates of continua. This new approach has the essential property of providing an expression of the driving force on cracks in accordance with the well-known formula of the energy release rate in thermoelasticity.