Abstract
In this paper we discuss how configurational forces can be computed in an efficient and robust manner when a constitutive continuum model of gradient-enhanced viscoplasticity is adopted, whereby a suitably tailored mixed variational formulation in terms of displacements and micro-stresses is used. It is demonstrated that such a formulation produces sufficient regularity to overcome numerical difficulties that are notorious for a local constitutive model. In particular, no nodal smoothing of the internal variable fields is required. Moreover, the pathological mesh sensitivity that has been reported in the literature for a standard local model is no longer present. Numerical results in terms of configurational forces are shown for (1) a smooth interface and (2) a discrete edge crack. The corresponding configurational forces are computed for different values of the intrinsic length parameter. It is concluded that the convergence of the computed configurational forces with mesh refinement depends strongly on this parameter value. Moreover, the convergence behavior for the limit situation of rate-independent plasticity is unaffected by the relaxation time parameter.
Highlights
Strain gradient effects in metals become important in situations when the deformation localizes within a capturing the “right” physics related to size effects is not the sole reason for employing the gradient theory
The phenomenological strain gradient theory introduced by Fleck and Hutchinson [9] is used in Xia and Hutchinson [10] in order to derive closed-form solutions for the near-tip fields, whereby it is required that the near-tip fields obtained for strain gradient plasticity should tend to the HRR fields from local theory at a distance from the crack-tip that is larger than the chosen internal length
Configurational forces have been computed via a gradientenhanced mixed-dual variational formulation
Summary
Capturing the “right” physics related to size effects is not the sole reason for employing the gradient theory. Examples are the thermodynamic force on a material inhomogeneity, cf Eshelby [14], or the energy release rate in fracture mechanics, cf Griffith [15] Computation of such forces for inelastic materials most often involves the evaluation of spatial gradients of internal variables. In Menzel et al [19], it is shown that shifting from a conventional displacement-based variational formulation to a mixed displacement–internal variable formulation adds nothing to the accuracy of the measured configurational forces-related quantities, while the conventional formulation results in computationally more efficient schemes. Configurational forces are computed based on a gradient-enhanced constitutive theory via a mixed-dual variational formulation. The coupled primary fields are the displacements along with a so-called micro-stress field, the latter being the stress measure which is energy-conjugated to the spatial gradient of the internal variables In this way, two interrelated goals are accomplished.
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