Background: Inverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought. Methods:In this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached. Results: In our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved. Conclusions:With the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.

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A geometrically linear continuum mechanics framework is proposed for gradient plasticity combining 'strain gradients' and, with a novel approach, 'stress gradients'. Thereby the duality of kinematic and kinetic quantities is exploited in view of the 'div-grad-curl orthogonality' in continuum eld theories. On the one hand the non-integrability of the plastic distortion results in the well-established dislocation density-often denoted as the geometrically-necessary-dislocation (GND) density-that enters the energy storage function. On the other handas entirely novel concept introduced in this contribution-the non-equilibrium of the plastic stress results in the disequilibrium density that parameterizes the dual dissipation potential within the convex analysis setting of plasticity. Consequently both, the dislocation density as well as the disequilibrium density contribute in modelling the size-dependent hardening state of a material in a continuum mechanics setting. The novel approach is eventually elucidated in much detail for the specic case of single crystal plasticity.

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