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The heat treatment of steels is a process of fundamental importance for manipulating the properties of the steel. We present a consistent thermodynamic framework for modelling the heat treatment process which is able to capture both displacive and diffusive mechanisms. The present formulation is applicable to the a grain size level of observation. Some salient features of the thermodynamic framework are highlighted, especially those associated with surface energies, and a number of numerical experiments are presented.

Using high-level abstractions, it is possible to efficiently automate the development of finite element models. This has advantages in terms of rapid development, a dramatic reduction in programming errors and offers the possibility of performing special optimisations to produced highly efficient code. The power of this concept is illustrated using tools from the FEniCS project for the simulation of a viscoelastic fluid in an Eulerian framework, and the benchmark problem of a sphere falling in a cylinder pipe is simulated.

Continuum damage models are widely used to represent the development of microscopic defects that coalesce into a macroscopic crack. The microscopic defects cause a progressive weakening or softening of the material (damage). Strain gradient-dependent terms have been included in some damage theories to regularize them, and thereby avoid a pathological mesh-dependence in the solution. A strain gradient-dependent damage model is considered here for the simulation of this feature in quasi-brittle materials. In the model considered, the damage parameter depends upon a regularized equivalent strain. The regularization is introduced through a dependency on the Laplacian of an equivalent strain measure. The introduction of the Laplacian of the strain leads to numerical difficulties as the governing differential equations are fourth-order, and additional boundary conditions must be specified. The application of such a model in a standard finite element framework requires C1 continuity of the shape functions. Here, a continuous/ discontinuous mixed Galerkin method is presented which avoids the need for high-order continuity. The formulation allows the use of C0 or C -1 interpolations for the regularized strain field and a 6C 0 interpolation of the displacement field. Numerical examples are presented to validate the formulation in one and two dimensions. Several interpolations are tested extensively in one dimension in order to provide guidance for the most appropriate formulations in two dimensions. The formulation is applied to crack propagation in a three-point bending test, with the computed result being independent of the discretization.

The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand C1 continuity of the shape functions. Here, recent developments in discontinuous Galerkin methods are explored and exploited for the solution of a prototype non-linear strain gradient-dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard C0 shape functions. The formulation is tested in one dimension for the simplest possible finite element formulation: continuous piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.

The numerical solution of strain gradient-dependent continuum problems has been hindered by continuity demands on the basis functions. The presence of terms in constitutive models that involve gradients of the strain eld means that the C0 continuity of standard nite element shape functions is insufcient. Despite a resurgence of research interest in strain gradient continuum models to represent micro-mechanical effects, a sound, effective and simple framework for the numerical solution strain gradient-dependent problems is lacking. Here, a formulation is presented which allows the use of C0 finite element shape functions for the solution of a prototype strain gradient-dependent damage model. The formulation is examined in two dimensions for the simulation of crack propagation. Particular attention is paid to the application of non-standard boundary conditions.