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We develop a multiscale thermomechanical model to analyze martensitic phase transformations from a cubic crystalline lattice to a tetragonal crystalline lattice. The model is intended for simulating the thermomechanical response of single-crystal grains of austenite. Based on the geometrically nonlinear theory of martensitic transformations, we incorporate microstructural effects from several subgrain length scales. The effective stiffness tensor at the grain level is obtained through an averaging scheme, and preserves crystallographic information from the lattice scale as well as the influence of volumetric changes due to the transformation. The model further incorporates a transformation criterion that includes a surface energy term, which takes into account the creation of interfaces between martensite and austenite. These effects, which are often neglected in martensitic transformation models, thus appear explicitly in the expression of the transformation driving force that controls the onset and evolution of the transformation. In the derivation of the transformation driving force, we clarify the relations between different combinations of thermodynamic potentials and state variables. The predictions of the model are illustrated by analyzing the response of a phase-changing material subjected to various types of deformations. Although the model is developed for cubic to tetragonal transformations, it can be adapted to simulate martensitic transformations for other crystalline structures.

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SUMMARYAn elasto-plastic material model is proposed that can be used to simulate the cyclic deterioration of railway tracks. The model describes the envelope of the irreversible, plastic material response generated during a cyclic loading process, thereby distinguishing the mechanisms of frictional sliding and volumetric compaction. The reversible response is represented by a pressure-dependent, hypo-elastic material law. After the numerical integration of the model is speciÿed, the model is calibrated on laboratory experiments and employed in a ÿnite-element case study of the long-term settlement behaviour of a railway track. The main features of the model are illustrated by comparing the computed response with the response obtained by in situ track measurements.

Summary The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.Keywords Strain-gradient Models, Higher-order Continuum, Microstructure, Wave Propagation, Stability IntroductionClassical continuum theories assume that the stresses in a material point depend only on the first-order derivative of the displacements, i.e. on the strains, and not on higher-order displacement derivatives. As a consequence of this limitation on the kinematic field, a classical continuum is not always capable of adequately describing heterogeneous phenomena. For instance, unrealistic singularities in the stress and/or strain field may occur nearby imperfections. Furthermore, severe problems in the simulation of localisation phenomena with classical continua have been encountered, such as loss of well-posedness in the mathematical description and pathological mesh-dependence in numerical simulations (see [25] for an overview). To avoid these types of deficiencies, it has been proposed to include higher-order strain gradients into the constitutive equations, so that the defects of the classical continuum may be successfully overcome, [4,12,17,19,22,24,25]. The second-order strain gradients that are normally used for these purposes introduce accessory material parameters that reflect the microstructural properties of the material. However, the second-gradient term is often postulated, rather than derived from the microstructure. Hence, this class of models can be denoted as phenomenological.

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SUMMARYA time integration scheme is presented for the martensitic phase transformation model developed in recent theoretical work of Turteltaub and Suiker (A multi-scale thermomechanical model for cubic to tetragonal martensitic phase transformations 2005; Transformation-induced plasticity in ferrous alloys 2005). The phase transformation model can be used for analysing transformation-induced plasticity (TRIP) phenomena in ferrous alloys. The microstructural information for the phase transformation model is provided by the crystallographic theory of martensitic transformations. The transformation characteristics depend on the specific transformation systems activated during a loading process. The time integration scheme is formulated within a framework of finite deformations, where the stressupdate algorithm is based on a fully implicit Euler backward discretization. A robust search algorithm is used for detecting the transformation systems activated during loading. The completion of the transformation process is prescribed by a constraint on the total martensitic volume fraction, which is accurately satisfied in the converged state using a sub-stepping algorithm. The computation of the consistent tangent operator is performed through a numerical differentiation method, which avoids the determination of extensive analytical derivatives and allows the model to be easily adapted if necessary. The ability of the algorithm to solve complex transformation-induced plasticity problems is illustrated with the aid of three-dimensional analyses, in which an aggregate of single-crystal grains of retained austenite embedded in a ferrite-based matrix is subjected to uniaxial tension. The response characteristics are in agreement with experimental findings of Jacques et al.