We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic oneand two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative K = 4 for bcc. A two-dimensional generalization suggests K 2D = 5. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical nonlinearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain.

The spontaneous nucleation of accelerating slip along slowly driven frictional interfaces is central to a broad range of geophysical, physical, and engineering systems, with particularly far‐reaching implications for earthquake physics. A common approach to this problem associates nucleation with an instability of an expanding creep patch upon surpassing a critical length Lc. The critical nucleation length Lc is conventionally obtained from a spring‐block linear stability analysis extended to interfaces separating elastically deformable bodies using model‐dependent fracture mechanics estimates. We propose an alternative approach in which the critical nucleation length is obtained from a related linear stability analysis of homogeneous sliding along interfaces separating elastically deformable bodies. For elastically identical half‐spaces and rate‐and‐state friction, the two approaches are shown to yield Lc that features the same scaling structure, but with substantially different numerical prefactors, resulting in a significantly larger Lc in our approach. The proposed approach is also shown to be naturally applicable to finite‐size systems and bimaterial interfaces, for which various analytic results are derived. To quantitatively test the proposed approach, we performed inertial Finite‐Element‐Method calculations for a finite‐size two‐dimensional elastically deformable body in rate‐and‐state frictional contact with a rigid body under sideway loading. We show that the theoretically predicted Lc and its finite‐size dependence are in reasonably good quantitative agreement with the full numerical solutions, lending support to the proposed approach. These results offer a theoretical framework for predicting rapid slip nucleation along frictional interfaces.

Understanding the dynamic stability of bodies in frictional contact steadily sliding one over the other is of basic interest in various disciplines such as physics, solid mechanics, materials science and geophysics. Here we report on a two-dimensional linear stability analysis of a deformable solid of a finite height H, steadily sliding on top of a rigid solid within a generic rate-and-state friction type constitutive framework, fully accounting for elastodynamic effects. We derive the linear stability spectrum, quantifying the interplay between stabilization related to the frictional constitutive law and destabilization related both to the elastodynamic bi-material coupling between normal stress variations and interfacial slip, and to finite size effects. The stabilizing effects related to the frictional constitutive law include velocity-strengthening friction (i.e. an increase in frictional resistance with increasing slip velocity, both instantaneous and under steady-state conditions) and a regularized response to normal stress variations. We first consider the small wave-number k limit and demonstrate that homogeneous sliding in this case is universally unstable, independently of the details of the friction law. This universal instability is mediated by propagating waveguidelike modes, whose fastest growing mode is characterized by a wave-number satisfying kH ∼ O(1) and by a growth rate that scales with H −1 . We then consider the limit kH → ∞ and derive the stability phase diagram in this case. We show that the dominant instability mode travels at nearly the dilatational wave-speed in the opposite direction to the sliding direction. In a certain parameter range this instability is manifested through unstable modes at all wave-numbers, yet the frictional response is shown to be mathematically well-posed. Instability modes which travel at nearly the shear wave-speed in the sliding direction also exist in some range of physical parameters. Previous results obtained in the quasi-static regime appear relevant only within a narrow region of the parameter space. Finally, we show that a finite-time regularized response to normal stress variations, within the framework of generalized rate-and-state friction models, tends to promote stability. The relevance of our results to the rupture of bi-material interfaces is briefly discussed.

We examine the interaction between precipitates and grain boundaries, which undergo shearcoupled motion. The elastic problem, emerging from grain boundary perturbations and an elastic mismatch strain induced by the precipitates, is analysed. The resulting free elastic energy contains interaction terms, which are derived numerically via the integration of the elastic energy density. The interaction of the shear-coupled grain boundary and the coherent precipitates leads to potential elastic energy reductions. Such a decrease of the elastic energy has implications on the grain boundary shape and also on the solubility limit near the grain boundary. By energy minimisation we are able to derive the grain boundary shape change analytically. We apply the results to the Fe-C system to predict the solubility limit change of cementite near an α-iron grain boundary. arXiv:1905.03068v1 [cond-mat.mtrl-sci]

Abstract:We present a selection of scale transfer approaches from the electronic to the continuum regime for topics relevant to hydrogen embrittlement. With a focus on grain boundary related hydrogen embrittlement, we discuss the scale transfer for the dependence of the carbon solution behavior in steel on elastic effects and the hydrogen solution in austenitic bulk regions depending on Al content. We introduce an approximative scheme to estimate grain boundary energies for varying carbon and hydrogen population. We employ this approach for a discussion of the suppressing influence of Al on the substitution of carbon with hydrogen at grain boundaries, which is an assumed mechanism for grain boundary hydrogen embrittlement. Finally, we discuss the dependence of hydride formation on the grain boundary stiffness.

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The formation of coherent precipitates is often accompanied by large elastic mismatch stresses, which suppress phase separation. We discuss the presence of interfaces as a mechanism for stress relaxation, which can lead to preferred zones of precipitation. In particular, we discuss the proximity of free surfaces and shear-coupled grain boundaries, for which we can obtain a substantial local energy reduction and predict the influence on the local precipitation thermodynamics. The latter case is accompanied by morphological changes of the grain boundary, which are less suitable for large-scale descriptions. For that purpose, we develop an effective description through an elastic softening inside the grain boundary and map the microscopic grain boundary relaxation to a mesoscopic elastic and phase field model, which also allows generalizing the description to multi-phase situations.

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