Celia Reina

Kinematic description of crystal plasticity in the finite kinematic framework: A micromechanical understanding of F=FeFp

Abstract The plastic component of the deformation gradient plays a central role in finite kinematic models of plasticity. However, its characterization has been the source of extended debates in the literature and many important issues still remain unresolved. Some examples are the micromechanical understanding of F = F e F p with multiple active slip systems, the uniqueness of the decomposition, or the characterization of the plastic deformation without reference to the so-called intermediate configuration. In this paper, we shed some light to these issues via a two-dimensional kinematic analysis of the plastic deformation induced by discrete slip surfaces and the corresponding dislocation structures. In particular, we supply definitions for the elastic and plastic components of the deformation gradient as a function of the active slip systems without any a priori assumption on the decomposition of the total deformation gradient. These definitions are explicitly and uniquely given from the microstructure and do not make use of any unrealizable intermediate configuration. The analysis starts from a semi-continuous mathematical description of the deformation at the microscale, where the displacements are considered continuous everywhere in the domain except at the discrete slip surfaces, over which there is a displacement jump. At this scale, where the microstructure is resolved, the deformation is uniquely characterized from purely kinematic considerations and the elastic and plastic components of the deformation gradient can be defined based on physical arguments. These quantities are then passed to the continuous limit via homogenization, i.e. by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. This continuum limit is computed for several illustrative examples, where the well-known multiplicative decomposition of the total deformation gradient is recovered. Additionally, by similar arguments, an expression of the dislocation density tensor is obtained as the limit of discrete dislocation densities which are well characterized within the semi-continuous model.

Derivation of F = F e F p as the continuum limit of crystalline slip

Abstract In this paper we provide a proof of the multiplicative kinematic description of crystal elastoplasticity in the setting of large deformations, i.e. F = F e F p , for a two dimensional single crystal. The proof starts by considering a general configuration at the mesoscopic scale, where the dislocations are discrete line defects (points in the two-dimensional description used here) and the displacement field can be considered continuous everywhere in the domain except at the slip surfaces, over which there is a displacement jump. At such scale, as previously shown by two of the authors, there exists unique physically based definitions of the total deformation tensor F and the elastic and plastic tensors F e and F p that do not require the consideration of any non-realizable intermediate configuration and do not assume any a priori relation between them of the form F = F e F p . This mesoscopic description is then passed to the continuum limit via homogenization, i.e. by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. We show for two-dimensional deformations of initially perfect single crystals that the classical continuum formulation is recovered in the limit with F = F e F p , det F p = 1 and G = Curl F p the dislocation density tensor.

Mesoscale computational study of the nanocrystallization of amorphous Ge via a self-consistent atomistic phase-field model

Abstract Germanium is the base element in many phase-change materials, i.e. systems that can undergo reversible transformations between their crystalline and amorphous phases. These materials are widely used in current digital electronics and hold great promise for the next generation of non-volatile memory devices. However, the ultra-fast phase transformations required for these applications can be exceedingly complex even for single-component systems, and a full physical understanding of these phenomena is still lacking. In this paper we study the growth of crystalline Ge from amorphous thin films at high temperature using phase-field models informed by atomistic calculations of fundamental material properties. The atomistic calculations capture the full anisotropy of the Ge crystal lattice, which results in orientation dependences for interfacial energies and mobilities. These orientation relations are then exactly recovered by the phase-field model at finite thickness via a novel parametrization strategy based on invariance solutions of the Allen–Cahn equations. By means of this multiscale approach, we study the interplay between nucleation and growth and find that the relation between the mean radius of the crystallized Ge grains and the nucleation rate follows simple Avrami-type scaling laws. We argue that these can be used to cover a wide region of the nucleation rate space, hence facilitating comparison with experiments.

Discrete Averaging Relations for Micro to Macro Transition

The well-known Hills averaging theorems for stresses and strains as well as the so-called Hill-Mandel principle of macrohomogeneity are essential ingredients for the coupling and the consistency between the micro and macro scales in multiscale finite element procedures (FE

Entropy production and the geometry of dissipative evolution equations

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Formation of Nanotwin Networks during High-Temperature Crystallization of Amorphous Germanium

). We show in this paper that these averaging relations hold exactly under standard finite element discretizations, even if the stress field is discontinuous across elements and the standard proofs based on the divergence theorem are no longer suitable. The discrete averaging results are derived for the three classical types of boundary conditions (affine displacement, periodic and uniform traction boundary conditions) using the properties of the shape functions and the weak form of the microscopic equilibrium equations. The analytical proofs are further verified numerically through a simple finite element simulation of an irregular representative volume element undergoing large deformations. Furthermore, the proofs are extended to include the effects of body forces and inertia, and the results are consistent with those in the smooth continuum setting. This work provides a solid foundation to apply Hills averaging relations in multiscale finite element methods without introducing an additional error in the scale transition due to the discretization.

Computing diffusivities from particle models out of equilibrium

Purely dissipative evolution equations are often cast as gradient flow structures, z ̇=K(z)DS(z), where the variable z of interest evolves towards the maximum of a functional S according to a metric defined by an operator K. While the functional often follows immediately from physical considerations (e.g., the thermodynamic entropy), the operator K and the associated geometry does not necessarily do so (e.g., Wasserstein geometry for diffusion). In this paper, we present a variational statement in the sense of maximum entropy production that directly delivers a relationship between the operator K and the constraints of the system. In particular, the Wasserstein metric naturally arises here from the conservation of mass or energy, and depends on the Onsager resistivity tensor, which, itself, may be understood as another metric, as in the steepest entropy ascent formalism. This variational principle is exemplified here for the simultaneous evolution of conserved and nonconserved quantities in open systems. It thus extends the classical Onsager flux-force relationships and the associated variational statement to variables that do not have a flux associated to them. We further show that the metric structure K is intimately linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed gradient flows, and that the proposed variational principle encloses an infinite-dimensional fluctuation-dissipation statement.

Broadband locally resonant metamaterials with graded hierarchical architecture

Germanium is an extremely important material used for numerous functional applications in many fields of nanotechnology. In this paper, we study the crystallization of amorphous Ge using atomistic simulations of critical nano-metric nuclei at high temperatures. We find that crystallization occurs by the recurrent transfer of atoms via a diffusive process from the amorphous phase into suitably-oriented crystalline layers. We accompany our simulations with a comprehensive thermodynamic and kinetic analysis of the growth process, which explains the energy balance and the interfacial growth velocities governing grain growth. For the 〈111〉 crystallographic orientation, we find a degenerate atomic rearrangement process, with two zero-energy modes corresponding to a perfect crystalline structure and the formation of a Σ3 twin boundary. Continued growth in this direction results in the development a twin network, in contrast with all other growth orientations, where the crystal grows defect-free. This particular mechanism of crystallization from amorphous phases is also observed during solid-phase epitaxial growth of 〈111〉 semiconductor crystals, where growth is restrained to one dimension. We calculate the equivalent X-ray diffraction pattern of the obtained nanotwin networks, providing grounds for experimental validation.

Slip-induced conservation laws for dislocation structures in the finite kinematic framework

A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have Gaussian fluctuations but it is otherwise allowed to undergo arbitrary out-of-equilibrium evolutions. This could be potentially relevant for particle data obtained from experimental applications. The key idea underlying the method is that finite, yet large, particle systems formally obey stochastic partial differential equations of gradient flow type satisfying a fluctuation–dissipation relation. The strategy is here applied to three classic particle models, namely independent random walkers, a zero-range process and a symmetric simple exclusion process in one space dimension, to allow the comparison with analytic solutions.

Variational coarse-graining procedure for dynamic homogenization

We investigate the effect of hierarchical designs on the bandgap structure of periodic lattice systems with inner resonators. A detailed parameter study reveals various interesting features of structures with two levels of hierarchy as compared with one level systems with identical static mass. In particular: (i) their overall bandwidth is approximately equal, yet bounded above by the bandwidth of the single-resonator system; (ii) the number of bandgaps increases with the level of hierarchy; and (iii) the spectrum of bandgap frequencies is also enlarged. Taking advantage of these features, we propose graded hierarchical structures with ultra-broadband properties. These designs are validated over analogous continuum models via finite element simulations, demonstrating their capability to overcome the bandwidth narrowness that is typical of resonant metamaterials.