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In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call "twinning with variable volume fraction." Two 1996 papers by Chopra et al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. The upper bounds are ansatz-based, providing full details about some (nearly) optimal microstructures. The lower bounds are ansatz-free, so they explain why no other arrangement of the two phases could be significantly better. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model presenting formation of microstructure depending on the parameters and the initial data. In particular, we investigate how the presence of stochastic perturbations affects this phenomenon in its asymptotic behavior. Two different sufficient conditions are provided in order to prevent the formation of microstructure: the first one for Stratonovich noise while the second for Itô noise. The main contribution of the paper is that these conditions are independent of the initial values unlike in the deterministic model. Thus, we can interpret our results as some kind of stabilization produced by both types of noise. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy Γ -converges to a limit functional comprised of two contributions: one is given by a constant c ∞ > 0 gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that c ∞ is reached when dislocations are evenly-spaced on the one dimensional circle. [ABSTRACT FROM AUTHOR]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants-so called crossing twins. Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632-1669, ; Proc. R. Soc. Edinb., Sect. A, Math. 142:273-327, ) as well as to a paper by Dolzmann and Müller (Meccanica 30:527-539, ). [ABSTRACT FROM AUTHOR]

A unified treatment of slip and twinning in Bravais lattices is given, focussing on the case of cubic symmetry, and using the Ericksen energy well formulation, so that interfaces correspond to rank-one connections between the infinitely many crystallographically equivalent energy wells. Twins are defined to be such rank-one connections involving a nontrivial reflection of the lattice across some plane. The slips and twins minimizing shear magnitude for cubic lattices are rigorously calculated, and the conjugates of these and other slips analyzed. It is observed that all rank-one connections between the energy wells for the dual of a Bravais lattice can be obtained explicitly from those for the original lattice, so that in particular the rank-one connections for fcc can be obtained explicitly from those for bcc. [ABSTRACT FROM AUTHOR]

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor $\mathbf{S}$ . Following Eshelby's seminal work, a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches. [ABSTRACT FROM AUTHOR]

The current paper addresses the effect of the hydrogen partial pressure on the microstructure and transformation of the AgO film. The transformation process and mechanism were also analyzed in detail. Increasing the hydrogen partial pressure can accelerate the transformation of AgO to Ag and lower the critical transformation temperature of the film due to the enhanced hydrogen reduction and to both of the lowered activation energy of the reaction of AgO with hydrogen and enhanced lattice strain of the AgO film. Hydrogen-involved reaction in the whole hydrogen annealing process is mainly hydrogen reduction reaction with AgO. Diffusion of hydrogen and gaseous HO molecules accompanies the whole hydrogen annealing process. [ABSTRACT FROM AUTHOR]

The computational cost of numerical methods in microscopic-scales such as molecular dynamics (MD) is a deterrent factor that limits simulations with a large number of particles. Hence, it is desirable to decrease the computational cost and run time of simulations, especially for problems with a symmetrical domain. However, in microscopic-scales, implementation of symmetric boundary conditions is not straight-forward. Previously, the present authors have successfully used a symmetry boundary condition to solve molecular flows in constant-area channels. The results obtained with this approach agree well with the benchmark cases. Therefore, it has provided us with a sound ground to further explore feasibility of applying symmetric solutions of micro-fluid flows in other geometries such as variable-area ducts. Molecular flows are solved for the whole domain with and without the symmetric boundary condition. Good agreement has been reached between the results of the symmetric solution and the whole domain solution. To investigate robustness of the proposed method, simulations are conducted for different values of affecting parameters including an external force, a flow density, and a domain length. The results indicate that the symmetric solution is also applicable to variable-area ducts such as micro-nozzles. [ABSTRACT FROM AUTHOR]

During deep bed filtration of suspensions and colloids in a porous medium, some particles are retained in the pores and form a fixed deposit. A one-dimensional mathematical model of filtration with particles of two types is considered. Exact solution is derived. The existence and the uniqueness of the solution are proved by the method of characteristics, and a solution in the form of a traveling wave is obtained. The profiles of total and partial retained concentrations, showing the dependence of the retained particles concentrations on the coordinate at a fixed time, are studied. It is shown by Taylor expansions that the retained profiles of large particles decrease monotonically, while the retained profiles of small particles are non-monotonic. At a short time, the profile of small particles decreases monotonically; with increasing time, a maximum point appears on it, moving from the inlet to the outlet of the porous medium. When the maximum point reaches the outlet, the profile becomes monotonically increasing. The condition for the non-monotonicity of the total retained profile is obtained. [ABSTRACT FROM AUTHOR]

We study some particular cases of the n -well problem in two-dimensional linear elasticity. Assuming that every well in U ⊂ R sym 2 × 2 belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull L e (U) for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in U . For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly contained in its convex hull C (U) . We extend this result to some particular configurations of n wells. Most of the proofs are constructive, and we also present explicit examples. [ABSTRACT FROM AUTHOR]

A model for deep bed filtration of a polydisperse suspension with small impurities in a porous medium is considered. Different suspended particles move with the same velocity as the carrier water and get blocked in the pore throats due to the size-exclusion mechanism of particle retention. A solution of the model in the form of a traveling wave is obtained. The global exact solution for a multiparticle filtration with one high concentration and several low concentrations of suspended particles is obtained in an explicit form. The analytic solutions for a bidisperse suspension with large and small particles are constructed. The profiles of the retained small particles change monotony with time. The global asymptotics for the filtration of a polydisperse suspension with small kinetic rates is constructed in the whole filtration zone. [ABSTRACT FROM AUTHOR]

We have performed a computer simulation of processes taking place at formation of energetic equilibrium oxygen-octahedral clusters in a ferroelectric phase of a lithium niobate crystal (LN, LiNbO3). In order to do so we used a semiclassical atomistic model with a qualitative approach. For the first time we have modeled lithium niobate structure on the basis of oxygen octahedra, while unit cells are usually used. We have established that there is an energy efficient cluster size with a specific structure that approaches a congruent LN structure. We have also proved that stoichiometric composition of LN leads to loss of electrical neutrality; thus achievement of demanded properties including photorefractive ones is possible by doping. We have chosen an ion with a charge 3+ to build a model. [ABSTRACT FROM AUTHOR]

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid–solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of Γ -convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases. [ABSTRACT FROM AUTHOR]

Abstract: We prove the existence of minimisers for a family of models related to the single‐slip‐to‐single‐plane relaxation of single‐crystal, strain‐gradient elastoplasticity with L p‐hardening penalty. In these relaxed models, where only one slip‐plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower‐semicontinuous along bounded‐energy sequences which satisfy the single‐plane condition, meaning precisely that this side condition should be preserved in the weak L p‐limit. This is done with the aid of an ‘exclusion’ lemma of Conti & Ortiz, which essentially allows one to put a lower bound on the dislocation energy at interfaces of (single‐plane) slip patches, thus precluding fine phase‐mixing in the limit. Furthermore, using div‐curl techniques in the spirit of Mielke & Müller, we are able to show that the usual multiplicative decomposition of the deformation gradient into plastic and elastic parts interacts with weak convergence and the single‐plane constraint in such a way as to guarantee lower‐semicontinuity of the (polyconvex) elastic energy, and hence the total elasto‐plastic energy, given sufficient ( p > 2) hardening, thus delivering the desired result. [ABSTRACT FROM AUTHOR]

The rotation $${{\mathrm{polar}}}(F) \in {{\mathrm{SO}}}(3)$$ arises as the unique orthogonal factor of the right polar decomposition $$F = {{\mathrm{polar}}}(F)\,U$$ of a given invertible matrix $$F \in {{\mathrm{GL}}}^+(3)$$ . In the context of nonlinear elasticity Grioli (Boll Un Math Ital 2:252-255, 1940) discovered a geometric variational characterization of $${{\mathrm{polar}}}(F)$$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $$\mu > 0$$ and $$\mu _c \ge 0$$ (Grioli: $$\mu = \mu _c$$ ). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field $$F :=\nabla \varphi : \Omega \rightarrow {{\mathrm{GL}}}^+(3)$$ and the microrotation field $$R: \Omega \rightarrow {{\mathrm{SO}}}(3)$$ . The corresponding set of non-classical energy-minimizing rotations represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping $$\varphi _\mathrm{nano}$$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns $${{\mathrm{rpolar}}}^\pm _{1,0}(F_\mathrm{nano})$$ with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored. [ABSTRACT FROM AUTHOR]

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field [ABSTRACT FROM AUTHOR]

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field [ABSTRACT FROM AUTHOR]

We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $${\sigma }$$ , and a measure of the bonding strength between the film and substrate, $${\gamma }$$ . We prove upper bounds on the minimum energy of the form $${\sigma }^a {\gamma }^b$$ and find that there are four different parameter regimes corresponding to different values of a and b and to different folding patterns of the film. In some cases, the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds. [ABSTRACT FROM AUTHOR]

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $$\Gamma $$ -convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $$SBV^p$$ functions whose jump sets have a prescribed orientation. [ABSTRACT FROM AUTHOR]

A nonlinear beam formulation is presented based on the Gurtin-Murdoch surface elasticity and the modified couple stress theory. The developed model theoretically takes into account coupled effects of the energy of surface layer and microstructures sizedependency. The mid-plane stretching of a beam is incorporated using von-Karman nonlinear strains. Hamilton's principle is used to determine the nonlinear governing equation of motion and the corresponding boundary conditions. As a case study, pull-in instability of an electromechanical nano-bridge structure is studied using the proposed formulation. The nonlinear governing equation is solved by the analytical reduced order method (ROM) as well as the numerical solution. Effects of various parameters including surface layer, size dependency, dispersion forces, and structural damping on the pullin parameters of the nano-bridges are discussed. Comparison of the results with the literature reveals capability of the present model in demonstrating the impact of nanoscale phenomena on the pull-in threshold of the nano-bridges. [ABSTRACT FROM AUTHOR]

This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy-Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress-strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross non-ellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures. [ABSTRACT FROM AUTHOR]

The equilibrium configurations of a row of uniformly distributed dislocation dipoles are first studied. The analysis is then generalised to study dipoles in two-dimensional rectangular periodic lattices. By examining the stability of the equilibrium configurations we find that the system may undergo a natural transition from the Taylor lattice to a row of dipole walls. This bifurcation may be involved in the transition from channel-vein to persistent slip band (PSB) structures in the early stage of metal fatigue. [ABSTRACT FROM AUTHOR]

We present an elementary method of explicit calculation of Young measures for a certain class of functions. This class contains, in particular, functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs, for instance, in nonlinear elasticity (solid–solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in this article makes no use of functional analytic tools. There is no need to use a generalized version of the Riemann–Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions. [ABSTRACT FROM AUTHOR]

We design a new mesoscopic thin-film model for shape-memory materials which takes into account thermomechanical effects. Starting from a microscopic thermodynamical bulk model, we guide the reader through a suitable dimension reduction procedure followed by a scale transition valid for specimen large in area up to a limiting model which describes microstructure by means of parametrized measures. All our models obey the second law of thermodynamics and possess suitable weak solutions. This is shown for the resulting thin-film models by making the procedure described above mathematically rigorous. The main emphasis is, thus, put on modeling and mathematical treatment of joint interactions of mechanical and thermal effects accompanying phase transitions and on reduction in specimen dimensions and transition of material scales. [ABSTRACT FROM AUTHOR]

The aim of the present article is a description of the many phenomenological aspects of fracture and of their relations with other inelastic phenomena, such as the formation of microstructure and the changes in the material's strength induced by plasticity and damage. The whole analysis is carried on using a single mathematical tool, the incremental energy minimization. The energy functional is assumed to be made of two parts, an elastic energy and a cohesive energy. The structure assumed for the cohesive energy term determines different modes of inelastic response and fracture. To avoid the heavy technical difficulties met in higher dimension, the whole analysis is one-dimensional. This article reflects the contents of six lectures delivered in a couple of occasions. The object of the first lecture is Griffith's theory of brittle fracture. In the second lecture, the Barenblatt-Dugdale regularization is discussed. In Lecture 3 it is shown that the concept of cohesive energy can be successfully used to describe the formation of microstructure. A description of the phenomenon of elastic unloading, based on the assumption of dissipativity of the cohesive energy, is given in Lecture 4. The last two lectures deal with the diffuse cohesive energy model. In it, the cohesive energy, instead of being defined on surfaces as usual, is supposed to be diffused over the volume. In particular, the non-local model discussed in Lecture 6 provides a comprehensive description of the strain-softening phenomenon. [ABSTRACT FROM AUTHOR]

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity. [ABSTRACT FROM AUTHOR]

This work presents the study of the synthesis and characterization of polycrystalline diamond (PCD) compacts using a bi-layered assembly method and solving the catastrophic nature of failure molding when using single-layered assembly and 0.5-1.5 μm diamond powder as starting materials. The well-sintered PCD compacts are prepared on WC-16wt% Co substrates at temperatures from 1350 °C to 1500 °C for 15 min at static high pressure of 5.2 GPa, using the bi-layered assembly method. According to our scanning electron microscope observation, the specimens sintered at 1450 °C have a homogeneous microstructure. The graphitization of the diamond appears at 1500 °C according to X-ray diffraction analysis. The specimens sintered at 1450 °C show the best performance according to the wear resistance test. [ABSTRACT FROM AUTHOR]