A dislocation density based material model to simulate the anisotropic creep behavior of single-phase and two-phase single crystals

The primary and secondary creep behavior of single crystals is observed by a material model using evolution equations for dislocation densities on individual slip systems. An interaction matrix defines the mutual influence of dislocation densities on different glide systems. Face-centered cubic (fcc), body-centered cubic (bcc) and hexagonal closed packed (hcp) lattice structures have been investigated. The material model is implemented in a finite element method to analyze the orientation dependent creep behavior of two-phase single crystals. Three finite element models are introduced to simulate creep of a γ′ strengthened nickel base superalloy in 〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 directions. This approach allows to examine the influence of crystal slip and cuboidal microstructure on the deformation process.     

Thermoelastic stresses in a cylinder or disk with cubic anisotropy

The thermoelastic stresses in a crystal in the shape of a circular cylinder or disk are considered. The crystal is a cubically-orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axisymmetry of the geometry and thermal loading. This problem is motivated by a thermoelastic model used for certain crystal growth processes. Two simplifying assumptions are made here: (a) the problem is two-dimensional with plane strain or plain stress conditions, and (b) the elastic properties do not depend on the temperature. A new Fourier-type perturbation method is devised and an analytic asymptotic solution of a closed form is obtained, based on the weak cubic anisotropy of the crystal as a perturbation parameter. A general solution technique is described which yields the asymptotic solution up to a desired order. Numerical results are presented for typical parameter values.     

Inhomogeneous elastic medium with nonlocal interaction

In [1] the author examined a macroscopically homogeneous elastic medium of simple structure with spatial dispersion. In that case the assumption of the existence of an elementary unit of length and long-range forces conditioned the nonlocalizability of the theory, and the macroscopic homogeneity was manifested in the invariance of the integral operators under shear (difference kernels).In this paper the more general model of an inhomogeneous elastic medium of simple structure with nonlocal interaction is constructed. In § 1 the existence of a symmetric stress tensor is proved with broad assumptions, and the corresponding operator Hooke's law is written down. As a corollary, the usual expression for the energy density is obtained. In §2 the case of point defects is considered. An explicit expression is found for the Green's tensor for a medium with point defects in terms of the Green's tensor for the homogeneous medium. With the help of the Green's tensor the self-energy of the defect and the energy of the interaction force are calculated.     

Energetics of misfit dislocation dipoles in anisotropic epitaxial films with nanoscale compositional modulation

The elastic strain and stress fields associated with nanoscale compositional modulation in an anisotropic epitaxial film on an anisotropic substrate are obtained by using Stroh formalism and the Eshelby-type inclusion method. The composition of the epitaxial film is considered to periodically fluctuate in a surface soft mode, with the amplitude of the composition modulation maximal near the growing surface and decreasing exponentially into the film. It has been experimentally observed that the composition modulation affects the formation of a new type of crystal defects, i.e., misfit dislocation dipoles, in III–V compound semiconductor materials. The formation energy of a misfit dislocation dipole under the elastic fields due to the composition modulation is calculated in this study. It is composed of the core and self energies of two dislocations, the interaction energy between two dislocations, and the interaction energies between the composition modulation and two dislocations. Numerical calculations are performed for a dislocation dipole in a lattice-matched Ga_0.5In_0.5P film on a GaAs substrate.     

An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene

Force multipoles are employed to represent various types of defects and physical phenomena in solids: point defects (interstitials, vacancies), surface steps and islands, proteins on biological membranes, inclusions, extended defects, and biological cell interactions among others. In the present work, we (i) as a prototype simple test case, conduct quantum mechanical calculations for mechanics of defects in graphene sheet and in parallel, (ii) formulate an enriched continuum elasticity theory of force dipoles of various anisotropies incorporating up to second gradients of strain fields (thus accounting for nonlocal dispersive effects) instead of the usual dispersion-less classical elasticity formulation that depends on just the strain (c.f. Peyla, P., Misbah, C., 2003. Elastic interaction between defects in thin and 2-D films. Eur. Phys. J. B. 33, 233-247). The fundamental Green's function is derived for the governing equations of second gradient elasticity and the elastic self and interaction energies between force dipoles are formulated for both the two-dimensional thin film and the three-dimensional case. While our continuum results asymptotically yield the same interaction energy law as Peyla and Misbah for large defect separations (∼1/rⁿ for defects with n-fold symmetry), the near-field interactions are qualitatively far more complex and free of singularities. Certain qualitative behavior of defect mechanics predicted by atomistic calculations are well captured by our enriched continuum models in contrast to classical elasticity calculations. For example, consistent with our atomistic calculations of defects in isotropic graphene, even two dilation centers show a finite interaction (as opposed to classical elasticity that predicts zero interaction). We explicitly find the physically consistent result that the self-energy of a defect is equivalent to half the interaction energy between two identical defects when they “merge” into each other. The atomistic, classical elastic and the enriched continuum predictions are thoroughly compared for two types of defects in graphene: Stone-Wales and divacancy.     

On the modelling of anisotropic material behaviour in viscoplasticity

A uniaxial viscoplastic deformation is motivated as a discrete sequence of stable and unstable equilibrium states and approximated by a smooth family of stable states of equilibrium depending on the history of the mechanical process. Three-dimensional crystal viscoplasticity starts from the assumption that inelastic shearings take place on slip systems, which are known from the particular geometric structure of the crystal. A constitutive model for the behaviour of a single crystal is developed, based on a free energy, which decomposes into an elastic and an inelastic part. The elastic part, the isothermal strain energy, depends on the elastic Green strain and allows for the initial anisotropy, known from the special type of the crystal lattice. Additionally, the strain energy function contains an orthogonal tensor-valued internal variable representing the orientation of the anisotropy axes. This orientation develops according to an evolution equation, which satisfies the postulate of full invariance in the sense that it is an observer-invariant relation. The inelastic part of the free energy is a quadratic function of the integrated shear rates and corresponding internal variables being equivalent to backstresses in order to consider kinematic hardening phenomena on the slip system level. The evolution equations for the shears, backstresses and crystallographic orientations are thermomechanically consistent in the sense that they are compatible with the entropy inequality. While the general theory applies to all types of lattices, specific test calculations refer to cubic symmetry (fcc) and small elastic strains. The simulations of simple tension and compression processes of a single crystal illustrates the development of the crystallographic axes according to the proposed evolution equation. In order to simulate the behaviour of a polycrystal the initial orientations of the anisotropy axes are assumed to be space-dependent but piecewise constant, where each region of a constant orientation corresponds to a grain. The results of the calculation show that the initially isotropic distribution of the orientation changes in a physically reasonable manner and that the intensity of this process-induced texture depends on the specific choice of the material constants.     

Material model describing the orientation dependent creep behavior of single crystals based on dislocation densities of slip systems

A material model is proposed which describes single crystal creep behavior by evolution equations for dislocation densities on individual slip systems. An interaction matrix determines the influence from one glide system to the other. Assuming a face centered cubic crystal, allowing deformation on octahedral glide planes and cube glide planes with a Burgers vector of the type a/2〈110〉, nine independent parameters of the interaction matrix can be distinguished. A parameter check of the nine independent parameters has been carried out, showing the influence of parameters on specific orientations of the load axis. If one assumes dislocation interaction of a glide system only with itself a smooth behavior is predicted with a maximum creep rate for [001] orientation, followed by [011] and [111]. If a strong interaction is assumed, the orientation dependent creep behavior is not at all smooth, instead it shows a sharp drop in creep rates mainly in symmetric positions of the standard orientation triangle. The orientations with highest creep rates are in this case those which favor single glide. Highly symmetric orientations, such as [001], [011] and [111] have strongly decreased stationary creep rates.     

Discrete Crystal Elasticity and Discrete Dislocations in Crystals

This article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals. The theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups, differential forms and operators, and a theory of integration of forms. In particular, we define the lattice complex of a number of commonly encountered lattices, including body-centered cubic and face-centered cubic lattices. We show that material frame indifference naturally leads to discrete notions of stress and strain in lattices. Lattice defects such as dislocations are introduced by means of locally lattice-invariant (but globally incompatible) eigendeformations. The geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations, such as the dislocation density tensor, the equation of conservation of Burgers vector, Kröner's relation and Mura's formula for the stored energy. We additionally supply conditions for the existence of equilibrium displacement fields; we show that linear elasticity is recovered as the Γ-limit of harmonic lattice statics as the lattice parameter becomes vanishingly small; we compute the Γ-limit of dilute dislocation distributions of dislocations; and we show that the theory of continuously distributed linear elastic dislocations is recovered as the Γ-limit of the stored energy as the lattice parameter and Burgers vectors become vanishingly small.     

A new mechanism of elastic energy relaxation in heteroepitaxy of monocrystalline films: Interaction of point defects and dilatation dipoles

A new method for growing a low-defect elastic-stress-free silicon carbide film on silicon substrates is theoretically developed and experimentally implemented. In this method, the relaxation of inevitable elastic stresses is attained by an essentially new mechanism, namely, by dilatation dipoles (stable complexes consisting of attracting dilatation centers) formed by a carbon atom in interstitial position and a silicon vacancy. The tensor Green function for elastic-anisotropic media is used to obtain the dependence of the point defect interaction energy on their mutual crystallographic location in silicon. It is shown that the situation where the dilatation dipole is perpendicular to the plane (111) is most efficient for a cubic crystal. In this case, practically the whole elastic energy of the film dilatation can be relaxed only at the expense of dipoles, and this must produce a high quality of silicon carbide films. The assembly of nanoscale silicon carbide films on a silicon substrate was realized for the first time in practice by synthesis of dilatation dipoles, which play the role of molecular seeds. Highly perfect carbide layers were grown on silicon substrate, and all of their basic characteristics were measured. Such films were for the first time used to produce a wide-band light-emitting diode structure on silicon.     

An anisotropic micromechanics model for predicting the rafting direction in Ni-based single crystal superalloys

An anisotropic micromechanics model based on the equivalent inclusion method is developed to investigate the rafting direction of Ni-based single crystal superalloys. The micromechanical model considers actual cubic structure and orthogonal anisotropy properties. The von Mises stress, elastic strain energy density, and hydrostatic pressure in different inclusions of micromechanical model are calculated when applying a tensile or compressive loading along the [001] direction. The calculated results can successfully predict the rafting direction for alloys exhibiting a positive or a negative mismatch, which are in agreement with pervious experimental and theoretical studies. Moreover, the elastic constant differences and mismatch degree of the matrix and precipitate phases and their influences on the rafting direction are carefully discussed.     

Theoretical properties of cubic crystals at arbitrary pressure—I. Density and bulk modulus

The theoretical elastic behaviour of simple monatomic cubic crystals at arbitrary pressure is to be presented in a series of papers. In the present paper, general expressions are derived for calculating the pressure P and the bulk modulus κ as a function of the all-round stretch λ for crystals in which the interatomic interaction energies are modelled by pairwise functions φ. With the aid of a particular family of functions φ, calculations are carried out for the three cubic structures, and a detailed study is made of the influence of crystal structure and explicit nature of φ upon the theoretical elastic behaviour under pressure loading. Calculations are also made of higher order derivatives of P(λ) and gk(λ) in the reference state (i.e. at λ = 1) and the “higher order moduli” thus calculated are used to formulate series expansion approximations to the functions P(λ) and κ(λ). Values of P(λ) and κ(λ) in the series approximations, based upon successively higher order moduli (evaluated in the reference state), are compared with the corresponding “exact” values evaluated in the current state. The theoretical results are useful as empirical relationships modelling the elastic behaviour of crystals at arbitrary pressure.     

Pair vs many-body potentials: Influence on elastic and plastic behavior in nanoindentation of fcc metals

Molecular-dynamics simulation can give atomistic information on the processes occurring in nanoindentation experiments. In particular, the nucleation of dislocation loops, their growth, interaction and motion can be studied. We investigate how realistic the interatomic potentials underlying the simulations have to be in order to describe these complex processes. Specifically we investigate nanoindentation into a Cu single crystal. We compare simulations based on a realistic many-body interaction potential of the embedded-atom-method type with two simple pair potentials, a Lennard-Jones and a Morse potential. We find that qualitatively many aspects of nanoindentation are fairly well reproduced by the simple pair potentials: elastic regime, critical stress and indentation depth for yielding, dependence on the crystal orientation, and even the level of the hardness. The quantitative deficits of the pair potential predictions can be traced back: (i) to the fact that the pair potentials are unable in principle to model the elastic anisotropy of cubic crystals and (ii) as the major drawback of pair potentials we identify the gross underestimation of the stable stacking fault energy. As a consequence these potentials predict the formation of too large dislocation loops, the too rapid expansion of partials, too little cross slip and in consequence a severe overestimation of work hardening.     

立方晶粒任意集合下的格林函数和有效弹性刚度张量

为了推导多晶体材料的有效弹性刚度张量,给出立方晶粒任意集合的格林函数封闭但近似的表达式,该格林函数表达式包含三个单晶弹性常数和多晶体材料五个织构系数,它考虑取向分布函数的影响直至织构系数的线性项,它适用于弱织构多晶体材料或具有弱各向异性晶粒的多晶体材料(如金属铝),它与Nishioka格林函数近似式的比较通过三个算例给出;Synge的格林函数积分式则直接通过数值计算完成,它可作为问题的精确解供参考．该文还简单介绍了多晶体材料有效弹性刚度张量的推导过程,并把所得结果和有限元计算结果进行比较。     

A relaxation method for the energy and morphology of grain boundaries and interfaces

The energy density of crystal interfaces exhibits a characteristic “cusp” structure that renders it non-convex. Furthermore, crystal interfaces are often observed to be faceted, i.e., to be composed of flat facets in alternating directions. In this work, we forge a connection between these two observations by positing that the faceted morphology of crystal interfaces results from energy minimization. Specifically, we posit that the lack of convexity of the interfacial energy density drives the development of finely faceted microstructures and accounts for their geometry and morphology. We formulate the problem as a generalized minimal surface problem couched in a geometric measure-theoretical framework. We then show that the effective, or relaxed, interfacial energy density, with all possible interfacial morphologies accounted for, corresponds to the convexification of the bare or unrelaxed interfacial energy density, and that the requisite convexification can be attained by means of a faceting construction. We validate the approach by means of comparisons with experiment and atomistic simulations including symmetric and asymmetric tilt boundaries in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. By comparison with simulated and experimental data, we show that this simple model of interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally induced morphological transitions.     

Nb3Sn单晶超导相转变时的弹性性能变化

Nb3Sn超导体失超，是超导磁体装备运行过程中的重要现象。失超，即超导体从超导相转变为正常相的过程；在强磁场超导磁体工程中，由于超高的储能密度，失超伴随着力/热/电等物理参量在瞬时的剧烈变化。失超瞬时，超导相转变的同时伴随着弹性力学性能的突变，研究超导相转变时弹性性能的变化是失超诱发应力跨尺度分析的关键。本文首先采用第一性原理方法计算了弹性常数随温度的变化规律，结果表明由于未考虑A15结构的超导材料在环境温度变化的情况下产生的特殊电子能带结构，基于准静态近似方法的材料弹性常数计算从0K外推至有限温度时，会导致模拟结果与实验观测结果出现定性上的差异；之后，基于晶格自由能函数，给出了描述立方相Nb3Sn单晶弹性性能随温度变化的解析模型，模型预测结果与实验观测结果定性吻合，初步实现了对Nb3Sn单晶超导相转变时弹性性能变化的理论描述和预测。研究结果对于超导体失超应力的跨尺度模拟及超导磁体的安全分析具有一定的理论参考价值。     

The computation of dynamical moduli of solids under pressure

On the basis of the theory of finite strains, expressions are obtained in general form for the effective adiabatic second order elastic constants of crystals of any symmetry in terms of the isothermal elastic constants of second, third, and higher orders in the free energy decomposition. These expressions are used in the case of crystals of cubic symmetry under hydrostatic conditions to find the elastic wave velocities in mono- and polycrystals, and their pressure dependences. The polycrystal was considered as an isotropic body consisting of a large number of cubic monocrystals. The isotropic elastic constants were calculated from theoretical and experimental results for monocrystals in the Voigt-Reuss-Hill approximation. A method of applying this approximation to thermodynamic effective second order elastic constants is proposed. The results of a computation are compared with data of experiments to measure the sound velocity in polycrystalline NaCl and CsCl specimens under pressures to 100 kbar. The results of this comparison are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 162–170, July–August, 1972.     

Perturbation analysis of Mode III interfacial cracks advancing in a dilute heterogeneous material

The paper addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. It is assumed that the bonding between dissimilar elastic materials is weak so that the interface is a preferential path for the crack. The perturbation analysis is made possible by means of the fundamental solutions (symmetric and skew-symmetric weight functions) derived in Piccolroaz et al. (2009). We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate “effective” tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. For a stable propagation of the crack, the perturbation of the stress intensity factor induced by the defects is then balanced by the elongation of the crack along the interface, thus giving an explicit asymptotic formula for the calculation of the crack advance. The method is general and applicable to interfacial cracks with general distributed loading on the crack faces, taking into account possible asymmetry in the boundary conditions.The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations. Numerical computations based on the explicit analytical formulae allows for the analysis of crack propagation and arrest.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.     

Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory

The deviation from the classical elastic characteristics induced by the free surface energy can be considerable for nanostructures due to the high surface to volume ratio. Consequently, this type of size dependency should be accounted for in the mechanical behaviors of nanoscale structures. In the current investigation, the influence of free surface energy on the nonlinear primary resonance of silicon nanoshells under soft harmonic external excitation is studied. In order to obtain more accurate results,the interaction between the first, third, and fifth symmetric vibration modes with the main oscillation mode is taken into consideration. Through the implementation of the Gurtin-Murdoch theory of elasticity into the classical shell theory, a size-dependent shell model is developed incorporating the effect of surface free energy. With the aid of the variational approach, the governing differential equations of motion including both of the cubic and quadratic nonlinearities are derived. Thereafter, the multi-time-scale method is used to achieve an analytical solution for the nonlinear size-dependent problem. The frequency-response and amplitude-response of the soft harmonic excited nanoshells are presented corresponding to different values of shell thickness and surface elastic constants as well as various vibration mode interactions. It is depicted that through consideration of the interaction between the higher symmetric vibration modes and the main oscillation mode, the hardening response of nanoshell changes to the softening one. This pattern is observed corresponding to both of the positive and negative values of the surface elastic constants and the surface residual stress.