In this paper we extend the classical method of lattice dynamics to defective crystals with partial symmetries. We start by a nominal defect configuration and first relax it statically. Having the static equilibrium configuration, we use a quasiharmonic lattice dynamics approach to approximate the free energy. Finally, the defect structure at a finite temperature is obtained by minimizing the approximate Helmholtz free energy. For higher temperatures we take the relaxed configuration at a lower temperature as the reference configuration. This method can be used to semi-analytically study the structure of defects at low but non-zero temperatures, where molecular dynamics cannot be used.
Effect of strain and oxygen vacancies on the structure of 180^o ferroelectric domain walls in PbTiO_3Submitted by Arash_Yavari on Tue, 2010-01-05 20:49.
In this paper, we study the effect of normal and shear strains and
oxygen vacancies on the structure of 180^o ferroelectric
domain walls in PbTiO_3.
This paper presents a geometric theory of the mechanics of growing bodies.
This paper is dedicated to the memory of Professor Jim Knowles.
In this paper we formulate a geometric theory of thermal stresses.
Given a temperature distribution, we associate a Riemannian
material manifold to the body, with a metric that explicitly
depends on the temperature distribution. A change of temperature
corresponds to a change of the material metric. In this sense, a
temperature change is a concrete example of the so-called
referential evolutions. We also make a concrete connection between
our geometric point of view and the multiplicative decomposition
of deformation gradient into thermal and elastic parts. We study
the stress-free temperature distributions of the
finite-deformation theory using curvature tensor of the material
The above article is an April Fool's joke. It reminded me of the recent "discoveries" in the mechanics community regarding stress tensor.
The fractal crack model described here incorporates the essential
features of the fractal view of fracture, the basic concepts of
the LEFM model, the concepts contained within the
Barenblatt-Dugdale cohesive crack model and the quantized
(discrete or finite) fracture mechanics assumptions. The
well-known entities such as the stress intensity factor and the
Barenblatt cohesion modulus, which is a measure of material
toughness, have been re-defined to accommodate the fractal view of
This paper revisits continua with microstructure from a geometric point of view. We model a structured continuum as a triplet of Riemannian manifolds: a material manifold, the ambient space manifold of material particles and a director field manifold. Green-Naghdi-Rivlin theorem and its extensions for structured continua are critically reviewed.
This paper studies the invariance of balance of
energy for a system of interacting particles under groups of
transformations. Balance of energy and its invariance is first
examined in Euclidean space. Unlike the case of continuous media,
it is shown that conservation and balance laws do not follow
from the assumption of invariance of balance of energy under
time-dependent isometries of the ambient space. However, the
postulate of invariance of balance of energy under arbitrary
diffeomorphisms of the ambient (Euclidean) space, does yield
the conservation laws. These ideas are then extended to the case
when the ambient space is a Riemannian manifold. Pairwise
This paper presents a geometric discretization of elasticity when
the ambient space is Euclidean. This theory is built on ideas from
algebraic topology, exterior calculus and the recent developments
of discrete exterior calculus. We first review some geometric
ideas in continuum mechanics and show how constitutive equations
of linearized elasticity, similar to those of electromagnetism,
can be written in terms of a material Hodge star operator. In the
discrete theory presented in this paper, instead of referring to
continuum quantities, we postulate the existence of some discrete
scalar-valued and vector-valued primal and dual differential forms
on a discretized solid, which is assumed to be a triangulated
In this paper we covariantly obtain the governing equations of linearized elasticity. Our motivation is to see if one can make a connection between (global) balance of energy in nonlinear elasticity and its counterpart in linear elasticity. We start by proving a Green-Naghdi-Rivilin theorem for linearized elasticity. We do this by first linearizing energy balance about a given reference motion and then by postulating its invariance under isometries of the Euclidean ambient space. We also investigate the possibility of covariantly deriving a linearized elasticity theory, without any reference to the local governing equations, e.g. local balance of linear momentum. In particular, we study the consequences of linearizing covariant energy balance and covariance of linearized energy balance.
This paper extends the recently developed theories of fracture
mechanics with finite growth (mainly the work of Pugno and Ruoff, 2004
on quantized fracture mechanics) to fractal cracks. One interesting
result is the prediction of crack roughening for fractal cracks.
This paper presents an anharmonic lattice statics analysis of 180 and 90 domain walls in tetragonal ferroelectric perovskites. We present all the calculations and numerical examples for the technologically important ferroelectric material PbTiO3. We use shell potentials that are fitted to quantum mechanics calculations. Our formulation places no restrictions on the range of the interactions. This formulation of lattice statics is inhomogeneous and accounts for the variation of the force constants near defects. The discrete governing equations for perfect domain walls are reduced using symmetry conditions. We 20 solve the linearized discrete governing equations directly using a novel method in the setting of the theory of difference equations.
This paper shows that the stress field in the classical theory of continuum mechanics
may be taken to be a covector-valued differential two-form. The balance laws and other funda-
mental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A
This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal (referential) variations are equivalent to those resulting from vertical (spatial) variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R^3. Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call ‘configurational inviscidity’. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the Doyle-Ericksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density.
This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogeneous force constants. Instead, it starts with an interatomic potential, possibly with in¯nite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton-Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.