V.K. Kalpakides

On Material Equations in Second Gradient Electroelasticity

The concern of this work is the derivation of material conservation and balance laws for second gradient electroelasticity. The conservation laws of material momentum, material angular momentum and scalar moment of momentum on the material manifold are derived using Noethers theorem and the exact conditions under which they hold are rigorously studied. The corresponding balance laws are also presented.

Canonical formulation and conservation laws of thermoelasticity without dissipation

Abstract This work is concerned with the derivation of conservation laws for the Green-Naghdi theory of nonlinear thermoelasticity without dissipation. The lack of dissipation allows for a variational formulation which is used for the application of Noethers theorem. The balance laws on the material manifold are derived and the exact conditions under which they hold are rigorously studied. Also, the relationship with the “classical” theory is examined.

On the configurational force balance in thermomechanics

The aim of this paper is a new formulation of the configurational force balance in the general framework of thermomechanics. To this end, invariance requirements of the configurational working are used, following an approach proposed by Gurtin. To define the notion of working in a more general framework, the ideas of Green and Naghdi concerning the basic postulates of thermomechanics are invoked and used consistently. The main results are that the configurational forces balance in the general setting of thermomechanics without the invocation of constitutive relations and the pseudo–momentum equation for classical and dissipationless thermoelasticity. Finally, we give an application to fracture theory. We show that in classical thermoelasticity the derived pseudo–momentum balance provides a crack–driving force consistent with the well–known expression of the energy–release rate.

A general theory for elastic dielectrics. II. The variational approach

Abstract In the present work a variational approach is used to derive a theory of elastic dielectrics with polarization gradient, quadrupole polarization and polarization inertia. The results (field equations, jump conditions and constitutive relations) are favorably compared with those of the approach followed in part I of this work. Finally, the relation of the present theory with existing ones is extensively discussed.

Isovector fields and similarity solutions of nonlinear thermoelasticity

The isovector fields and the similarity solutions associated with the non-linear equations of dynamical thermoelasticity are derived by exterior calculus technique. Also, the general form of the free energy function associated with the obtained isovector fields is given and the reduction of the initial system of partial differential equations to a system of ordinary differential equations is presented.

On the symmetries and similarity solutions of one-dimensional, non-linear thermoelasticity

The homogeneous, one-dimensional, non-linear thermoelasticity is exploited from the point of view of symmetries and similarity solutions. Special cases of free energy function and conductivity function are considered and the corresponding admitted symmetry group of transformations is derived. Also, the similarity solutions, if any, for every symmetry group are provided. Finally, by virtue of the obtained similarity solutions, the system of PDEs is converted to a system of ODEs.

On isovector fields and similarity solutions of generalized dynamic thermoelasticity

The purpose of the present work is the investigation of the isovector fields as well as the similarity solutions of the PDEs describing the generalized dynamic thermoelasticity. The adopted methodology and solution techniques belong totally to the analytical realm, while special treatment of the reduced differential equations resulting from similarity solution adoption has been realized.

A general theory for elastic dielectrics – Part I. The vectorial approach

Abstract In this paper a full-dynamic theory for elastic dielectrics is presented by a systematic application of thermomechanical balance laws. Polarization gradient, quadrupole polarization and polarization inertia are taken into account in order for a general theory to be obtained. The constitutive relations are produced by using the second law of thermodynamics in the form of Clausius–Duhem inequality. In addition, a new formulation of the internal equation of motion is presented and compared with that of previous investigations.

Configurational Forces in Continuous Theories of Elastic Ferroelectrics

Domain walls in a ferroelectric crystal are considered as sharp interfaces, so their motion is governed by field equations, jump conditions and an appropriate kinetic relation between the domain wall velocity and the driving force. In this article, a regularized version of the sharp-interface theory in ferroelectrics is presented, by introducing a level set function that changes sign from domain to domain smoothly and thus eliminating discontinuities. It is proved that considering level set functions as constitutive variables in the energy functional, the driving forces that move domain walls are configurational forces obeying the canonical momentum equation. A new, recently proposed differential equation is used to describe the evolution of the level set function which keeps level set function closer to a signed distance function as possible. Theoretical considerations and numerical simulations show that configurational forces are closely related to the level set description of sharp interface theories in solids. Moreover, it is displayed that in-homogeneity forces drive the system successfully to the typical domain structure of elastic ferroelectrics.

The Use of Material Forces to Improve the Finite Element Solution in Elasticity

In this work, the use of material forces to improve the finite element solution in elastostatics is proposed. It is well known that in the case of a homogeneous elastic body, in which there are no configurational changes, the corresponding material forces are identically zero. Nevertheless, the approximate solutions obtained by the finite element method do not fulfil exactly the governing equations of equilibrium. Due to this fact, non-zero pseudo-material forces appear. In general, the magnitude and the direction of pseudomaterial forces at any discretization node depend on the error of the approximate solution at that point. Hence, any procedure, which minimizes the pseudo-material forces by rearranging the interior nodes of the mesh, can lead to an improvement of the discretization solution. Here, a procedure of this type is proposed and is implemented on particular problems of two-dimensional elastostatics.