Modeling Of Electro Elastic Materials Using Implicit Theories

This proposal develops a mathematical model for various electro-active materials using implicit constitutive theories and solve some boundary value problems for different practical cases. There are some materials such as quartz, PZT etc. which react to electric fields by showing deformation and electric field which may be large. Some of these materials are called piezoelectric, but there are many other classes with different names (see in particular the book of Maugin [1]). Such materials show small deformations, and that is the reason that even today, people use the classical linearized theory to model their behavior. Classical constitutive relations like the linearized elastic model is explicit model such as Hooke’s law [2] which shows explicit relations between the stress and kinematical quantities, with in the context of fully nonlinear theory developed by Cauchy [3] and within the class of elastic bodies where in the stress can be derived from a stored energy as presented by Green [4]. For example, in pure linear elasticity the stress is a linear function of the strain. Assuming this kind of constitutive relations results in models with flaws like (i) Strains exhibit no limiting behavior for large stress values. Strains grow larger as stresses take higher values, thus posing difficulties in important problems of linear mechanics. For example, in a continuum media with crack the classical theory fails providing strains with values approaching infinity near the crack tip. (ii)Assuming stress as a function of strain, it is evident that the cause (stress) is defined by the result (strain).

This appears contradictory in the sense of causality [5]. In consideration of the above, to get rid of these problems, Rajagopal [6-8] and Rajagopal and Srinivasa [9, 10] presented a general framework for elasticity. Therefore, Rajagopal [6-8] extended the class of such elastic bodies by introducing implicit constitutive relations between stresses and the deformation gradient and the possibility to have a stored energy function depending on stress and deformation gradient. Such an implicit approach to the response of solids after linearization implies that even though the strains are extremely small the relationship between the stress and the linearized strain is nonlinear. Thus, it is easily understood that the classical Cauchy or Green elastic bodies represent a subclass with minimum expectations for advanced research. Even the introduction of gradient theories has its limitations, when working with this subclass. Similarly, the classical theory of electro elasticity assume that the stress explicitly in terms of strain and electric field, and also for the electric behavior they assume electric displacement explicitly in terms of strain and electric field. Voigt [11] developed the first general theoretical model for piezoelectric materials, see details in the the books by Cady [12] and Katzir [13] that provide thorough review of piezoelectricity. Voigt assumed linear relations between the stresses, the electric field, the strains and the polarization [12, 13]. Such a model are suitable when the strains are infinitesimal small, but in particular as soon as the electric independent variable is also small. However, for many practical problems the electric variable is not necessarily small, and what happens with many such materials is that, for example, when the electric field is large enough, a strong nonlinear behavior is observed for the material, for both the electric response but also for the deformation of the body, even though the strain are still small, of the order of less than 1%. The interaction of deformable bodies under influence of electromagnetic fields has attracted the attention of several researchers for a long period of time. The development of theories capable of accounting for the different phenomena with regard to the interaction of deformable bodies with electromagnetic fields has not been a simple task.

The main problem has been the difficulty of separating the different contributions due to the electromagnetic fields in the internal stresses. Different expressions for the body forces, body couples and stresses have been proposed (see [14] for details; see also [15–17]). One of the early treatments related to the behavior of electrical-field-dependent materials undergoing large deformations is ch. F of the authoritative book by Truesdell & Toupin [18]; other important treatments can be found in the monograph by Hutter et al. [19], the compact review article by Pao [14], the book by Maugin [1] and there is lots of article [20] by Maugin. In the references [14, 19] mentioned earlier, they find mainly general expressions for the body forces and body couples, which are generated when electromagnetic fields are applied. When one is interested in studying the behavior of a body made of different materials subject to external stimuli, one needs to specify constitutive assumptions concerning the relationship between the stresses, the strains and the electrical variables. In the case of electroelasticity, one of the main constitutive assumptions has been that the stresses can be expressed as functions of the strains and the electric field. Not many researchers have studied such nonlinear problems, few to mention are Lax, Maugin, Nelson and Tiersten [20-22], and they considered implicit models of the form in which stress is a function of strain and electric field, and similarly electric displacement is a function of strain and electric field. Lax and Nelson proposed some polynomials for the constitutive equations. However, if the strains are small, from the physical point of view whenever you have function of strain, that function will become a linear function after using Taylor expansion. However, that is a physical condition which is closely connected with the idea of working with the small strain tensor. From mathematical point of view we can write stress is a function of small strain, but from the physical point of view that does not make any sense. To get rid of these problems, Rajagopal [6-8] propounded a general framework for elasticity. Bustamante and Rajagopal [23-25] propounded some problems for elastic bodies using implicit theories and solved some boundary value problems as well. Furthermore, Bustamante and Rajagopal [26, 27] proposed a general framework for electro elasticity in the form of implicit constitutive theories by introducing the implicit constitutive laws in which different special subclasses of electro-elastic bodies were examined and solved some boundary value problems for electro elstic bodies. Moreover, Bustamante and Rajagopal [27] also performed a similar investigation for magneto-elastic bodies. Further, Bustamante and Rajagopal presented one special case in which the strains are small while the electric field is not small.

Therefore, In this proposal, first of all, we will extend the work of Bustamante and Rajagopal for electro active materials in the absence of temperature using implicit constitutive theories. We will develop constitutive relations for electro active materials like quartz and PZT etc. using implicit theory, where deformations are small and electric field are large enough by considering electric interaction in the absence of time, magnetic field and temperature using particular constitutive relations where stress is a function of strain and electric field and electric displacement is a function of stress and electric field. In the first approach, we will neglect the dissipation due to deformation and also due to electric interaction. This generalization involves two sets of implicit constitutive relations, one between the stress, the Cauchy–Green tensor and the electric field and an implicit relation between the stress, the electric field and the electric displacement field. The obtained constitutive relations are capable of describing the nonlinear response that is observed in piezoelectric bodies while the classical small displacement gradient theories are in capable of describing such behavior. Some boundary value problems will be solved for different practical cases in the context of this theory. Therefore, this proposal also devoted to study the boundary value problems for practical problems such as the inflation of a tube with some radial electric field,some 2D problems, such as the extension of a plate with a hole, elliptical hole and a crack and vibration of a plate etc. in order to check the efficacy of proposed model. In all the boundary value problems, we will study strains are small and the stresses and the electric field may be large. One more important thing is, the relationship between the linearized strain and stress would be nonlinear, in addition to the nonlinear relation to the electric field. The proposed coupled nonlinear differential equations will be solved using finite element method.

The results will be calculated for different variables such as stresses, strains, electric field and electric displacement etc. to show the influence of different physical parameters such as mechanical load, electric field etc. The constitutive relations allow for the nonlinear coupling between the stresses, linearized strain, electric field and the electric displacement which is an impossibility within the context of theories for electro-elastic materials that are currently available. Now, the above theory so far does not consider the thermodynamics yet, because that is kind of complicated. Since many piezoelectric materials shows quite strong coupling with temperature. Therefore, we will develop a possible extension of the above implicit theories for modeling the behavior of thermo-electro-elastic bodies. To extend the case of electro-elastic deformations, the restrictions for a body to be elastic, based on the 1st law of thermodynamics including electro-elastic interactions.

Therefore, for that we need to extend the theory thermoelasticity, considering now electric interaction as well. In this proposal, we will extend the implicit constitutive theory proposed by Bustamante and Rajagopal [29] for thermo-elastic bodies to the thermo-electro-elastic response of materials and generalize the implicit model proposed by Bustamante and Rajagopal [26, 27] to describe the electro-elastic bodies when thermal effects come into play. Again, for simplicity we wil not consider mechanical dissipation and electrical dissipation, but we would consider dissipation due to heat transfer. The thermoelastic behavior studied by Lord and Shulman [30] as well as the other researchers such as Ezzat [30], consider the behavior of Cauchy elastic bodies (or the sub-class of Green elastic bodies) with the inclusion of thermal effects. In this proposal, we will investigate the response of a new class of electro elastic bodies that are not necessarily Cauchy electro elastic bodies. To propose some implicit constitutive relationship between the stress and the deformation gradient and the electric displacement vector when thermal effects are included.

The constitutive relation is a nonlinear relationship between the linearized strain and the Cauchy stress. This extension will propose some implicit constitutive relations between the stress, the Cauchy–Green tensor, the electric field, heat flux vector and temperature to describe the thermoelectroelastic bodies. The obtained constitutive relations are capable of describing the nonlinear response that is observed in piezoelectric bodies while the classical small displacement gradient theories are incapable of describing such behavior. Some boundary value problems will be solved for different practical cases in the context of this theory to check the efficacy of the proposed constitutive model. We will solve several boundary value problems which are quasi-static, and time dependent. The problems based on fracture mechanics would be a special case here. Again, to solve the boundary value problems, numerical method such as finite element method (FEM), will be applied and comparison will be made with homotopy analysis method to check the efficiency of the method. In recent years, implicit constitutive theory getting more attention due to more general framework for elasticity.

One important thing is that the constitutive relationship between the stresses and linearized strain is nonlinear. The present classical theory is incapable to describing such nonlinear behavior of elastic bodies while constitutive relations are capable of describing such behavior. Based on the literature review, we can see there is lot of new improvements in implicit theory for elastic bodies other than this project. Thus, we believe that in the future this fashion will continue. Therefore, it assumes that very interesting to do research in this area for long term (3 years or more) and reporting new contributions to enhance the knowledge and application in the different field of engineering.