scholarly journals On a new class of electro-elastic bodies. II. Boundary value problems

2013 ◽  
Vol 469 (2155) ◽  
pp. 20130106 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Electric Field ◽  
Boundary Value Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Electric Displacement ◽  
Cylindrical Tube ◽  
The Body ◽  
Elastic Bodies ◽  
Nonlinear Relation ◽  
Elastic Slab

In part I of this two-part paper, a new theoretical framework was presented to describe the response of electro-elastic bodies. The constitutive theory that was developed consists of two implicit constitutive relations: one that relates the stress, stretch and the electric field, and the other that relates the stress, the electric field and the electric displacement field. In part II, several boundary value problems are studied within the context of such a construct. The governing equations allow for nonlinear coupling between the electric and stress fields. We consider boundary value problems wherein both homogeneous and inhomogeneous deformations are considered, with the body subject to an electric field. First, the extension and the shear of an electro-elastic slab subject to an electric field are studied. This is followed by a study of the problem of a thin circular plate and a long cylindrical tube, both subject to an inhomogeneous deformation and an electric field. In all the boundary value problems considered, the relationships between the stress and the linearized strain are nonlinear, in addition to the nonlinear relation to the electric field. It is emphasized that the theories that are currently available are incapable of modelling such nonlinear relations.

scholarly journals Implicit constitutive relations for nonlinear magnetoelastic bodies

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140959 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Boundary Value Problems ◽  
Large Deformations ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Small Displacement ◽  
Elastic Bodies ◽  
Classical Models ◽  
Linearized Model ◽  
Implicit Constitutive Relations

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.

scholarly journals Solution of inverse non-stationary boundary value problems of diffraction of plane pressure wave on convex surfaces based on analytical solution

2020 ◽  
Vol 18 (4) ◽  
pp. 676-680
Author(s):  
Olga Egorova ◽  
Ko Ye
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problems ◽  
Pressure Wave ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
The Body ◽  
Parabolic Cylinder ◽  
Pressure Jump ◽  
Stationary Boundary ◽  
Special Cases

Research in the field of unsteady interaction of shock waves propagating in continuous media with various deformable barriers are of considerable scientific interest, since so far there are only a few scientific works dealing with solving problems of this class only for the simplest special cases. In this work, on the basis of analytical solution, we study the inverse non-stationary boundary-value problem of diffraction of plain pressure wave on convex surface in form of parabolic cylinder immersed in liquid and exposed to plane acoustic pressure wave. The purpose of the work is to construct approximate models for the interaction of an acoustic wave in an ideal fluid with an undeformable obstacle, which may allow obtaining fundamental solutions in a closed form, formulating initial-boundary value problems of the motion of elastic shells taking into account the influence of external environment in form of integral relationships based on the constructed fundamental solutions, and developing methods for their solutions. The inverse boundary problem for determining the pressure jump (amplitude pressure) was also solved. In the inverse problem, the amplitude pressure is determined from the measured pressure in reflected and incident waves on the surface of the body using the least squares method. The experimental technique described in this work can be used to study diffraction by complex obstacles. Such measurements can be beneficial, for example, for monitoring the results of numerical simulations.

Power-Formula Viscoplasticity, Its Modification and Some Applications

1984 ◽  
Vol 106 (4) ◽  
pp. 383-387 ◽  
Author(s):  
Yu Chen
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Boundary Value Problems ◽  
Physical Phenomenon ◽  
Boundary Value ◽  
The Body ◽  
Dislocation Dynamics ◽  
Basic Relation ◽  
Mathematical Form ◽  
Rate Dependent

A power viscoplastic constitutive equation was first proposed by Bodner and Partom in 1972. This specific formula has its origin in the physical phenomenon of dislocation dynamics and due to the simplicity of its mathematical form, it is a useful constitutive equation in solving boundary value problems. In this paper a brief review of the basic relation is given, followed by discussion of some relatively less known features of this formula. The body of the paper deals with the modification of the equation to explore its potential in several directions. The first modification consists of making the “threshold stress” strain rate dependent. The second modification aims of modeling strain rate history dependency. Due to space the formulation of the boundary value problems of uniaxial testing and its implementation through finite element programs will not be reported here. Results demonstrating the effect of strain rate and strain-rate history are presented. They are in good qualitative agreement with available experimental data.

Electrical structures of interfaces: a Painlevé II model

2010 ◽  
Vol 466 (2119) ◽  
pp. 2117-2136 ◽  
Author(s):  
L. Bass ◽  
J. J. C. Nimmo ◽  
C. Rogers ◽  
W. K. Schief
Keyword(s):  
Electric Field ◽  
Boundary Value Problems ◽  
Field Distribution ◽  
Electric Field Distribution ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Physical Parameters ◽  
Airy Functions ◽  
Ion Concentrations ◽  
Painlevé Ii

A Painlevé II model derived out of the classical Nernst–Planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x >0 occupied by an electrolyte. For privileged flux ratios of the ion concentrations, the auto-Bäcklund transformation admitted by the Painlevé II equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems. These representations involve, in turn, either Yablonskii–Vorob’ev polynomials or classical Airy functions. The requirement that the electric field distribution and ion concentrations in these representations be non-singular imposes constraints on the physical parameters. These are investigated in detail along with asymptotic properties.

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