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 On a new class of electroelastic bodies. I

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120521 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Large Deformations ◽  
Constitutive Relations ◽  
Small Displacement ◽  
Electrical Field ◽  
New Class ◽  
Constitutive Relationships ◽  
Elastic Bodies ◽  
Biological Matter ◽  
Implicit Constitutive Relations

Implicit constitutive relations are proposed for large deformations of electroelastic bodies, and approximations to these are developed within the context of small displacement gradients. The resultant theories lead to the interesting situation wherein the constitutive relationships are nonlinear though the strain is ‘linearized’. In the absence of the effects due to the electrical field, the models reduce to a class of constitutive relations that have been studied recently, which have applications in the fracture of, and propagation of cracks in, brittle elastic bodies. The current class of electroelastic bodies have applications in a variety of important areas such as the response of piezoelectric bodies, electro-sensitive elastomers and biological matter.

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 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.

Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory

2020 ◽  
Vol 73 (1) ◽  
pp. 51-75
Author(s):  
S M Mkhitaryan ◽  
M S Mkrtchyan ◽  
E G Kanetsyan
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Canonical Forms ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Hadamard Finite Part ◽  
Elastic Bodies

Summary The exact solutions of a class of hypersingular integral equations with kernels $\left( {s-x} \right)^{-2}$, $\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\left( {\sinh \frac{s-x}{2}} \right)^{-2},\cos \frac{s-x}{2}\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\cosh \frac{s-x}{2}\left( {\sinh \frac{s-x}{2}} \right)^{-2}$ are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.

Verification of Plasticity Theory with Isotropic Hardening and Additive Decomposition of Left Deformation Tensor Using Digital Image Correlation System

2015 ◽  
Vol 240 ◽  
pp. 61-66 ◽  
Author(s):  
Marcin Gajewski ◽  
Cezary Ajdukiewicz ◽  
Andrzej Piotrowski
Keyword(s):  
Digital Image Correlation ◽  
Boundary Value Problems ◽  
Digital Image ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Experimental Tests ◽  
Image Correlation ◽  
Deformation Tensor ◽  
Isotropic Hardening ◽  
Complex Boundary

The development of measurement methods, and in particular digital image correlation (DIC) systems, which are designed to measure of entire displacements and deformations fields, opens up new areas of research. In general, the materials constitutive relations are formulated in such a way that material parameters could be determined with relatively simple experimental tests carried out on samples with uniform (approximately) stress and strain fields. Then it is possible to apply them to complex boundary value problems formulated e.g. in the small or large deformation theories. The application of DIC allows to verify the accuracy of their predictions by comparing the results of the experiment with solutions to boundary value problems obtained using the finite element method (FEM).

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