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K R Rajagopal - One of the best experts on this subject based on the ideXlab platform.
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well posedness of the problem of non penetrating cracks in Elastic Bodies whose material moduli depend on the mean normal stress
International Journal of Engineering Science, 2019Co-Authors: Hiromichi Itou, Victor A Kovtunenko, K R RajagopalAbstract:Abstract The well-posedness of the problem of non-penetrating crack in an Elastic body whose material moduli depends on the mean normal stress is studied. The type of models considered are based on a new implicit theory for the response of Elastic Bodies. Bodies defined by these implicit theories are more general than Cauchy Elastic Bodies. The problem is studied within the context of a variational inequality.
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a note on the linearization of the constitutive relations of non linear Elastic Bodies
Mechanics Research Communications, 2018Co-Authors: K R RajagopalAbstract:Abstract Within the context of the non-linear theory of Cauchy Elastic Bodies (hence Green Elastic Bodies which are a sub-set of Cauchy Elastic Bodies wherein the stress is derivable from a potential), linearization with regard to the gradient of displacement, in the sense that the squares of the norms of the gradient of displacement can be neglected in comparison tothe norm of the gradient of displacement, leads inexorably to the classical linearized Elastic model. It is however common, especially in work related to inElastic Bodies, to see expressions for the Cauchy stress as a nonlinear function of the linearized strain. Even though such models are outside the purview of purely Elastic response, the nonlinear relationship between the stress and the linearized strain is also often assumed to hold in the Elastic range also. While the linearized strain being a nonlinear function of the stress has no basis within the context of the classical Cauchy Elasticity theory, we show that a proper justification can be provided for such models within the context of the new class of constitutive relations that have been developed to describe the response of Elastic Bodies by Rajagopal [19], and these models can be generalized to also describe the inElastic response in the small strain regime.
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on the consequences of the constraint of incompressibility with regard to a new class of constitutive relations for Elastic Bodies small displacement gradient approximation
Continuum Mechanics and Thermodynamics, 2016Co-Authors: R Bustamante, K R RajagopalAbstract:Recently, there has been an interest in the development of implicit constitutive relations between the stress and the deformation gradient, to describe the response of Elastic Bodies as such constitutive relations are capable of describing physically observed phenomena, in which classical models within the construct of Cauchy Elasticity are unable to explain. In this paper, we study the consequences of the constraint of incompressibility in a subclass of such implicit constitutive relations.
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solutions of some boundary value problems for a new class of Elastic Bodies undergoing small strains comparison with the predictions of the classical theory of linearized Elasticity part i problems with cylindrical symmetry
Acta Mechanica, 2015Co-Authors: R Bustamante, K R RajagopalAbstract:There is considerable evidence that shows that for a large class of materials the relationship between the stress and the strain is nonlinear even in the range of strain that is considered small enough for the classical linearized theory of Elasticity to be applicable (see Saito et al. in Science 300:464–467, 2003; Li et al. in Phys Rev Lett 98:105503, 2007; Talling et al. in Scr Mater 59:669–672, 2008; Withey et al. in Mater Sci Eng A 493:26–32, 2008; Zhang et al. in Scr Mater 60:733–736, 2009). A proper description of the experiments requires an alternative theory which when linearized would allow the possibility of such a nonlinear relationship between the stress and the strain. Recently, such a theory of Elastic Bodies has been put into place (see Rajagopal in Appl Math 48:279–319, 2003; Bustamante in Proc R Soc A 465:1377–1392, 2009; Rajagopal in Math Mech Solids 16:536–562, 2011). In this paper, we consider a special class of Bodies that belong to the new generalization of response relations for Elastic Bodies that have a nonlinear relationship between the linearized strain and the stress. We use the special class of Bodies that exhibit limited small strain to study two boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing of a tube. The results that we obtain for the models under consideration are markedly different from the predictions of the classical linearized Elastic model with regard to the same boundary value problems.
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on the determination of semi inverse solutions of nonlinear cauchy Elasticity the not so simple case of anti plane shear
International Journal of Engineering Science, 2015Co-Authors: Edvige Pucci, K R Rajagopal, Giuseppe SaccomandiAbstract:Abstract We provide a systematic and complete analysis of the overdetermined problem that one obtains while considering the balance equations of unconstrained isotropic nonlinear Cauchy Elastic Bodies undergoing anti-plane shear deformations.
A M Khludnev - One of the best experts on this subject based on the ideXlab platform.
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semirigid inclusions in Elastic Bodies mechanical interplay and optimal control
Computers & Mathematics With Applications, 2019Co-Authors: A M Khludnev, T. S. PopovaAbstract:Abstract The paper concerns an analysis of an equilibrium problem for 2D Elastic body with two semirigid inclusions. It is assumed that inclusions have a joint point, and we investigate a junction problem for these inclusions. The existence of solutions is proved, and different equivalent formulations of the problem are proposed. We investigate a convergence to infinity of a rigidity parameter of the semirigid inclusion. It is proved that in the limit, we obtain an equilibrium problem for the Elastic body with a rigid inclusion and a semirigid one. A parameter identification problem is investigated. In particular, the existence of a solution to a suitable optimal control problem is proved.
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junction problem for rigid and timoshenko Elastic inclusions in Elastic Bodies
Mathematics and Mechanics of Solids, 2017Co-Authors: A M Khludnev, Luisa Faella, T. S. PopovaAbstract:This paper concerns an equilibrium problem for a two-dimensional Elastic body with a thin Timoshenko Elastic inclusion and a thin rigid inclusion. It is assumed that the inclusions have a joint point and we analyze a junction problem for these inclusions. The existence of solutions is proved and the different equivalent formulations of the problem are discussed. In particular, the junction conditions at the joint point are found. A delamination of the Elastic inclusion is also assumed. In this case, the inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between the crack faces. We investigate the convergence to infinity and zero of a rigidity parameter of the Elastic inclusion. It is proved that in the limit, we obtain a rigid inclusion and a zero rigidity inclusion (a crack).
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rigidity parameter identification for thin inclusions located inside Elastic Bodies
Journal of Optimization Theory and Applications, 2017Co-Authors: A M KhludnevAbstract:The paper is concerned with an identification of a rigidity parameter for thin inclusions located inside Elastic Bodies. It is assumed that inclusions cross an external boundary of the Elastic body. In addition to this, a delamination of the inclusions is assumed thus providing a crack between inclusions and the Elastic body. To exclude a mutual penetration between crack faces, inequality-type boundary conditions are imposed. We consider Elastic inclusions as well as rigid and rigid-Elastic inclusions. To find a solution of the problem formulated, we solve an optimal control problem. A cost functional characterizes a displacement of the external part of the inclusion, and a rigidity parameter serves as a control function. We prove a solution existence of the problems formulated.
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on delaminated thin timoshenko inclusions inside Elastic Bodies
Mathematical Methods in The Applied Sciences, 2016Co-Authors: Hiromichi Itou, A M KhludnevAbstract:In the paper, we consider equilibrium problems for 2D Elastic Bodies with thin inclusions modeled in the frame of Timoshenko beam theory. It is assumed that a delamination of the inclusion takes place thus providing a presence of cracks between the inclusion and the Elastic body. Nonlinear boundary conditions at the crack faces are imposed to prevent a mutual penetration between the faces. Different problem formulations are analyzed: variational and differential. Dependence on physical parameters characterizing the mechanical properties of the inclusion is investigated. The paper provides a rigorous asymptotic analysis of the model with respect to such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Also anisotropic inclusions with parameters are analyzed when parameters tend to zero and infinity. In particular, in the limit, we obtain the so called semi-rigid inclusions. Copyright © 2014 John Wiley & Sons, Ltd.
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thin inclusions in Elastic Bodies crossing an external boundary
Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2015Co-Authors: A M KhludnevAbstract:We consider an equilibrium problem for 2D Elastic body with a thin inclusion crossing an external boundary at zero angle. It is assumed that the inclusion is delaminated, therefore a crack between the inclusion and the body is considered. To prevent a mutual penetration between crack faces, inequality type boundary conditions are imposed at the crack faces. We analyze Elastic inclusions as well as rigid inclusions. Passages to limits are investigated as a rigidity parameter of the inclusion goes to infinity. Theorems of existence and uniqueness are proved.
Bernd G. Schmidt - One of the best experts on this subject based on the ideXlab platform.
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dynamical Elastic Bodies in newtonian gravity
arXiv: General Relativity and Quantum Cosmology, 2011Co-Authors: Lars Andersson, Todd A Oliynyk, Bernd G. SchmidtAbstract:Well-posedness for the initial value problem for a self-gravitating Elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the Elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed.
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rotating Elastic Bodies in einstein gravity
Communications on Pure and Applied Mathematics, 2009Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that, given a stress-free, axially symmetric Elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary, axisymmetric solution to the Einstein equations coupled to the equations of relativistic Elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency. © 2009 Wiley Periodicals, Inc.
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rotating Elastic Bodies in einstein gravity
arXiv: General Relativity and Quantum Cosmology, 2008Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that, given a stress-free, axially symmetric Elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary axisymmetric solution to the Einstein equations coupled to the equations of relativistic Elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency.
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static self gravitating Elastic Bodies in einstein gravity
Communications on Pure and Applied Mathematics, 2008Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that given a stress-free Elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic Elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries. c � 2008 Wiley Periodicals, Inc.
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celestial mechanics of Elastic Bodies
Mathematische Zeitschrift, 2008Co-Authors: Robert Beig, Bernd G. SchmidtAbstract:We construct time independent configurations of two gravitating Elastic Bodies. These configurations either correspond to the two Bodies moving in a circular orbit around their center of mass or strictly static configurations.
R Bustamante - One of the best experts on this subject based on the ideXlab platform.
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on the consequences of the constraint of incompressibility with regard to a new class of constitutive relations for Elastic Bodies small displacement gradient approximation
Continuum Mechanics and Thermodynamics, 2016Co-Authors: R Bustamante, K R RajagopalAbstract:Recently, there has been an interest in the development of implicit constitutive relations between the stress and the deformation gradient, to describe the response of Elastic Bodies as such constitutive relations are capable of describing physically observed phenomena, in which classical models within the construct of Cauchy Elasticity are unable to explain. In this paper, we study the consequences of the constraint of incompressibility in a subclass of such implicit constitutive relations.
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solutions of some boundary value problems for a new class of Elastic Bodies undergoing small strains comparison with the predictions of the classical theory of linearized Elasticity part i problems with cylindrical symmetry
Acta Mechanica, 2015Co-Authors: R Bustamante, K R RajagopalAbstract:There is considerable evidence that shows that for a large class of materials the relationship between the stress and the strain is nonlinear even in the range of strain that is considered small enough for the classical linearized theory of Elasticity to be applicable (see Saito et al. in Science 300:464–467, 2003; Li et al. in Phys Rev Lett 98:105503, 2007; Talling et al. in Scr Mater 59:669–672, 2008; Withey et al. in Mater Sci Eng A 493:26–32, 2008; Zhang et al. in Scr Mater 60:733–736, 2009). A proper description of the experiments requires an alternative theory which when linearized would allow the possibility of such a nonlinear relationship between the stress and the strain. Recently, such a theory of Elastic Bodies has been put into place (see Rajagopal in Appl Math 48:279–319, 2003; Bustamante in Proc R Soc A 465:1377–1392, 2009; Rajagopal in Math Mech Solids 16:536–562, 2011). In this paper, we consider a special class of Bodies that belong to the new generalization of response relations for Elastic Bodies that have a nonlinear relationship between the linearized strain and the stress. We use the special class of Bodies that exhibit limited small strain to study two boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing of a tube. The results that we obtain for the models under consideration are markedly different from the predictions of the classical linearized Elastic model with regard to the same boundary value problems.
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direct determination of stresses from the stress equations of motion and wave propagation for a new class of Elastic Bodies
Mathematics and Mechanics of Solids, 2015Co-Authors: R Bustamante, D SfyrisAbstract:For a new class of Elastic Bodies, where the linearized strain tensor is given as a function of the Cauchy stress tensor, the problem of considering unsteady motions is studied. A system of partial differential equations that only depends on the stress tensor is found from the equation of motion, which is a system of six partial differential equations for the six components of the stress tensor. A simple boundary value problem is solved for a 1D bar using exact and numerical methods.
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a numerical study of Elastic Bodies that are described by constitutive equations that exhibit limited strains
International Journal of Solids and Structures, 2014Co-Authors: Alejandro Ortizbernardin, R Bustamante, K R RajagopalAbstract:Abstract Recently, a very general and novel class of implicit Bodies has been developed to describe the Elastic response of solids. It contains as a special subclass the classical Cauchy and Green Elastic Bodies. Within the class of such Bodies, one can obtain through a rigorous approximation, constitutive relations for the linearized strain as a nonlinear function of the stress. Such an approximation is not possible within classical theories of Cauchy and Green Elasticity, where the process of linearization will only lead to the classical linearized Elastic body. In this paper, we study numerically the states of stress and strain in a finite rectangular plate with an elliptic hole and a stepped flat tension bar with shoulder fillets, within the context of the new class of models for Elastic Bodies that guarantees that the linearized strain would stay bounded and limited below a value that can be fixed a priori, thereby guaranteeing the validity of the use of the model. This is in contrast to the classical linearized Elastic model, wherein the strains can become large enough in the body leading to an obvious inconsistency.
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on a new class of electro Elastic Bodies ii boundary value problems
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2013Co-Authors: R Bustamante, K R RajagopalAbstract: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 c...
Robert Beig - One of the best experts on this subject based on the ideXlab platform.
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rotating Elastic Bodies in einstein gravity
Communications on Pure and Applied Mathematics, 2009Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that, given a stress-free, axially symmetric Elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary, axisymmetric solution to the Einstein equations coupled to the equations of relativistic Elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency. © 2009 Wiley Periodicals, Inc.
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rotating Elastic Bodies in einstein gravity
arXiv: General Relativity and Quantum Cosmology, 2008Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that, given a stress-free, axially symmetric Elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary axisymmetric solution to the Einstein equations coupled to the equations of relativistic Elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency.
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static self gravitating Elastic Bodies in einstein gravity
Communications on Pure and Applied Mathematics, 2008Co-Authors: Lars Andersson, Robert Beig, Bernd G. SchmidtAbstract:We prove that given a stress-free Elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic Elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries. c � 2008 Wiley Periodicals, Inc.
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celestial mechanics of Elastic Bodies
Mathematische Zeitschrift, 2008Co-Authors: Robert Beig, Bernd G. SchmidtAbstract:We construct time independent configurations of two gravitating Elastic Bodies. These configurations either correspond to the two Bodies moving in a circular orbit around their center of mass or strictly static configurations.
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celestial mechanics of Elastic Bodies
arXiv: General Relativity and Quantum Cosmology, 2006Co-Authors: Robert Beig, Bernd G. SchmidtAbstract:We construct time independent configurations describing a small Elastic body moving in a circular orbit in the Schwarzschild spacetime. These configurations are relativistic versions of Newtonian solutions constructed previously by us. In the process we simplify and sharpen previous results of ours concerning Elastic Bodies in rigid rotation.