A note on the linearized finite theory of elasticity
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Abstract:
In this paper we solve a simple dead load problem for an isotropic constrained material according to the linearized finite theory of elasticity. We show that the solution of such a problem can be obtained by linearizing with re- spect to the displacement gradient the solution of the corresponding problem in finite elasticity for an isotropic material subject to the same constraint, exactly as occurs for the constitutive equations of the two theories. On the contrary, the solution of the same dead load problem provided by the classical linear elasticity for constrained materials can be obtained by the solution of the corresponding problem for the unconstrained linear elastic material for limiting behaviour of suitable elastic moduli. The same applies for the constitutive equations of the classical linear elasticity for constrained materials: they are derived by those of the linear elasticity for unconstrained materials for limiting values of some elastic modulus. Finally we compare the solutions in finite elasticity, linearized finite theory of elasticity, classical linear elasticity for constrained materials and we show that they are in agreement with different hypotheses on the prescribed loads.Keywords:
Elasticity
Linear elasticity
Limiting
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In this paper there are presented two nonlinear models within the area of linearized elasticity and some applications for them. First model aims to describe the degradation of linearized elastic solid by considering the Hooke's law with material moduli which depends on a concentration of a diffusing fluid. This model is solved numerically on a square sample with a elliptic hole and the fluid diffuses through the hole. Second model introduces constitutive relation where strain is nonlinear function of the stress into the framework of linearized elasticity. This model can be used to model materials for which maximal strain is a priori bounded. It has been recently shown that such models can be justified by means of implicit constitutive theory. Using this model is studied square sample with a V-notch subject to anti-plane stress. Described problems are solved using finite element method.
Elasticity
Linear elasticity
Nonlinear elasticity
Hooke's law
Plane stress
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Abstract We obtain linear elasticity as $$\Gamma $$ Γ
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Linear elasticity
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Elasticity
Linear elasticity
Representation
Hooke's law
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Elasticity
Linear elasticity
Theory of Constraints
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Orthotropic material
Elasticity
Linear elasticity
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One possible method is proposed for reducing the problem of isotropic hereditary elasticity to solving a set of similar quasi-static problems in the theory of elasticity and thermoelasticity. The representability of the solution of the problem of linear hereditary elasticity in the form of the sum of solutions of three problems is substantiated: the linear theory of elasticity for imaginary bodies-incompressible and having a zero Poisson’s ratio and stationary uncoupled thermoelasticity for a body whose properties do not depend on temperature. The shear and bulk relaxation kernels are considered independent; the viscoelastic Poisso ratio is time dependent. Two theorems that reduce solutions of the general quasi-static problem of linear viscoelasticity theory to a solution of the corresponding problem of elasticity theory are proved. These theorems hold if one of the following conditions is satisfied: 1) the material is close to a mechanically uncompressible material; 2) the mean stress is zero; 3) the shift and volume hereditary functions are equal. The theorems provide free direct and inverse transforms between solutions of viscoelasticity and elasticity problems, which make them convenient in applications. They have been applied to solutions of problems on the pure torsion of a prismatic viscoelastic solid with an arbitrary simply connected cross section. Some examples describing the obtained results have been considered.
Elasticity
Linear elasticity
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Constant (computer programming)
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It has been shown previously, that Laplacian based explicit and implicit gradient elasticity models can be derived as particular cases of Mindlin's gradient elasticity and Mindlin's micro-structured (or equivalently Eringen's micromorphic) elastic materials. Also, they can be established as counterparts of viscoelastic solids, in view of an analogy to linear viscoelasticity based on mechanical elements. The present paper aims to compare responses predicted by the gradient elasticity counterpart of the three-parameter viscoelastic solid with those predicted by the gradient elasticity counterpart of the viscoelastic Kelvin model and by the classical elasticity. The comparison is mainly based on closed form solutions of one-dimensional problems in statics and dynamics. Specifically, the analysis includes static loading, axial vibrations and tension step pulse loading of a bar.
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