Finite elasticity

2020 ◽  
pp. 611-636
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Finite Deformation ◽  
Constitutive Relations ◽  
Finite Elasticity ◽  
Important Case ◽  
Material Symmetry ◽  
Isotropic Materials ◽  
Frame Invariance ◽  
Stress Tensors ◽  
Kinematic Measures ◽  
Material Frame

This chapter provides an overview of the theory of finite elasticity including a review of finite deformation kinematic measures, stress tensors for finite deformation, the concept of material frame invariance and change of frame, and constitutive relations. Material symmetry is discussed, and the important case of isotropy is presented. Special considerations needed for incompressible isotropic materials are elaborated upon.

The Complementary Energy Theorem in Finite Elasticity

1965 ◽  
Vol 32 (4) ◽  
pp. 826-828 ◽  
Author(s):  
Mark Levinson
Keyword(s):  
Finite Deformation ◽  
Degrees Of Freedom ◽  
Finite Elasticity ◽  
Strain Tensor ◽  
Complementary Energy ◽  
Elastic Continuum ◽  
Stress Tensors ◽  
Elastic Systems ◽  
Lagrange Strain ◽  
Castigliano’S Theorem

Other investigators have extended the complementary energy theorem (Castigliano’s theorem) to cover the finite deformation of elastic systems with a finite number of degrees of freedom (structures) and they then have indicated that the extension of the theorem to cover the finite deformation of an elastic continuum involved certain unstated difficulties. The present paper shows that when the strain tensor and Trefftz stress tensor, the usual choice of conjugate deformation and stress tensors, are chosen to characterize the finite deformation of an elastic continuum, one cannot establish a strict complementary energy theorem. It is then shown that a strict complementary energy theorem for the finite deformation of an elastic continuum can be established if what Fritz John calls the Lagrange strain and Lagrange stress tensors are used as the conjugate deformation and stress tensors characterizing the deformation.

A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I – reconsideration of curvature-based flexoelectricity theory

2021 ◽  
pp. 108128652110015
Author(s):  
YL Qu ◽  
GY Zhang ◽  
YM Fan ◽  
F Jin
Keyword(s):  
Classical Theory ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Constitutive Relations ◽  
Stress Theory ◽  
Isotropic Materials ◽  
Flexoelectric Effect ◽  
Beam Bending ◽  
Gradient Theories ◽  
Anisotropic And Isotropic Materials

A new non-classical theory of elastic dielectrics is developed using the couple stress and electric field gradient theories that incorporates the couple stress, quadrupole and curvature-based flexoelectric effects. The couple stress theory and an extended Gauss’s law for elastic dielectrics with quadrupole polarization are applied to derive the constitutive relations of this new theory through energy conservation. The governing equations and the complete boundary conditions are simultaneously obtained through a variational formulation based on the Gibbs-type variational principle. The constitutive relations of general anisotropic and isotropic materials with the corresponding independent material constants are also provided, respectively. To illustrate the newly proposed theory and to show the flexoelectric effect in isotropic materials, one pure bending problem of a simply supported beam is analytically solved by directly applying the formulas derived. The analytical results reveal that the flexoelectric effect is present in isotropic materials. In addition, the incorporation of both the couple stress and flexoelectric effects always leads to increased values of the beam bending stiffness.

Finite elasticity of elastomeric materials

2020 ◽  
pp. 637-654
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Free Energy ◽  
Constitutive Relations ◽  
Finite Elasticity ◽  
Energy Functions ◽  
Slightly Compressible ◽  
Elastomeric Materials ◽  
Volumetric Response

This chapter presents several technologically important constitutive relations for elastomeric materials. In particular, the Neo-Hookean, Mooney-Rivlin, Ogden, Arruda-Boyce, and Gent free energy functions are discussed in the context of incompressible response. Extensions to the slightly compressible case are also detailed, this includes a presentation of a number of possible volumetric response relations and their properties.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190407
Author(s):  
Bob Svendsen
Keyword(s):  
Angular Momentum ◽  
Constitutive Relations ◽  
Volume Averaging ◽  
Mass Point ◽  
Momentum Balance ◽  
Spatial Averaging ◽  
Material Frame Indifference ◽  
Phase Space Transport ◽  
Polar Continua ◽  
Material Frame

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

scholarly journals On Dynamic Extension of a Local Material Symmetry Group for Micropolar Media

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1632
Author(s):  
Victor A. Eremeyev ◽  
Violetta Konopińska-Zmysłowska
Keyword(s):  
Energy Density ◽  
Kinetic Energy ◽  
Symmetry Group ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Constitutive Relations ◽  
Kinetic Energy Density ◽  
Material Symmetry ◽  
Local Material ◽  
Micropolar Media

For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two kinematically independent fields, that are translation vector and orthogonal microrotation tensor. In other words, in micropolar continua we have six degrees of freedom (DOF) that are three DOFs for translations and three DOFs for rotations. So the corresponding kinetic energy density nontrivially depends on linear and angular velocity. Here we define the local material symmetry group as a set of ordered triples of tensors which keep both kinetic energy density and strain energy density unchanged during the related change of a reference placement. The triples were obtained using transformation rules of strain measures and microinertia tensors under replacement of a reference placement. From the physical point of view, the local material symmetry group consists of such density-preserving transformations of a reference placement, that cannot be experimentally detected. So the constitutive relations become invariant under such transformations. Knowing a priori a material’s symmetry, one can establish a simplified form of constitutive relations. In particular, the number of independent arguments in constitutive relations could be significantly reduced.

scholarly journals A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

2020 ◽  
Vol 25 (12) ◽  
pp. 2222-2230
Author(s):  
David Cichra ◽  
Vít Průša
Keyword(s):  
Constitutive Relation ◽  
Finite Deformation ◽  
Constitutive Relations ◽  
Plastic Response ◽  
Inelastic Response ◽  
Perfectly Plastic ◽  
Novel Approach ◽  
Elastic Perfectly Plastic ◽  
Rate Type ◽  
Thermodynamic Basis

Implicit rate-type constitutive relations utilising discontinuous functions provide a novel approach to the purely phenomenological description of the inelastic response of solids undergoing finite deformation. However, this type of constitutive relation has so far been considered only in the purely mechanical setting, and the complete thermodynamic basis is largely missing. We address this issue, and we develop a thermodynamic basis for such constitutive relations. In particular, we focus on the thermodynamic basis for the classical elastic–perfectly plastic response, but the framework is flexible enough to describe other types of inelastic response as well.

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