Strain Measures
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Keywords:
Finite strain theory
Infinitesimal strain theory
Hyperelastic material
Strain rate tensor
Cauchy elastic material
Hyperelastic material
Finite strain theory
Infinitesimal strain theory
Cauchy elastic material
Elasticity
Strain energy
Ogden
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As a notion of strain compatible with the theory of plasticity having the plastic potential of von Mises type with the normality principle as flow rule, the strain tensor is defined by integrating the rate of deformation tensor considering the material spin with the rotation tensor. For a group of elastic-plastic continua exhibiting infinitesimal elastic and finite plastic deformation, elastic-plastic decomposition of the strain tensor is established, and it is proved that the elastic strain tensor has the exact physical meaning and that the stress tensor can be calculated through the generalized Hooke's law with the elastic strain tensor without any ambiguity.
Cauchy elastic material
Strain rate tensor
Infinitesimal strain theory
Levy–Mises equations
Hyperelastic material
Hooke's law
Finite strain theory
Viscous stress tensor
Cartesian tensor
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It is widely accepted that any finite strain recorded in the field may be interpreted in terms of the simultaneous combination of a pure shear component with one or several simple shear components. To predict strain in geological structures, approximate solutions may be obtained by multiplying successive small increments of each elementary strain component. A more rigorous method consists in achieving the simultaneous combination in the velocity gradient tensor, but solutions already proposed in the literature are valid for some special cases only and cannot be used, e.g., for the general combination of a pure shear component and six elementary simple shear components. In this paper, we show that the combination of any strain components is very simple, both analytically and numerically. The finite deformation tensor is given by D = exp ( L Δ t ), where L Δ t is the time‐integrated velocity gradient tensor. This method makes it possible to predict finite strain for any combination of strain components. Reciprocally, L Δ t = ln ( D ), which allows us to unravel the simplest deformation history that might have generated a given finite deformation. Given the strain ellipsoid only, it is still possible to constrain the range of compatible deformation tensors and thus the range of strain component combinations. Interestingly, certain deformation tensors, though geologically sensible, have no real logarithm and so cannot be explained by a deformation history implying strain rate components with a common time dependence. This implies significant changes of stress field or material rheology during deformation.
Finite strain theory
Infinitesimal strain theory
Strain rate tensor
Ellipsoid
Pure shear
Viscous stress tensor
Velocity gradient
Strain (injury)
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Hyperelastic material
Hooke's law
Cauchy elastic material
Infinitesimal strain theory
Finite strain theory
Elasticity
Strain rate tensor
Linear elasticity
Viscous stress tensor
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Infinitesimal strain theory
Finite strain theory
Viscous stress tensor
Hyperelastic material
Strain rate tensor
Cauchy elastic material
Plane stress
Dilatant
Strain (injury)
Isochoric process
Strain energy
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The behavior of foams is typically rate-dependent and viscoelastic. In this paper, multiplicative decomposition of the deformation gradient and the second law of thermodynamics are employed to develop the differential constitutive equations for isotropic viscoelastic foams experiencing finite deformations, from a phenomenological point of view, i.e. without referring to micro-structural viewpoint. A model containing an equilibrium hyperelastic spring which is parallel to a Maxwell model has been utilized for introducing constitutive formulation. The deformation gradient tensor is decomposed into two parts: elastic deformation gradient tensor and viscoelastic deformation gradient tensor. A strain energy function is presented for the equilibrium spring as a function of the invariants of the left Cauchy-Green stretch tensor to obtain equilibrium stress components. Also, a strain energy function is presented for the intermediate spring as a function of the invariants of elastic deformation gradient tensor to determine overstress components. The constants of the strain energies are calculated by using nonlinear regulation numerical methods and by comparing with the experimental data obtained from uniaxial tension tests. The developed finite deformation constitutive equations are derived such that for every admissible process, the second law of thermodynamics is satisfied.
Hyperelastic material
Cauchy elastic material
Finite strain theory
Infinitesimal strain theory
Strain rate tensor
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Hyperelastic material
Cauchy elastic material
Infinitesimal strain theory
Strain rate tensor
Finite strain theory
Levy–Mises equations
Elastic energy
Hooke's law
Viscous stress tensor
Strain energy
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The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.
Hyperelastic material
Cauchy elastic material
Infinitesimal strain theory
Finite strain theory
Orthotropic material
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Hyperelastic material
Cauchy elastic material
Infinitesimal strain theory
Strain rate tensor
Strain energy
Finite strain theory
Strain (injury)
Stress–strain curve
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Abstract. Image correlation techniques have provided new ways to analyse the distribution of deformation in analogue models of tectonics in space and time. Here, we demonstrate, using a new version of our software package (TecPIV), how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also to calculate the finite displacements and finite strain tensor. We illustrate with synthetic images how the algorithm produces maps of the velocity gradients, small-strain tensor components, incremental or instantaneous principal strains and maximum shear. The incremental displacements can then be summed up with Eulerian or Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state and decomposed into the finite rigid-body rotation and left or right finite-stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterisation of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data and help produce conceptual mechanical models.
Finite strain theory
Infinitesimal strain theory
Strain rate tensor
Ellipsoid
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Citations (29)