On the stress dependence of the elastic tensor

Geophysical Journal International ◽

10.1093/gji/ggaa591 ◽

2020 ◽

Author(s):

Matthew Maitra ◽

David Al-Attar

Keyword(s):

Finite Elasticity ◽

Deformation Gradient ◽

Unique Function ◽

Numerical Calculations ◽

Inverse Modelling ◽

Stress Dependence ◽

Equilibrium Stress ◽

Small Perturbations ◽

Elastic Tensor ◽

Reference Body

Summary The dependence of the elastic tensor on the equilibrium stress is investigated theoretically. Using ideas from finite elasticity, it is first shown that both the equilibrium stress and elastic tensor are given uniquely in terms of the equilibrium deformation gradient relative to a fixed choice of reference body. Inversion of the relation between the deformation gradient and stress might, therefore, be expected to lead neatly to the desired expression for the elastic tensor. Unfortunately, the deformation gradient can only be recovered from the stress up to a choice of rotation matrix. Hence it is not possible in general to express the elastic tensor as a unique function of the equilibrium stress. By considering material symmetries, though, it is shown that the degree of non-uniqueness can sometimes be reduced, and in some cases even removed entirely. These results are illustrated through a range numerical calculations, and we also obtain linearised relations applicable to small perturbations in equilibrium stress. Finally, we make a comparison with previous studies before considering implications for geophysical forward- and inverse-modelling.

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Stresses in Ventricular Wall

Journal of Applied Mechanics ◽

10.1115/1.3423806 ◽

1976 ◽

Vol 43 (2) ◽

pp. 194-197 ◽

Cited By ~ 28

Author(s):

H. Demiray

Keyword(s):

Left Ventricle ◽

Elasticity Theory ◽

Tangential Stress ◽

Finite Elasticity ◽

Spherical Geometry ◽

Elastic Stiffness ◽

Numerical Calculations ◽

Ventricular Wall ◽

Material Constants ◽

Stress Gradients

Assuming an idealized geometry, i.e., a spherical geometry, for the left ventricle, the ventricular wall stresses and elastic stiffness are investigated by use of a finite elasticity theory. The values of material constants are obtained via comparison of analytical results with experiments. The numerical calculations indicate that the endocardial layers experience very large tangential stress gradients which may be the cause of ischemia of left ventricle.

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ON THE REPRESENTATION OF ENERGY AND MOMENTUM IN ELASTICITY

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202500000136 ◽

2000 ◽

Vol 10 (02) ◽

pp. 203-216 ◽

Cited By ~ 2

Author(s):

P. PODIO-GUIDUGLI ◽

S. SELLERS ◽

G. VERGARA CAFFARELLI

Keyword(s):

Stress Response ◽

Kinetic Energy ◽

Total Energy ◽

Hyperbolic System ◽

A Priori ◽

Finite Elasticity ◽

Deformation Gradient ◽

Quasilinear Hyperbolic System ◽

Standard Format ◽

Energy And Momentum

In order to clarify common assumptions on the form of energy and momentum in elasticity, a generalized conservation format is proposed for finite elasticity, in which total energy and momentum are not specified a priori. Velocity, stress, and total energy are assumed to depend constitutively on deformation gradient and momentum in a manner restricted by a dissipation principle and certain mild invariance requirements. Under these assumptions, representations are obtained for energy and momentum, demonstrating that (i) the total energy splits into separate internal and kinetic contributions, and (ii) the momentum is linear in the velocity. It is further shown that, if the stress response is strongly elliptic, the classical specifications for kinetic energy and momentum are sufficient to give elasticity the standard format of a quasilinear hyperbolic system.

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Experimentally Evaluating the Equilibrium Stress in Shear of Glassy Polycarbonate

Journal of Engineering Materials and Technology ◽

10.1115/1.2345445 ◽

2006 ◽

Vol 128 (4) ◽

pp. 537-542 ◽

Cited By ~ 9

Author(s):

Mehrdad Negahban ◽

Ashwani Goel ◽

Pierre Delabarre ◽

Ruqiang Feng ◽

Amy Dimick

Keyword(s):

Plastic Strain ◽

Strain Rate ◽

Mechanical Response ◽

Deformation Gradient ◽

Back Stress ◽

Glassy Polymers ◽

Plastic Strain Rate ◽

Flow Rule ◽

Strain Rates ◽

Equilibrium Stress

One group of models proposed for characterizing the mechanical response of glassy polymers is based on a structure that resembles finite plasticity. In most cases, a constitutive equation for stress is proposed, which depends on the elastic deformation gradient, supplemented by a flow rule for the plastic deformation, which depends on the “over stress.” The over stress is a properly invariant difference between the stress and the back stress (equilibrium stress). The back stress represents conditions under which relaxation events should stop and the material should be able to carry an applied load indefinitely without a need to change the strain. Questions that arise in using these models are whether such equilibrium stresses exist, how can they be evaluated, and what experiments can be used to characterize the flow rule. One challenge in accurately evaluating the locus of equilibrium conditions is the fact that the relaxation process substantially slow down around these points, and, therefore, a method that does not directly require being at the equilibrium is desirable. Focusing on shear, a thermodynamic theory for characterizing the response of glassy polymers, similar to models currently used for this purpose, is developed, and using this model it is shown that one can set up a method to calculate the plastic strain rate. This method is based on evaluating the slope of stress-strain response under conditions of similar elastic and plastic strain, but different strain rates. Since the equilibrium stress occurs when the plastic strain rate goes to zero, the evaluated plastic strain rates allow evaluation of the needed information for developing the flow rule and obtaining the back stress. This method is used to evaluate the plastic strain rate and back stress at room temperature for polycarbonate. The evaluated results match well with results obtained by direct probing of the equilibrium stress, in which one searches for points at which the stress remains constant at a constant strain over long durations. The method proposed looks promising in evaluating the back stress of glassy polymers. The added advantage of this method is that it also provides a map of plastic strain rate and tangent modulus over a large range of loading conditions.

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