Thermodynamic model of SMA pseudoelasticity based on multiplicative decomposition of deformation gradient tensor

Supplementary Text 1: Material modeling and characterization We used the following incompressible neo-Hookean material model to define the instantaneous constitutive behavior of the shells, = tr − 3, (S1) where W is the strain energy density function, µ is the shear modulus, F is the deformation gradient tensor. To describe the viscoelastic behavior of the shells, Prony series were used and the shear modulus µ can be expressed as = 1 − ∑ 1 − ⁄ , (S2) where µ0 is the instantaneous shear modulus, n is the number of the series terms, is the dimensionless relaxation modulus, t is the time, and τi is the relaxation time constant. Here we characterize the viscoelastic properties of the silicone rubber (Dragon SkinTM30) and urethane rubber (VytaFlexTM 20). We modeled their viscou

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Path-integrated Lagrangian measures from the velocity gradient tensor

Nonlinear Processes in Geophysics ◽

10.5194/npg-20-987-2013 ◽

2013 ◽

Vol 20 (6) ◽

pp. 987-991 ◽

Cited By ~ 6

Author(s):

V. Pérez-Muñuzuri ◽

F. Huhn

Keyword(s):

Deformation Gradient ◽

Fluid Flows ◽

Gradient Tensor ◽

Deformation Gradient Tensor ◽

Velocity Gradient Tensor ◽

Finite Time Lyapunov Exponent ◽

Spatial Fields ◽

Unsteady Fluid ◽

Time Periodic ◽

Double Gyre

Abstract. Spatial maps of the finite-time Lyapunov exponent (FTLE) have been used extensively to study LCS in two-dimensional dynamical systems, in particular with application to transport in unsteady fluid flows. We use the time-periodic double-gyre model to compare spatial fields of FTLE and the path-integrated Eulerian Okubo–Weiss parameter (OW). Both fields correlate strongly, and by solving the dynamics of the deformation gradient tensor, a theoretical relationship between both magnitudes has been obtained. While for long integration times more and more FTLE ridges appear that do not seem to coincide with the stable manifold, ridges in the field of path-integrated OW represent fewer additional structures.

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Kinematics

Continuum Mechanics of Solids ◽

10.1093/oso/9780198864721.003.0003 ◽

2020 ◽

pp. 35-66

Author(s):

Lallit Anand ◽

Sanjay Govindjee

Keyword(s):

Deformable Body ◽

Deformation Gradient ◽

Volumetric Strain ◽

Decomposition Theorem ◽

Important Case ◽

Gradient Tensor ◽

Deformation Gradient Tensor ◽

Small Strains ◽

Deformation Kinematics ◽

Small Deformations

This chapter develops the necessary mathematics for describing general deformations that a solid body may undergo, a topic known as kinematics. Definitions of motion, displacement, velocity, and acceleration which are vectors, and the deformation gradient and displacement gradient which are tensors are given. The mapping of material vectors by the deformation gradient tensor as a basic concept in describing the large deformation kinematics of a deformable body is presented. The powerful polar decomposition theorem is discussed and applied to the deformation gradient tensor to show that it can be decomposed into a stretch followed by a rotation, or a rotation followed by a stretch. Non-linear measures of strain are defined in terms of the stretch tensors. The important case of small deformations, which results in linear measures of strain, is discussed. For small strains the important decomposition of the state of strain that separates a volumetric strain from a non-volumetric or deviatoric strain is presented.

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