Douglas--Randers manifolds with vanishing stretch tensor

2015 ◽  
Vol 86 (3-4) ◽  
pp. 423-432 ◽  
Author(s):  
AKBAR TAYEBI ◽  
TAYEBEH TABATABAEIFAR
Keyword(s):  
Stretch Tensor

scholarly journals An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach to martensitic transformations

2020 ◽  
Vol 25 (6) ◽  
pp. 1219-1246 ◽  
Author(s):  
Anup Basak ◽  
Valery I Levitas
Keyword(s):  
Phase Field ◽  
Kinematic Model ◽  
Martensitic Transformations ◽  
Order Parameters ◽  
Field Approach ◽  
Natural Logarithm ◽  
Stretch Tensor ◽  
Second Derivatives ◽  
Derivatives Of ◽  
Martensitic Variants

A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase field studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented.

General Derivations for Conjugate Strains of Eshelby-Like Stress Tensors

Volume 1 ◽  
2004 ◽  
Author(s):  
K. Behfar ◽  
A. Sheshmani ◽  
R. Naghdabadi
Keyword(s):  
Stress Tensor ◽  
General Class ◽  
General Type ◽  
Principal Axis ◽  
Stretch Tensor ◽  
Stress Tensors ◽  
Definition Of ◽  
General Relations ◽  
The Right ◽  
Conjugate Stress

In this paper, a general type of Eshelby-like stress tensor is defined which is based on the right stretch tensor and is equal to the product of a general class of strain and the corresponding conjugate stress tensor. The Eshelby-like stress tensor depending on the fact that from which side the stress tensor is multiplied by, is categorized into the right-weighted and left-weighted ones. General relations for conjugate strains of Eshelby-like stress tensors are investigated using the method, based on the definition of energy conjugacy and Hill’s principal axis method.

Intrinsic Decomposition of The Stretch Tensor for Fibrous Media

2010 ◽  
Author(s):  
David C. Kellermann ◽  
Jane W. Z. Lu ◽  
Andrew Y. T. Leung ◽  
Vai Pan Iu ◽  
Kai Meng Mok
Keyword(s):  
Fibrous Media ◽  
Stretch Tensor
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