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Anisotropic stiffness of isotropic material

Dear colleagues,


Consider a simple non-linear elastic material with stress given as

σ = D(εdev) εdev + B εiso

where εdev is the norm of εdev, D is a function of εdev and B is constant. The material is isotropic since the principal directions of  σ and ε will coincide.

If we differentiate σ wrt ε to obtain the material stiffness the form of the stiffness tensor will be

Zhigang Suo's picture

Scalar done wrong

Update on 9 April 2016.  At the bottom of this post, I attach a pdf file of my notes on scalar.

Notes on scalars now forms part of my notes on linear algebra.

Zhigang Suo's picture

Textbook on linear algebra

Linear algebra is significant to many aspects of mechanics.  For some years I have been using the book by Shilov.  But this book may or may not be a good one to recommend to a student, depending on his or her prior experience.  On StackExchange Mathematics, there are several excellent threads discussing textbooks of linear algebra.  A particular recommendation was made for

Questions about tensors and matrices

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I am having questions with tensors. I know that not all matrices are tensor. Tensor is a homogeneous, linear and vector valued function. So my questions are:


  1. Are all 3*3 matrices tensors?
  2. Are all anisymmetric second order tensors orthogonal?



principal directions for 4th order tensor

Is it possible to find eigenvalues and principal directions for a 4th order tensor? How?

For a zero order tensor? for a first order tensor? for a third order tensor.........

many thanks

Sia Nemat-Nasser's picture

On the nature of the Cauchy stress tensor

Checking the iMechanica web, I notice some discussion about the Cauchy stress tensor, whether it is covariant or contravariant.  Well, a tensor is neither covariant nor contravariant, while it can be expressed by its covariant, contravariant, or mixed *components* with respect to any arbitrary coordinate system. See the attached short explanation.

Derivative of Logarithmic Strain

Some of you probably work on problems that involve moderately large strains. An useful strain measure for such problems in the logarithmic or Hencky strain. In particular, if you deal with the numerics of large strain simulations, you will often need to compute the material time derivatives of logarithmic strains.

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