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# tensor

## Anisotropic stiffness of isotropic material

Thu, 2015-01-22 06:26 - sykledustDear colleagues,

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

**σ** = D(εdev) **ε**dev + B **ε**iso

where εdev is the norm of **ε**dev, 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

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## Scalar done wrong

Tue, 2013-12-31 02:17 - Zhigang Suo**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.

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## Textbook on linear algebra

Fri, 2013-12-27 18:39 - Zhigang SuoLinear 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

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## Questions about tensors and matrices

Sun, 2013-12-01 17:36 - APHello,

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:

- Are all 3*3 matrices tensors?
- Are all anisymmetric second order tensors orthogonal?

Thanks.

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## principal directions for 4th order tensor

Fri, 2009-02-13 11:12 - whitmanIs 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

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## On the nature of the Cauchy stress tensor

Sun, 2008-11-23 11:41 - Sia Nemat-NasserChecking 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.

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## Derivative of Logarithmic Strain

Sun, 2007-07-01 20:21 - Biswajit BanerjeeSome 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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