How Does A Strain Tensor Transform?

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SUMMARY

The discussion centers on the transformation of strain tensors as described in Landau and Lifshitz's theory of elasticity. The participant struggles with the transformation law T_hk=T(dx_i/dx_h)(dx_j/dx_k) and questions the validity of the strain tensor's classification due to perceived discrepancies in the transformation. Another contributor clarifies that the strain tensor is defined for Cartesian coordinates and that correct transformation laws apply only under global rotations represented by orthogonal matrices.

PREREQUISITES
  • Understanding of tensor notation and operations
  • Familiarity with the theory of elasticity as presented in Landau and Lifshitz's texts
  • Knowledge of orthogonal matrices and their properties
  • Basic concepts of coordinate transformations in physics
NEXT STEPS
  • Study the transformation laws for tensors in different coordinate systems
  • Explore the properties of orthogonal matrices in the context of physics
  • Review the definitions and applications of strain tensors in elasticity theory
  • Investigate examples of tensor transformations in various physical scenarios
USEFUL FOR

Students and professionals in physics, particularly those focusing on elasticity theory, tensor calculus, and coordinate transformations. This discussion is beneficial for anyone seeking to deepen their understanding of strain tensors and their mathematical properties.

antarctic
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Hello All,
I am trying to learn the theory of elasticity from Landau's book and from the very beginning I've run into trouble. As I learned it, a (0,2) tensor T_ij obeys the transformation law T_hk=T(dx_i/dx_h)(dx_j/dx_k). But I do not see how a strain tensor can be transformed in such a manner (I'm getting two extra terms at the end that don't seem to add to 0). How can you justify it being called a tensor then? Or does the transformation work, and I'm just not seeing something?
Any help appreciated.
 
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Hi,
I suspect that L&L have defined the strain tensor for Cartesian coordinates only. You will only get the correct transformation laws for global rotations x'^i=R^i_{\phantom{i}j}x^j where R is an orthogonal matrix.
 

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