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What does operator ： mean in COMSOL equations？
Posted 10.05.2015, 02:45 GMT-4 Parameters, Variables, & Functions, Structural Mechanics Version 5.0 1 Reply
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I found this operator in equation views of poroelastic material model which is showing at the attached picture. It seems that the operator ' : ' connects two tensor (C and epsilon). But the tensor 'C' has 6 rows and 6 columns while the tensor 'epsilon' has 3 rows and 3 columns. It doesn't seem to be a common multiplication.
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1 Reply Last Post 11.05.2015, 08:41 GMT-4
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Posted:
9 years ago
Hi,
strain (epsilon) and stress (s) are symmetric 3x3 tensors. Basically elasticity (C) is tensor of rank 4
and dimensions 3x3x3x3. Since this is to complicated visualise many people use the following transformation. Due to symmetry stress and strain have only 6 independent values that can be written as a 6-dimensional vector. Only then C appears to be a (symmetric) 6x6 matrix (with 21 independent values).
Hence you can read the colon as the multipliateon of a rank-4-tensor with a rank-2-tensor resulting
in a rank-2-tensor. Or, as a matrix-vector-product in six dimensions.
Hope that make it clear
Regards
Jens
strain (epsilon) and stress (s) are symmetric 3x3 tensors. Basically elasticity (C) is tensor of rank 4
and dimensions 3x3x3x3. Since this is to complicated visualise many people use the following transformation. Due to symmetry stress and strain have only 6 independent values that can be written as a 6-dimensional vector. Only then C appears to be a (symmetric) 6x6 matrix (with 21 independent values).
Hence you can read the colon as the multipliateon of a rank-4-tensor with a rank-2-tensor resulting
in a rank-2-tensor. Or, as a matrix-vector-product in six dimensions.
Hope that make it clear
Regards
Jens