scholarly journals Handbook of bi-dimensional tensors: Part I: Harmonic decomposition and symmetry classes

2016 ◽  
Vol 22 (9) ◽  
pp. 1847-1865 ◽  
Author(s):  
N Auffray ◽  
B Kolev ◽  
M Olive
Keyword(s):  
General Problem ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Strain Gradient Elasticity ◽  
Symmetry Classes ◽  
Different Types ◽  
Tensor Spaces ◽  
Harmonic Decomposition ◽  
Physical Laws ◽  
Physical Phenomena

To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in [Formula: see text] using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

scholarly journals Classification of first strain-gradient elasticity tensors by symmetry planes

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210165
Author(s):  
Hung Le Quang ◽  
Qi-Chang He ◽  
Nicolas Auffray
Keyword(s):  
Strain Gradient ◽  
Gradient Elasticity ◽  
Elasticity Tensor ◽  
Constitutive Law ◽  
Sixth Order ◽  
Strain Gradient Elasticity ◽  
Present Contribution ◽  
Symmetry Classes ◽  
Elasticity Tensors

First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

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