New Explicit Expression of Barnett-Lothe Tensors for Anisotropic Linear Elastic Materials |
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Authors: | TCT Ting |
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Institution: | (1) Department of Civil and Materials Engineering, University of Illionis at Chicago, 842 West Taylow Street (M/C246), Chicago, IL 60607-7023, USA |
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Abstract: | The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic
elastic materials 1–3]. They also appear in certain three-dimensional problems 4, 5]. The algebraic representation of H,
L, S requires computation of the eigenvalues pv(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing
p
v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee 6] have recently
obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the
algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form.
The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes
the result attractive to numerical computation. Even though the tensor H given in 6] is in terms of the elastic stiffnesses
Cμ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′
μv
, the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of
L, S that remain valid for the degenerate cases p
1
p
2 and p1=p2 = p
3. One may want to compute H, L, S using either C
μv
or s′
μv
v, but not both. We show how an expression in Cμ v can be converted to an expression in s′
μv
v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in Cμ v and s′
μv
v.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | elasticity anisotropic Green's function |
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