Abstract
Under mechanical deformation, most materials exhibit both elastic and fluid (or plastic) responses. No existing formalism derived from microscopic principles encompasses both their fluid-like and solid-like aspects. We define the statistical texture tensor to quantify the intuitive notion of stored deformation. This tensor links microscopic and macroscopic descriptions of the material, and extends the definition of elastic strain.
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\(\stackrel{\vrule height.4pt width3.5mm}{\overline{M}}\) distinguishes the anisotropy of rank two tensors (``optical anisotropy''). For instance, \(\stackrel{\vrule height.4pt width3.5mm}{\overline{ M}}\) distinguishes between birefrigent and non-birefringent networks. In an ``optically'' isotropic state, \(\stackrel{\vrule height.4pt width3.48mm}{\overline{M}}\) has two identical eigenvalues and reduces to a scalar: \(\stackrel{\vrule height.4pt width3.48mm}{\overline{M}}(\vec{R})=M(\vec{ R})\stackrel{\vrule height.4pt width1.68mm}{\overline{I}}_{d}\), where \(\stackrel{\vrule height.4pt width1.68mm}{\overline{I}}_{d}\) is the identity tensor in d dimensions. Since the trace of \(\stackrel{\vrule height.4pt width3.48mm}{\overline{M}} \) is \(\left\langle ℓ _{i}ℓ _{i}\right\rangle =\left\langle ℓ ^{2}\right\rangle \), then \(M=\left\langle ℓ ^{2}\right\rangle /d\). Note, however, that \(\stackrel{\vrule height.4pt width3.48mm}{\overline{M}}\) does not distinguish the anisotropy of rank four tensors (``mechanical anisotropy''): for instance, \(\stackrel{\vrule height.4pt width3.48mm}{ \overline{M}}\) does not necessarily distinguish between crystals and glasses.
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U.M.R. n° 5819: CEA, CNRS and Univ. Joseph-Fourier
U.M.R. n° 5588: CNRS and Univ. Joseph-Fourier Address for correspondence: graner@ujf-grenoble.fr Fax: (+33) 4 76 63 54 95, BP 87, 38402
We thank M. Asipauskas, S. Courty, B. Dollet, F. Elias, S. Ifadir, E. Janiaud and G. Porte for discussions. YJ is supported by the US DOE under contract W-7405-ENG-36. JAG acknowledges support from NSF, DOE and NASA contracts NSF-DMR-0089162, DOE-DE-FG0299ER45785 and NASA-NAG3-2366, the French Ministére de l'Enseignement Supérieur, and hospitality at the LSP.
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Aubouy, M., Jiang, Y., Glazier, J. et al. A texture tensor to quantify deformations. GM 5, 67–70 (2003). https://doi.org/10.1007/s10035-003-0126-x
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DOI: https://doi.org/10.1007/s10035-003-0126-x