# Why elastic materials are discribed by tensors?

I am starting to read about elasticity of thin surfaces and I don't understand why tensors play such a major part?

What are the tensors describing about the material?

And just to clarify - Is there some calculation often done with 2nd order tensors (describing 3d materials) that is common for elastic materials?

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In order to understand where tensors appear, it may help to use the Cartesian of suffix notation for vectors. In that notation, a vector $\vec{F}$ is written as $F_i$, with $i = 1,2,3$. If a vector $\vec{G}$ is proportional to another vector $\vec{F}$ then the most general way of writing that dependence is $G_i = \tau_{ij} F_j$. This allows for the case that $\vec{G}$ is not parallel to $\vec{F}$. If we represent $G_i$ or $F_j$ as column matrices, then $\tau_{ij}$ is a square matrix connecting the two. The square matrix represents a tensor.
An example that appeals someone with a background in electrodynamics is, $\vec{D}$ is proportional to $\vec{E}$. In the case of an isotropic dielectric, $\vec{D} = \epsilon\vec{E}$, where the scalar $\epsilon$ is the material's permittivity. If the material is anisotropic, then the relation is $D_i = \epsilon_{ij}E_j$, where $\epsilon_{ij}$ is the second order permittivity tensor. Another easy to understand relation is between angular momentum and angular velocity via the moment of inertia tensor.
In continuum mechanics, we relate stress $\tau_{ij}$, a second order tensor, with strain $\gamma_{ij}$, another second order tensor. The most general relationship between them is $\tau_{ij} = A_{ijkl}\gamma_{kl}$, where $A_{ijkl}$ is the elasticity modulus, a fourth order tensor.
1. You meant perpendicular, not proportional right? 2. when you write G_i = \tau _ij F_j what are you describing exactly? A transformation of one tensor to another? –  GuySoft May 11 '13 at 13:29
1. No, I meant proportional, the way $\vec{D}$ depends on $\vec{E}$. 2. You can also view $G_i = \tau_{ij}F_j$ as a transformation of a (1st order) tensor $F_i$ to another one $G_i$. But that's a mathematical way of looking at it. I would prefer to interpret it more like relation between $\vec{E}$ and $\vec{D}$, –  Amey Joshi May 11 '13 at 13:35