Is young's modulus and "elasticity tensor" the same thing? Is young's modulus and "elasticity tensor" the same thing?
Particularly,
If I'm given definitions
$$E_{ijkl}=X(x) \bar{E}_{ijkl}$$
and
$$\frac{E_i}{E_o}=p^n \iff E_i=p^n E_o$$
where $p^n$ and $X(x)$ have the same "function" in rationing the density of what's applied to the rhs of it. $E_{ijkl}$ is an elasticity tensor and $\bar{E}_{ijkl}$ is a constant-valued elasticity tensor. $E_i$ is intermediate and $E_o$ is original Young's moduli.
So are these two expressions equivalent (without considering that $X(x)$ is discrete indicator function and $p^n$ is continuous). They are used to ration "density" in material density method in shape optimization.
 A: I don't understand what your $p^n$ and $X(x)$ are supposed to be, but the answer to the basic question is "no". Young's modulus is only part of the information contained in the elasticity tensor, even in the simplest case of an isotropic material with constant elastic properties.
The fourth-order elasticity tensor $E_{ijkl}$ is the general form of the 3-dimensional relationship between stress and strain in the material. 
There are a lot of symmetry relationships between the individual components of the $E_{ijkl}$, because of the symmetry of the stress and strain tensors, but for anisotropic materials it contains 21 independent components (compared with the 81 components in an arbitrary 4th-order tensor). For an isotropic material, all the components can be expressed in terms of just two independent parameters.
On the other hand, "Young's modulus" is a scalar that describes the relation between the stress and strain in one particular direction. Of course for an isotropic material, the relationship is the same for all directions within the material, but Young's modulus doesn't include the fact that a stress in one direction may cause a strain in a different direction. That interaction can (and usually does) occur even in isotropic materials, and it is what Poisson's ratio measures.
One possible choice for the "two independent parameters" to create the full elasticity tensor for an isotropic material are Young's modulus and Poisson's ratio, but there are several other choices that are commonly used in continuum mechanics.
