# Isotropic materials and levels of stress

I was wondering if I was to have two objects both made of isotropic materials such as steel and aluminum and the two objects were the same geometry and I was to apply the same load and boundary conditions to each, would one of the materials have a higher stress?

My understanding is that because stress = load / area then the stress for steel would be exactly the same as for aluminum.

But then there is the relation between Young's modulus $E = \frac{\mbox{stress}}{\mbox{strain}}$. So it is difficult for me to justify my first thought.

It would be interesting to hear some thoughts on this.

• The way you have described the problem is a bit odd, as loads are a type of boundary condition. If the BCs are instead equal strains, then the answer is no, but that is only due to difference in modulus as you noted. Apr 26 '16 at 12:10

In general the answer is yes, however for particular cases the stresses could be the same.

Your stress "definition" (stress = load/area) is valid for uni-axial stress state, for example a rod subjected to an axial force. In this case, within the elastic range stresses in isotropic materials would be the same.

Note that this valid only if we accept some common engineering assumptions such as using engineering stresses. This means that we do not take into account how the cross-section changes under the loading due to Poisson effect. However, if we take this into account and dealing with true stresses there will be a difference due to the difference in Poisson's ratios.

Even when engineering stresses are used there are cases where the stresses will be different due to the different Poisson's ratio. For example internal forces and stresses in a plate are influenced by the Poisson's ratio.

Another example is geometrical non-linearity where the change of global geometry under loading influences the distribution of internal forces, stresses (the change in cross-section is typically not considered). Since the deformations are dependent on the Young's modulus you will get different stresses for different materials. A sagging cable is simple example for this.

• To sumarize: for simple structures (beams, frames - under small deformations), then yes, the stresses will be the same (but strains will be different). For just about anything more complicated (plates, solids - anything under large deformations), then no, the stresses will be different.
– Wasabi
Apr 26 '16 at 10:35