# Does same stress always produce same strain?

I am reading about pure bending from Beer Johnston's mechanics of materials. There, a prismatic member with a plane of symmetry was subjected to equal and opposite couples M and M' acting in that plane. It was said about this member that since the bending moment M is same in any cross section, the member will bend uniformly. I could not understand this.

Is it because that stress will be same on each cross section and therefore since same stresses cause same deformation, it will bend uniformly?

• The same stress always causes the same strain in this situation, but note this is not always true. For example a temperature change can cause strain (thermal expansion, i.e. a change of length) without any stress, or if the object is constrained so it can't move, it will cause stress (which may be big enough to crack or break the object) with no strain. Also if the stress is large enough, there may be plastic deformation, where the object does not return to its initial shape when the loads are removed - as a simple demonstration, bend a paper clip! Mar 21, 2019 at 9:30

$$\frac{1}{\rho}= \frac{d\phi}{d_x}=\frac{d^2w}{d_x^2}= - \frac{M}{EI}$$
$$\therefore \ \text{for a constant moment we have a constant radius.}$$