IDEALIZED STRESS–STRAIN RESPONSE AND YIELDING
Description;
If we apply an incremental vertical load, DP, to a deformable cylinder (Figure) of cross-sectional area A, the cylinder will compress by, say, Dz and the radius will increase by Dr. The loading condition we apply here is called uniaxial loading. The change in vertical stress is
Forces and displacements
on a cylinder.
The vertical and radial strains are, respectively,
where Ho is the original length and ro is the original radius. In Equations. a negative sign should be inserted for expansion and a positive sign for compression. Thus, for radial expansion, Equation should have a negative sign. It is not included here for generality. The ratio of the radial (or lateral) strain to the vertical strain is called Poisson’s ratio, n, defi ned as
Typical values of Poisson’s ratio for soil . We can plot a graph of sz 5 SDsz versus εz 5 SDεz. If, for equal increments of DP, we get the same value of Dz, then we will get a straight line in the graph of sz versus εz, as shown by OA in Figure 7. 5. If at some stress point, say, at A (Figure), we unload the cylinder and it returns to its original confi guration, the material comprising the cylinder is called a linearly elastic material. Suppose for equal increments of DP we get different values of Dz, but on unloading the cylinder it returns to its original confi guration. Then a plot of the stress–strain relationship will be a curve, as illustrated by OB in Figure 7. 5. In this case, the material comprising the cylinder is called a nonlinearly elastic material. If we apply a load P1 that causes a displacement Dz1 on an elastic material and a second load P2 that causes a displacement Dz2
Linear and nonlinear stress–strain
curves of an elastic material.
then the total displacement is Dz 5 Dz1 1 Dz2. Elastic materials obey the principle of superposition.
The order in which the load is applied is not important; we could apply P2 fi rst and then P1, but the fi nal displacement would be the same. Some materials—soil is one of them—do not return to their original confi gurations after unloading. They exhibit a stress–strain relationship similar to that depicted in Figure 7.6, where OA is the loading response, AB the unloading response, and BC the reloading response. The strains that occur during loading, OA, consist of two parts—an elastic or recoverable part, BD, and a plastic or unrecoverable part, OB. Such material behavior is called elastoplastic. Part of the loading response is elastic, the other plastic.