12.4 Elasticity and plasticity  (Page 2/10)

For stresses beyond the elastic limit, a material exhibits plastic behavior    . This means the material deforms irreversibly and does not return to its original shape and size, even when the load is removed. When stress is gradually increased beyond the elastic limit, the material undergoes plastic deformation. Rubber-like materials show an increase in stress with the increasing strain, which means they become more difficult to stretch and, eventually, they reach a fracture point where they break. Ductile materials such as metals show a gradual decrease in stress with the increasing strain, which means they become easier to deform as stress-strain values approach the breaking point. Microscopic mechanisms responsible for plasticity of materials are different for different materials.

We can graph the relationship between stress and strain on a stress-strain diagram    . Each material has its own characteristic strain-stress curve. A typical stress-strain diagram for a ductile metal under a load is shown in [link] . In this figure, strain is a fractional elongation (not drawn to scale). When the load is gradually increased, the linear behavior (red line) that starts at the no-load point (the origin) ends at the linearity limit at point H . For further load increases beyond point H , the stress-strain relation is nonlinear but still elastic. In the figure, this nonlinear region is seen between points H and E . Ever larger loads take the stress to the elasticity limit E , where elastic behavior ends and plastic deformation begins. Beyond the elasticity limit, when the load is removed, for example at P , the material relaxes to a new shape and size along the green line. This is to say that the material becomes permanently deformed and does not come back to its initial shape and size when stress becomes zero.

The material undergoes plastic deformation for loads large enough to cause stress to go beyond the elasticity limit at E . The material continues to be plastically deformed until the stress reaches the fracture point (breaking point). Beyond the fracture point, we no longer have one sample of material, so the diagram ends at the fracture point. For the completeness of this qualitative description, it should be said that the linear, elastic, and plasticity limits denote a range of values rather than one sharp point.

The value of stress at the fracture point is called breaking stress (or ultimate stress ). Materials with similar elastic properties, such as two metals, may have very different breaking stresses. For example, ultimate stress for aluminum is $2.2\times {10}^8\text{Pa}$ and for steel it may be as high as $20.0\times {10}^8\text{Pa},$ depending on the kind of steel. We can make a quick estimate, based on [link] , that for rods with a ${1\text{-in}}^2$ cross-sectional area, the breaking load for an aluminum rod is $3.2\times {10}^4\text{lb},$ and the breaking load for a steel rod is about nine times larger.