Plastic limit theorems in continuum mechanics provide two bounds that can be used to determine whether material failure is possible by means of

plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...

deformation for a given external loading scenario. According to the theorems, to find the range within which the true solution must lie, it is necessary to find both a stress field that balances the external forces and a velocity field or flow pattern that corresponds to those stresses. If the upper and lower bounds provided by the velocity field and stress field coincide, the exact value of the collapse load is determined.

Limit theorems

The two plastic limit theorems apply to any elastic-perfectly plastic body or assemblage of bodies. Lower limit theorem:

If an equilibrium distribution of
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure.

Upper limit theorem:

The body (or bodies) will collapse if there is any
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pattern of plastic deformation for which the rate of work done by the external loads exceeds the internal plastic dissipation.

References

{{reflist Continuum mechanics Physics theorems