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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 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. [2] Upper limit theorem:
A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied.
The elastic limit is, therefore, the lowest stress point at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on the equipment used and operator skill. For elastomers, such as rubber, the elastic limit is much larger than the proportionality limit. Also, precise ...
The equations that govern the deformation of jointed rocks are the same as those used to describe the motion of a continuum: [13] ˙ + = ˙ = = ˙: + = where (,) is the mass density, ˙ is the material time derivative of , (,) = ˙ (,) is the particle velocity, is the particle displacement, ˙ is the material time derivative of , (,) is the Cauchy stress tensor, (,) is the body force density ...
The point on the shock Hugoniot at which a material transitions from a purely elastic state to an elastic-plastic state is called the Hugoniot elastic limit (HEL) and the pressure at which this transition takes place is denoted p HEL. Values of p HEL can range from 0.2 GPa to 20 GPa. Above the HEL, the material loses much of its shear strength ...
where is the elastic stiffness tensor, is the closest point projection of the stress state on to the boundary of the region that bounds all possible elastic stress states. The quantity P σ {\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}} is typically found from the rate-independent solution to a plasticity problem.
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Total strain energy theory – This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension.
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