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Stable States and Paths of Structures with Plasticity or Damage

Stability conditions for irreversible structural systems are formulated on the basis of the second law of thermodynamics. It is found that distinction must be made between stable equilibrium states and stable equilibrium paths. An equilibrium state is stable if any admissible deviation from it leads to a decrease of the internal entropy of the structure. Among all the equilibrium paths emanating from a bifurcation point, the stable path is that which maximizes the increment of internal entropy. These criteria are expressed in terms of the second-order incremental work, and distinction between load control and displacement control is made. Shanley’s perfect elastoplastic column is analyzed as an example. It is found that the undeflected states of the column are stable up to the reduced modulus load. However, the undeflected stable states above the tangent modulus load are not reachable by a continuous loading process. The stable path is such that the deflection becomes nonzero as soon as the tangent modulus load is exceeded. Uniaxial strain-softening must localize right after the peak stress state even though the limit of stable homogeneous strain states occurs only later at a finite descending slope. The results indicate a need to include checks for path stability in inelastic finite element programs, especially those for damage with strain-softening.