J--integral conservation. It is pointed out that due to the stress drop and strain softening of brittle material damage localization will occur ahead of a macroscopic crack under external loads. The method of calculating the length of damage localization band is given. The shakedown theory of structures with damage is developed also in this thesis. The close relationship between damage of materials and shakedown of structures is revealed. The ductile damage factor is adopted as the control parameter of the failure criterion of elastic-plastic structures subjected to variable loads. A mathematical programming method is presented to calculate the lower bound of safety load domain and the upper bound of damage factor in either elastic-perfectly plastic or strain hardening structures at shaked
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