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By M.P. Cherkasova

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Extra resources for Collected Problems in Numerical Methods

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As a consequence, damage cannot increase in the circled element during error estimation. When estimating the error for the circled element, the nonlocal state variable YH, rather than the local state variable YH, is representative of its mechanical properties. For this reason, a modification of the nonlocal damage model is proposed here, see Table 2. The difference resides in the way the nonlocal state variable Yh is computed. By means of a first-order Taylor expansion, the local state variable Yh is expressed as Error Estimation for Nonlocal Damage Models YH plus an error term ey.

The problem has been investigated with several discretiz- Figure 2. Compression of a plate with an initial imperfection ations, where the expected mesh-objectivity of the result has been confirmed. The development of the shear bands and the localization process in the softening branch is illustrated in Figure 3, where the effective plastic strain is depicted. It can be noticed that the softening branch is modelled up to complete failure, where the deformation Gradient-enhanced Ductile Damage 33 Figure 3.

99999 is set. In this manner, zero stiffness is avoided and it is not necessary to remove the fully damaged elements from the mesh. It is assumed that D depends on a state variable Y, which in turn depends on the strains: The basic idea of nonlocal damage models is averaging the state variable Y in the neighbourhood of each point. In this manner, the nonlocal state variable Y is obtained: The weight function a, which depends on the distance d to the point under consideration, is typically the Gaussian where the characteristic length lc is a material parameter of the nonlocal damage model, which acts as a localization limiter and can be associated to the grain size [PIJ 91].

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