To explore time dependent (dynamic) fracture propagation in a two-dimensional isotropic continuum, we use certain results from the theory of irreversible phenomena. The body's elastic strain energy density is assumed to be geometrically nonlinear and dependent on the strain gradient. In the description of microstructured media, such generalized continua frequently appear. The size of internal boundary layers in these materials is determined by an intrinsic length scale. The non-locality offered by this internal length scale, in particular, prevents deformation concentration, which is common when dealing with local models and leads to mesh reliance. To characterize the internal state of structural degradation of the system, a scalar Lagrangian damage field with a range of zero to one is introduced. All standard Lamé and second-gradient elastic coefficients are assumed to decrease as damage rises, and to be locally zero if damage reaches one. This last circumstance is linked to the propagation of cracks. In the instance of a notched rectangular/square specimens exposed to extension/compression loading tests, numerical solutions are offered, and fracture propagation is examined.
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