Boundary Layer Effects in a Finite Linearly Elastic Peridynamic Bar

Abstract

The peridynamic theory is an extension of the classical continuum mechanics theory. The peridynamic governing equations involve integrals of interaction forces between near particles separated by finite distances. These forces depend upon the relative displacements between material points within a body. On the other hand, the classical governing equations involve the divergence of a tensor field, which depends upon the spatial derivatives of displacements. Thus, the peridynamic governing equations are valid not only in the interior of a body, but also on its boundary, which may include a Griffith crack, and on interfaces between two bodies with different mechanical properties. Near the boundary, the solution of a peridynamic problem may be very different from the classical solution. In this work, we investigate the behavior of the displacement field of a unidimensional linearly elastic bar of length L near its ends in the context of the peridynamic theory. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young's modulus E in the classical theory and is given by different expressions found in the literature. We find that, depending on the expression of C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in the classical linear elasticity. In spite of the above, we show that the peridynamic displacement field converges to its classical counterpart as a length scale, called peridynamic horizon, tends to zero.

Homogenization of the 1D Peri-static/dynamic Bar with Constant Micromodulus

“…Aguiar et al [10] investigated the behavior of the displacement field near the ends of a peridynamic bar and found that depending on the micromodulus, the displacement field could become unbounded near the bar ends.…”

Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus

“…Aguiar et al [12] investigated the behavior of the displacement field near the ends of a peridynamic bar and found that depending on the micromodulus, the displacement field could become unbounded near the bar ends.…”

Fatigue degradation strategies to simulate crack propagation using peridynamic based computational methods

“…One of the significant features of the peridynamic approach is that it allows for spontaneous formation, interaction, and growth of discontinuities in a consistent framework (Xu et al 2008). Using this theory, promising results have been attained in elastic bar (Aguiar et al 2018) dynamic brittle fracture in glass Bobaru 2010, 2011;Bobaru and Ha 2011;Hu et al 2013), fiber-reinforced composites (Ghajari et al 2014;Hu et al 2012), functionally graded materials (Cheng et al 2015) and corrosion damage (Chen and Bobaru 2015;Chen et al 2016).…”