Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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722 BAHATTIN KILIC AND ERDOGAN MADENCI Figure 14. Discretization of the plate with a circular cutout. The validity of the peridynamic predictions is first established in the absence of failure by comparing the steady-state displacements against the peridynamic theory and the finite element method using AN- SYS, a commercially available program. In the case of no failure, the plate is subjected to a uniaxial tension of 6000 lb. Also, the plate is free of any displacement constraints. Both the peridynamic and finite element models are constructed by utilizing the same discretization as that of the peridynamic model shown in Figure 14. In the case of the peridynamic model, the tensile load is applied through a uniform body force over the volumes having the length 0.05 in. Hence, the value of the body force corresponds to 2.0 × 10 5 lb/in 3 . In the case of the finite element model, the SOLID45 brick elements of ANSYS are utilized to approximate the displacements. The uniaxial tension is applied through surface tractions of 1.0 × 10 4 lb/in 2 . In the case of coupled analysis, the tensile load is applied by using uniform point forces at the nodes located at x = ±a/2 and their values are determined by dividing applied tension by the number of nodes located at x = ±a/2. Figure 15 shows the horizontal and vertical displacements along the bottom line of the plate. The comparison of horizontal displacements indicates a close agreement (Figure 15, left). Although there is also good correlation of the vertical displacements obtained from the coupled analysis, the peridynamic theory, and the finite element method (Figure 15, right), the deviation is more significant for vertical displacements than for horizontal displacements. The vertical displacements from the coupled analysis and peridynamic methods are in agreement near the cutout since both methods use the peridynamic theory in this region. However, the finite element results display a larger deviation because the Poisson’s ratio of 0.25 is not exactly satisfied for thin plates, as explained in [Kilic 2008]. For the same reason, a small jump is observed near the boundary between regions 1 and 2. The plate with a circular cutout is further investigated for failure simulation. Most experiments involving failure are performed under quasistatic conditions. Therefore, it may not be practical to solve quasistatic problems using standard explicit time integrators because explicit time integrations often require very small time steps, limiting the total time of the simulation. Therefore, the adaptive dynamic relaxation is used to simulate failure.
COUPLING OF PERIDYNAMIC THEORY AND THE FINITE ELEMENT METHOD 723 Figure 15. Horizontal (left) and vertical (right) displacements along the bottom line. As explained in [Kilic 2008], the loading rate might significantly affect the failure patterns since the dynamic problems locally reach higher displacement gradients than under quasistatic conditions because of the traveling elastic waves. In order to circumvent premature failure due to high local displacement gradients, the tensile load is gradually applied by incrementing the displacement at each iteration of the adaptive dynamic relaxation with a constant value of 3.125 × 10 −8 in. This is a small enough value that does not create high local displacement gradients. As evidenced with this type of loading, the plate is subjected to constant displacement, and that they are applied on nodes located at x = ±a/2. Furthermore, the resulting damage pattern in the peridynamic region is illustrated in Figure 16 as a contour plot of local damage given by (5). As expected, the crack initiates at the cutout, near which high displacement gradients exist (Figure 16, top left), and then propagates toward the edges of the plate (top right), until ultimate failure of the plate (bottom). The local damage index in Figure 14 is computed by (5). The local damage being 0.5 is fairly good indicator of material separation. In Figure 16, the local damage value of 0.38 indicates extensive damage not necessarily material separation. 5. Conclusions Coupling between the peridynamic theory and finite element method was presented in order to take advantage of both methods. Since the finite element is numerically more efficient than the peridynamic theory, the regions without failure were simulated using the finite element method. Furthermore, the peridynamic theory was utilized for the regions where material failure was expected. The coupling was introduced by defining an overlap region. Both the peridynamic and finite element equations were used within this region. Furthermore, the displacement and velocity fields were determined using finite element equations in the overlap region. These fields were then utilized to compute the body force densities using the peridynamic theory. Finally, these body force densities served as external forces for finite elements in the overlap region.
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