Mechanics modeling using a spring network
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I. INTRODUCTION
In this short paper, we apply the spring network model introduced in the previous paper1 to a number of problems which have been analyzed by traditional continuum and fracture mechanics approaches. The problems studied here are thus necessarily such that the springs represent regions much larger than the discrete elements discussed in the previous paper, and hence represent the average properties of a microscopically disordered system. The point of carrying out these calculations, which are somewhat analogous to coarsemesh finite element calculations, is to show that the network model can predict some important aspects of crack growth problems in excellent agreement with far more detailed calculations. This agreement provides us with considerable additional confidence that the network model is a useful tool for analyzing crack growth problems in systems where the explicit disorder on a small length scale is important. The specific problems studied below are (1) the scaling of stress intensity with crack size at the tip of a single slit crack, (2) the scaling of the stress intensity with size and separation of two colinear cracks, (3) the shielding of crack-tip stress by a zone of modified modulus (microcrack shielding), and (4) wake growth and toughening in both microcrack and transformation-toughened materials. From the results of the latter problem, we can also directly simulate R-curves for these toughened materials. The increasing complexity of these problems corresponds to the increasing importance of interactions between a crack and its environment. We start with no interactions (one crack) and end with the interactions between a crack and microcracked/dilated regions induced by the presence of the crack and which then affect the propagation of the main crack. 554
http://journals.cambridge.org
J. Mater. Res., Vol. 5, No. 3, Mar 1990
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The network model used here is a 2d triangular lattice of nodes, with each node connected to its nearneighbor nodes by linearly elastic springs (see Fig. 1) with a failure threshold stress or strain. The springs are characterized by a modulus a, a zero-force length r0, and a failure displacement rc or failure strain (rc - ro)/ro, and there are no angle-dependent forces (see Fig. 2). For a given set of spring parameters, say a crack formed by setting a = 0 for c successive springs, we perform tensile tests as follows. The vertical positions of the nodes at the bottom and top surfaces are
FIG. 1. Network of springs (solid lines) connected at node points (dots) and geometry used for tensile tests: free side boundaries and a uniaxial stress or strain applied along y. © 1990 Materials Research Society
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W. A. Curtin and H. Scher: Mechanics modeling using a spring network
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DISPLACEMENT FIG. 2. Force-displacement characteristic of a spring representing a brittle element. The zero-force length is r0, the linear slope a determines the elastic modulus, and the maximum displacement rc determ
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