The explicit Lagrangian method is a new numerical analytical method different from the implicit difference method. It divides the computational domain into several quadrilateral elements, each of which is endowed with physical-mechanical properties, and simulates the three-dimensional mechanical behavior of rock or other materials. In the simulation, the grid cell can be deformed with the deformation of the material [47]. This algorithm is very suitable for simulating large deformation for rock. Figure 4 shows such a specific loop calculation relationship. This relation first calls the equation of motion to derive new velocities and displacements from stresses and external forces. The strain rate is derived from the velocity, and new stresses are derived from the strain rate. For each cycle of the cycle, we use a time step.
The magnitude of the separation is governed by the cohesive law until the cohesive strength of the cracked element is zero, after which the phantom and the real nodes move independently. To have a set of full interpolation bases, the part of the cracked element that belongs in the real domain, , is extended to the phantom domain, . Then the displacement in the real domain, , can be interpolated by using the degrees of freedom for the nodes in the phantom domain, . The jump in the displacement field is realized by simply integrating only over the area from the side of the real nodes up to the crack; i.e., and . This method provides an effective and attractive engineering approach and has been used for simulation of the initiation and growth of multiple cracks in solids by Song (2006) and Remmers (2008). It has been proven to exhibit almost no mesh dependence if the mesh is sufficiently refined.
Activation Simulation Mechanical 2008 Crack
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The simulation of structures with unstable propagating cracks is challenging and difficult. Nonconvergent behavior may occur from time to time. Localized damping is included for the XFEM-based LEFM approach by using the viscous regularization technique. Viscous regularization damping causes the tangent stiffness matrix of the softening material to be positive for sufficiently small time increments.
232. McDonald, S.A., Coban, S.B., Sottos, N.R. and Withers, P.J., Tracking capsule activation and crack healing in a microcapsule-based self-healing polymer, Nature Research, 9, 177773 (2019). DOI: 10.1038/s41598-019-54242-7 2ff7e9595c
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