Phase field models are promising to tackle various fracture problems where a diffusive crack is introduced and modelled using the phase variable. Owing to the non-convexity of the energy functional, the derived partial differential equations are usually solved in a staggered manner. However, this method suffers from a low convergence rate, and a large number of staggered iterations are needed, especially at the fracture nucleation and propagation. In this study, we propose novel staggered schemes, which are inspired by the fixed-stress split scheme in poromechanics. By fixing the stress when solving the damage evolution, the displacement increment is expressed in terms of the increment of the phase variable. The relation between these two increments enables a prediction of the displacement and the active energy based on the increment of the phase variable. Thus, the maximum number of staggered iterations is reduced, and the computational efficiency is improved. We present three staggered schemes by fixing the first invariant, second invariant, or both invariants of the stress, denoted by S1, S2, and S3 schemes. The performance of the schemes is then verified by comparing with the standard staggered scheme through three benchmark examples, i.e., tensile, shear, and L-shape panel tests. The results exhibit that the force-displacement relations and the crack patterns computed using the fast schemes are consistent with the ones based on the standard staggered scheme. Moreover, the proposed S1 and S2 schemes can largely reduce the maximum number of staggered iterations and total CPU time in all benchmark tests. The S2 scheme performs comparably except in the shear test, where the underlying assumption is violated in the region close to the crack.
翻译:由于能量功能的不调和,衍生的局部差分方程式通常以错开的方式解决。但是,这种方法的趋同率较低,需要大量错开迭代,特别是在断裂核核分离和传播方面。在本研究中,我们提出新的错开办法,这些办法受固态螺旋裂裂分法的启发,采用阶段变异法的模型。在解决损坏演变时,变异增加主要表现为阶段变异的增量。这两种增量之间的关系使得人们可以根据阶段变异的增量来预测变异率和主动能量。因此,错开迭迭代的最大次数减少,计算效率得到提高。我们提出三种错开的错开办法,先是修复第一个变异性、第二个变异性或压力的变异性,然后用S1、S2和S3的变异性变异变法来表示。这两种变异变法之间的关系关系关系关系,在Slder-C2和S3级变异的变异变法中,在Slight-Creal 变变变变法中,在Straal-real roal tal tal tal tal tal test test test sal tal test sal tal test sal tal test sal tal tal tal test sal sal test sal sal sal 和slateslt sal talt sal 和S- sal test sal ex sal test saltaltaltal tal test sal test saltaltal resslate sal tal restaltaltaltal 和stal_ ress suptal 和s 和stal sal_ ressal sal sal sal sal sal restal resstal resstal sal ress ress restal restal restal ress restal restal restal restal tal restal restal restal restal restal ressl) restal ress restal 和Sl) 和s