Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. Finally, we show the evolution of the spacing between peaks for all solutions as a function of that parameter.

中文翻译：

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土工材料压实带的自动自适应稳定有限元和连续分析

在压缩蠕变下，经历内部传质过程的粘塑性固体可以适应奇异的 cnoidal 波解，因为材料在驻波极限处不稳定性。当加载速率明显快于材料扩散内部扰动的能力时，就会出现这些不稳定性，从而导致局部失效特征（例如裂纹和压实带）。这些通常在流体中发现的 cnoidal 波具有很强的非线性，会产生周期性模式。由于解的奇异性，该理论的适用性目前是有限的。此外，实用的仿真工具需要适当的正则化来克服奇点带来的挑战。我们专注于使用基于最近自适应稳定有限元方法的非线性方法对控制方程进行数值处理。这种自动细化方法提供了驱动网格自适应性的误差估计，这是解决手头问题的关键特征。我们将这种自适应策略的性能与分析和标准有限元解决方案进行了比较。然后，我们研究了扩散比的敏感性、控制过程的参数，并确定了具有多个应力峰值的多个可能的解决方案。最后，我们展示了所有解的峰间距随该参数的变化。我们将这种自适应策略的性能与分析和标准有限元解决方案进行了比较。然后，我们研究了扩散比的敏感性、控制过程的参数，并确定了具有多个应力峰值的多个可能的解决方案。最后，我们展示了所有解的峰间距随该参数的变化。我们将这种自适应策略的性能与分析和标准有限元解决方案进行了比较。然后，我们研究了扩散比的敏感性、控制过程的参数，并确定了具有多个应力峰值的多个可能的解决方案。最后，我们展示了所有解的峰间距随该参数的变化。