Abstract We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco- elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix–vector operations for each time-step, allowing us to reduce the computational complexity from O ( N 3 ) to O ( N 2 ) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

中文翻译：

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具有异常材料记忆相关损伤的热力学一致分数粘弹塑性模型

摘要 我们开发了一种热力学一致的分数粘弹塑性模型，并结合了异常材料的损伤。该模型将 Scott-Blair 流变元素用于粘弹性/塑料部件。本构方程是通过 Scott-Blair 元素的亥姆霍兹自由能势以及依赖于记忆的分数屈服函数和耗散不等式获得的。通过分数损伤能量释放率引入了依赖于记忆的 Lemaitre 型损伤。对于所得非线性方程组的时间分数积分，我们开发了一阶半隐式分数返回映射算法。我们还开发了分数损伤能量释放率的有限差分离散化，这导致每个时间步长的 Hankel 型矩阵向量运算，允许我们通过使用快速傅立叶变换将计算复杂度从 O ( N 3 ) 降低到 O ( N 2 )。我们的数值结果表明，由于异常塑性滑移和整体损伤能量释放率之间的竞争，粘弹塑性的分数阶在损伤演化中起着至关重要的作用。