Computational homogenization of inelastic conventional and gradient-extended microstructures by a shear band approach

该建议致力于一个数值有效的方法计算周期性微结构的平均应力响应。该方法利用了这样一个事实，即许多不均匀的微观结构，有和没有梯度效应主要变形的剪切带的形成。通过在这些带上引入少量的自由度，获得了计算上非常便宜的模型，对于广泛的微结构具有令人惊讶的准确性。因此，该方法允许有效的双尺度模拟，其中微观结构模型连接到宏观有限元模型的每个积分点。与经典的FE²方法相比，微观模型的自由度要小得多。这使得非常复杂的宏观结构的快速双尺度模拟成为可能。该模型的微观应变是分段常数。因此，微结构内的应力计算的数量显著小于许多其他降阶方法，例如，本征正交分解（POD）因此，在某些情况下，该方法的性能可能上级这些后者的方法。作为另一个优点，该方法的实现相当简单，并且不需要输入某些数据对象，如有限元刚度矩阵或残差向量，这些数据对象是POD所需的，但在有限元程序中并不总是容易访问。由于尺寸效应在许多微观结构中起着重要的作用，因此提出了将模型扩展到梯度塑性的概念。

This proposal is dedicated to a numerically efficient approach for the computation of the average stress response of periodic microstructures. The method exploits the fact that many heterogeneous microstructures with and without gradient effects mainly deform by the formation of shear bands. By introducing a small number of degrees of freedom on these bands, a computationally very cheap model is obtained being surprisingly accurate for a wide range of microstructures. Thus, the approach allows for efficient two-scale simulations, where a microstructural model is attached to each integration point of a macroscopic finite element model. In contrast to the classical FE²-method, the microscopic model has significantly less degrees of freedom. This makes the fast two-scale simulation of very complex macroscopic structures possible. The microscopic strains of the model are piecewise constant. Thus, the number of stress computations within the microstructure is significantly smaller than in many other order reduction methods being, e.g., based on the proper orthogonal decomposition (POD). As a result, the performance of the method may in certain situations be superior to these latter approaches. As another advantage, the implementation of the method is rather simple and does not require the input of certain data objects like the finite element stiffness matrix or the residual vector, which are needed for the POD but are not always easy to access in finite element programs. Since size effects play a significant role in many microstructures, a concept for the model extension to gradient plasticity is developed.