Finite element modelling of wave dispersion with dynamically consistent gradient elasticity
Finite element modelling of wave dispersion with dynamically consistent gradient elasticity
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DOI：
10.1007/s00466-008-0347-2
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发表时间：
2009-05
影响因子：
4.1
通讯作者：
T. Bennett;H. Askes
中科院分区：
文献类型：
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作者：
T. Bennett;H. Askes
最近开发的梯度弹性模型，能够描述波的色散和微观结构的影响的离散化方面进行了研究。该模型包括高阶刚度项以及高阶惯性项。为了避免在数值实现中需要$${\fancyscript{C}^1}$$ -连续插值，场方程使用最近开发的算子分裂重写。在空间上使用有限元离散方程，在时间上使用Newmark时间积分器离散方程。首先推导了条件稳定时间积分器的临界时间步长。接下来，连续介质的色散行为与离散介质的色散行为进行比较，这允许制定关于如何选择元件尺寸和时间步长的规则。这些规则，然后验证在一维酒吧的例子，并在一个二维的波传播的例子。因此，必须选择大致等于最小的固有长度尺度的单元尺寸，而最佳的时间步长从单元尺寸和波速的比率，并从长度尺度的比率。
The discretisation aspects are investigated of a recently developed gradient elasticity model that is capable of describing wave dispersion and microstructural effects. The model includes higher-order stiffness terms alongside higher-order inertia terms. To avoid the need for $${\fancyscript{C}^1}$$ -continuous interpolations in a numerical implementation, the field equations are rewritten using a recently developed operator split. The equations are discretised in space using finite elements and discretised in time using the Newmark time integrator. Firstly, the critical time step is derived for use with conditionally stable time integrators. Next, the dispersion behaviour of the continuum is compared with the dispersion behaviour of the discretised medium, which allows the formulation of rules on how to select the element size and the time step size. These rules are then verified in a one-dimensional bar example and in a two-dimensional example of wave propagation. It follows that the element size must be chosen roughly equal to the smallest of the intrinsic length scales, while the optimal time step follows from the ratio of element size and wave velocity and from the ratio of the length scales.