Finite elements for gradient elasticity in statics and dynamics

力学模型用于模拟材料和结构的行为。最简单的力学模型是所谓的经典弹性：如果施加载荷，那么材料或结构就会变形，如果去除载荷，那么变形就会消失，而不会损失任何能量。这个理论对于描述材料和结构的整体力学行为是非常强大的，但如果发生局部扰动，则不那么强大。原因是每种材料都是异质的：混凝土是由小块岩石浇铸在水泥中组成的，钢是由晶体组成的，木材是由纤维组成的。经典弹性理论并不承认这些不均匀性，当然，人们可以选择分别对每一种不均匀性建模，但这将导致一个非常复杂和低效的模型。相反，可以使用改进的弹性理论，其中包括异质性的影响。后者通常被称为梯度弹性理论，它们可以用来描述冲击波如何在混凝土结构和土壤中传播，预测大型钢海洋结构裂缝尖端周围的变形，或者进行纳米尺度测试的强度预测。通常，弹性方程不能用手求解-这适用于经典弹性，更适用于梯度弹性。因此，必须使用计算机方法，在其中找到近似（而不是精确）的解决方案。最流行的方法，也是许多数学软件中可用的方法，是有限元法。经典弹性力学有许多好的有限元。一些有限元已经制定了梯度弹性，但他们不是很有效：所需的计算机能力通常是显着高于经典弹性。因此，尽管梯度弹性理论具有以简单方式解决复杂问题的潜力，但它的广泛应用仍然受到阻碍。本研究的目的是制定和实施梯度弹性有限元。新的有限元应该比现有的方法更有效，它们应该被设计用于标准的有限元软件。而以前的实现梯度弹性集中在智能有限元，这项研究的主要目的是找到合适的格式梯度弹性本身，这样，随后的有限元实现将更加简单。

Mechanical models are used to simulate the behaviour of materials and structures. The simplest mechanical model is so-called classical elasticity: if a load is applied, then a deformation of the material or the structure results, and if the load is removed, then the deformation disappears without any loss of energy. This theory is very powerful for the description of the global mechanical behaviour of materials and structures, but less so if local perturbations occur. The reason is that every material is heterogeneous: concrete consists of little pieces of rock cast in cement, steel is built up from crystals, and timber consists of fibers. Classical elasticitydoes not recognise these heterogeneities.Of course, one could choose to model each of these heterogeneitiesseparately, but that would lead to a very complicated and inefficient model. Instead, improved elasticity theories can be used which incorporate the effects of the heterogeneity. The latter are commonly known as gradient elasticity theories, and they can be used to describe how shock waves propagate through concrete structures and soils, to predict the deformation around the tips of cracks in large steel offshore structures or to make strength predictions of nano-scale tests.Generally, the elasticity equations cannot be solved by hand - this holds for classical elasticity and more so for gradient elasticity. Therefore, computer methods must be used, in which an approximate (rather than exact) solution is found. The most popular method, that is also available in many commerical software, is the finite element method. Many good finite elements exist for classical elasticity. Some finite elements have been formulated for gradient elasticity, but they are not very efficient: the required computer power is normally significantly higher than that for classical elasticity. Therefore, a wide-spread application of gradient elasticity is still hampered, despite the potential of gradient elasticity to solve complicated problems in a simple manner. This research aims at formulating and implementing finite elements for gradient elasticity. The new finite elements should be more efficient than the existing methods, and they should be designed for use in standard finite element software. Whereas previous implementations of gradient elasticity have focussed on smart finite elements, this research will aim primarily at finding suitable formats of gradient elasticity itself, such that the subsequent finite element implementation will be more straightforward.