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产生变形，如果去除载荷，那么变形就会消失，而不会损失任何能量。这一理论对于描述材料和结构的整体力学行为是非常有力的，但如果发生局部扰动就不那么有效了。原因是每种材料都是异质的：混凝土是由水泥浇铸的小块岩石组成的，钢铁是由晶体组成的，木材是由纤维组成的。经典弹性不承认这些异质性。当然，可以选择分别对这些异构性进行建模，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来考虑异质性的影响。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，预测大型海上钢结构裂缝尖端周围的变形，或进行纳米级测试的强度预测。一般来说，弹性方程不能用手工求解，这对经典弹性力学是如此，对梯度弹性力学更是如此。因此，必须使用计算机方法，在其中找到近似（而不是精确）解。最流行的方法是有限元法，这在许多商业软件中也是可用的。经典弹性力学有许多好的有限元。一些梯度弹性的有限元已经建立，但它们不是很有效：所需的计算机能力通常比经典弹性的要高得多。因此，尽管梯度弹性具有以简单的方式解决复杂问题的潜力，但它的广泛应用仍然受到阻碍。本研究旨在建立和实现梯度弹性有限元。新的有限元方法应该比现有的方法更有效，并且应该设计成在标准有限元软件中使用。鉴于以前的梯度弹性实现主要集中在智能有限元上，本研究将主要着眼于寻找梯度弹性本身的合适格式，以便随后的有限元实现将更加直接。

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.