基于Willis方程的残余应力梯度场反演方法

In the recent study of elastic metamaterials, it was found that the classic linear elastic equations are not form-invariant. Thus, they cannot accurately describe the linear elastic behaviors of inhomogeneous materials. At the same time, the form-invariance of the Willis equations has been proved mathematically and it reveals that the Gradient of Initial Stresses (GIS) in the Willis equations is an important reason for the accurate description of inhomogeneous materials, based on the continuum mechanics theory. However, experimental evidences are still needed to prove the validity of the Willis equations and hence, expand their applications in scientific studies and engineering practices. To obtain these experimental evidences, this project will firstly develop the numerical simulation methods and the corresponding inverse methods based on the Willis equations to indentify the distribution of GIS. Then, large amount of ultrasonic and quasi-static tests will be conducted to identify the distributions of the GIS inside specimens, which will be used to demonstrate the validity of the Willis equations. Upon obtaining the reliable experimental evidence, this inversion scheme will be used to investigate the influence of the surface curvature on the distribution of the GIS near the surface.

近年来，人们在研究弹性波超材料时发现经典的线弹性力学方程不具有任意变换下的形式不变性，因而不能精确地描述非均匀材料的线弹性力学行为。同时，人们不但从数学上证明了Willis方程具有形式不变性，而且还根据连续介质理论发现初应力梯度项的存在是该方程能更加精确地描述非均匀介质的重要原因。不过，这些理论还需要得到实验验证才能使人真正信服，并在科学研究和工程实践中发挥更大的作用。为了获得Willis方程的实验证据，本项目将首先发展基于Willis方程的数值模拟方法和相应的初应力梯度场反演方法。在此基础上，通过开展大量的超声波和准静态实验来反演试件中的初应力梯度分布，并以此来检验Willis方程的正确性。在获得可靠实验证据的基础上，本项目还将用这套实验和反演方法来探究曲率对表面附近初应力梯度分布的影响规律。

经典线弹性力学方程不具有任意变换下的形式不变性，因而不能精确地描述非均匀材料的线弹性力学行为，从而大大延缓了弹性波超材料研究的进展。在本项目中，我们从能量的角度出发，进一步证明了包含初应力梯度和位移耦合项的Willis方程是经典线弹性理论放松均匀性假定后的自然扩展，并开展了旋转弹簧实验进行了验证。在此新方程的基础上，我们提出了一种无损检测方法，可以精确地辨识缺口附近的全场初应力及其梯度的分布。另外，我们还提出了一种近似方法来减弱初应力梯度的影响，从而可以设计出弯曲器、黑洞等弹性波超材料来有效地控制低频、宽带、复杂模式弹性波的传播。随后，我们根据位移耦合形式的Willis方程发展出一套基于割线刚度的几何非线性有限元方法，可以自动地估计初始缺陷对结构屈曲载荷的影响。最后，我们还讨论了准脆性材料断裂过程区内的应力梯度对多尺度破坏的影响。这些成果不但为位移耦合形式的Willis方程找到了坚实的理论依据和实验证据，而且初步展示了该方程的广阔的实际应用前景。