岩石应变软化本构模型建立及NR-AL法求解研究

针对岩土工程材料应变软化问题及有限元对其数值计算时切线刚度矩阵负定造成求解困难的问题进行研究。建立了基于Drucker-Prager(D-P)强度准则的岩石弹塑性应变软化本构模型,本构积分算法采用一种完全隐式返回映射算法,它具有无条件稳定和精确的特点,详细论述了如何进行本构模型的程序化求解;考虑弧长法在判断切线刚度矩阵正定性导致效率低的缺点,在弹塑性增量有限元方程的迭代计算中尝试采用Newton-Raphson法和arc-length法(NR-AL法)联合迭代求解的思路,即在结构未达到极限荷载前采用NR迭代法,而当结构接近极限荷载时转换为AL法控制迭代,从而使结构越过峰值点进入软化区直至破坏,NR-AL法汲取了2者迭代求解中具有的优势;利用C++语言对所建应变软化模型的本构求解和弹塑性增量有限元方程迭代求解过程给予程序实现,应用所编程序进行数值计算,分析了D-P理想弹塑性模型、应变软化模型、应变硬化模型计算的应力-应变曲线的区别,同时将应变软化模型计算结果与试验数据进行了对比。研究结果表明:所建应变软化本构模型可以较好地模拟岩石材料的峰后软化特性,能够揭示峰后应变软化特性和破坏机制,同时NR-AL法能够求解由于应变软化造成的负刚度问题,也克服了单独使用弧长法时判断切线刚度矩阵正定性效率低的缺点。

Establishing strain softening constitutive model of rock and solution of NR-AL method

Strain softening problem in geotechnical engineering and the difficult solution problem of the finite element numerical calculation due to the negative tangent stiffness of strain softening model are studied. An elastoplastic strain softening constitutive model of rock is established based on the Drucker-Prager strength criteria. A fully implicit return mapping algorithm which has characteristics of the unconditional stability and precision is used to solve the constitutive equation, and how the programmed constitutive model to be solved is discussed in detail. Then, the shortcomings of low efficiency of the arc-length method in judging stiffness matrix is considered, Newton-Raphson scheme and arc-length method (NR-AL method) are combined to iteratively solve the calculation of elastoplastic incremental finite element equations. Namely Newton-Raphson scheme is used before the structure reaching the limit load, and when the structure is close to the limit load, turning to the arc-length method, so that the structure can go over the peak point into the softening phase until destruction. NR-AL method has the advantages in the iterative solution. A program of the built strain softening model and elastoplastic incremental finite element to solve the constitutive equation for the iterative process is compiled using C++ language. The program is applied to numerical calculation, and the stress-strain curves of the idealized elastoplastic model, strain softening and strain hardening model based on the Drucker-Prager strength criteria are comparatively analyzed. The results show that the strain softening constitutive model can simulate the characteristics the post-peak softening of rock material well, and it can reveal the features of the post-peak strain softening and failure mechanism. NR-AL method can solve the negative stiffness problem caused by strain softening and also overcome the shortcomings of low efficiency in judging stiffness matrix using the arc-length only.