Researches On The Nonlinear Complementary Method For3D Elastoplasticity Problems And Contact Problems

Elastoplasticity problems and contact problems are the two most important nonlinear problems in geotechnical engineering. Some problems, such as the singularity problem in Mohr-Coulomb criterion and the sliding direction problem in the three-dimensional contact, has not yet been solved satisfactorily. The two problems can all be attributed to the complementary problem.The research in this paper starts from their complementary problems, and the main content and research results are obtained as follows:1. Mohr-Coulomb criterion lead to convergence difficulties in the numerical calculation due to the presence of singularity. Based on Koiter’s rule in the principal stress space, the multi-yield surface of Mohr-Coulomb criterion is expressed by complementary model. Furthermore, it is described by Fischer-Burmeister complementarity function, which can be solved by Newton algorithm easily. The singularity problem that arises from the Mohr-Coulomb criterion is overcomed, and the error that arises from the traditional smoothness of potential function is also avoided.2. Based on the equivalent complementary model in smooth contact problems, a new model is proposed, which is equivalent to the original complementary model when the parameter in which approaches to zero. Derivative of the new model exists and continuous, so the corresponding Jacobian matrix is non-singular. Therefore, the conventional Newton method or Newton family algorithms can be used directly.3. The constraint function method proposed by Bathe is extended to3D frictional contact problem. The stability problem that arise from the periodicity of direction angle is solved by replacing the sliding direction angle with sliding direction vector. By this method, the normal constraint and tangential constraint can be approached by the same function for3D frictional contact problem.4. To verify the proposed method, some typical three-dimensional contact problems in geotechnical engineering are given: the rock uniaxial test, the contact problem between rigid foundation and elastic foundation, are analyzed.5. Brazilian test is simulated by the proposed method, the results show that some stress concentration exists at the contact area between the pressing plate and the specimen, to eliminate the stress concentration, arc-shaped plate with1.02times the specimen radius is suggested. 6. Using the method described above, the results of linear element and quadratic element in strength reduction technique by finite element method are compared. The results show that the error using quadratic element is aboutl/7of the maximum error using linear element. The quadratic element converges faster than linear element as the number of elements increases. The linear element will overestimate the safety factor, so it is dangerous for engineering practice. quadratic element is recommended in strength reduction technique by finite element method.