The 12 th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India On Finite Element Implementation for Cam Clay Model Dipika Devi Research Scholar, Dept. of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, India Arbind K. Singh Dept. of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati -781 039, India Keywords: cam clay model, differential algebraic equation, object oriented programming ABSTRACT: Cam clay models are one of the popularly used soil plasticity models. Finite element implementation of these models requires integration of constitutive relations and load stepping scheme. There have been considerable advances in the integration of constitutive relations. These methods can be grouped as explicit and implicit methods. The superiority of one over the other lies in the context of convergence, accuracy, stability, consistency and simplicity of implementation. In this paper these issues have been addressed in the context of solution of algebraic-differential equations. This approach is extension of the work of Ellsiepen and Hartmaan to modified Cam clay model. Treating nonlinear problem of Cam clay plasticity in this way facilitates to apply all the numerical algorithm of mathematics for the solution of algebraic differential equations. Numerical analysis of this approach gives the better proof for convergence and stability. This approach combines the time integration method of differential-algebraic equation and a multi-level Newton method to solve nonlinear system of algebraic equations. Consistent tangent operator can be better explained which provides the quadratic rate of convergence of multi-level Newton method. The general case of diagonally implicit Runge-Kuttta methods can be applied which is stiffly accurate and has better stability property. Also, adaptive time steps can be better implemented which is more efficient than backward Euler method. Some of the issues of concern are integration in stress space, strain driven and eigen space. The finite element programs developed in academic institutions as well as in industries are based on procedural programming. The procedural programming has its inherent limitation with respect to code reuse, maintainability, extendibility and modification. Object oriented programming has its superiority over the procedural programming in terms of encapsulation of data, inheritance, and polymorphism. By defining appropriate classes, modified Cam clay model has been implemented in the finite element framework using object-oriented programming. The class hierarchy has been identified in a manner such that the method which is used repeatedly will remain at the top of the class tree. This abstraction allows the code re-usability. Thus the framework has been developed which exploits the commonalities in the methodology between different analysis, constraints and equation solving algorithms. Also, the framework ensures that no modifications are required to previously implemented functions and classes when the code is extended. The object-oriented implementation for plane strain case has been carried out. The implemented code has been verified using various test problems and some practical problems such as footing, have been solved. 1 Introduction For reliable numerical predictions of performance of practical geotechnical problems, constitutive equations, which can model accurately the behaviour of soils, are essential. The most widely used generalized models for soil behaviour are Cam clay models and Modified Cam clay models which are formulated using the conceptual framework of critical state soil mechanics. In order to implement these models in non-linear finite element analysis for solution of boundary value problems, the constitutive equations must be integrated numerically over finite time steps. These methods used to integrate the constitutive equations have direct impact on the overall performance of the non-linear finite element analysis and has been active area of research currently (Simo and Taylor, 1985; Simo and Taylor, 1986; Wang and Atluri, 1994; Nazem et al., 2006; Clausen et al., 2006). There have been considerable advances in the methods of integration and superiority of one over the other lies in the context of convergence, accuracy, stability, consistency and simplicity of performance. These integration algorithms can be grouped as explicit and implicit methods. Both explicit and implicit methods have been used to integrate many advanced constitutive models in soil mechanics (Sloan et al., 2001; Wang et al., 2004; Borja and Lee, 1990; Borja, 1991; Luccioni et al., 2000; Luccioni et al., 2001; Kojic et al., 2002; Borja et al., 2003; Zhao et al., 2005; Simo and Hughes, 1998; Singh and Pandey, 1999; Simo and Taylor, 1986; Zhang 1995). 462