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Lagrangian and Eulerian Forms of Finite Plasticity

  • Author(s): de Vera, Giorgio Tantuan;
  • Advisor(s): Casey, James;
  • et al.

Over the past half century, much work has been published on the theory of elastic-plastic materials undergoing large deformations. However, there is still disagreement about certain basic issues. In this dissertation, the strain-based Lagrangian theory developed by Green, Naghdi and co-workers is re-assessed. Its basic structure is found to be satisfactory. For the purpose of applications, the theory is recast in Eulerian form. Additionally, a novel three-factor multiplicative decomposition of the deformation gradient is employed to define a unique intermediate configuration. The resulting theory of finite plasticity contains an elastic strain tensor measured from the intermediate stress-free configuration. The constitutive equations involve the objective stress rate of the rotated Cauchy stress, which can be expressed in terms of the rate of deformation tensor. In their general forms, the Lagrangian theory can be converted into the Eulerian theory and vice versa. Because the Green-Naghdi theory has a strain measure that represents the difference between total strain and plastic strain, rather than representing elastic strain, it does not lend itself to a physically realistic linearization for the case of small elastic deformations accompanied by large plastic deformations. The proposed theory is well suited to describe this case as it can be linearized about the intermediate configuration while allowing the plastic deformations to be large. Differences between the linearized Green-Naghdi theory and the new theory are illustrated for uniaxial tension and compression tests.

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