Theoretical and practical aspects of finite elements in the context of some problems of solid mechanics.

*(English)*Zbl 0701.73058This paper is concerned with the use of computational modelling to produce approximations to the solutions of certain linear and nonlinear problems of solid mechanics.

The aims are as follows: to describe the forms of the mathematical models of the problems, indicating the major differences between the various contexts and explaining what assumptions have been made in these concerning the physics; to desribe the application of the finite element method using Galerkin techniques for each problem and to give some theoretical finite element error estimates where this is possible. In the context of error estimates the phenomenon of superconvergence is introduced.

We have shown the applicability and state of knowledge in the use of finite element methods in the context of linear elasticity, elastoplasticity and viscoelasticity. The major differences between these result from the different constitutive relations required. In the simplest case of linear elasticity no time or history effects are involved and the theory is well developed. However with the more complicated materials the models may involve time and history effects. In the elastoplastic model the resulting equations are nonlinear. This model involves a “time” parameter as a result of the incremental loading procedure, as well as indirectly in the integration of the hardening rule. The history affects the value of the hardening parameters. In the viscoelastic model time and history effects are explicitly involved through the constitutive equations. In both these cases the theory concerning error estimates is much less developed than for the case of linear elasticity.

The aims are as follows: to describe the forms of the mathematical models of the problems, indicating the major differences between the various contexts and explaining what assumptions have been made in these concerning the physics; to desribe the application of the finite element method using Galerkin techniques for each problem and to give some theoretical finite element error estimates where this is possible. In the context of error estimates the phenomenon of superconvergence is introduced.

We have shown the applicability and state of knowledge in the use of finite element methods in the context of linear elasticity, elastoplasticity and viscoelasticity. The major differences between these result from the different constitutive relations required. In the simplest case of linear elasticity no time or history effects are involved and the theory is well developed. However with the more complicated materials the models may involve time and history effects. In the elastoplastic model the resulting equations are nonlinear. This model involves a “time” parameter as a result of the incremental loading procedure, as well as indirectly in the integration of the hardening rule. The history affects the value of the hardening parameters. In the viscoelastic model time and history effects are explicitly involved through the constitutive equations. In both these cases the theory concerning error estimates is much less developed than for the case of linear elasticity.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74Hxx | Dynamical problems in solid mechanics |