An Eulerian-Lagrangian finite element method for modeling crack growth in creeping materials
Lee, Hae Sung
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Permalink
https://hdl.handle.net/2142/20376
Description
Title
An Eulerian-Lagrangian finite element method for modeling crack growth in creeping materials
Author(s)
Lee, Hae Sung
Issue Date
1991
Doctoral Committee Chair(s)
Haber, Robert B.
Department of Study
Civil and Environmental Engineering
Discipline
Civil Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Engineering, Civil
Language
eng
Abstract
Ductile, history-dependent material behavior governs crack growth in metal structures that are exposed to high temperatures over extended periods, such as nuclear power plants and gas turbines. This study is concerned with the development of finite element solution methods for the analysis of quasi-static, ductile crack growth in history-dependent materials. The mixed Eulerian-Lagrangian description (ELD) kinematic model is shown to have several desirable properties for modeling inelastic crack growth. Accordingly, a variational statement based on the ELD for history-dependent materials is developed and a new moving-grid finite element method based on the variational statement is presented. The moving-grid finite element method is applied to the analysis of transient, quasi-static, mode-III crack growth in creeping materials.
The class of history-dependent constitutive models considered here involves rate-form evolution equations for the nonlinear strain. The combination of the rate-based constitutive model with the ELD introduces convective terms that transform the governing equations to a mixed, elliptic-hyperbolic system. The hyperbolic nature of the governing equations leads to two major difficulties. First, if a single-field variational formulation is attempted, then the convective terms introduce a regularity condition that is difficult to satisfy in finite element methods. Second, some kind of stabilization scheme must be used to obtain oscillation-free numerical solutions.
A mixed variational method is developed to address these problems, in which the displacement and the nonlinear strain appear as independent fields. The displacement field is modeled by $H\sp1$ functions and the nonlinear strain field is modeled by piecewise-continuous $L\sb2$ functions to satisfy the Babuska-Brezzi conditions. A generalized Petrov-Galerkin method (GPG) is developed that simultaneously stabilizes the solution and relaxes the regularity condition of the mixed variational statement to admit $L\sb2$ basis functions for the nonlinear strain field. Several existing stabilization schemes, such as the SUPG method, the Galerkin/least-squares method and the discontinuous Galerkin method, occur as special cases of the GPG method. A new moving-grid finite element method is developed by combining the GPG method with the ELD kinematic model.
Quasi-static, mode-III crack growth in creeping materials under small-scale-yielding (SSY) conditions is considered. After a detailed discussion of previous asymptotic and numerical solutions for this class of problems, the GPG/ELD moving-grid finite element formulation is used to model a transient crack-growth problem. The GPG/ELD results compare favorably with previously-published numerical results and the asymptotic solutions.
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