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DEM: A new computational approach to sheet metal forming problems
Author(s) -
Cavendish James C.,
Wenner Michael L.,
Burkhardt John,
Hall Charles A.,
Rheinboldt Werner C.
Publication year - 1986
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620230507
Subject(s) - discretization , finite element method , mathematics , algebraic equation , displacement field , nonlinear system , backward differentiation formula , field (mathematics) , ordinary differential equation , system of linear equations , differential algebraic equation , mathematical analysis , differential equation , physics , quantum mechanics , pure mathematics , thermodynamics
Many current approaches to finite element modelling of large deformation elastic—plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three‐field formulation in which displacements, stresses and effective strain are dependent variables. Next, the formulation is converted to a one‐field displacement formulation by an algebraic time discretization which uses a low order explicit time‐stepping procedure to integrate the equations. This approach does not produce approximations which satisfy the discrete equilibrium equations at all times and, moreover, the advantage of the single‐field algebraic formulation is realized at the expense of very small time steps needed to produce stability and accuracy in the numerical calculations. This paper describes a variant of the mixed method in which all three field variables (displacements, stresses and effective strain) are given finite element representations. The discrete equilibrium equations then generate a nonlinear system of algebraic equations whose solutions represent a manifold, while the constitutive equations form a system of ordinary differential equations. A commercially available, variable time step/variable order code is then used to integrate this differential/algebraic system. When applied to the problem of hydrostatic bulging of a membrane, the new approach requires far less computer time than the incremental method.