Continuum Mechanics, Nonlinear Finite Element Techniques and Computational Stability
This three lectures course will give a modern concept of finite-element- analysis in nonlinear solid mechanics using material (Lagrangian) and spatial (Eulerian) coordinates. Elastic response of solids is treated as an essential example for the geometrica
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P. Wriggers T. H. Darmstadt, Darmstadt, Germany
Abstract This three lectures course will give a modern concept of finite-element- analysis in nonlinear solid mechanics using material (Lagrangian) and spatial (Eulerian) coordinates. Elastic response of solids is treated as an essential example for the geometrically and material nonlinear behavior. Furthermore a brief introduction in stability analysis and the associated numerical algorithms will be given. A main feature of these lectures is the derivation of consistent linearizations of the weak form of equilibrium within the same order of magnitude, taking also into account the material laws in order to get Newton-type iterative algorithms with quadratic convergence. The lectures are intended to introduce into effective discretizations and algorithms based on a well founded mechanical and mathematical analysis.
1 Survey of nonlinear continuum mechanics 1.1 Introductory Remarks In this section a short summary of the continuum mechanics of solids is given. In detail we will discuss the ingredients which must be provided for any theory used E. Stein (ed.), Progress in Computational Analysis of Inelastic Structures © Springer-Verlag Wien 1993
P. Wriggers
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for the description of an isothermal mechanical process. These are the kinematical relations, the balance laws and the constitutive equations. We will restrict ourselves to the Lagrangian and spatial description of the motion and the related strain measures. Furthermore we will introduce the weak forms of the balance laws which will be the basis for the finite element formulation derived in section 2. As an example for constitutive laws the isotropic hyperelastic materials of Neo-Hookean type have been chosen to show besides geometrically nonlinear effects also the implications of nonlinear material behavior. However due to the restricted space only a few equations and discussions concerning continuum mechanics can be presented. Thus, the reader who wishes to get a deeper insight may consult standard textboo""o_n continuum mechanics and the mathematical background (Malvern {1969), True1lell and Noll {1965), Mar&den and Hughe& (1983), Ogden {1984}, Oiarlet {1987)).
1.2 Kinematics Let B be the reference configuration of a continuum body with particles X E B. A deformation is described by the one-to-one mapping tp which defines the current configuration tp(B) as shown in Fig. 1.
Fig. 1 Motion of a body B. A particle X hi the undeformed configuration is decribed by the position vector X = XA eA. At time t the position occupied by the parti~e X is given by x
= tp (X, t) = z; ei.
{1.1)
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Continuum Mechanics, ...
The deformation gradient F is defined as the tensor which associates an infinitesimal vector dX at X with a vector dx at x: dx = FdX. Therefore, the components of F are the partial derivatives :;~ = Zi,A· With {1.1) we obtain
= FiA ei ® eA = Gradx, FiA = Zi,A. F
(1.2)
By J = det F we define the determinant of the deformation gradient. Throughout our considerations we restrict the deformation su
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