Nonlinear behavior geometric nonlinearity

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

The above problems of fabrication and performance present a challenging task of identification of the governing material mechanisms. Use of nonlinear finite element analysis enables close simulation of actual thermal and mechanical loading conditions when combined with measurable geometrical and material parameters. As we continue to investigate real phenomena, we need to incorporate non-linearities in behavior into carefully refined models in order to achieve useful descriptions of structural responses. 

Geometric nonlinearity occurs if the relationships of strains and displacements are nonlinear with the stresses and forces. This can lead to changes in structural behavior and loss of structural stabihty. Examples of geometric nonlinearity include buckling and large displacement problems. 

The nonlinearity of the response, in general, is due to two different causes The first one lies in the fact that the body under consideration changes its shape and size by the application of an external stimulus. Therefore, in the course of this process all quantities that are referred to unit volume (in fact obtained by dividing it by the volume of the body), or to unit area of a surface will be influenced by the continuous change of volume or of surface area, leading to a deviation from truly linear behavior. Effects of this sort are called geometrical nonhnearities. 

Fiber-based frame analysis is one of the most advanced methodologies to model the nonlinear behavior of beams and columns under combined axial and bending loads. The Mid-America Earthquake Center analysis environment ZEUS-NL (Elnashai et al., 2002), is a compntational tool for the analysis of two and three dimensional frames. In ZEUS-NL, elements capable of modeling material and geometric nonlinearity are available. The forces and moments at a section are obtained by integrating the inelastic responses of individnal fibers. The Eularian approach towards geometric nonlinearity is employed at the element level. Therefore, full account is taken of the spread of... 

Formulations for beams considering both constitutive and geometric nonlinearity are rather scarce most of the geometrically nonlinear models are limited to the elastic case, Ibrahimbegovic (1995) and the inelastic behavior has been mainly restricted to plasticity, Simo et al. (1984). Recently, Mata et al. (2007b, 2008a) have extended the geometrically exact formulation for beams due to Reissner-Simo (Reissner 1973, Simo 1985, Simo Vu-Quoc 1988) to an arbitrary distribution of composite materials on the cross sections for the static and dynamic cases. 

Mata, R, Oiler, S. Barbat, A.H. 2007b. Static analysis of beam structures under nonlinear geometric and constitutive behavior. Computer Methods in Applied Mechanics and Engineering 196 4458-4478. 

Once the optimization procedure has been used to determine more geometrical configurations of the panels, their behavior has been numerically evaluated it consists essentially of the link between the top displacement and the applied shear load. Again, the models include the nonlinear... 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

The results obtained for film-substrate systems in Section 2.5 and in the present section provide connections between substrate curvature and film mismatch strain that define boundaries between regimes of behavior. For the case of a very thin film on a relatively thick substrate, it was shown in Section 2.5.1 that the response is linear with spherical curvature for normalized mismatch strain in the range 0 < em < 0.3. Furthermore, for a circular substrate, the response is geometrically nonlinear but axially symmetric for... 

Fig. 2.31. An illustration of a substrate curvature map for the case when = 0.01 and Mf/Ms = 1. The regimes of behavior are separated by curves which represent the locus of conditions for which geometrically nonlinear effects come into play and for which asymmetric bifurcation occurs in a circular substrate.

This chapter deals with the transverse or out-of-plane deflection of a thin film, and it includes quantitative descriptions of the phenomena associated with the buckling, bulging or peeling of a film from its substrate. A common thread throughout the discussion is that system behavior extends beyond the range of geometrically linear deformation. Consequently, aspects of finite or nonlinear deformation must be incorporated to capture essential features of behavior. The progression of delamination associated with trans-... 

If the deformations are not kept small, but are carried to the point where the elastic behavior is nonlinear, equations 38 and 39 do not hold. For soft polymeric solids, deviations from linearity appear sooner i.e., at smaller strains) in extension than in shear, because of the geometrical effects of Hnite deformations. At substantial deformations, the relations between creep and recovery are much more complicated than those given above, and require formulation by nonlinear constitutive relationships. 

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