Elastic response, nonlinear

Calculations of this type are carried out for fee, bcc, rock salt, and hep crystal structures and applied to precursor decay in single-crystal copper, tungsten, NaCl, and LiF [17]. The calculations show that the initial mobile dislocation densities necessary to obtain the measured rapid precursor decay in all cases are two or three orders of magnitude greater than initially present in the crystals. Herrmann et al. [18] show how dislocation multiplication combined with nonlinear elastic response can give some explanation for this effect. 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

The work of the present section shows that shock-compression experiments provide an effective method for determination of higher-order elastic properties and that, by the same token, the effects of nonlinear elastic response should generally be taken into account in investigations of shock compression (see, e.g., Asay et al. [72A02]). Fourth-order contributions are readily apparent, but few coefficients have been accurately measured. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

A natural extension of linear elasticity is h rperelasticity.l l H rperelasticity is a collective term for a family of models that all have a strain energy density that only depends on the applied deformation state. This class of material models is characterized by a nonlinear elastic response, and does not capture yielding, viscoplasticity, or time-dependence. The strain energy density is the energy that is stored in the material as it is deformed, and is typically represented either in terms of invariants... 

Figure 3. Stress-strain curves in nonlinear elastic and nonlinear viscoelastic responses...

While considering tendons and ligaments as simple nonlinear elastic elements (Table 48.6) are often sufficient, additional accuracy can be obtained by incorporating viscous damping. The quasi-hnear viscoelastic approach [Fung, 1981] introduces a stress relaxation function, G(t), that depends only on time, is convoluted with the elastic response, T (A,), that depends only on the stretch ratio, to yield the complete stress response, K X, t). To obtain the stress at any point in time requires that the contribution of all preceding deformations be assessed ... 

Hyperelastic models are often used to represent the behavior of crosslinked elastomers, where the viscoelastic response can sometimes be neglected compared with the nonlinear elastic response. Because UHMWPE behaves differently than do elastomers, there are only a few specific cases when a hyperelastic representation is appropriate for UHMWPE simulations. One such case is when the loading is purely monotonic and at one single loading rate. Under these conditions it is not possible to distinguish between nonlinear elastic and viscoplastic behavior, and a hyperelastic representation might be considered. Note that if a hyperelastic model is used in an attempt to capture the... 

In 1973, Hart-Smith took this a stage further by considering the plastic deformation of an adhesive in addition to the elastic response. This work, coupled with the finite element analysis technique provided the platform for calculation of stress distribution in complex nonlinear joints. The finite element approach to stress analysis is very convenient because the number of elements can be increased in areas of significant stress change. 

There are a number of methods used to do the finite element analysis. Among the commercially available programs are NASTRAN, ANSYS, STARDYNE, and MARC-CDC (see bibliography). All of the programs are capable of generating networks for use in structural analysis on plastics parts provided that assumptions are made with respect to the elastic response of the materials. The last program listed is a nonlinear program and can accommodate the nonlinear stress-strain performance of plastics. 

The Second Edition has added a section to Chapter 5 describing the finite element concept and how it can be applied to the design of plastics structures. The limitation of the program for the nonlinear elastic response of polymer-based materials is pointed out and several useful finite element programs are suggested for design use. 

An alternative nonlinear static procedure was proposed by Aydinoglu (2003). The procedure, called Incremental Response Spectrum Analysis (IRSA) does not properly belong to the pushover methods. The procedure is displacement-based, it uses the equal displacement rule and the structure nonlinear behavior is modeled as piece-wise linear. The initial modes are computed and the elastic response spectrum for the initial structure is carried out. In the first stop the spectral ordinates are scaled to the formation of the first plastic hinge, which corresponds in the piece-wise linear capacity curve, to the first change of stiffness. The updated modal quantities are computed and additional spectral ordinates are computed up to the formation of the second plastic hinge. The procedure continues until the entire spectral ordinates are applied. The method does not require any transformation to an equivalent SDOE system. 

Multi-mode pushover analysis procedure IRSA (Incremental Response Spectrum Analysis) has been introduced by the first author to enable the two and three dimensional nonlinear analyses of buildings and bridges (Aydmoglu 2003). The practical version of the procedure (Aydmoglu 2004, 2007) works directly with smoothed elastic response spectrum and makes use of the well-known equal displacement rule to scale modal displacement increments at each piecewise linear step of an incremental application of linear Response Spectrum Analysis (RSA). In this paper, main steps of IRSA are summarized and its performance is evaluated on two example bridges under three different ground motions. 

For nonlinear response history analyses (NRHA), three different earthquake records are employed with characteristics listed in Table 1. These records are appropriately scaled to match a smoothed elastic response spectrum as shown in Figure 3. In the analysis PERFORM-3D structural analysis program (CSI 2006) has been utilized. 

Eq. (1.47) deserves some discussion. We see that a real chain has an elastic response which is significantly more nonlinear than an ideal chain. This appears on the plot of ip(x) shown qualitatively in Rg. 1.11. 

In a continuum analysis, Li et al. (2002) adopted an approach whereby the nonlinear elastic response of the material was based on the known homogeneous deformation response of the single crystal of interest. The stress field throughout the performing elastic crystal is then determined incremen-... 

Non-Newtonian fluids have both viscous and elastic properties, and they are called viscoelastic fluids. An example is so-called "silly putty," which is made from poly-dimethyl-siloxane (silicone). It flows like a liquid out of the container, but when it forms a ball, it behaves as elastic, i.e., it bounces back. The crucial factor determining the viscous and elastic behavior is the time period of the force applied short force pulse leads to elastic response, whereas long-lasting force causes flow. The viscoelasticity in polymers is due to shear-induced entanglements and nonlinear behavior of tire chains, coils. A well-known natural viscoelastic material is for, example, the egg white, which springs back when a shear force is released. A polymer resembles both liquid and solids. 

In more realistic networks, of course, there is another source for the onset of the nonlinear elastic response, namely chemical short paths. Nevertheless such investigations show the relevance of disorder on the actual elastic properties. 

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