Stress nonlinear elastic

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

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. 

To describe properties of solids in the nonlinear elastic strain state, a set of higher-order constitutive relations must be employed. In continuum elasticity theory, the notation typically employed differs from typical high pressure science notations. In the present section it is more appropriate to use conventional elasticity notation as far as possible. Accordingly, the following notation is employed for studies within the elastic range t = stress, t] = finite strain, with both taken positive in tension. 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

Fig. 23. Deformation and recurrent deformation at constant stress as a function of time, (a) total deformation at high stress (nonlinear behavior, relaxation time rai), (a ) deformation at low stress, (b) viscous flow, (b ) viscous flow at low stress, (c) purely elastic deformation for high stress, and for low stress (c ), (d) and (d ) recurrent effects (diffusion process)...

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

When compression and/or shear rise sufficiently, the symmetrical elasticity of the unit cell collapses, and the cell walls begin to buckle (Figure 6.26). The response to increased stress no longer produces a linear strain, and the material begins to respond to nonlinear elastic behavior, characterized by region B in Figure 6.22. 

Recent studies show that nonlinear stress response can be represented in terms of a time-dependent phase angle difference 8 between the nonlinear elastic stress and viscoelastic stress. This approach appears to have an advantage over the numerical treatments used by previous workers because it provides some information about the reversible structural changes which occur during the cycle. However, such analyses require instruments which can subject the specimen simultaneously to two cyclic deformations of different amplitude and frequency. 

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

Nonlinear Elastic Stress Response to the Sinusoidal Straining. When a nonlinear elastic body whose modulus varies by Equation 4 is subjected to cyclic straining, the stress response would be ... 

Referring to Figure 5, we can take the minimum strain point as the reference point and apply Equation 5 to establish the nonlinear elastic stress response (Tei y) to the sinusoidal straining ... 

In Figure 5, the elastic stress curve falls between the strain curve and the viscoelastic stress curves during the contracting phase, showing that a part of the phase angle difference between the strain and viscoelastic stress is contributed by the nonlinear elasticity of the specimen. 

Phase Angle Difference between the Nonlinear Viscoelastic Stress and Nonlinear Elastic Stress. The phase angle difference 8 between the nonlinear viscoelastic stress and nonlinear elastic stress is defined by the relation ... 

A quantitative analysis of stress waves in terms of nonlinear elastic, nonlinear viscous, and reversible strain-induced energy eflFects will be presented in a subsequent publication. 

Although the fibril extension stress can be predicted from the nonlinear elastic properties of the adhesive, in practice the important property that one wishes to predict is the adhesion energy rather than simply the plateau stress of the fibrillar zone. This prediction would require a better understanding of which molecular features control the detachment of the fibril from the surface, once it is... 

However, for PSA layers we need to introduce two modifications which complicate the analysis the adhesives are both viscoelastic and strained in the nonlinear elastic regime. In other words the term Gc will include a dissipative term and the term E should be replaced with a high-strain equivalent controlled by the nonlinear elastic properties as shown in Fig. 22.14 and 22.15. As a result, the crack front will not have the same shape as the classical interfacial crack and the exact nature of the stress distribution at the crack tip will be unknown. 

The nonlinear elastic properties can be described by both the Mooney-Rivlin model and the molecularly based slip-tube model. Both of these models stress the fact that the low-strain modulus of the adhesives is controlled by the entanglement structure of the isoprene -i- resin phase, while the high-strain modulus is controlled by the physical crossHnk structure. The incorporation of diblocks in the adhesive dramatically reduces the density of crossHrrks and causes a more pronounced softening in the high-strain part of the stress-strain curve. 

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 ... 

The most important coupling to deformations of the network is the one that is linear in both the strain of the network and the nematic order parameter. As has been discussed earlier in this section this leads to the consequence that the strain tensor can be used as an order parameter for the nematic-isotropic transition in nematic sidechain elastomers, just as the dielectric or the diamagnetic tensor are used as macroscopic order parameters to characterize this phase transition in low molecular weight materials. But it has also been stressed that nonlinear elastic effects as well as nonlinear coupling terms between the nematic order parameter and the strain tensor must be taken into account as soon as effects that are nonlinear in the nematic order parameter are studied [4, 25]. So far, no deviation from classical mean field behavior concerning the critical exponents has been detected in the static properties of this transition and correspondingly there are no reports as yet discussing static critical fluctuations. 

The damping function, g(s), in Eq. (6.30) accounts for lack of proportionality between stress and strain. The product, g(e)e, quantifies the nonlinear elasticity (g(e) = 1 for linear viscoelastic behavior). Separability of time and strain is illustrated for 1,4-polyisoprene in Figures 6.4 and 6.5 the time-dependence of the stress relaxation is the same for shear strains of varying amplimde and for different modes of deformation (Fuller, 1988). 

However, (7.30) can be used not only to find Young s modulus. It also describes the nonlinear elasticity, which takes up quite a lot of room on the stress versus strain curve. (In Figure 7.1, it spans from point A where the elasticity ceases being linear up to point B where the reversibility is lost.) What is more. Equation (7.30) is just as good for uni-axial compression. You only need to bear in mind that A will be less than one in this case. Another warning is that when compressed along the x-axis, the sample will... 

We do not usually think of ceramics as being elastic but do not confuse local stresses with macroscopic behavior. Macroscopically the ceramic may be brittle, but if the atoms move slightly off their perfect-crystal sites because of an applied force, that is an elastic deformation. We can also have anisotropic elasticity (which becomes important for noncubic ceramics) and nonlinear elasticity, but usually approximate to the simple case. 

The elastic relations introduced above are always defined at zero strain (or stress). As elastic strains increase in magnitude, the relations become progressively nonlinear and ultimately lead to tensile or volumetric de-cohesion or ideal shear, not considering plastic yielding, which can intervene much earlier. 

Kang H, Wen Q, Janmey PAetal (2009) Nonlinear elasticity of stiff filament networks strain stiffening, negative normal stress, and filament alignment in fibrin gels. J Phys Chem B... 

They used an anisotropic stress loading on the material and achieved a more anisotropic material response. While an anisotropic, nonlinear elastic-plastic model would be best to model skin, the preceding may be used as an intermediate step in FEA. 

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