Nonlinear elasticity

As the current pulse is largely dominated by the stress differences, a short duration current pulse is observed upon loading with a quiescent period during the time at constant stress. With release of pressure upon arrival of the unloading wave from the stress-free surface behind the impactor, a current pulse of opposite polarity is observed. The amplitude of the release wave current pulse provides a sensitive measure of the elastic nonlinearity of the target material at the peak pressure in question. 

Let us begin by considering an elastic, nonlinear viscous material whose total strain rate under uniaxial tension loading conditions is characterized by the following equation ... 

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. 

Stergiopulos, N., Meister, J.-J., and Westerhof, N. 1995b. Scatter in input impedance spectrum may result from the elastic nonlinearity of the arterial wall. Am. J. Physiol. 269 H1490-H1495. 

Program name Material properties Linear elastic Analysis linear visco- elastic Nonlinear visco- elastic Geometric nonlinearity Loading Time function Nonlinear diffusion... 

Polymer melts in shear flow exhibit normal stresses as well as shear stresses. These are associated with the melt elasticity (nonlinear viscoelasticity) and are the cause of the Weissenberg rod climbing effect [46]. The stress tensor in the shear flow defined by Eq. 1.26 is of form ... 

Solution of the Fokker-Planck-Kolmogorov Equation for an SDOF Elastic Nonlinear Second-Order System... 

Several solutions of the FPK equation for nonlinear elastic systems with n degrees of freedom (nDOF) subjected to random excitation have been developed in the past (Piszczec and Niziol 1986). Let us consider the following set of n equations for an nDOF system, characterized by elastic nonlinear properties ... 

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

Of course, the above independence takes place provided that / = 0 in the domain with the boundary C. The integral of the form (4.100) is called the Rice-Cherepanov integral. We have to note that the statement obtained is proved for nonlinear boundary conditions (4.91). This statement is similar to the well-known result in the linear elasticity theory with linear boundary conditions prescribed on S (see Bui, Ehrlacher, 1997 Rice, 1968 Rice, Drucker, 1967 Parton, Morozov, 1985 Destuynder, Jaoua, 1981). 

We have to note that the result is obtained for nonlinear boundary conditions (4.128). The well-known path independence of the Rice-Cherepanov integral was previously proved in elasticity theory for linear boundary conditions a22 = 0,ai2 = 0 holding on Ef (see Parton, Morozov, 1985). 

Polarization which can be induced in nonconducting materials by means of an externally appHed electric field is one of the most important parameters in the theory of insulators, which are called dielectrics when their polarizabiUty is under consideration (1). Experimental investigations have shown that these materials can be divided into linear and nonlinear dielectrics in accordance with their behavior in a realizable range of the electric field. The electric polarization PI of linear dielectrics depends linearly on the electric field E, whereas that of nonlinear dielectrics is a nonlinear function of the electric field (2). The polarization values which can be measured in linear (normal) dielectrics upon appHcation of experimentally attainable electric fields are usually small. However, a certain group of nonlinear dielectrics exhibit polarization values which are several orders of magnitude larger than those observed in normal dielectrics (3). Consequentiy, a number of useful physical properties related to the polarization of the materials, such as elastic, thermal, optical, electromechanical, etc, are observed in these groups of nonlinear dielectrics (4). 

Liquid crystal polymers are also used in electrooptic displays. Side-chain polymers are quite suitable for this purpose, but usually involve much larger elastic and viscous constants, which slow the response of the device (33). The chiral smectic C phase is perhaps best suited for a polymer field effect device. The abiHty to attach dichroic or fluorescent dyes as a proportion of the side groups opens the door to appHcations not easily achieved with low molecular weight Hquid crystals. Polymers with smectic phases have also been used to create laser writable devices (30). The laser can address areas a few micrometers wide, changing a clear state to a strong scattering state or vice versa. Future uses of Hquid crystal polymers may include data storage devices. Polymers with nonlinear optical properties may also become important for device appHcations. 

The design of shape-memory devices is quite different from that of conventional alloys. These materials are nonlinear, have properties that are very temperature-dependent, including an elastic modulus that not only increases with increasing temperature, but can change by a large factor over a small temperature span. This difficulty in design has been addressed as a result of the demands made in the design of compHcated smart and adaptive stmctures. Informative references on all aspects of SMAs are available (7—9). 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

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. 

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. 

Asay et al. [24] investigate further the effects of nonlinear elasticity on micromechanical interpretation of decaying elastic shock fronts. Values of (Til, I, r/Cj, and which represent the highest Mg" " impurity concentration are shown in Table 7.1 for D = 0.1 GPa. 

That fraction of the applied work which is not consumed in the elastic-plastic deformation remains to create the new crack surface, i.e., the crack driving force. Therefore, a nonlinear fracture toughness, G, may be defined as follows ... 

The J value is defined as the elastic potential difference between the linear and nonlinear elastic bodies with the same geometric variables [52,53]. The elastic potential energy for a nonlinear elastic body is expressed by ... 

At low strains there is an elastic region whereas at high strains there is a nonlinear relationship between stress and strain and there is a permanent element to the strain. In the absence of any specific information for a particular plastic, design strains should normally be limited to 1%. Lower values ( 0.5%) are recommended for the more brittle thermoplastics such as acrylic, polystyrene and values of 0.2-0.3% should be used for thermosets. 

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. 

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