Deformation large

As in the solution to exercise 4 a), we again consider the deformation of a brick with edge lengths / to the new lengths li + A/j. This time, however, we want to account for large deformations. 

In practice, the individual polymer coils wiU rarely be fuUy extended. In most thermoplastic samples, the polymer chains wiU be entangled and the degree to which full extension can be achieved is influenced by the presence of these physical entanglements. In cross-linked rubbers, the chemical cross-links wiU determine the extent to which the chains between the cross-links can be extended. However, the concept of the transformation of a coiled chain to one that is fuUy extended is a good starting point for consideration of the changes in shape which occur in elastomeric materials. 

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For large deformations or for networks with strong interactions—say, hydrogen bonds instead of London forces—the condition for an ideal elastomer may not be satisfied. There is certainly a heat effect associated with crystallization, so (3H/9L) t. would not apply if stretching induced crystal formation. The compounds and conditions we described in the last section correspond to the kind of system for which ideality is a reasonable approximation. 

Little error is introduced using the idealized stress—strain diagram (Eig. 4a) to estimate the stresses and strains in partiady plastic cylinders since many steels used in the constmction of pressure vessels have a flat top to their stress—strain curve in the region where the plastic strain is relatively smad. However, this is not tme for large deformations, particularly if the material work hardens, when the pressure can usuady be increased above that corresponding to the codapse pressure before the cylinder bursts. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

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. 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

A number of other indifferent stress rates have been used to obtain solutions to the simple shear problem, each of which provides a different shear stress-shear strain response which has no latitude, apart from the constant Lame coefficient /r, for representing nonlinearities in the response of various materials. These different solutions have prompted a discussion in the literature regarding which indifferent stress rate is the correct one to use for large deformations. 

In fact, as Atluri [17] has pointed out, the hypoelastic equation of grade zero has inadequate latitude to represent realistic nonlinear response of various materials in large deformations, and it is necessary to use a hypoelastic equation of at least grade one to do so. If the grade is one, then, continuing to use Jaumann s stress rate and nondimensionalizing the stress as before, the isotropic representation (A.92) may be used in (5.112) with d = A and s = B to obtain... 

A ubiquitous feature accompanying large deformations in inelastic materials is the appearance of various instabilities. For example, plastic deformation may lead to shear banding, and the development of damage frequently leads to the formation of fault zones. As remarked in Section 5.2.7, normality conditions derived from the work assumption may imply stability which is too strong for such cases. Physical instabilities are likely to be associated with loss of normality and violation of the work assumption. 

Kinematical relations in large deformations are given here for reference. Most of the material is well known, and may be extracted or deduced from the comprehensive expositions of Truesdell and Toupin [19], Truesdell and Noll [20], or other texts in continuum mechanics, where further details may be found. 

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

This chapter is a brief diseussion of large deformation wave codes for multiple material problems and their applications. There are numerous other reviews that should be studied [7], [8]. There are reviews on transient dynamics codes for modeling gas flow over an airfoil, incompressible flow, electromagnetism, shock modeling in a single fluid, and other types of transient problems not addressed in this chapter. 

Large Deformation Wave Codes quantity advected... 

With the advances in computing hardware that have occurred over the last decade, three-dimensional computational analyses of shock and impact problems have become relatively common. In Lagrangian calculations, element erosion schemes have provided a means for handling the large deformations and material failure that is often involved, and Fig. 9.28 shows results of a penetration calculation which makes use of this methodology [68]. 

Goudreau, G.L. and Hallquist, J.O., Synthesis of Hydrocode and Finite Element Technology for Large Deformation Lagrangian Computation, Lawrence Livermore Laboratory, University of California Preprint No. UCRL-82858, Livermore, CA, 13 pp., August 1979. 

Large deformation contacts and finite size effects... 

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