Small deformations, elastic behavior

Elastic Behavior The assumption that displacement strains will produce proportional stress over a sufficiently wide range to justify an elastic-stress analysis often is not valid for nonmetals. In brittle nonmetallic piping, strains initially will produce relatively large elastic stresses. The total displacement strain must be kept small, however, since overstrain results in failure rather than plastic deformation. In plastic and resin nonmetallic piping strains generally will produce stresses of the overstrained (plasfic) type even at relatively low values of total displacement strain. 

Designers of most structures specify material stresses and strains well within the pro-portional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or process-wise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

Deformation contributes significantly to process-flow defects. Melts with only small deformation have proportional stress-strain behavior. As the stress on a melt is increased, the recoverable strain tends to reach a limiting value. It is in the high stress range, near the elastic limit, that processes operate. 

Although a key characteristic of the mechanical behavior of rubber-like materials is their ability to undergo large elastic deformations, we will present here some important results from the theory of linear elasticity [1], which is valid only for small deformations. These serve our present purposes better than the nonlinear theory, because of their simpler character and physical transparency. 

Let s start by looking at a simple polymer, polyethylene, that has a lot going on in its stress/strain plots (Figure 13-38). Flexible, semi-crystalline polymers such as this (where the T of the amorphous domains is below room temperature) usually display a considerable amount of yielding or cold-drawing, as long as they are not stretched too quickly. For small deformations, Hookean elastic-type behavior (more or less) is observed, but beyond what is called the yield point irreversible deformation occurs. 

If we have a model for linear elastic behavior, we must surely have one for Newtonian viscous flow and we do, the dashpot shown also in Figure 13-87. This is simply a piston in a cylinder that can be filled with various Newtonian fluids, each with a different value of the viscosity. Pulling (or pushing) on the piston causes it to move, as the fluid flows past the small gap between the piston and the cylinder walls, but the rate of deformation will depend on the viscosity of the fluid. (Some students who are a bit slow on the uptake or, more probably, trying to give us a hard time, ask what happens when the piston clunks to a stop at the bottom of the cylinder or pops out of the end don t be too literal minded here, this is just a picture representing a type of behavior )... 

At low force, the powder particles remain fixed in position and the particles deform elastically. This takes place over a very small range of deformation for diy ceramic powders. This behavior is referred to as a compact body deformation. This deformation can be estimated from the elastic properties of the particles, the void fraction of the powder packing, and the nature of the liquid or binder occupying the voids [71]. 

Elastoplastic materials Elastoplastic materials deform elastically for small strains, but start to deform plastically (permanently) for larger ones. In the small-strain regime, this behavior may be captured by writing the total strain as the sum of elastic and plastic parts (i.e., e = e -I- gP, where e and gP are the elastic and plastic strains, respectively). The stress in the material is generally assumed to depend on the elastic strain only (not on the plastic strain or the strain rate), and hence, no unique functional relationship exists between stress and strain. This fact also implies that energy is dissipated during plastic deformation. The point at which the material starts to deform plastically (the yield locus) is usually specified via a yield condition, which for one-dimensional plasticity may be stated as (38)... 

In industrially interesting flows a non-linear-elastic behavior is generally observed, since the deformation y and deformation rate y are not small and the above statements do not apply. In this case the following procedure provides a solution. 

The elastic behavior upon applied shear stress is primarily typical in the case of solids. The nature of elasticity is in the reversibility of small deformations of interatomic (or intermolecular) bonds. In the limit of small deformations the potential energy curve is approximated by a quadratic parabola, which corresponds to a linear t(y) dependence. Elasticity modulus of solids depends on the type of interactions. For molecular crystals it is 109 N m 2, while for metals and covalent crystals it is 1011 N m"2 or higher. The value of elasticity modulus is only weakly dependent (or nearly independent) on temperature. 

At small strains, polymers (both amorphous and crystalline) show essentially linear elastic behavior. The strain observed in this phase arises from bond angle deformation and bond stretching it is recoverable on removing the applied stress. The slope of this initial portion of the stress-strain curve is the elastic modulus. With further increase in strain, strain-induced softening occurs, resulting in a reduction of the instantaneous modulus (i.e., slope decreases). Strain-softening phenomenon is attributed to uncoiling... 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

As normally prepared, molecular networks comprise chains of a wide distribution of molecular lengths. Numerically, small chain lengths tend to predominate. The effect of this diversity on the elastic behavior of networks, particularly under large deformations, is not known. A related problem concerns the elasticity of short chains. They are inevitably non-Gaussian in character and the analysis of their conformational statistics is likely to be difficult. Nevertheless, it seems necessary to carry out this analysis to be able to treat real networks in an appropriate way. 

The word vixt oeUislic encompasses many fluids that exhibit both elasticity (solidlike behavior) and flow (liquid-like behavior) when sheared. Most concentrated pastes, emulsions, and gels are viscoelastic. Under small deformations, viscoelastic fluids literally behave as elastic solids under higher deformations they flow as liquids. 

The initial part of the curve, OA in Fig. 1, is the characteristic linear-elastic behavior of the material, i.e., the extension that occurs is fully reversible and the relationship between the force and the extension is linear. At an atomic level the bonds between the atoms of the crystal structure are just flexing. The extension in this region is however very small and can only be measured using special extensometers. This linearity ceases at point A and the material starts to behave irreversibly, i.e., permanent or plastic deformation occurs. This phenomenon is known as yielding. In this region the atoms take up new position relative to each other by the mechanism of dislocation activation. 

The rheological behavior of a viscoelastic material can be investigated by applying a small-amplitude sinusoidal deformation. The behavior can be described by a mechanical model, called the Maxwell model [33], consisting of an elastic spring with the Hookean constant, G , and a dashpot with the viscosity, r/<,. The variation of storage modulus (G ) and loss modulus (G") with shear frequency, O), are given by the equations... 

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