The Deformation of an Elastic Solid

In several of the following chapters we consider the behaviour of solid polymers subject to large deformations and show also that in general these materials are viscoelastic, which means that stress (or strain) varies with time. As a starting point, however, we need to consider a polymer as a linear elastic solid when a load is applied the deformation is instantaneous, after which it remains constant until the load is removed, when the recovery is instantaneous and complete linearity means that stress and strain are always proportional to one another. 

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It is convenient to use a simple weightless Hookean, or ideal, elastic spring with a modulus G and a simple Newtonian (fluid) dashpot or shock absorber having a liquid with a viscosity of 17 as models to demonstrate the deformation of an elastic solid and an ideal liquid, respectively. The stress-strain curves for these models are shown in Figure 14.1. 

The equations for the deformation of an elastic solid given in Table... 

Deformation of an elastic solid through which a mass point of the solid with co-ordinates Xi, X2, X-i in the undeformed state moves to a point with co-ordinates xi, X2, X3 in the deformed state and the deformation is defined by... 

Mumaghan FD (1937) Finite deformations of an elastic solid. Am J Math 49 235-260 Niesler H, Jackson I (1989) Pressure derivatives of elastic wave velocities from ultrasonic interferometric measurements on jacketed poly crystals. J Acoust Soc America 86 1573-1585 Nomura M, Nishizaka T, Hirata Y, Nakagiri N, Fujiwara H (1982) Measurement of the resistance of Manganin under liquid pressure to 100 kbar and its application to the measurement of the transition pressures of Bi and Sn. Jap J Appl Phys 21 936-939 Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford... 

The loading rate and time affect the deformations in polymeric solids. As already pointed out, viscoelastic materials show simultaneously the face of an elastic solid and that of a flowing viscous liquid. This implies that the simplest constitutive relation for a polymeric solid should, in general, contain time and frequency as variables in addition to stress and strain. 

Although we have implied that the stress being applied to the material is the cause and the deformation (strain or strain rate) is the effect, the reverse is equally valid. A material may be deformed in a specific manner, and the stress which is required to achieve this deformation is that which must overcome the internal forces in the material which act to resist the deformation. For an elastic solid, these stresses depend only on the magnitude of the deformation, whereas for a viscous fluid, they depend only on the rate of deformation. 

Work must be done by the external forces acting on a body when the latter deforms and in the case of an elastic solid all this work is stored as potential energy of the distorted solid, or strain energy. The whole of this stored energy may be recovered when the external forces are removed from the elastic solid reversibly. 

Note 3 The deformation gradient tensor for the simple shear of an elastic solid is... 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

An element of an elastic solid body under a shear stress deforms as shown in Figure 10.4. The shear strain (deformation)... 

A most ingenious use for soap films has been discovered by Griffith and Taylor.1 The equations representing the deformation under torsion of an elastic, solid bar of any cross-section are of the same form as those for the displacement of a soap film stretched over a hole in a flat plate, the hole being of the same shape as the section of the bar. The mathematical solution of these equations may be difficult, but it is easy to measure the displacement of the soap film hence by forming a soap film on a box, in the lid of which is a hole of the same shape as the bar, and measuring the contour lines of the film when pressure is applied inside the box, by means of a spherometer, the effect of torsional stress on bars of the most complicated section may be ascertained. 

Definition of stress and strain permits derivation of the equation of motion for elastic deformations of a solid, in particular wave motion. Figure 2.4 shows an elemental volume of an elastic solid. The stresses that exert forces in the x direction of each face are indicated, with the assumption that stress has only changed a small amount AT,- across the elemental lengths Ax, Ay, Az. The force exerted on each face is the product of the stress component indicated times the area over which the stress acts. The summation of all of the x-directed forces acting on the cube is thus... 

The effect on resilience (approximate rate of recovery from deformation) of reducing is more complex. At relatively low degrees of cross linking, the system exhibits rubbery elasticity. As decreases due to further cross linking, T increases and as it approaches the test temperature, a point of maximum damping is achieved. Here the resilience is at a minimum. Further decrease in Me increases resiliency until the sample become an elastic solid. 

Failure. When a stress, however small, is applied to a Newtonian liquid, it flows, implying that all bonds between the constituting molecules frequently break, while new ones are formed. When an increasing stress is applied to a piece of an elastic solid, it becomes deformed, and when a certain stress is reached, the test piece starts to fracture. Macroscopic fracture is characterized by... 

Such process is referred to as the elastic aftereffect and can be found in solid-like systems that reveal an elastic behavior. Elastic aftereffect is mechanically reversible the removal of applied stress results in gradual decrease of strain to zero due to the energy stored in the elastic element. The object thus restores its original shape. At the same time, in contrast to the case of a truly elastic body, the deformation of an object that follows Kelvin s model is thermodynamically irreversible due to the dissipation of energy in the... 

In 1956 Thompson and Woods reported that dynamic experiments in extension indicated that orientation increased the temperature of the p transition, about 80°C, for oriented crystalline fibres, and reduced the drop in modulus occurring at higher temperatures. Subsequently nuclear magnetic resonance was used to demonstrate that orientation reduced molecular mobility above the glass transition temperature. Measurements of dynamic extensional and torsional moduli of hot stretched filaments and films were reported in 1963 by Pinnock and Ward, who found that the relations between measured compliances below the glass transition temperature were consistent with the deformation of an incompressible elastic solid. 

An adhesive is often subjected to a rupture test, in which the stress response of the material is measured in order to determine the utility of the adhesive. In such a test, one or a combination of several different modes of deformation—shear, extension, compression, torsion, or flexure—can be important. While one of these modes may resemble the application of interest more closely than the other modes, the knowledge obtained regarding material behavior from the different tests is similar in some cases, the information is the same. In other words, the information gathered in one experiment can often be predicted from the results of the other experiments. Although this is a gross simplification, one can, for purposes of illustration, cite the behavior of linearly elastic solids and purely viscous Newtonian liquids. While the former material is characterized by its elastic modulus, the behavior of the latter is determined by the (shear) viscosity. In the case of incompressible Hookean solids, the modulus of elasticity is three times the shear modulus. (See also Chapter 2 by Krieger.)... 

A portion of an elastic solid is shown in Figure 8.24 in its natural reference configuration. For plane deformation, the material occupies the area R. The portion of the bounding surface with the potential for shape change is S. The remainder of the boundary S" (shown dashed) represents workless constraints. Material points are located in R with respect to a fixed frame of reference by means of a set of rectangular base vectors and coordinates Xk. 

The prime notation G and G" is conventional and does not mean differentiation. G (ft>) is known as the storage modulus, and G" co) is known as the loss modulus. The nomenclature follows from our understanding of classical materials. A linear elastic material (a Hookean solid) is a material for which the stress is proportional to the strain, and the deformation is completely recoverable that is, the energy required for displacement is stored elastically, and the body returns to its undeformed shape when the stress is removed. The stress for a linear elastic material will be in phase with the strain in an oscillatory experiment. Hence, G defines the magnitude of an elastic response to a deformation. A linear viscous material (a Newtonian liquid) is a material for which the stress is proportional to the strain rate, and the deformation is completely nonrecoverable that is, the energy required for displacement is dissipated, and the body remains deformed when the stress is removed. The stress for a linear viscous material will be in phase with the strain rate, or 90° out of phase with the strain. 

Now consider a small strain deformation of unit volume of an elastic solid occurring under adiabatic conditions. The first law of thermodynamics gives... 

The liquid is an example of a viscous fluid while the rubber band is an example of an elastic solid. There are materials with mechanical responses that span the entire range between these extremes. Mechanical constitutive equations are relations between the dynamics (stresses and their time rates of change) and kinematics (deformations and their time rates of change) of materials. As such they provide closure to the balance equations (see Section 3), thereby allowing the solution to a specific mechanical problem involving a specific material. 

Upon removal of the load (or stress) at time t, a sample corresponding to the Maxwell model will retract by a value equal to its elastic contribution (so = an,olE), but will be permanently strained by a value e(t) = (an,o/il)F In a creep experiment, such a sample behaves at the onset like an elastic solid and then like a viscous liquid thus it exhibits the characteristics of a viscoelastic liquid. However, this Maxwell model also predicts a linear deformation as a function of time when subjected to a constant stress which is not realistic, because no such example could be found in the field of polymers (see Figure 12.10). 

Rheology has been properly defined as the study of the flow and deformation of materials, with special emphasis being usually placed on the former [37]. There exist many fluids whose flow cannot be described by the linear response of the Newtonian flow equations. These materials are called as complex fluids, or non-Newtonian materials, since they display behaviors that range from that of viscous liquids to that of an elastic solid to some combination of the two. PE in aqueous dispersions exhibit behaviors of complex fluids [38]. 

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