Hookean elastic solid

On the other hand, if the top plate moves but its displacement is not directly proportional to the applied force (it may be either more or less than proportional to the force), the material is said to be a nonlinear (i.e., non-Hookean) elastic solid. It can be represented by an equation of the form... 

A spring with a modulus of G, and a dashpot containing a liquid with a viscosity of rj, have been used as models for Hookean elastic solids and Newtonian liquids, respectively. In these models, the spring stores energy in a reversible process, and the dashpot dissipates energy as heat in an irreversible process. Figure 5.3 is a stress-strain curve for a typical elastomer the straight... 

The most straightforward rheological behaviour is exhibited on the one hand by Newtonian viscous fluids and on the other by Hookean elastic solids. However, most materials, particularly those of a colloidal nature, exhibit mechanical behaviour which is intermediate between these two extremes, with both viscous and elastic characteristics in evidence. Such materials are termed viscoelastic. 

For Hookean elastic solids the stress and strain are in phase, whereas for purely viscous liquids the strain lags 90° behind the applied stress. 

Polymeric (and other) solids and liquids are intermediate in behavior between Hookean, elastic solids, and Newtonian, purely viscous fluids. They often exhibit elements of both types of response, depending on the time scale of the experiment. Application of stresses for relatively long times may cause some flow and permanent deformation in solid polymers while rapid shearing will induce elastic behavior in some macromolecular liquids. It is also frequently observed that the value of a measured modulus or viscosity is time dependent and reflects the manner in which the measuring experiment was performed. Tliese phenomena are examples of viscoelastic behavior. 

These materials exhibit both viscous and elastic properties. In a purely Hookean elastic solid, the stress corresponding to a given strain is independent of time, whereas for viscoelastic substances the stress will gradually dissipate. In contrast to purely viscous liquids, on the other hand, viscoelastic fluids flow when subjected to stress, but part of their deformation is gradually recovered upon removal of the stress. 

Viscoelastic material such as polymers combine the characteristics of both elastic and viscous materials. They often exhibit elements of both Hookean elastic solid and pure viscous flow depending on the experimental time scale. Application of stresses of relatively long duration may cause some flow and irrecoverable (permanent) deformation, while a rapid shearing will induce elastic response in some polymeric fluids. Other examples of viscoelastic response include creep and stress relaxation, as described previously. 

Elastic effects can and do occur in molten or thermally softened polymer systems. This is a case of the fluid having both the characteristics of a Newtonian liquid and a Hookean elastic solid. The elastic effects will generally manifest themselves in entrance or exit situations. Hence, they can become important in capillaries. However, it should be pointed out that normally elastic effects are compensated for in the entrance length correction. 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

Using the negative sign convention, the equation for this model can be written by simply combining the rheological equation for a Hookean linear elastic solid... 

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. 

Rigid Solid (Euclidian) Linear Elastic Solid (Hookean) Nonlinear Elastic Solid (Non-Hookean) Visco-Elastic Fluids and Solids (Non-Linear) Nonlinear Viscous Fluid (Non-Newtonian) Linear Viscous Fluid (Newtonian) Inviscid Fluid (Pascalian)... 

A practical explanation for the velocity-porosity variations is provided by a simple elastic theory (Wood, 1941) where seawater-saturated, unconsolidated marine sediments are considered to be nonrigid systems, consisting of discrete, noninteracting mineral grains suspended in seawater (Hamilton, 1971,1972). Sound velocity would only depend on the relative proportion of solid and fluid, and their respective compressibilities and densities, expressed through Hookean elastic equations (except for attenuation which must be treated viscoelastically) (Hamilton, 1972,1980). 

An interesting three-parameter model (the Burger model has four parameters) was proposed by Hsueh [6] and is shown in Fig. 3b. He demonstrated that for a Hookean elastic element (Ei) in series with a Kelvin solid (E2,ry), the stress-strain rate relations for constant strain rate and constant stress creep tests are,... 

Guo, Z., Peng, X. and Moran, B. (2006). Mechanical response of neo-hookean fiber reinforced incompressible nonlinearly elastic solids, International Journal of Solids and Structures 44, pp. 1949-1969. 

For Newtonian behaviour r = ri where r is the shear stress, 77 is the viscosity and y is the strain rate. Kaolinite dispersions are cohesive sediments and show shear thinning behaviour ie rj decreases as y increases. The extrapolated value of r at zero shear rate is called the Bingham yield stress, Tg. For a Hookean solid r = G y, where G is the modulus of rigidity. Most substances are neither purely elastic (solid-like) nor purely viscous (liquidlike) but are viscoelastic. Kaolinite is in this category. For viscoelastic behaviour... 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

The coefficients A, /r are called Lame s constants. Then the response of the linear elastic solid, called the Hookean solid, is written as... 

A special case is the linearly elastic solid, which obeys Hooke s law of proportionality between stress and strain. For the Hookean solid the shear modulus is a material constant, which may depend on pressure and temperature but is independent of the magnitude of the shear strain. Most elastic solids obey Hooke s law for small shear strains, but become non-Hookean at large strains. Energy expended to deform an elastic solid is conserved, and may be recovered when the solid returns to its original shape upon gradual removal of the stress. The energy w stored per unit volume is... 

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.)... 

When this model is subjected to a constant stress, the response includes an instantaneous elastic strain caused by spring 1, retarded elastic strain by the Kelvin component, viscous flow by dashpot 1, instantaneous elastic strain on unloading from spring 1, retarded strain recovery from the Kelvin element and permanent deformation in dashpot 1. The multiparameter model response is shown in Figure 4.13. This model can be described as the combined response of a Hookean elastic element, a Kelvin retarded-elastic solid and a Newtonian viscous fluid. 

Viscoelasticity refers to rheological behavior that is a combination of viscous (liquid-like) and elastic (solid-like) behavior. Ideal elastic behavior is called Hookean, where the stress is directly proportional to the strain. A Hookean solid deforms as long as stress is applied. Once stress is removed, it fuUy recovers its original shape. This behavior can be modeled by a spring that stores energy under deformation and then releases it Ideal viscous behavior is called Newtonian , where the stress is directly proportional to the rate of strain. 

If the material achieves an equilibrium configuration (constant strain) in response to a constant stress t, the material is called an elastic solid. If the magnitude of the shear strain is proportional to the magnitude of the shear stress, the material is a linear elastic solid (sometimes called a Hookean solid), and the proportionally constant is the shear modulus G ... 

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. 

The neo-Hookean model gives a good but not perfect fit to tensile data on real rubber samples. As shown in Figures 1.1.2 and 1.6.1, tensile stress deviates from the model at high extensions. Is there some logical way to generalize the idea of an elastic solid to better describe experimental data ... 

In a number of polymer processing operations, such as blow molding, film blowing, and thermoforming, deformations are rapid and the polymer melt behaves more like a crosslinked rubber than a viscous liquid. Figure I.l showed typical deformation and recovery of a polymeric liquid. As the time scale of the experiment is shortened, the viscoelastic liquid looks more and more like the Hookean solid. In Chapters 3 and 4 we develop models for the full viscoelastic response, but in many cases of rapid deformations, the simplest and often most realistic model for the stress response of these polymeric liquids is in fact the elastic solid. 

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