Elastic response

For electromechanical and actuation applications, the following parameters are critical maximum electric-induced strain response (Xm), block force (Fb) and elastic energy density, including the volumetric elastic energy density (Wy) and gravimetric elastic 

The Xm of an E-M material is directly related to the displacement generated in an actuator. For most piezoelectric ceramics and polymers, Xm is about 0.1-0.2 %, while the newly developed E-M polymers exhibit a strain response of more than 5%, in some cases achieving as much as 100 %. This makes it possible to create actuators that exhibit a giant displacement. measures the maximum force needed to maintain the zero displacement when the material is under electric field. For an E-M polymer with a linear elastic response, Fb can be expressed as Fb=Fxj . Due to their low Young s modulus, E-M polymers usually exhibit a small block force compared to E-M ceramics. 

The elastic energy density characterize the elastic energy stored in the E-M materials and are defined as Wv = and Wg = where p is the density of the material. Wy is related to the volume of the device, while Wg is related to the mass of the device. A device made of an E-M material with a higher elastic energy density would have a smaller size/mass. As shown in Tables 16.1 and 16.2, the newly developed electrostrictive polymers exhibit a much higher elastic energy density than piezoelectric ceramics and polymers. 

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A considerable number of experimental extensions have been developed in recent years. Luckliam et al [5] and Dan [ ] review examples of dynamic measurements in the SFA. Studying the visco-elastic response of surfactant films [ ] or adsorbed polymers [7, 9] promises to yield new insights into molecular mechanisms of frictional energy loss in boundary-lubricated systems [28, 70]. 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Many types of hardness tests have been devised. The most common in use are the static indentation tests, eg, Brinell, Rockwell, and Vickers. Dynamic hardness tests involve the elastic response or rebound of a dropped indenter, eg, Scleroscope (Table 1). The approximate relationships among the various hardness tests are given in Table 2. 

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

After impact the first bead assumes a velocity 2v, due to its rigid elastic response. This is the instantaneous particle velocity that the bead acquires. The first bead travels across the gap d and impacts the second bead. The only way by which momentum and energy can be simultaneously conserved is for the first bead to come to rest at the instant the second bead acquires a velocity... 

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. 

We imagine a finite-duration shock pulse arriving at some point in the material. The strain as a function of time is shown as the upper diagram in Fig. 7.11 for elastic-perfectly-plastic response (solid line) and quasi-elastic response generally observed (dash-dot line). The maximum volume strain = 1 - PoIp is designated... 

Most materials scientists at an early stage in their university courses learn some elementary aspects of what is still miscalled strength of materials . This field incorporates elementary treatments of problems such as the elastic response of beams to continuous or localised loading, the distribution of torque across a shaft under torsion, or the elastic stresses in the components of a simple girder. Materials come into it only insofar as the specific elastic properties of a particular metal or timber determine the numerical values for some of the symbols in the algebraic treatment. This kind of simple theory is an example of continuum mechanics, and its derivation does not require any knowledge of the crystal structure or crystal properties of simple materials or of the microstructure of more complex materials. The specific aim is to design simple structures that will not exceed their elastic limit under load. 

As of this time, no one has solved the problem of the effect of asperities on a curved surface nor has anyone addressed the issue of crystalline facets. Needless to say, the problem of asperities on an irregular surface has not been addressed. However, Fuller and Tabor [118] have proposed a model that addresses the effects of variations of asperity size on adhesion for the case of planar surfaces. Assuming elastic response to the adhesion-induced stresses, they treated surface roughness as a random series of asperities having a Gaussian height distribution (f> z) and standard deviation o. Accordingly,... 

BRs were found to have a rate-sensitive mechanical response with very low tensile and shear strengths [63]. The stress-strain curves of the adhesives were characterized by an initial elastic response followed by a region of large plastic flow. 

The work of the present section shows that shock-compression experiments provide an effective method for determination of higher-order elastic properties and that, by the same token, the effects of nonlinear elastic response should generally be taken into account in investigations of shock compression (see, e.g., Asay et al. [72A02]). Fourth-order contributions are readily apparent, but few coefficients have been accurately measured. 

J. M. Whitney and J. E. Ashton, Effect of Environment on the Elastic Response of Layered Composite Piates, AIAA Journal, September 1971, pp. 1708-1713. 

No material is perfectly elastic in the sense of strictly obeying Hooke s law. Polymers, particularly when above their glass transition temperature, are certainly not. For these macromolecular materials there is an element of flow in their response to an applied stress, and the extent of this flow varies with time. Such behaviour, which may be considered to be a hybrid of perfectly elastic response and truly viscous flow, is known as viscoelasticity. 

The conhned liquid is found to exhibit both viscous and elastic response, which demonstrates that a transition from the liquid to solid state may occur in thin hlms. The solidihed liquid in the him deforms under shear, and hnally yields when the shear stress exceeds a critical value, which results in the static friction force required to initiate the motion. 

The aim for tree breeders and forest managers is to define and grow a plantation which will be elastic in its response to the large stresses induced by high wind speeds. Petty Swain (1985) have established models of the stress-strain responses of forest trees which may be used to define the sizes and morphologies of trees, for a defined range of wind speeds and elastic responses. A typical response of a plantation grown spruce tree to wind speed is shown on Fig. 2. This is a classic stress/strain curve, with an... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

According to the Brice model, for an isovalent series of ions with charge n+ and radius n entering crystal lattice site M, the partition coefficient,, can be described in terms of three parameters (Fig. 3) , the radius of that site the elastic response... 

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