Modes of deformation

It is rare to be able to observe elastic deformations (which occur for instance during earthquakes) since by definition an elastic deformation does not leave any record. However, many subsurface or surface features are related to the other two modes of deformation. The composition of the material, confining pressure, rate of deformation and temperature determine which type of deformation will be initiated. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

Until now we have restricted ourselves to consideration of simple tensile deformation of the elastomer sample. This deformation is easy to visualize and leads to a manageable mathematical description. This is by no means the only deformation of interest, however. We shall consider only one additional mode of deformation, namely, shear deformation. Figure 3.6 represents an elastomer sample subject to shearing forces. Deformation in the shear mode is the basis... 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

Additional complications can occur if the mode of deformation of the material in the process differs from that of the measurement method. Most fluid rheology measurements are made under shear. If the material is extended, broken into droplets, or drawn into filaments, the extensional viscosity may be a more appropriate quantity for correlation with performance. This is the case in the parting nip of a roUer in which filamenting paint can cause roUer spatter if the extensional viscosity exceeds certain limits (109). In a number of cases shear stress is the key factor rather than shear rate, and controlled stress measurements are necessary. 

Another property pecuHar to SMAs is the abiUty under certain conditions to exhibit superelastic behavior, also given the name linear superelasticity. This is distinguished from the pseudoelastic behavior, SIM. Many of the martensitic alloys, when deformed well beyond the point where the initial single coalesced martensite has formed, exhibit a stress-induced martensite-to-martensite transformation. In this mode of deformation, strain recovery occurs through the release of stress, not by a temperature-induced phase change, and recoverable strains in excess of 15% have been observed. This behavior has been exploited for medical devices. 

Since these assumptions are not always justified for plastics, the classical equations cannot be used indiscriminately. Each case must be considered on its merits and account taken of such factors as mode of deformation, service temperature, fabrication method, environment and so on. In particular it should be noted that the classical equations are derived using the relation. 

It should also be noted that in this case the material was loaded in compre-sion whereas the tensile creep curves were used. The vast majority of creep data which is available is for tensile loading mainly because this is the simplest and most convenient test method. However, it should not be forgotten that the material will behave differently under other modes of deformation. In compression the material deforms less than in tension although the efrect is small for strains up to 0.5%. If no compression data is available then the use of tensile data is permissible because the lower modulus in the latter case will provide a conservative design. 

This is the governing equation of the Maxwell Model. It is interesting to consider the response that this model predicts under three common time-dependent modes of deformation. 

It is interesting to observe that as well as the expected axial and transverse strains arising from the applied axial stress, we have also a shear strain. This is because in composites we can often get coupling between the different modes of deformation. This will also be seen later where coupling between axial and flexural deformations can occur in unsymmetric laminates. Fig. 3.17 illustrates why the shear strains arise in uniaxially stressed single ply in this Example. 

Assuming that the different modes of deformation are separable then considering equilibrium of forces in regard to the shear stress only, gives... 

Measured curvatures for a 100-mm laminate (a single closed circle on each curve) and eleven 150-mm laminates (a closed circle for the mean with a range for all other values shown) are shown in Figure 6-26. The quantitative comparison is only fair as might be expected for an approximate approach. However, the qualitative agreement as to mode of deformation is correct. 

Since these assumptions are not always justifiable when applied to plastics, the classic equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of such factors as the time under load, the mode of deformation, the service conditions, the fabrication method, the environment, and others. In particular, it should be noted that the traditional equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. From the review in Chapter 2 it should be clear that the modulus of a plastic is generally not a constant. Several approaches have been used to allow for this condition. The drawback is that these methods can be quite complex, involving numerical techniques that are not attractive to designers. However, one method has been widely accepted, the so-called pseudo-elastic design method. 

Note 4 Unlike the strain in forced uniaxial extensional oscillations, those in forced flexural deformations are not homogeneous. In the latter modes of deformation, the strains vary from point-to-point in the specimen. Hence, the equation defining the displacement y in terms of the amplitude of applied force (/q) caimot be converted into one defining strain in terms of amplitude of stress. 

The characteristics of particulate filled polymers are determined by the properties of their components, composition, structure and interactions [2]. These four factors are equally important and their effects are interconnected. The specific surface area of the filler, for example, determines the size of the contact surface between the filler and the polymer, thus the amount of the interphase formed. Surface energetics influence structure, and also the effect of composition on properties, as well as the mode of deformation. A relevant discussion of adhesion and interaction in particulate filled polymers cannot be carried out without defining the role of all factors which influence the properties of the composite and the interrelation among them. 

Tensile stages are used to observe the deformation mechanisms of materials in tension [1,45]. A common use of tensile stages in the study of polymers is to study the tensile failure of fibers and yarns [45]. Other modes of deformation such as binding and shearing can be studied with suitably modified stages [1]. 

As shown in Fig. 1, a cubic body of material under consideration is deformed in the directions of orthogonal axes Xt. If this mode of deformation, the coordinate axes coincide with the principal strain axes. In the principal stresses af corresponding to the principal strains are measured as functions of stretch ratios X, in the directions of Xh W can be calculated from... 

With this condition, there are a great many possible choices for the form of W as a function of Our ultimate purpose in the phenomenologic study of rubber elasticity is to find out its form applicable for an accurate and coherent description of the elastic behavior of rubber-like materials under various modes of deformation. We may use /j, J2, and J3 for the set of /<, which are defined by... 

The stress-strain relations for some special cases of biaxial defonnation are derived from Eqs. (13) to (15) in the following way. Strip biaxial extension of incompressible material is defined as the mode of deformation in which one of the Xj, say X2, is kept at unity, while the other, Xt, varies. This deformation is also called pure shear . We have for it ... 

From a relevant biaxial extension experiment one can determine the function w(X) constituting the Valanis-Landel expression for W, as was done, for example, by Jones and Treloar. However, the w function so obtained is of little value unless it is tested with the stress-strain relations for other modes of deformation. This kind of test was carried out by the present authors49) and is described below. 

Fig. 3. The normal modes of deformation and the corresponding magnetostriction modes for cubic and uniaxial...

The principal physical structural parameters that control the modes of deformation and failure and mechanical response of epoxies are (1) macroscopic inhomogenieties such as microvoids or concentrations of unreacted monomer, (2) the glassy-state free volume and (3) the crosslinked network structure characteristics. 

The measurement of extension (or other mode of deformation) is an essential part of several tests, notably tensile or compression stress/strain properties and also thermal expansion. The precision required must be specified in the individual test method and is unlikely to be the same as that required for test piece dimensions. The method of measurement will also be dependent on the test in question and particular techniques will be given in most cases. Hence, the requirements for specific tests will be discussed in the relevant sections in later chapters. 

Stress/strain relationships are commonly studied in tension, compression, shear or indentation. Because in theory all stress/strain relationships except those at breaking point are a function of elastic modulus, it can be questioned as to why so many modes of test are required. The answer is partly because some tests have persisted by tradition, partly because certain tests are very convenient for particular geometry of specimens and partly because at high strains the physics of rubber elasticity is even now not fully understood so that exact relationships between the various moduli are not known. A practical extension of the third reason is that it is logical to test using the mode of deformation to be found in practice. 

Data can be obtained from tests in uniaxial tension, uniaxial compression, equibiaxial tension, pure shear and simple shear. Relevant test methods are described in subsequent sections. In principle, the coefficients for a model can be obtained from a single test, for example uniaxial tension. However, the coefficients are not fully independent and more than one set of values can be found to describe the tension stress strain curve. A difficulty will arise if these coefficients are applied to another mode of deformation, for example shear or compression, because the different sets of values may not be equivalent in these cases. To obtain more robust coefficients it is necessary to carry out tests using more than one geometry and to combine the data to optimize the coefficients. 

The test conditions of temperature, strain rate and level of strain should reflect those that will be seen in service. This might involve making tests at more than one temperature and strain rate, although modulus is relatively insensitive to strain rate. With respect to strain level, BS 903-5 points out the need to take into account the fact that local strains may be rather higher than the overall strain. When data is obtained using more than one mode of deformation, the test conditions should be consistent with respect to strain levels, strain rates etc. It is also self evident that the test pieces should be produced in the same manner and their state of cure should be equal -... 

The results of dynamic tests are dependent on the test conditions test piece shape, mode of deformation, strain amplitude, strain history, frequency and temperature. ISO 4664 gives a good summary of basic factors affecting the choice of test method. Forced vibration, non-resonant tests in simple shear using a sinusoidal waveform are generally preferred for design data as... 

The Yerzley oscillograph is specified in ASTM D94519 and is shown schematically in Figure 9.7. It consists of a horizontal beam pivoted so as to oscillate vertically and in so doing deform the test piece mounted between the beam and a fixed support. A pen attached to one end of the beam records the decaying train of oscillations on a revolving drum chart. The dynamic deformation of the test piece can be superimposed on a static strain and the mode of deformation can be either shear or compression. The mass and, hence, the inertia of the beam can be varied by attached weights. 

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