Strain history

Adachi, K., 1983. Calculation of strain histories in Protean coordinate systems, Rheol. Acta 22, 326-335. 

A minimum of 10 to 35 parts carbon black to 100 parts of mbber is required to obtain a resistivity in the order of 10 Q-cm. At that loading the carbon black particles, which have an average radius of 10 nm, form grapelike aggregates that provide continuous paths for the electrical current. Special purpose mbbers containing even more carbon black have resistances as low as 1 Q-cm (129). The electrical resistivity of mbber with carbon black is sensitive to strain history, probably because of temporary dismptions of the continuity of the carbon black aggregates. 

In six-dimensional strain space, may be viewed as the inner produet of the normal to the elastic limit surface and the tangent to the strain history , see Fig. 5.1. Its value is negative, zero, or positive depending on whether i, points inward, along the tangent, or outward to the elastic limit surface. Four cases may be distinguished. 

Unloading. The strain lies on the elastic limit surface = 0, but the tangent to the strain history points inward into the elastic region < 0. It is assumed that k — 0. The material is said to be unloading and the elastic limit surface is stationary. 

In a given motion, a particular material particle will experience a strain history The stress rate relation (5.4) and flow rule (5.11), together with suitable initial conditions, may be integrated to obtain the eorresponding stress history for the particle. Conversely, using (5.16) instead of (5.4), may be obtained from by an analogous ealeulation. As before, may be represented by a continuous curve, parametrized by time, in six-dimensional symmetric stress spaee. 

In order to give a geometrieal interpretation of A it is neeessary to return for a moment to strain spaee shown in Fig. 5.1. The strain traverses its history with a veloeity given by the veetor e. When the strain history interseets the elastie limit surfaee and inelastie loading oeeurs, the elastie limit surfaee moves in sueh a way that the strain remains on it. 

During inelastie loading > 0 and the elastie limit surfaee in strain spaee is always moving outward in the direetion of the strain history. However, in stress spaee three eases may be distinguished. 

If a motion is specified with satisfies the continuity condition, the velocity, strain, and density at each material particle are determined at each time t throughout the motion. Given the constitutive functions (e, k), c(e, k), b( , k), and a s,k) with suitable initial conditions, the constitutive equations (5.1), (5.4), and (5.11) may be integrated along the strain history of each material particle to determine its stress history. If the density, velocity, and stress histories are substituted into (5.32), the history of the body force at each particle may be calculated, which is required to sustain the motion. Any such motion is termed an admissible motion, although all admissible motions may not be attainable in practice. 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

Figure 30.14 shows an interesting aspect of RPA-FT experiments, i.e., the capability to quantify the strain sensitivity of materials through parameter B of ht Equation 30.3. As can be seen, curatives addition strongly modifies this aspect of nonlinear viscoelastic behavior, with furthermore a substantial change in strain history effect. Before curatives addition, mn 2 data show very lower-strain... 

Along the mixing line, significant changes occur on Q1/Q2. At dump, data are scattered and there is little, if any, effect of the test frequency. After curatives addition, the test frequency significantly affects the Q1/Q2 signature, as expected since quarter cycle integration provides all-inclusive parameters (i.e., main torque component and all harmonics). One notes also a net strain history effect, i.e., run 1 and mn 2 data do not superimpose. 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Whilst the flow curves of materials have received widespread consideration, with the development of many models, the same cannot be said of the temporal changes seen with constant shear rate or stress. Moreover we could argue that after the apparent complexity of linear viscoeleastic systems the non-linear models developed above are very poor cousins. However, it is possible to introduce a little more phenomenological rigour by starting with the Boltzmann superposition integral given in Chapter 4, Equation (4.60). This represents the stress at time t for an applied strain history ... 

Note 4 For cases where there is a dependence of stress on strain history the following constitutive equation may be used, namely... 

The dynamic mechanical properties of elastomers have been extensively studied since the mid-1940s by rubber physicists [1], Elastomers appear to exhibit extremely complex behavior, having time-temperature- and strain-history-dependent hyperelastic properties [1]. As in polymer cures, DMA can estimate the point of critical entanglement or the gel point. 

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... 

Fritton, S.P., McLeod, KJ. and Rubin, C.T. (2000) Quantifying the strain history of bone spatial uniformity and self-similarity of low magnitude strains. Journal of Biomechanic 33 317-325... 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

D. C. Bogue, An Explicit Constitutive Equation Based on an Integrated Strain History, Ind. Eng. Chem. Fundam., 5, 253-259 (1966) also I. Chen and D. C. Bogue, Time-Dependent Stress in Polymer Melts and Review of Viscoelastic Theory, Trans. Soc. Rheol., 16, 59-78 (1972). 

If we accept the premise that the total strain is a key variable in the quality of laminar mixing, we are immediately faced with the problem that in most industrial mixers, and in processing equipment in general, different fluid particles experience different strains. This is true for both batch and continuous mixers. In the former, the different strain histories are due to the different paths the fluid particles follow in the mixer, whereas in a continuous mixer, superimposed on the different paths there is also a different residence time for every fluid particle in the mixer. To quantitatively describe the various strain histories, strain distribution functions (SDF) were defined (56), which are similar in concept to the residence time distribution functions discussed earlier. 

Batch Mixers In a batch mixer the shear rates throughout the volume are not uniform, and neither are the residence times of various fluid particles in the various shear-rate regions. Consequently, after a given time of mixing, different fluid particles experience different strain histories and accumulate different shear strains y. The SDF, g(y) dy, is defined as the fraction of the fluid in the mixer that has experienced a shear strain from y to y + dy. Alternatively, it is the probability of a fluid particle fed to the mixer to accumulating a shear strain of y in time t. By integrating g(y) dy, we get ... 

For the case of sinusoidal strain history Equation l can be transformed to yield an expression for the complex modulus, G (jw) ... 

FTMA has great potential for applications in conjunction with a coupled theory of the effects of both temperature and strain histories on mechanical material properties. This is beyond the scope of present day theoretical capabilities (31.31) and the exact constitutive equation for a given material has to be determined experimentally. 

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