Maxwell relaxation time

The driving force for relaxation is spring, and the viscosity controls the rate. The Maxwell relaxation time is given by,... 

Both, G and G" can be expressed in terms of frequency co and Maxwell relaxation time Tjjj by,... 

The temperature dependence of the Maxwell relaxation time is approximated by the expression... 

The Maxwell relaxation time is commonly evaluated from the first normal stress difference measurement. 

The source time constant (excluding electrode and cable resistance and capacitance) is the Maxwell relaxation time 500 s, and this determines the theoretical lii t for the low frequency roll-off of the sensor. Tbe mcdumical resonance frequency of the radial vibration mo determines tbe high-frequency cutoff, which is tipic y in the kilohertz range. 

Mass spectroscopy, silicate, 167-170 Mass transfer coefEcient, 791, 796 Material coordinate, 441 Maxwell relaxation time, 397 Mean field theory, 315, 331 Mean free path, 290... 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

This equation, based on the generalized Maxwell model (e.g. jL, p. 68), indicates that G (o) can be determined from the difference between the measured modulus and its relaxational part. A prerequisite, however, is that the relaxation spectrum H(t) should be known over the entire relaxation time range from zero to infinity, which is impossible in practice. Nevertheless, the equation can still be used, because this time interval can generally be taken less wide, as will be demonstrated below. 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

Suppose the multiple Maxwell model which describes the material we are interested in is composed of m processes each with an elasticity Gj, a viscous process with a viscosity rjj and a corresponding relaxation time ty. We can form the relaxation function by adding all these models together ... 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

Figure 4.15 The stress growth function for a Maxwell model with a relaxation time tr...

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

Note 5 The relaxation spectrum (spectrum of relaxation times) describing stress relaxation in polymers may be considered as arising from a group of Maxwell elements in parallel. 

Since polymers are viscoelastic solids, combinations of these models are used to demonstrate the deformation resulting from the application of stress to an isotropic solid polymer. Maxwell joined the two models in series to explain the mechanical properties of pitch and tar (Figure 14.2a). He assumed that the contributions of both the spring and dashpot to strain were additive and that the application of stress would cause an instantaneous elongation of the spring, followed by a slow response of the piston in the dashpot. Thus, the relaxation time (t), when the stress and elongation have reached equilibrium, is equal to rj/G. 

Thus, according to Equation 14.8 for the Maxwell model or element, under conditions of constant strain, the stress will decrease exponentially with time and at the relaxation time t = T, s will equal 1/e, or 0.37 of its original value, So-... 

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

Apparently the stress relaxation proceeds in two phases, each with a stress decrease of 1 MPa at a strain e = 1. It seems logical to think of a parallel arrangement of two Maxwell elements, both with a spring constant E = I MPa, but with relaxation times which differ by a factor 10,000. Inspection of the stress values (with a = (To exp(-i/T)) easily results in T = sec, = 10,000 sec. The viscosities are then r = vE) 10 and 10 Pa s. 

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