Compliance time-dependent

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. 

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

Figure 3.11 Time-dependent shear compliance [as J(t)/J(°°)] versus time (as t/r) (a) linear coordinates and (b) log-log coordinates.

For small shear strains we can define a time-dependent compliance (reciprocal modulus) by the equation... 

The creep (/) at time / depends on the compliance function J(t), which is a characteristic of the polymer at a given temperature, and on the initial stress time scale has to be employed in J (i.e., ( -0 the time over which that stress was applied. Furthermore, while (0 for any load is given by the product AO17, the stress of concern is the incremental added stress or... 

K(l) is the function defining the time dependence of the creep. The constant ac is a critical stress characteristic of the material, and at stresses greater than (r< the creep compliance increases rapidly with stress.. ... 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Covariates affected by treatment allocation. Variables measured after randomisation (e.g. compliance, duration of treatment) should not be used as covariates in a model for evaluation of the treatment effect as these may be influenced by the treatment received. A similar issue concerns late baselines , that is covariate measures that are based on data captured after randomisation. The term time-dependent covariate is sometimes used in relation to each of the examples above. 

Notice that the compliance is inversely proportional to the modulus. Equations for the time-dependent strain can be developed for the four-element model shown above, or for any combination of elements that provide a useful model. The corresponding time-dependent compliance can then be determined using Eq. (5.75), or the time-dependent modulus using an analogous equation. 

Finally, the time-dependent stiffnesses in Equation 8.40 can then be obtained through inversion of the compliance matrix following the quasielastic approximation ... 

Integral representations for the time dependent compliance and modulus may be written down similarly as above (9). The creep compliance function is given by... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

The deformation of a material when subjected to a constant stress is, as discussed, usually time-dependent. At times of c. 10 6 s and less all materials, including liquids, have shear compliances (i.e shear/shear stress) of c. 10 n to 10"9 m2 N-1. This is because there is only sufficient time available for an alteration of interatomic distances and bending of bond angles to take place, and the response of all materials is of the same order of magnitude in this respect. Hie time required for the various structural units of a material to move into new positions relative to one another depends on the size and shape of the units and the strength of the bonds between them. 

Viscoelasticity is termed linear when the time-dependent compliance (strain/stress) of a material is independent of the magnitude of the applied stress. All materials have a linearity limit (see Table 9.2). 

The time dependent compliance of a Voigt-Kelvin element is... 

The results of creep experiments are usually expressed in the quantity creep compliance, the time-dependent quotient of strain and stress. 

TABLE 13.18 The basic elastic constants gd and ech, the highest filament values of the modulus, and the strength, average values of the creep compliance, j[t) (ratio of time dependent creep and local stress), at 20 °C and the interchain bond for a variety of organic polymeric fibres (after Northolt et al., 2005)... 

Other elastic moduli or compliances, e.g., bulk, dilatation, etc., may be described as above. Note that the use of i = V-1 in the above expressions implies a time dependence (Some authors prefer +j the choice is... 

The following simple calculation illustrates the very significant temperature and time dependence of viscoelastic properties of polymers. It serves as a convenient, but less accurate, substitute for the accumulation of the large amount of data needed for generation of master curves. Suppose that a value is needed for the compliance (or modulus) of a plastic article for 10 years service at 25°C. What measurement time at 80°C will produce an equivalent figure We rely here on the use of a shift factor, aj, and Eq. (11 -39). Assume that the temperature dependence of the shift factor can be approximated by an Arrhenius expression of the form... 

Region B-C corresponds to a time dependent retarded elastic region with a compliance Jr. In this region the bonds break and reform, but all of them do not break and reform at the same rate. The equation for this part using mean values for the parameters is ... 

Rheological properties of mayonnaise have been studied using different rheological techniques steady shear rate-shear stress, time dependent shear rate-shear stress, stress growth and decay at a constant shear rate, dynamic viscoelastic behavior, and creep-compliance viscoelastic behavior. More studies have been devoted to the study of rheological properties of mayonnaise than of salad dressings, probably because the former is a more stable emulsion and exhibits complex viscous and viscoelastic rheological behavior. 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

In the case of harmonic time-dependence of the variables (i.e., a,y, e i a exp(/difference between the relaxed and unrelaxed compliances ... 

Needless to say, the economic consequences of poor compliance will sooner or later attract serious attention of insurers and other payors for healthcare. Prescription drugs, after all, are a principal interventional arm of modem medicine, and their actions are invariably dose- and time dependent, so their ineffective or suboptimal dosing represents an inefficiency in medical care that is potentially remediable. In considering this prospect, one should recall the words of one of the pioneers in compliance research, Stefan Norell, who wrote in... 

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