Instantaneous compliance

Since the stress is constant, it follows that so also is the creep rate. The creep compliance at time t, 7, can be considered to consist of three terms, an instantaneous compliance, Jq, a term covering a variety of retardation processes, xj/it), and a viscous term, t/rj. These are related by ... 

The creep curves of Fig. 4 show the three regions described above. The instantaneous compliance can be used to calculate the instantaneous modulus only for cases where is fairly large(to give a measurable Gq) This is the case with the sample containing 30 g dm bentonite which gives J = 1.1 X 10 m N corresponding to a G of 91 N m. With the sample containing 45 g dm bentonite,... 

A region (A-B) of instantaneous compliance Jo in which the bonds between the different structural units are stretched elastically. In this region, if the stress was... 

The terms Jo, J, and J2, are the instantaneous compliance, and the compliances associated with retarded elastic behavior, respectively ti = r]]/G and 12 = m/Gi are the retardation times associated with retarded elasticity r], t]2, and are the viscosities associated with Newtonian flow. 

The compliance J t) at any time was described by a six-element model made up of an instantaneous compliance, two Kelvin-Voigt bodies, and Newtonian compliance (Sherman 1970) ... 

The first two terms on the right-hand side of equation [12.6] are viscoelastic terms proposed by Schapery, where e represents uniaxial kinematic (or total) strain at time t, o is the Cauchy stress at time t, is the instantaneous compliance and AD(r[i ) is a transient creep compliance function. The factor g defines stress and temperature effects on the instantaneous elastic compliance and is a measure of state dependent reduction (or increase) in stiffness. Transient compliance factor gi has a similar meaning, operating on the creep compliance component. The factor gj accounts for the influence of loading rate on creep. The function i ) represents a reduced timescale parameter defined by ... 

J = Ji+Ji+ h- Js can be considered as zero for a crosslinked or highly crystalline polymers. Table 7.7 lists creep-related data of PU/C20A nanocomposites. The instantaneous compliance decreases with the addition of clay, and the nanocomposites observes the nearly same equilibrium compliance except 5wt% C20A, which is similar to the equilibrium stress during stress relaxation. It is also similarly found that both the creep rate and retardant time increases with the addition of clay, which should also be the result of enhanced phase microseparation. 

Each element is stressed by the stress which also acts on the instantaneous compliance The strains in the elements all add to give the total strain. 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

If the instantaneous compliance Dj is defined as the compliance term multiplying the instantaneous stress a, and the remaining terms in Eq. (36) as hereditary strains E, then... 

Hence, Eq. (37) expresses Schapery s single integral constitutive law in terms of a stress operator that includes instantaneous compliance and hereditary strains. 

For a viscoelastic system, the compliance shows a rapid elastic response Jo at t —> 0 [instantaneous compliance Jo = 1/Gq, where Go is the instantaneous modulus, which is a measure of the elastic (i.e. solid-hke ) component]. At 0, J increases slowly with time and this corresponds to the retarded response ( bonds are broken and reformed but not at the same rate). Above a certain time period (which depends on the system), the compliance shows a hnear increase with time (i.e. the system reaches a steady state with constant shear rate). If, after the steady state is reached, the stress is removed elastic recovery occurs and the strain changes sign. 

In constant stress (creep) measurements, one applies the stress (that is kept constant at each measurement) in small increasing increments. If the stress applied is below the yield stress, the system behaves as a viscoelastic solid. In this case, the strain shows a small increase at zero time and this strain remains virtually constant over the duration of the experiment (near zero shear rate). When the stress is removed, the strain returns back to zero. This behaviour will be the same at increasing stress values, provided the applied stress is stiH below the yield stress. Any increase in stress will be accompanied by an increase in strain at zero time. However, when the stress exceeds the yield stress, the system behaves as a viscoelastic liquid. In this case, the strain rapidly increases at zero time, giving a rapid elastic response characterised by an instantaneous compliance Jo (the compliance is simply the ratio between the strain and applied stress, Pa ). At time larger than zero, the strain shows a gradual and slow increase with time. This is the region of retarded response (bonds are broken and reformed at different rates). Ultimately,... 

In this case an instantaneous compliance Jg must be added to allow for the possibility of a discrete contribution with t = 0. (Although Jg may be inaccessible experimentally, its presence must be inferred or else instantaneous deformation would require inflnite stress." )... 

The stress-strain relations (6.25) are of the familiar form employed in linear viscoelasticity, except that the retardation spectra incorporate now aging affects, and all instantaneous compliances age with time. Recalling (6.18) for a discrete spectrum of retardation times Xr, expressions (6.25) read... 

In Eq. 59, t) is also called the dynamic viscosity. In Eq. 55, 7o is the instantaneous compliance (sometimes called the glassy compliance and denoted by 7g) and is associated with solid-like elastic behavior while 1 /cor] is associated with liquidlike viscous behavior, which leaves 7 and 7" as the terms that are thus associated with viscoelastic behavior. Again, the reader should note that some authors introduce the dynamic creep components differently, by burying these terms into the definitions of 7 and 7". If one goes along with the latter definitions, Eq. 55 becomes... 

Likewise, one can also define the instantaneous compliance or storage compliance this corresponds to the ratio of the in-phase strain component to the stress the loss compliance [/"( )] then corresponds to the ratio of the out-of-phase component of the strain lagging behind the stress to the latter. g (oi) and / (co) reflect the propensity of a sample to retain a supplied mechanical energy and to restore it in the form of an elastic strain and on the other hand, reflect the... 

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