Stress relaxation after a step strain

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7  

Stress relaxation in step strain experiments on a viscoelastic solid (upper curve) and a viscoelastic liquid (lower curve). The dashed lines show the value of the stress at the relaxation time t of the liquid. The solid has the same relaxation time. 

Notice that the above equation is simply a time-dependent generalization of Hooke s law [Eq. (7.98)]. For viscoelastic solids, G(t) relaxes to a finite value, called the equilibrium shear modulus G q (see Fig. 7.22, top curve) — 

For viscoelastic liquids, the Maxwell model can be used to qualitatively understand the stress relaxation modulus. In the step strain experiment, the total strain 7 is constant and Eqs (7.101)-(7.103) can be combined to give a first order differential equation for the time-dependent strain in the viscous element  

Combined with the initial condition of no strain in the viscous element when the strain is first applied [7v(0) = 0] allows integration of this differential equation  

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Next we consider a polymer melt of high molecular weight in which entanglement is very important. To calculate G(t)> it is convenient to consider the stress relaxation after a step strain. Suppose at t = 0 a shear strain y is applied to the system in equilibrium. The strain causes the deformation of the molecular conformation, and creates the stress, which relaxes with time as the conformation of polymers goes back to... 

As discussed in Section 3.2, stress relaxation after a step strain yq is the fundamental way in which we define the relaxation modulus. The relaxing stress data illustrated in Figures 3.1.2 and 3.1.3 can be used to determine Gif) directly... 

Even very stiff torsion bars must twist slightly to record the torque. With high viscosity polymer melts, this small twist can lead to significant errors in the strain or transient strain rate imposed on a sample (Gottlieb and Macosko, 1982). For example, consider stress relaxation after a step strain of 100% on a polymer melt with Go = 10 Pa (see Figure 3.3.6). Using a 0.1 rad, 25 mm diameter cone, to achieve a strain of 100% requires an initial torque of 0.4 N-m. However, if the transducer stiffness is 10 N-m/rad then it will twist 0.04 rad, and the true strain in the sample will only be 60%. There are also transient errors, For very viscous samples, the parameter C in eq. 8.2.1 dominates the damping of the transducer, and it can be used to estimate the time constant of the error ... 

Fig. 7.2. Explanation of the stress relaxation after small step strain, (a) Before deformation, the conformation of the tube is in equilibrium, (b) Immediately after the deformation, the whole tube is deformed. The deformed part is indicated by the oblique lines. For small strain, the contour length of die tube is unchanged, (c) At a later time t, the chain is pardy confined in a deformed tube.

Ilg. 7J2. Explanation of the stress relaxation after large step strain, (a) Before deformation the conformatian of the fnimitive chain is in equilibrium (r = —0). (b) Immediately after deformation, the primitive chain is in the afiindy deformed conformation (t = -1-0). (c) After time Tj, the primitive chain contracts along the tube and recovers the eqi brium contour length (t Tj,). (d) After the time Xj, the primitive chain leaves the deformed tube by reptation (t Xa). The oblique lines indicates the deformed part of the tube. Reproduced from ref. 107. 

Relaxation After a Step Strain for the Lodge Equation Calculate the relaxation of the shear stress and the first normal stress... 

The application of stress relaxation is shown in Figure 3. The relaxation modulus (G) is determined after a step strain as a function of time. A step strain is applied to the sample causing a stress. The modulus is measured as the stress relaxes. The stress relaxation modulus shows how molecular weight affects the relaxation process as a function of time as depicted in Figure 4. 

Analysis of the distribution of lifetimes for the bridges can be used to deduce their affect on the shear stress relaxation after a unit shear strain [40]. A similar approach has been used to study the dynamic response of triblock copolymers, adsorbed via their terminal blocks between two parallel plates, when they are subjected to step and sinusoidal shear [41]. 

Fig. 17 Shear stress relaxation modulus for unfilled LDPE and a series of LDPE/LDH nanocomposite melts after a step strain...

Stress relaxation. In a stress-relaxation test a plastic is deformed by a fixed amount and the stress required to maintain this deformation is measured over a period of time (Fig. 2-33) where (a) recovery after creep, (b) strain increment caused by a stress step function, and (c) strain with stress applied (1) continuously and (2) intermittently. The maximum stress occurs as soon as the deformation takes place and decreases gradually with time from this value. From a practical standpoint, creep measurements are generally considered more important than stress-relaxation tests and are also easier to conduct. 

The imposed strain is denoted in Figure 3.72(a) and the measured stress relaxation is shown in Figure 3.72(b) (where G, the relaxation modulus, is defined as a t)lyo, where a t) is the stress at time t after the step strain and yo is the magnitude of the step strain.)... 

The simplest viscoelastic response is the direct measurement of G t) itself. This measurement is done by monitoring the relaxation of the stress induced by the application of a step strain at some initial time, t = 0. As shown in Fig. 4.5, we let the applied strain reach a constant value Aq in a very short period of time e. A perfect step strain is made when e —> 0. We further assume that the strain being applied within the period e changes with time linearly. That is, the rate-of-strain is the constant, A = Ao/e, from t = —t to t = 0, and is zero before t = —e and after t = 0. Then Eq. (4.22) can be written as... 

After the application of a strain step att = to, both viscoelastic liquid and solid show a nonlinear delay in stress relaxation as a function of their viscoelastic properties see Fig. lb(4). For a viscoelastic liquid, a delayed but complete stress relaxation takes place if the period of observation is sufficiently long. However, for a viscoelastic solid, a delayed and partial stress relaxation takes place even after a long period of observation. 

Doi and Edwards [24] have extended the work of de Gennes, and have derived mathematical expressions for features sueh as the stress relaxation that occurs after a large strain. Their explanation for the physical situation is illustrated in Figure 6.19, in which the hatched area indicates the deformed part of the tube. Here (a) represents the tube before deformation, when the conformation of the primitive chain is in equilibrium. The deformation is considered to be affine, so that each molecule deforms to the same extent as the macroscopic body. In (b) the situation immediately after the step deformation is given, with the primitive chain... 

Step-strain stress-relaxation measurements have been frequently used to determine Sr(X) for polymer melts > . Equation (6) shows that if separability of time and strain effects is possible for the melt under consideration, the stress after a step elongational strain can be factored into a time-dependent function, the linear shear relaxation modulus G(t), and a strain-dependent function, the nonlinear strain measure Sr(X). Also other types of experiment may be oerformed to obtain Sr(X), such as constant-strain-rate experiments "", creep under constant stress and constant-stretching-rate experiments but these methods require more involved analytical and/or numerical calculations. 

After the application of a strain step at f = fo, both viscoelastic liquid and solid show a non-linear delay in stress relaxation as a function of their viscoelastic properties, see... 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

In the limit of linear stress-strain relations, the relaxation modulus does not depend on the initial deformation step and the rheological properties are only described by transient functions. Equation 9.11 suggests that the relaxation modulus describes the stress relaxation after the onset of a step function shear strain. In viscoelastic liquids of entangled solutions of rod-shaped micelles, an applied stress is always relaxing to zero after inflnite long periods of time. 

We have carried out standard rheometric tests as done many times in the literature for entangled polymer solutions. These experiments include startup shear, large amplitude oscillatory shear (LAOS) and large step strain. In terms of the rheological features, we observed the same as others. For example, there is a stress overshoot in startup shear in the stress plateau region the apparent G can drop below G" at frequencies of the elastic plateau and amplitudes around and above 100% and relaxation modulus decreases in time after large step strains. 

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