Viscoelastic stress response

According to this representation, the nonlinear viscoelastic stress response resulting from sinusoidal strain ... 

The most characteristic features of viscoelastic materials are that they exhibit a time dependent strain response to a constant stress (creep) and a time dependent stress response to a constant strain (relaxation). In addition when the... 

Typical for the spectroscopic character of the measurement is the rapid development of a quasi-steady state stress. In the actual experiment, the sample is at rest (equilibrated) until, at t = 0, oscillatory shear flow is started. The shear stress response may be calculated with the general equation of linear viscoelasticity [10] (introducing Eqs. 4-3 and 4-9 into Eq. 3-2)... 

Figure 4.6 An oscillating strain and the stress response for a viscoelastic material...

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Due to the viscoelastic nature of the material the stress response, after the application of the oscillatory shear strain, is also a sinusoidal but out of phase relative to the strain what can be represented by equation (2.7) as... 

Fig. E3.3 The schematic stress response of elastic, a viscous, and a viscoelastic hody to a sinusoidally applied strain.

When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent of the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. In the following, we mainly discuss carbon black filled rubbers because carbon black is the most widespread filler in rubber products, as for example, automotive tires and vibration mounts. The presence of carbon black filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why carbon black filled rubbers are considered as nonlinear viscoelastic materials. The term non-linear viscoelasticity will be discussed later in more detail. 

Let us assume that a sinusoidal shear strain z t) = Sq sin cot is imposed on a viscoelastic solid, where 8q and co are, respectively, the amplitude and frequency of the perturbing strain. A dynamic shear strain is illustrated in Figure 6.1. Experimentally one observes that the shear stress (response) is... 

The linear viscoelastic connection between stress (response) and velocity of strain (stimulus) can be written [132]... 

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 ... 

Remember that a viscoelastic flnid has two components related to y by Eq. 6.1 and y by Eq. 6.2. Erom Eq. 6.5, it is clear that for such dynamic oscillatory displacement, the measnred stress response has two components an in-phase component (sincot) and an ont-of-phase component (coscot). Viscoelastic materials prodnce this two-component stress response when they undergo mechanical deformation becanse some of the energy is stored elastically and some is dissipated or lost. The stress response, which is in-phase with the mechanical displacement, defines a storage or elastic modulus, G, and the out-of-phase stress response defines a loss or viscous modulus, G"". The storage modulus (G ) provides information about the fluid s elasticity and network structure. 

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy is to perform a thought experiment in which a step function in strain is applied, e t) = Cq H t), where H t) is the Heaviside step function, and the stress response a t) is measured. Then a stress relaxation modulus can be defined by E t) = <7(t)/ o Note that does not have to be infinitesimal due to the assumed superposition principle. To develop a model capable of predicting the stress response from an arbitrary strain history, start by decomposing the strain history into a sum of infinitesimal strain increments ... 

Recent studies show that nonlinear stress response can be represented in terms of a time-dependent phase angle difference 8 between the nonlinear elastic stress and viscoelastic stress. This approach appears to have an advantage over the numerical treatments used by previous workers because it provides some information about the reversible structural changes which occur during the cycle. However, such analyses require instruments which can subject the specimen simultaneously to two cyclic deformations of different amplitude and frequency. 

When Ay is very small (Ay < 0.1%), the stress response caused by the sinusoidal straining given by Equation 1 is approximately sinusoidal, and the viscoelastic behavior falls in the region of linear viscoelasticity. In this case, the phase angle difference 8 between the stress wave and strain wave is constant throughout the cycle, and the stress response can be expressed by ... 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

While considering tendons and ligaments as simple nonlinear elastic elements (Table 48.6) are often sufficient, additional accuracy can be obtained by incorporating viscous damping. The quasi-hnear viscoelastic approach [Fung, 1981] introduces a stress relaxation function, G(t), that depends only on time, is convoluted with the elastic response, T (A,), that depends only on the stretch ratio, to yield the complete stress response, K X, t). To obtain the stress at any point in time requires that the contribution of all preceding deformations be assessed ... 

The Stress relaxation experiment consists of applying a constant relative strain. Yii. and recording the shear. stress response, as shown in Fig. 12. Typical relaxation curves for elastic solids, viscoelastic solids or liquids, and Newtonian liquids are shown in the latter figure. The clastic solid is able to store energy and, in consequence, can maintain deformation and does not relax under the applied strain. The other extreme is the Newtonian flow that relaxes completely and flows. 

Miehe, C. and Keck, J. (2000) Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled mbbety polymers. Experiments, modeling and algorithmic implementation. Journal of the Mechanics and Physics of Solids, 48, 323-365. 

It is known that a viscoelastic fluid, e.g., a solution with a trace amount of highly deformable polymers, can lead to elastic flow instability at Reynolds number well below the transition number (Re 2,000) for turbulence flow. Such chaotic flow behavior has been referred to as elastic turbulence by Tordella [2]. Indeed, the proper characterization of viscoelastic flows requires an additional nondimensional parameter, namely, the Deborah number, De, which is the ratio of elastic to viscous forces. Viscoelastic fluids, which are non-Newtonian fluids, have a complex internal microstructure which can lead to counterintuitive flow and stress responses. The properties of these complex fluids can be varied through the length scales and timescales of the associated flows [3]. Typically the elastic stress, by shear and/or elongational strains, experienced by these fluids will not immediately become zero with the cessation of fluid motion and driving forces, but will decay with a characteristic time due to its elasticity. 

This eifect was investigated for the viscoelastic/viscoplastic response using a test program where the preload was applied at varying stress levels. Variation of viseoplastie strain was observed with change in preloading mode. ... 

The above properties are static physical properties which are determined with a linearly increasing applied force. Polymeric materials, including structural adhesives, have another important set of physical properties due to the fact that these materials behave in a manner that is not only elastic, but also viscoelastic in response to an applied stress. Viscous response may be treated by means of the linear constitutive equation formalism. Thus for a polymeric body, following FerryEq. (14) may be written ... 

Also note that the hydrostatic pressure is indeterminate because the K-BKZ is an incompressible material model. As in finite elasticity theory, the material parameters need to be obtained and, in principle, the stress response to any deformation history can be obtained. Unlike linear viscoelasticity, the integration must be carried out from —00 to t, which can lead to difficulties in numerical computer codes. This aspect of the K-BKZ theory has been discussed by (62) Larson, among others. 

G. B. McKenna and L. J. Zapas, The Normal Stress Response in Nonlinear Viscoelastic Materials-Some Experimental Findings J. Rheol. 24, 367-377 (1980). 

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