Oscillatory deformation viscoelasticity

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

A linear viscoelastic constitutive model of dilute emulsion viscoelastic properties was proposed by Oldroyd [111, 112]. The model considered low deformation of monodispersed drops of one Newtonian liquid in another, with an interphase. Choi and Schowalter [113] extended their cell model to dilute emulsions with Newtonian matrix and viscoelastic drops under infinitesimally small oscillatory deformation. Oldroyd s model was modified by Palierne [126, 127] for dilute viscoelastic hquids emulsions with polydispersed spherical drops (thus, subject to small deformations) with constant interfacial tension coefficient, Vu, at concentrations below that where the drop-drop interactions start complicating the flow field, that is, < 0.1 ... 

While the Choi and Schowalter [113] theory is fundamental in understanding the rheological behavior of Newtonian emulsions under steady-state flow, the Palierne equation [126], Eq. (2.23), and its numerous modifleations is the preferred model for the dynamic behavior of viscoelastic liquids under small oscillatory deformation. Thus, the linear viscoelastic behavior of such blends as PS with PMMA, PDMS with PEG, and PS with PEMA (poly(ethyl methacrylate))at <0.15 followed Palierne s equation [129]. From the single model parameter, R = R/vu, the extracted interfacial tension coefficient was in good agreement with the value measured directly. However, the theory (developed for dilute emulsions) fails at concentrations above the percolation limit, 0 > (p rc 0.19 0.09. 

Dynamic mechanical methods (typically oscillatory parallel plate rheometry) are commonly used to measure the dynamic mechanical properties from the liquid state to the solid state. By using small-amplitude oscillatory deformations (linear viscoelastic regime), the dynamic storage and loss moduli can be obtained. From these quantities, the viscosity and modulus can be calculated (71) (see Dynamic Mechanical Analysis). 

An extensional deformation is also known as a tensile deformation. If an oscillatory deformation is applied to a viscoelastic material, a complex modulus with components E and E" can be defined, in analogy to the dynamic shear moduli G and G" (cf. Eq. 1.37). 

In the rheological structure of most food systems there is a viscous element present, and the deformation curves are often highly influenced by the rate of the imposed strain. This is due to the fact that the material relaxes (or flows) while tested under compression and the resultant deformation of this flow is dependent on the nature of the viscous element (Szczesniak, 1963 Peleg and Bagley, 1983). In the viscoelastic food systems, where during processing it is caused to oscillate sinusoidally, the strain curve may or may not be a sine wave. In cases when a periodic oscillatory strain is applied on a food system like fluid material, oscillating stress can be observed. The ideal elastic solid produces a shear stress wave in phase with... 

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. 

The effects of secondary aggregation of small particle carbon blacks on the elastic modulus at small strains are large. They have been studied primarily in dynamic oscillatory loading experiments and are discussed in Section VII, dealing with viscoelastic behavior. The effects of prior deformation on stress-strain relationships (stress softening) are also time-dependent phenomena, consideration of which is postponed to a later point in this review. 

Concentrated suspensions commonly display viscoelatic behavior. The viscoelastic properties can be measured by oscillatory tests (26). Comparing with steady shear measurements, oscillatory measurements are made under small deformations, at which the suspension structure is only slightly perturbed. Hence, oscillatory measurements are suitable for correlating rheological behavior with structural data and interparticle potentials, even for strongly flocculated systems that show irreversible changes when subjected to large deformations. 

DMA is based on the viseoelastie response of a material subjected to a small oscillatory strain imposed by flexural bending, but can also have the capability of other deformations such as shear. The viscoelasticity of the material is separated into the two components of modulus, E comprising a real part, the elastic modulus E ) and an imaginary part, which is the damping or viscous component ( ") ... 

DMA measures the viscoelastic properties of a sample using either transient or dynamic oscillatory tests. Transient tests include creep and stress relaxation. In creep, a stress is applied to the sample and held constant, while deformation is measured versus time. After some time, the stress is removed and the recovery measured. In stress relaxation, a deformation is applied to the sample and held constant the degradation of the stress required to maintain the deformation is measured versus time. The most common test is the dynamic oscillatory test, where a sinusoidal stress (or strain) is applied to the material and the resultant sinusoidal strain (or stress) is measured. Also measured is the phase difference, 8, between the two sine waves. The phase lag will be 0° for a purely elastic material and 90° for a purely viscous material. Viscoelastic materials such as polymers will exhibit an intermediate phase difference. Since modulus equals stress divided by strain, the complex modulus, E, can be calculated. From E and 8, the storage modulus, E, the loss modulus, E", and tan 8 can be calculated ... 

---