Elastic, defined Loss modulus

The point at which the elastic and loss modulus intersects is defined as the gel... 

The storage modulus is proportional to the amount of energy which is stored in the material elastically, whereas the loss modulus corresponds to the energy that is dissipated during one load cycle. Both quantities are combined in the damping factor tan 8 which is defined as... 

It is necessary to state more precisely and to clarify the use of the term nonlinear dynamical behavior of filled rubbers. This property should not be confused with the fact that rubbers are highly non-linear elastic materials under static conditions as seen in the typical stress-strain curves. The use of linear viscoelastic parameters, G and G", to describe the behavior of dynamic amplitude dependent rubbers maybe considered paradoxical in itself, because storage and loss modulus are defined only in terms of linear behavior. 

Through use of classical network theories of macromolecules, G has been shown to be proportional to crosslink density by G = nKT -i- Gen, where n is the nnmber density of crosslinkers, K is the Boltzmann s constant, T is the absolnte temperature, and Gen is the contribution to the modulus because of polymer chain entanglement (Knoll and Prud Homme, 1987). The loss modulus (G") gives information abont the viscous properties of the fluid. The stress response for a viscous Newtonian fluid would be 90 degrees out-of-phase with the displacement but in-phase with the shear rate. So, for an elastic material, all the information is in the storage modulus, G, and for a viscous material, aU the information is in the loss modulus, G". Refer to Eigure 6.2, the dynamic viscosities p and iT are defined as... 

These differences are shown in the following examples where measurements of the dynamic moduli, G and G" are used to monitor the structure of gel networks. Measurements are performed by imposing an oscillatory shear field on the material and measuring the oscillatory stress response. The stress is decomposed into a component in phase with the displacement (which defines the storage modulus G ) and a component 90 out of phase (which defines the loss modulus G"). The value of G indicates the elastic and network structure in the system (15, 17, 18) and can be interpreted by using polymer kinetic theories. 

These are most easily represented by the equation E = E + iE". where E is the ratio between (the amplitude of the in-phase stress component strain, a/e) and E" is the loss modulus (the amplitude of the out-of-phase component. strain amplitude). Similarly for (7 and K and the ratio between the Young s modulus E and the shear modulus C includes Poisson s ratio u, for an isotropic linear elastic solid with a uniaxial stress. (Poisson s ratio is more correctly defined as minus the ratio of the perpendicular. strain to the plane strain, or for one orthogonal direction 22 which equals the. 3.3 strain if the sample is... 

The quantities E and G refer to quasi-static measurements. When cyclic motions of stress and strain are involved, it is more convenient to use dynamical mechanical moduli. The complex Young s modulus is then defined as = " + iE", where E is the storage modulus and " the loss modulus. The storage modulus is a measure of the energy stored elastically during deformation the loss modulus is a measure of the energy converted to heat. Similar definitions hold for G, J, and other mechanical properties. 

The prime notation G and G" is conventional and does not mean differentiation. G (ft>) is known as the storage modulus, and G" co) is known as the loss modulus. The nomenclature follows from our understanding of classical materials. A linear elastic material (a Hookean solid) is a material for which the stress is proportional to the strain, and the deformation is completely recoverable that is, the energy required for displacement is stored elastically, and the body returns to its undeformed shape when the stress is removed. The stress for a linear elastic material will be in phase with the strain in an oscillatory experiment. Hence, G defines the magnitude of an elastic response to a deformation. A linear viscous material (a Newtonian liquid) is a material for which the stress is proportional to the strain rate, and the deformation is completely nonrecoverable that is, the energy required for displacement is dissipated, and the body remains deformed when the stress is removed. The stress for a linear viscous material will be in phase with the strain rate, or 90° out of phase with the strain. 

A number of physical parameters can be measured as a way to characterize polymers. The specific tests that are done are normally dictated by the end use of a particular part. We have defined many of these parameters in the previous few chapters, such as elastic (or Young s) modulus, E, shear (or Hooke s) modulus, G, storage modulus (E or G ), loss modulus (E or G ), tan 6, which all apply to solids, as well as the viscosity used for polymer melts (and solutions) as described with polymer rheology. In addition to these terms, you may also run across hardness (how resistant the polymer is to penetration by a needle), toughness... 

Sharp increases in both the storage modulus (G ) and loss modulus (G") of PAL-PLX-PAL aqueous solutions were observed as the temperature increased (Fig. 11). G and G" are an elastic component and a viscous component of the complex modulus of a system, respectively. When G is greater than G", the system is considered to be a gel, and the crossover point was defined as the sol-to-gel transition temperature. The sol-to-gel transition temperatures defined by the test tube inverting method coincided with those defined by dynanfic mechanical analysis of G and G" within 2 to 3 °C. By varying the polymer concenlration, not only sol-to-gel transition temperature but also modulus of the gel could be controlled. The control of gel modulus (G ) has a significant effect on 3D cell culture as well as the differentiation of the stem cell. In the case of chondrocytes, the modulus of 300-2,500 Pa showed a cytocompatible microenvironment for proliferation of the cells. The gel prepared from 10.0 wt% aqueous solution of PAL-PLX-PAL formed a gel with a G of 380 Pa at 37 °C, thus being recommendable as a 3D culture matrix for chondrocytes. 

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

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. 

E, G, B and v are functions of both the temperature and frequency (rate) of measurement. They are often treated as complex (dynamic) properties. The real portion quantifies the energy which is reversibly stored by the "elastic" component of the deformation. The imaginary portion quantifies the energy lost ( dissipated ) by the "viscous" component of the deformation. For example, equations 11.8 and 11.9 define the complex Young s modulus E, its real and imaginary components E and E", and the mechanical loss tangent tan g under uniaxial tension. 

Equations 11.8 and 11.9 are isomorphous to equations 9.9 and 9.10 which define the storage and loss components of the complex dielectric constant . Similar equations are also used to define the complex bulk modulus B, the complex shear modulus G, and the complex Poisson s ratio v, in terms of their elastic and viscous components. The physical mechanism giving rise to the viscous portion of the mechanical properties is often called "damping" or "internal friction". It has important implications for the performance of materials [8-15],... 

The theoretical and practical significance of polymer dynamic/ mechanical properties is well appreciated [180,183]. The dynamic elastic modulus and mechanical losses are the most sensitive indicators for all forms of polymer molecular mobility, especially for the glasslike state. The dynamic elastic modulus is the primary index for polymer deformation properties. Apart from their purely theoretical interest, understanding of the mechanisms of molecular motion within polymers and of mechanical losses may have significant practical importance in properly defining other polymer mechanical properties. 

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