Shear modulus, dissipative

Fig. 17 (a) Storage shear modulus (G ) vs. temperature, (b) Dissipative shear modulus (G") of the mechanical p and a relaxations vs. temperature for P(VDF-TrFE)/Ni nanowires from 0 vol.% to 10 vol.% (reproduced with permission of Elsevier, A. Lonjon et al.. Journal of Non-Crystalline Solids [34])... 

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

Tan landa, a damping term, is a measure of the ratio of energy dissipated as heat to the maximum energy stored in the material during one cycle of oscillation. For small to medium amounts of damping. G is the same as the shear modulus measured by other methods at comparable time scales. The loss modulus G" is directly proportional to the heat H dissipated per... 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

A perfectly elastic solid subjected to a non-destructive shear force will deform almost instantaneously an amount proportional to its shear modulus and then deform no further, strain energy being stored in the bonds of the material. A fluid, on the other hand, continues to deform under the action of a shear stress, the energy imparted to the system being dissipated as flow. 

The concepts of inter-particle bonding, net work structure, and viscous dissipation, as well as texture maps should be applicable to all structured dispersions, such as cosmetics and other consumer products. The vane yield stress test is a versatile test in which a fluid food is subjected to small deformations during the initial stages and large deformations during the latter stages of the experiment. From the former set of linear data, a shear modulus (G) of the sample can be estimated. 

Rheological and elastic properties under flow and deformations are highly characteristic for many soft materials like complex fluids, pastes, sands, and gels, viz. soft (often metastable) solids of dissolved macromolecular constituents [1]. Shear deformations, which conserve volume but stretch material elements, often provide the simplest experimental route to investigate the materials. Moreover, solids and fluids respond in a characteristically different way to shear, the former elastically, the latter by flow. The former are characterized by a shear modulus Go, corresponding to a Hookian spring constant, the latter by a Newtonian viscosity r]o, which quantifies the dissipation. 

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. 

For RDX the Poisson ratio is approximated as v =. 5. The length d is the molecular spacing, d = 5.8 x 10 ° m, and the Burgers length, b = d. R is the radius of the dislocation core which is taken as R = 2d. The shear wave speed is Vo == 2 x 10 m/s and the density of RDX is p 1.8 x 10 kg/m. The nominal shear modulus for RDX is G = 4 GPa. Within the heavily deformed shear bands the lattice potential will be reduced which will reduce the shear modulus so that the calculation will over estimate the energy dissipation and temperature in the shear band. 

Equations 15 and 17 constitute a homogeneous system of 2N + 2 linear equations. A nontrivial solution for the set of amplitudes only exists if the determinant of this equation system vanishes. The search for the zeros of the determinant as a function of frequency will in general be carried out numerically. The zeros define the resonance frequencies. Since, for a real material, the shear modulus always contains a dissipative component, G", the resonance frequencies are complex (where the imaginary part is the halfband-half-width, r). 

The variables A/, /o, Aw, Apie o. and Fq represent, respectively, the crystal s measured frequency shift, initial resonant frequency, mass change, surface area, shear modulus, and density [6]. TSM resonators immersed in liquid present a special case because acoustic energy dissipation through the fluid lowers Q as well. Additional formulas need to be applied to account for the liquid s viscosity and loading [6]. Figure 2 shows a comparison between shifts induced by solid and hquid loading on resonators. 

For the volume fractions presented in Figs. (3-6) the shear modulus is on the order of lOdyn/cm and the sound velocity V(= /s/p) = l-5cm/s. The microscopic relaxation time T(-rj/E) 1-10 ms, and the attenuation length A.[= (ImK) =2F/0 t1 1-10cm. For frequencies below IkHz the dissipation is small and the shear waves are propagating. The dimensions of the measuring cell encourage the formation of standing waves. 

Fig. 3. Illustration of the changes in absolute shear modulus G and dissipation factor tan S with frequency at constant temperature for a simple amorphous polymer. Models show the changes in viscoelastic state...

We are now in a position to illushate the various regions of viscoelastic behaviour with an idealized curve of, for example, the magnitude of the absolute dynamic shear modulus G against frequency (log co) at a constant standard temperature (Fig. 3). On the same abscissa, the energy dissipation (tan 5) and applicable mechanical models are also shown. 

The QCM-D response to mass uptake on the crystal oscillator is reflected in the changes in both the resonant frequency (Afi and dissipation factor (AD) at different overtones. In contrast to OWLS, the QCM-D approach is sensitive to viscoelastic properties and the density of any mass coupled to the mechanical oscillation of the quartz crystal. In this case, the adsorbed mass consists of the PLL- -PEG copolymer along with solvent molecules associated with it. A Voigt-based model was therefore used in the analysis (software Q-tools, version 2.0.1), where the adsorbed layer was represented as a homogeneous, viscoelastic film characterized by shear viscosity ((/shear), shear modulus (Eshear), and film thickness (htum) (20—22). 

Because quartz crystals can vibrate with minimal energy dissipation, they can be used to build very stable oscillator circuits [26]. The quartz oscillator has a strong preference to vibrate at a characteristic resonant frequency (fo)> which depends on the shear modulus density (p ), and the thickness of the quartz (t,). 

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