Complex Young modulus

Fig. 13. a Complex piezoelectric strain constant (20 Hz), b Complex Youngs modulus (10 Hz, strained along 45° direction to draw-axis), and complex dielectric constant (1 kHz) of uniaxially drawn poly(y-methyl L-glutamate) film (a-helical form) plotted against temperature. Draw-ratio = 2. Drawn after Date, Takashita, and Fukada [J. Polymer Sd. A-2,8,61 (1970)] by permission of John Wiley Sons,... 

The complex Young modulus ( ) consisting of contributions of a storage modulus ), and of a loss modulus E"), is measured (eq. 3.1). The complex modulus reflects the inherent viscoelastic nature of the polymer, where stress and strains will be out of phase with one another. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Thus, FTMA determines complex modulus as the transfer function between input strain and output stress. A prerequisite is that the Fourier transform of y(t) must exist. White no se should suffice since it contains all frequencies. Note that G (jw) in Equation 10 will be the complex Young s modulus if a(t) and 7(t) are the normal stress and normal strain, respectively and the complex shear modulus if they are the shear stress and shear strain. 

E oo is the in-phase complex Young s modulus at infinite frequency or low temperature. 

E Complex Young s modulus, with real (elastic) and imaginary (viscous) components. 

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. 

Figure 9 shows the temperature dependence of the elastic part of the complex Young s modulus E a>) for various poly(vinyl methyl ether) hydrogel samples in water at a selected frequency of 20.1 rad/s. The polymer was cross-linked by electron beam irradiation (see Sect. 2.4, chapter Synthesis of hydrogels ). These data were compared with the temperature shrinking behaviour of sample PVME 20/80... 

If we consider the bone being driven by a strain at a frequency u, with a corresponding sinusoidal stress lagging by an angle 5, then the complex Young s modulus E (a>) may be expressed as... 

Complex dielectric constant n. Vectorial sum of the dielectric constant and the loss factor analogous to complex shear modulus and to complex Young s modulus. 

Complex Young s modulus Vectorial sum of Young s modulus and the loss modulus analogous to the complex dielectric constant. 

A suitable property to describe the mechanical behaviour of a pol3mier under time-dependent enforcement is the complex Young s modulus. For a polymer, which, in general, is composed of crystalline and amorphous parts (for a comprehensive description of polymer... 

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. 

A comprehensive review of measurement techniques is presented by Capps (167), who also gives data for the complex Young s modulus for a range of polymers. This data includes the rubbery, transition, and glassy regions, and parameters for time-temperature superposition (eq. 45). The measiuement techniques fall broadly into three categories wave propagation methods, resonance methods, and forced-vibration nonresonance methods. The resonance and forced-vibration... 

Fig. 7. Schematic diagram for the single cantilever beam apparatus for measurement of the complex Young s modulus. (A forced vibration nonresonance method.)...

The quantities E and G refer to quasistatic measurements. When cyclical or repetitive motions of stress and strain are involved, it is more convenient to talk about dynamic mechanical moduli. The complex Young s modulus has the formal definition... 

All the dynamic measurements thus far mentioned furnish the complex Young s modulus in the fiber direction. The shear modulus at right angles can be measured by torsional vibrations of the fiber itself, using a light crossbar or disc to provide the amount of inertia for a compound oscillating system in forced or free vibration. An example is shown in Rg. 7-7, where the forced vibrations are driven by an electrostatic device. Here the analog of equation 13 of Chapter 6 is ... 

Young s (tensile) modulus [Pa] tensile storage modulus [Pa] tensile loss modulus [Pa] complex tensile modulus [Pa] stiffness tensor [Pa] force [N]... 

A complex Young s modulus ( ) reflects the contribution of both storage ( ) and loss ( ") components to the stiffness of material, as follows ... 

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