Shear and Young’s modulus

In the active probe approach, SFM acquires both static and dynamic mechanical properties (Sect. 2.2.2). The former includes the shear and Young s modulus (G,E) as well as the surface indentation and contact area (S,a). Dynamic meas-... 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

Tests by Roe et al. [63] with unidirectional jute fiber-reinforced UP resins show a linear relationship (analogous to the linear mixing rule) between the volume content of fiber and Young s modulus and tensile strength of the composite over a range of fiber content of 0-60%. Similar results are attained for the work of fracture and for the interlaminate shear strength (Fig. 20). Chawla et al. [64] found similar results for the flexural properties of jute fiber-UP composites. 

Fig. 7.11. Normalized interface shear stress distributions along the fiber length for composites with and without PVAL coating coating thickness t = 5 pm and Young s modulus ratio of coating to matrix...

Note 3 The Lame constant, (/), is related to the shear modulus (G) and Young s modulus (E) by the equation... 

Note 4 Loose ends and ring structures reduce the concentration of elastically active network chains and result in the shear modulus and Young s modulus of the rubbery networks being less than the values expected for a perfect network structure. 

The fundamental shear and Young s moduli are the slopes of the shear and tension/compression stress/strain curves at the origin. The relationships given above are an attempt, with theoretical justification, to describe the shapes of the stress/strain curves at higher strains. Appreciation of this may avoid confusion between the absence of a single modulus figure for rubber whilst such values are quoted. 

Dynamic mechanical characteristics, mostly in the form of the temperature response of shear or Young s modulus and mechanical loss, have been used with considerable success for the analysis of multiphase polymer systems. In many cases, however, the results were evaluated rather qualitatively. One purpose of this report is to demonstrate that it is possible to get quantitative information on phase volumes and phase structure by using existing theories of elastic moduli of composite materials. Furthermore, some special anomalies of the dynamic mechanical behavior of two-phase systems having a rubbery phase dispersed within a rigid matrix are discussed these anomalies arise from the energy distribution and from mechanical interactions between the phases. 

The temperature dependence of the elastic constants of polycrystalline TiAl was studied (Schafrik, 1977), and Young s modulus = 174 GPa, shear modulus = 70 GPa, and Poisson s ratio = 0.23 were found for stoichiometric TiAl at room temperature, i.e. the elastic constants are larger and Poisson s ratio is smaller than for TijAl. Elastic constants and fault energies of TiAl have been studied theoretically by first-principles total-energy investigations and have been discussed with respect to mechanical behavior (Yoo et al., 1991 Lee and Yoo, 1990 Fu and Yoo, 1990 Yoo, 1991 Woodward etal., 1993 Yoo and Fu, 1991, 1993). Young s moduli for... 

The modulus is nearly the same, whether the composite contains rubber spheres or spherical voids, since the rubber shear and Young s moduli are... 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

Stiffness and energy dissipative characteristics of the soil as often represented by secant shear (or Young s) modulus and viscous or Coulomb damping. 

Since G is known from the shear wave velocity measurement, K can be calculated. With K and G known, both Poisson s ratio (i ) and Young s modulus (E) can be calculated using appropriate relations for isotropic elastic constants. Two such relations, one for v and one for E, are given below ... 

The shear modulus and Young s modulus are related as follows... 

For an isotropic medium such as polysilicon, the shear modulus G and Young s modulus E are related by... 

The models developed here are visualized in tension, with tensile stress cr, tensile strain s, and Young s modulus E. However, the same theory holds true for pure shear (viscometric) deformation, where a shear stress t results in a shear strain y with proportionality constant G (Hooke s modulus), rj represents the Newtonian (shear) viscosity, while the elongational (Trouton) viscosity is given by rjg. 

A measurement of the size dependence of the hardness, shear stress, and elastic modulus of copper nanoparticles, as shown in Fig. 28.8a, confirmed this expectation. The shear stress and elastic modulus of Cu reduce monotonically with the solid size but the hardness shows the strong IHPR. Therefore, the extrinsic factors become dominant in the plastic deformation of nanocrystals, which triggers the HPR and IHPR being actually a response to the contacting detection. However, as compared in Fig. 28.8b and c, the hardness and Young s modulus of Ni films are linearly interdependent. This observation indicates that extrinsic factors dominating the IHPR of nanograins contribute little to the nanoindentation test of film materials. 

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