Effective Young’s modulus

Young s modulus in the second in-plane principal direction (= t, transverse Young s modulus) effective Young s modulus... 

Consider a sphere of diameter D of material 1 in contact with a planar surface of material 2. The reaction of the load P is determined by an effective Young s modulus E, which is defined as... 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

On the basis of what has been discussed, we are in the position to provide a unified understanding and approach to the composite elastic modulus, yield stress, and stress-strain curve of polymers dispersed with particles in uniaxial compression. The interaction between filler particles is treated by a mean field analysis, and the system as a whole is macroscopically homogeneous. Effective Young s modulus (JE0) of the composite is given by [44]... 

Fig. 25. Effective Young s modulus for silica filled epoxy 46]. Circles are experimental data [45]...

According to Reuss, the effective Young s modulus of an elastically isotropic solid is given by ... 

Here E is an effective Young s modulus of the polymer (E = E for plane stress and E/(l — f) for plane strain v = Poisson s ratio) and a is the half length of the craze. The craze displacement profile is computed from... 

For a bilayer of total thickness 50 A the effective Young s modulus is of order 4x 10 dyn cm when an interfacial energy ofy 50 erg cm is taken. Experimentally, it is only possible to measure 7/q. Though experimental determinations of Y vary greatly, Requena et al. have recently argued that the black lipid membranes have a value of Y close to 2 x 10 dyn cm-. The theoretical value of Yto 200 dyn cm may also be compared with measured values in the range 51-357 dyn cm for red cell membranes. ... 

Figure 4 Brice model fits to 1+, 2+, and 3+ cation partitioning between plagioclase (Ang9) and silicate melt in the system diopside-albite-anorthite (Blundy and Wood, 1994). Note that the best-fit curve becomes tighter as the charge on the cation increases from 1 + through 3+. This is due to increase in effective Young s modulus of the site with increasing charge (see text).

Epoxy-amine liquid prepolymers are extensively appHed to metallic substrates and cured to obtain painted materials or adhesively bonded stractures. Overall performances of such systems depend on the interphase created between the organic layer and the substrates. When epoxy-amine Hquid mixtures are appHed to a more or hydrated metaUic oxide layer (such as Al, Ti, Sn, Zn, Fe, Cr, Cu, Ag, Ni, Mg, or E-glass), amine chemical sorption concomitant with metaUic surface dissolution appear, leading to the organometaUic complex or chelate formation [1, 2]. Furthermore, when the solubility product is exceeded, organometaUic complexes may crystaUize. These crystals induce changes of mechanical properties (effective Young s modulus, residual stresses, practical adhesion, durability, etc.). 

Fig. 8.4. The effective Young s modulus E = i/su as a function of stress orientation. Units are 10 dyne cm . ...

Twisting the cords reduces the effective Young s modulus further. McKenna et al. (2004) analysed twisted yam or strand. Each fibre follows a helical path of constant pitch and constant radius r. The heHx angle 9 increases from zero at r = 0 to a maximum of a at the outer radius. The Young s modulus of the strand depends on the average value of cos 9. For the twisted strand, the effective Yormg s modulus depends on cos a, with a typical value of 0.41 when a = 50°. For the rope core, in which both the... 

Tanpichai, S., Quero, F., Nogi, M., Yano, H., Young, R.J., Lindstrom, T., et al., 2012. Effective Young s modulus of bacterial and microfibrillated cellulose fibrils in fibrous networks. Biomacromolecules 13 (5), 1340—1349. 

Here Y, I, C, pm, and A denote the effective Young s modulus, the area moment of inertia, the internal damping ratio, the density, and the cross-sectional area of the IPMC beam, respectively, and f z, t) is the distributed force density acting on the beam. 

Figure 13.15 Tip displacement as a function of applied voltage for a Nafion -based IPMC. The effective Young s modulus is 0.5 GPa. The actuator size is 5 mm X 0.2 mm x 20 mm.

Fig.6. Hardness H, effective Young s modulus E, elastic recovery We and H/E ratio of the Zr-Al-0 films with low ratio Zr/All sputtered on Si(lOO) substrate at Us=Ufi, Ts=500°C, Pt=1 Pa, and and 2 A as a function of partial pressure of oxygen po2-MM, TM and OM are the metallic, transition and oxide modes of sputtering, respectively [21].

Bertolazzi et al. have recently been reported similar measurements of the stiffness and strength of monolayers of M0S2. They found the material to have an effective Young s modulus of 270 100 GPa with an average... 

A. Mathur, J. Erlebacher, Size dependence of effective Young s modulus of nanoporous gold. Appl. Phys. Lett. 90(6), 061910 (2007)... 

Adams and Coppendale (1976, 1977) used a dynamic technique, for the axial butt joint, using the adhesive layer as a spring in an axially vibrating system. By comparing the frequency reduction when having an adhesive layer instead of a solid bar, they were able to calculate E, the effective Young s modulus. They then tested the same bars in torsional oscillation and obtained the shear modulus G by the same frequency change technique. From E and G, it is possible to calculate Poisson s ratio v, and hence to obtain E. 

---