Metals elastic modulus values

Bar chart of room-temperature stiffness (i.e., elastic modulus) values for various metals, ceramics, polymers, and composite materials. 

Currently, there is a trend of low dielectric constant (low-k) interlevel dielectrics materials to replace Si02 for better mechanical character, thermal stability, and thermal conductivity [37,63,64]. The lower the k value is, the softer the material is, and therefore, there will be a big difference between the elastic modulus of metal and that of the low-k material. The dehiscence between the surfaces of copper and low-k material, the deformation and the rupture of copper wire will take place during CMP as shown in Fig. 28 [65]. 

Most engineering materials, particularly metals, follow Hooke s law by which it is meant that they exhibit a linear relationship between elastic stress and strain. This linear relationship can be expressed as o = E where E is known as the modulus of elasticity. The value of E, which is given by the slope of the stress-strain plot, is a characteristic of the material being considered and changes from material to material. 

The tensile strength and elastic modulus of metals decrease with increasing temperature. For example, the tensile strength of mild steel (low carbon steel, C < 0.25 per cent) is 450 N/mm1 2 3 4 at 25°C falling to 210 at 500°C, and the value of Young s modulus... 

In the discrete lattice model, discussed above, each bond is identical, having identical threshold values for its failure. In the laboratory simulation experiments (discussed in the previous section) on metal foils to model such systems, holes of fixed size are punched on lattice sites and the bonds between these hole sites are cut randomly. If, however, the holes are punched at arbitrary points (unlike at the lattice sites as discussed before), one gets a Swiss-cheese model of continuum percolation. For linear responses like the elastic modulus Y or the conductivity E of such continuum disordered systems, there are considerable differences (Halperin et al 1985) and the corresponding exponent values for continuum percolation are higher compared to those of discrete lattice systems (see Section 1.2.1 (g)). We discuss here the corresponding difference (Chakrabarti et al 1988) for the fracture exponent Tf. It is seen that the fracture exponent Tf for continuum percolation is considerably higher than that Tf for lattice percolation Tf = Tf 4- (1 -h x)/2, where x = 3/2 and 5/2 in d = 2 and 3 respectively. 

The nanohardness of the films can be considered relatively high (10-14 GPa) below 200°C, tending to the value of metallic Ni (2 GPa) above 600°C. The elastic modulus, giving a maximum value at 200°C of 130 GPa, shows the same behavior as the nanohardness. 

The elastic behavior upon applied shear stress is primarily typical in the case of solids. The nature of elasticity is in the reversibility of small deformations of interatomic (or intermolecular) bonds. In the limit of small deformations the potential energy curve is approximated by a quadratic parabola, which corresponds to a linear t(y) dependence. Elasticity modulus of solids depends on the type of interactions. For molecular crystals it is 109 N m 2, while for metals and covalent crystals it is 1011 N m"2 or higher. The value of elasticity modulus is only weakly dependent (or nearly independent) on temperature. 

In Equations (5.33) and (5.34) the indices 1 and 2 refer to the upper and lower members, respectively, of a single-lap joint and s and p refer to the side and centre plates, respectively, of a double-lap joint. The lap thickness is denoted by t, elastic modulus of members and fasteners by E and Ef respectively, and fastener diameter by db. The empirical constants a and b depend on the material of the members and the type of fastener. For bolted metallic members (a=2/3, b=3.0), for riveted metallic members (a=2/5, b=2.2), and for bolted carbon FRF members (a=2/3, b=4.2). It is recommended that the values valid for carbon FRF members be used for glass FRF joint composites until specific data are available. 

The Y value of the as-prepared aerogels was found to be 9.6 x 10 and 2.6 x 10 times smaller than those of Iron and Indium, respectively. The Y values of these aerogels are closer to those of rubber when compared with the metals mentioned above. It is only 2.6 X 10 times smaller than the Y value of rubber which is quite rarely observed in case of silica aerogels. This improvement in elastic modulus of superhydrophobic flexible aerogels with those of native silica aerogels is roughly two orders of magnitude. 

Values of linear thermal expansion coefficients of commonly used non-metallic thin film, interlayer or substrate materials are given in Table 2.1 over a broad range of temperatures of practical interest. Table 2.2 provides corresponding values of linear thermal expansion coefficients for polycrys-talline metals. Table 2.3 lists the room temperature values of elastic modulus and Poisson s ratio for a wide variety of polycrystalline and amorphous materials with isotropic elastic properties, which are commonly used as thin films, interlayers or substrates. The anisotropic elastic properties of cubic and hexagonal single crystals are given in Tables 3.1 and 3.2, respectively, in the next chapter. 

A thin film of a polycrystalline fee metal with 111 fiber texture was deposited onto a relatively thick Si substrate. Nonsymmetric x-ray diffraction measurements showed that the (242) and (242) planes had d—spacing values of 0.8268 and 0.8274 A, respectively. The thin film material is known to be nearly isotropic. The elastic modulus and Poisson ratio of the film material are 70 GPa and 0.33, respectively. 

Elastic behavior is typical of solid bodies. The nature of elasticity is related to the reversibility of small deformations of interatomic bonds. Within the limits of small deformations, the potential energy curve can be approximated with a quadratic parabola, which conforms to a linear relationship between y and t. The magnitudes of the elastic modulus G depend on the nature of the interactions in the solid body. For molecular crystals, G is 10 N/m, while for metals and covalent crystals, the value of G is on the order of 10" N/m. The elastic modulus G reveals either a weak dependence from temperature or no temperature dependence at all. 

Calculated and experimental values of some thermomechanical parameters for refractory carbides are compared in Table 2.1. Best agreement has been obtained for the lattice constants (the difference from the experimental values being not more than 0.05 au). Other parameters derived from the calculations are much less accurate, but in most cases they reproduce well the trends observed in experimental data. The most important of these is the increase in chemical bonding strength, when going from iva subgroup metal carbides to the carbides of va and via metals. This follows from the increase of the elasticity modulus and the hydrostatic breakdown tension values in this direction. 

Fig. 27.2 provides a comparison of different plastics and metals with respect to yield stress and elastic modulus. The plotted values for the various types of plastic are median values of hundreds of grades while the plotted values for the metals correspond to Aluminum 6061-0, Titanium 6-4, tempered AISI... 

The microhardness technique is used when the specimen size is small or when a spatial map of the mechanical properties of the material within the micron range is required. Forces of 0.05-2 N are usually applied, yielding indentation depths in the micron range. While microhardness determined from the residual indentation is associated with the permanent plastic deformation induced in the material (see section on Basic Aspects of Indentation), microindentation testing can also provide information about the elastic properties. Indeed, the hardness to Young s modulus ratio HIE has been shown to be directly proportional to the relative depth recovery of the impression in ceramics and metals (2). Moreover, a correlation between the impression dimensions of a rhombus-based pyramidal indentation and the HIE ratio has been found for a wide variety of isotropic poljuneric materials (3). In oriented polymers, the extent of elastic recovery of the imprint along the fiber axis has been correlated to Young s modulus values (4). 

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