Voigt elastic stiffnesses

Because of the asumed transverse isotropy it follows that = 1/2 (Cu - Cn). The terms Qj are the elastic stiffnesses expressed in the contracted (Voigt) notation. 

By using the relations between the elastic-stiffness coefficients in the cubic class from Table 10.3 in Eq. 10.19, the Voigt approximation of the Young s modulus is obtained for a material with cubic symmetry ... 

Using the relations between the elastic-stiffness coefficients from Table 10.3 in Eqs. 10.20 and 10.26, one may also derive the Voigt and Reuss approximations for the rigidity modulus of a cubic monocrystal. These are given by Eqs. 10.35 and 10.36, respectively ... 

Use the following values of the elastic-stiffness constants and the elastic-comphance constants (Kisi and Howard, 1998) for tetragonal zirconia monocrystals to determine the Voigt-Reuss-HiU averages for the Young s modulus, E, the shear modulus, G, and the bulk modulus, B. 

Figure 14.5 Draw ratio dependence of the elastic stiffnesses of Vectra B. The solid and dashed curves denote, respectively, the Voigt and Reuss bounds calculated according to the aggregate model. (Adapted from [10] by permission of the Society of Plastic Engineers.)...

Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

Many ceramics are used in a random polycrystalline form and thus, it is useful to be able to predict the elastic constants from those of the single crystals. The approaches outlined in the last two sections are used for this procedure by considering the random polycrystal as an infinite number of phases with all possible orientations. For example, Voigt and Reuss used a technique based on averaging the stiffness or compliance constants and obtained upper and lower bounds. The Voigt upper bounds for the bulk (B) and shear (/i) moduli of the composite can be written as... 

The stiffness of a DMO composite of AI2O3-AI with 22% alloy and 4% porosity (231 GPa) was modeled successfully [60] by assuming the metal and the ceramic skeleton to deform equally in series and in parallel, that is, by taking a Reuss-Voigt average. The additional effect of isolated pores could be included by using empirical expressions derived from data on porous alumina. The elastic modulus... 

For an isotropic aggregate, the stiffness averaging procedure had been proposed by Voigt, ° and the compliance averaging procedure by Reuss, many years previously. Each had been used to compare the elastic constants of single crystals with those of an isotropic aggregate of single crystals (see for example Ref. 12). 

The results of such calculations for semi-crystalline polyethylene have been reviewed elsewhere [37]. A rather wide range of predicted values is obtained, due to the choice of force constants and also to sensitivity to detailed assumptions on the unit cell structure. In spite of these limitations the principal predictions for the elastic anisotropy are clear. These include the anticipated high values for C33 and the very low values for the shear stiffnesses C44, C55 and cee, which reflect the major differences between bond stretching and bond bending forces that control C33 and the intermolecular dispersion forces that determine the shear stiffnesses. It is therefore of value to compare such theoretical results with those obtained experimentally. Table 7.3 shows results for polyethylene where data for the orthorhombic unit cell at 300 K are used to calculate these constants for an equivalent fibre (Voigt averaging procedure see Section 7.5.2 below) compared with ultrasonic data for a solid sheet made by hot compaction. It can be seen that... 

Finally, it is of interest to compare the theoretical values for a uniaxially oriented sheet (calculated by averaging the stiffness values using the Voigt averaging scheme) with those obtained for a die-drawn rod and a sheet made by hot compaction of high modulus polyethylene fibres (Table 8.4). It can be seen that although, as expected, these materials have not reached full axial orientation so that the experimental values of C33 are much less than the theoretical value, the patterns of anisotropy are very similar, and some of the values for the other elastic constants are surprisingly close. 

Voigt [Voigt 1889, 1910] has shown that, under the assumption that the strain inside the material is uniform (isostrain assumption), the effective elastic moduli of a dense (i.e. pore-free) polycrystalline material, e g. a densely sintered ceramic, composed of crystallites of arbitrary symmetry can be ealeulated from the 9 elastic constants (stiffnesses) C, C22, C33,... 

Table 7 lists the Voigt boimds (subscript V), Reuss boimds (subscript R) and VRH averages of the effective elastic moduli of polyciystalline alumina and tetragonal zirconia (t-Zr02), calculated from the components of the respective stiffness and compliance matrices due to [Wachtman et al. 1960, Kisi Howard 1998], listed in Table 2. For the effective Poisson ratio, for which the Voigt-Reuss boimds do not hold, only a range of values is given, cf. the discussion in [Pabst et al. 2004a].

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