Hooke cubic crystal

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

A body will obey Young s modulus only if it is stretched or compressed within its elastic limit if this limit is exceeded, structural failure ensues. For a one-dimensional system, or for a cubic crystal, Young s modulus reduces to the Hooke s law constant kH ... 

Let us consider cubic crystals in some more detail. From Table 2.3, it is found that three elastic constants are needed for cubic crystals, c, c,2and c. This is a result of the three independent modes of deformation in this crystal system. The first mode is dilatation by a hydrostatic stress (o-=cT =a-2=o-3). For this case, using Eq. (2.53) and Table 2.3, Hooke s Law for cr, is given by... 

As with other anisotropic materials, Hooke s Law for cubic crystals may be expressed in terms of the engineering elastic constants. Equation (2.57) can be written as... 

In order to describe the behavior of an anisotropic material. Hook s law can be written in completely general terms [42]. This requires that six strain components and 36 elastic coefficients be known in order to calculate the stress. S)mmietry considerations are such that even for the least S5Tnmetrical crystal structure, trichnic, the number of independent elastic constants required is reduced to 21. For metals, whose crystal systems all exhibit relatively high degrees of symmetry, the number of constants is further reduced to 12. Anisotropic materials such as magnesium, zinc, and tin require at least five elastic constants. Materials with cubic crystal structures require three independent elastic constants, while a truly isotropic material requires only two elastic constants. 

Another important special case of a homogeneous deformation (i.e. y is positionally constant) in which now, however, pressure anisotropies are effective, is the uniaxially stressed cubic crystal. Let us assume this time that there is tensile stress in the x-direction. There it holds for small effects (i.e. Hooke s law fulfilled) that s = P. dcxx/dsxx const = xx/ xx nnd also dsyy/dsxx — ds z/dsxx yy/sxx zz/ xx const. 

Regarding the historic explanations, the students were neither convinced by the idea of hooks and eyes (Lenkipp, Demokrit), nor by the idea of cubic particles (Hairy). However, the historic arguments were used by the students to discuss their own ideas of bonds between the particles and to discuss the cubic shape of some crystals. 

Rizwan, S.B. Boyd, B.J. Rades, T. Hook, S. Bicontinuous cubic liquid crystals as sustained delivery systems for peptides and proteins. Expert Opin. Drug Deliv. 2010, 7 (10), 1133—1144. 

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