Generalized Hookes Law

The generalized Hooke s law assumes that each of the nine components of the stress tensor is linearly related to each of the nine components of the strain tensor and vice versa. For example 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows  

Cllll ll+ C1211S12+ Ci3ii8i3 + O = C2111E21+ C2211 22+ e23n 23 + 

Clearly, similar relations may be written for other stress components. Succinctly, Eq. (1.66) may be presented as  

Note that Cyti = Cjiu = cyi = c uj. .. etc. for the other possible combinations. 

Furthermore, instead of the y s, the shear components are often given as 84, 5 and sg. Thus, one can write Eq. (1.70) compactly by replacing the y s with s  

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