Elastic Moduli Converter E G K

For a homogeneous, isotropic, linearly elastic solid the entire stress-strain response is fixed by exactly two independent constants, so any pair among Young's modulus E, shear modulus G, bulk modulus K and Poisson's ratio nu determines the others. E measures resistance to uniaxial stretching, G resistance to shape change at constant volume, K resistance to uniform compression, and nu the lateral contraction accompanying axial extension. The identities E = 2G(1 + nu) = 3K(1 - 2nu) = 9KG/(3K + G) follow from splitting any strain into a volumetric part governed by K and a deviatoric part governed by G. The first Lame parameter lambda = K - 2G/3 and the P-wave modulus M = K + 4G/3 are the forms seismologists prefer, because Vp = sqrt(M/rho) and Vs = sqrt(G/rho). Stability requires K > 0 and G > 0, confining nu between -1 and 0.5; at nu = 0.5 the material is incompressible and the conversion to K diverges. Representative values are 200 GPa at nu = 0.29 for steel, 70 GPa at 0.33 for aluminium, 40 to 70 GPa at about 0.25 for granite, 10 to 40 GPa for sandstone and 5 to 30 GPa for shale. A Vp/Vs ratio of sqrt(3), roughly 1.732, corresponds exactly to nu = 0.25, the Poisson solid used as a seismological reference. The conversions assume isotropy, so they are only approximate for bedded shale or foliated gneiss, where stiffness varies with direction by a factor of two or more, and for fluid-saturated rock, where Gassmann fluid substitution is required instead.

Thomas Young described the modulus that carries his name in his 1807 Course of Lectures on Natural Philosophy, though Giordano Riccati had measured an equivalent quantity in 1782 and Leonhard Euler had used the idea earlier still. The general theory of elasticity was built between 1821 and 1828 by Claude-Louis Navier and Augustin-Louis Cauchy, with Cauchy introducing the stress tensor and the notion of elastic constants. Simeon Denis Poisson published the lateral-contraction ratio in 1829, predicting from a molecular model that it must equal one quarter for every solid. That prediction set off the rari-constant versus multi-constant dispute of the mid-1800s. George Green showed in 1837, working from a strain-energy function rather than molecular forces, that an isotropic solid requires two independent constants rather than one, and measurements on glass and metals confirmed that Poisson's ratio varies from material to material. Gabriel Lame codified the modern notation in his 1852 Lecons sur la theorie mathematique de l'elasticite des corps solides, giving the parameters lambda and mu still used in seismology. Converting among the moduli became routine geophysical practice once seismic velocities were tied to them, and rock physics work through the twentieth century, including Fritz Gassmann's 1951 fluid substitution relations, made moving between E, G, K and nu a daily task in reservoir characterization.