The Debate about the Number of Elastic Constants

We take for granted the two constants that define the elastic response of a material - Young's modulus E and Poisson ratio v. But did you know that for 50 years, the number of these elastic constants was hotly debated?

When we think of how any material will deform in response of some loading, two classic material parameters come to mind: Young's modulus E and Poisson's ratio v. The former measures a material's stiffness, i.e. how much stress is required to deform it to a particular strain, while the latter measures how much a material deforms perpendicular to the loading direction. Both are vital and often the first properties people consider when determining the applications of a material. And with these two parameters, one can derive all other parameters for an isotropic material, including shear modulus G (describes resistance to shear deformation), bulk modulus K (describes resistance to uniform compression), and the Lamé parameters (used in constitutive relations).

Today any engineer will tell you both E and v are needed to describe an isotropic material (a material with properties independent of orientation). In FEM, these two constants are often the first material inputs. But when the theory of elasticity was being formed in the 19th century, the number of elastic constants that define an isotropic material was actually a hotly debated topic. In fact, the more common view, held by famous mechanicians Navier (1785-1836), Cauchy (1789-1857), and Lamé (1795-1870), was that only one constant, the stiffness E, was needed to fully define any material. In fact, even Poisson (1781-1840), whose name we remember in the definition of v, was also of this uniconstant view.

To explain the controversy, we have to dig into the history of elastic theories. In 1822, Cauchy was the first to introduce the notion of stress in his paper to the Academy of Sciences. His theory, based on the intramolecular forces between particles, contained a complete description of kinematics, equilibrium, and constitutive relations. For a general material, this theory required 15 elastic constants and for an isotropic material, it required only one constant, namely the stiffness E. Although at the time Navier and Cauchy were aware of the ratio of lateral to longitudal contraction (what we now call the Poisson's ratio), there was no need for a second elastic constant in the theory because experiments conducted by Poisson showed a constant ratio of 1/4 for all isotropic materials.

This theory remained unchallenged for six years until 1828, when the British mathematician George Green (1793-1841) published a competing elastic theory, based on the principle of conservation of energy. You might not have heard Green's full name, but you've likely heard of Green's functions, named in his honor. While there was significant overlap between Green's and Cauchy's theories, the key difference was that Green derived a total of 21 elastic constants for a general material and 2 constants for an isotropic material.

This paper started a controversy that gave rise to two schools of thought regarding the number of elastic constants. To resolve the controversy, engineers turned to experiments: If one could measure different values of v for different isotropic materials, then the uniconstant theory which assumed a constant value of v could be disproven. But the problem lie in proving that the material being tested was isotropic. Experiments with materials such as steel, glass, wood, rubber, and jelly clearly showed different values of Poisson's ratio, but proponents of the uniconstant theory argued that those materials were not isotropic and hence the results could still be explained by Cauchy's theory with its 15 elastic constants for a general material.

The debate continued for over 50 years, until it was finally put to bed in 1889, when the German physicist W. Voigt (1850-1919) published his results on experiments with prismatic bars cut from crystals. By using crystals which were clearly isotropic, Voigt showed that Poisson's ratio did indeed vary for isotropic materials and the uniconstant theory was not sufficient to describe their behavior. Green's theory superseded Cauchy's and the requirement of two elastic constants for an isotropic material, E and v, became widely used.

References:

History of Strength of Materials, Timoshenko 1953.