Feynman lectures - elastic materials - eqn 39.22

[lwymarie]

I'm reading Feynman Lectures on Physics Volume 2 Chapter 39 on elastic materials. Equation 39.22 relates the elements in the elasticity tensor to the Young's modulus and the Poisson's ratio. I have no clue on how to get these relations myself.

For example, for C_xxxx, I think it should tell how the material is elongated in the x-direction in response to a stress in the x-direction. Hence I would suppose C_xxxx is just the Young's modulus Y, because this relation is just like F / A = Y * (delta l) / l.

Please kindly show me directions of how to derive these relations. You can find a copy of Feynman Lectures Vol 2 Ch 39 in http://student.fizika.org/~jsisko/Knjige/Opca%20Fizika/Feynman%20Lectures%20on%20Physics/Vol%202%20Ch%2039%20-%20Elastic%20Materials.pdf

 

[LightHero]

. Equation 39.22 is on page 10.The derivation of the elements in the elasticity tensor can be found in many standard texts on solid mechanics, such as Timoshenko's Theory of Elasticity. The basic idea is that a stress in a given direction will cause a strain in the same direction. This strain can be determined by applying Hooke's Law, which states that the strain is proportional to the stress. The proportionality constant in this relationship is the Young's modulus (E) for a given material. Once the strain is known, the components of the elasticity tensor can be calculated using the following equations:C_xxxx = EC_yyyy = EC_zzzz = EC_xxyy = C_xyxy = C_xzxz = C_yxzy = E * vWhere v is the Poisson's ratio.These equations are derived by considering a material that is subjected to a uniform tensile or compressive load, and then calculating the resulting strain. By applying Hooke's Law, the components of the elasticity tensor can be determined.