Deformation of an elastic solid

The stress state associated with an elemental cube may be defined in terms of the six independent quantities in Eq. (4), where the first subscript refers to the direction of the normal to the plane on which the stress acts and the second subscript refers to the direction of the stress (at equilibrium the net torque is zero so that r,) = 7ji). 

The deformation may likewise be defined in terms of the components of engineering strain [Eq. (5)]. 

The generalized form of Hooke s law, which proposes a linear relationship between stress and strain at vanishingly small strains, is then given by Eq. (6). 

In an isotropic solid, there is no coupling between tensile and shear stresses, and Hooke s law takes the familiar form of Eq. (7), where v is Poisson s ratio and G is the shear modulus. 

Moreover, it is easily shown that Eq. (8) holds, so that a complete description of the small strain response requires only two independent parameters. 

---

Murnaghan, F. D. "Finite Deformation of an Elastic Solid" Dover Publications, Inc. New York, 1967. 

Deformation of an elastic solid through which a mass point of the solid with co-ordinates Xi, X2, X-i in the undeformed state moves to a point with co-ordinates xi, X2, X3 in the deformed state and the deformation is defined by... 

It is convenient to use a simple weightless Hookean, or ideal, elastic spring with a modulus G and a simple Newtonian (fluid) dashpot or shock absorber having a liquid with a viscosity of 17 as models to demonstrate the deformation of an elastic solid and an ideal liquid, respectively. The stress-strain curves for these models are shown in Figure 14.1. 

Murnaghan, F. D. Finite deformation of an elastic solid. New York John Wiley 1951. 

The equations for the deformation of an elastic solid given in Table... 

Mumaghan FD (1937) Finite deformations of an elastic solid. Am J Math 49 235-260 Niesler H, Jackson I (1989) Pressure derivatives of elastic wave velocities from ultrasonic interferometric measurements on jacketed poly crystals. J Acoust Soc America 86 1573-1585 Nomura M, Nishizaka T, Hirata Y, Nakagiri N, Fujiwara H (1982) Measurement of the resistance of Manganin under liquid pressure to 100 kbar and its application to the measurement of the transition pressures of Bi and Sn. Jap J Appl Phys 21 936-939 Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford... 

FIGURE 4-8 Deformation of an elastic solid (at constant stress)... 

Mumaghan, F.D., 1951, Finite Deformation of an Elastic Solid (Wiley, New York). 

---