Elastoplastic Wire

Abstract

In order to understand the basic ideas of elasticity and plasticity theory, we discuss in this chapter the simplest situation by choosing the wire as an example. Special emphasis will be placed on a comparison of important constitutive laws (stress—strain relations). We stress the possibility that plastic behavior may be described by multivalued constitutive equations, i.e., more precisely, by subgradients. The calculus of subgradients has been discussed in Part III.

Ut tensio sic vis.¹

Robert Hooke, De Potentia Restitutiva, (London, 1678)

The first mathematician to consider the nature of resistance of solids to rupture was Galileo (1638)…. He endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem and, in particular, the determination of this axis is known as Galileo’s problem.

The history of the theory of elasticity started from Galileo’s question. Undoubtedly, the two great landmarks are the discovery of Hooke’s law in 1660 (published in 1678), and the formulation of the general equations by Navier (1821). Hooke’s law provided the necessary experimental foundation for the theory….

In the interval between the discovery of Hooke’s law and that of the general differential equations of elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo’s problem, and the related theories of the vibrations of bars and plates and the stabty of columns.

The first investigation of any importance is that of the elastic line or elastica by Jacob Bernoulli (1705), in which the resistance to bending is a number proportional to the curvature of the rod when bent….

David Bernoulli suggested to Euler (by letter in 1742) that the differential equation of the elastica could be found by making the square of the curvature taken along the rod a minimum; and Euler (1744) was able to obtain the differential equation and to classify various solutions of it (an early study of elliptic integrals and elliptic functions)….

Navier (1821) was the first to investigate the general equations of equilibrium and vibration of elastic solids. He set out from the Newtonian conception of the constitution of bodies, i.e., bodies are made up of small parts called “molecules” which act upon each other by means of central forces….

The studies of Cauchy (1789–1857) in elasticity were first prompted by his being a member of the commission appointed to report upon a memoir by Navier on elastic plates which was presented to the Paris Academy in August, 1820. By the autumn of 1822, Cauchy had discovered most of the elements of the pure theory of elasticity (published in 1827). He had introduced the notion of stress. He had also shown how to introduce both the stress tensor and the strain tensor…. He had determined the equations of motion (or equilibrium) by which the stress components are connected with the forces…. By means of relations between stress components and strain components, he had eliminated the stress components from the equations of motion and equilibrium, and had arrived at equations in terms of the displacement…. Cauchy obtained his stress—strain relation (constitutive law) for isotropic materials by means of two assumptions:

(i) that the relations in question are linear; and

(ii) that the principal planes of stress are normal to the principal axes of strain.

The experimental basis on which these assumptions can be made to rest is the same as that on which Hooke’s law rests, but Cauchy did not refer to it. The methods used in these investigations are quite different from those of Navier’s memoir (1821). In particular, no use is made of material points and central forces. The resulting equations differ from Navier’s in one important respect: Navier’s equations contain a single constant to express the elastic behavior of an isotropic body, while Cauchy’s contain two such constants (today called the constants of Lamé (1795–1870))….

Green (1793–1841) was dissatisfied with the hypothesis on which the theory of elasticity was based, and he sought a new foundation in his paper (1839). Starting from what is now called the “principle of minimal elastic potential energy” he propounded a new method of obtaining the basic equations. The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier’s discovery of the basic equations. Green supposed the stored energy function (density of the elastic potential energy) to be capable of being expanded in powers and products of the components of strain…. From this principle Green deduced the equations of elasticity for anisotropic bodies, containing in the general case 21 constants. In the case of isotropy there are two constants, and the equations are the same as those of Cauchy…. (Green followed the pattern of the famous “La Mécanique Analytique” of Lagrange (1788). Green’s stored energy function corresponds to the Lagrangian function in mechanics.)

The history of the mathematical theory of elasticity shows cleariy that the development of the theory has not been guided exclusively by considerations of its utility for technical mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in natural philosophy than in material progress, in trying to understand the worid rather than in trying to make it more comfortable.

A. Love (1906)

The mechanics of continua, which is based on Cauchy’s (1827) general notion of stress, has been applied so far only to liquid and solid elastic bodies. In regard to plastic deformations. Saint Venant (1864) (based on experiments of Tresca (1864)) has sketched a theory which, however, does not yield the necessary number of equations in order to completely determine the motion.

But this paper leads to a complete system of equations of motion for plastic bodies.

Richard von Mises (1913)

The mathematical theory of plasticity owes its development to the demand for more realistic methods to determine the safety factors of structures or machine parts, and to the need for better control in technological forming processes such as rolling, drawing, and extruding.

William Prager and Philip Hodge (1951)