Elasticity and Plasticity

The Mathematical Theory of Elasticity and The Mathematical Theory of Plasticity

J. N. Goodier, P. G. Hodge, Jr.

• Publisher: Dover Publications
• Year: 2016
• ISBN:
  9780486810478
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This volume comprises two classic essays on the mathematical theories of elasticity and plasticity by authorities in this area of engineering science. Undergraduate and graduate students in engineering as well as professional engineers will find these works excellent texts and references.
The Mathematical Theory of Elasticity covers plane stress and plane strain in the isotropic medium, holes and fillets of assignable shapes, approximate conformal mapping, reinforcement of holes, mixed boundary value problems, the third fundamental problem in two dimensions, eigensolutions for plane and axisymmetric states, anisotropic elasticity, thermal stress, elastic waves induced by thermal shock, three-dimensional contact problems, wave propagation, traveling loads and sources of disturbance, diffraction, and pulse propagation. The Mathematical Theory of Plasticity explores the theory of perfectly plastic solids, the theory of strain-hardening plastic solids, piecewise linear plasticity, minimum principles of plasticity, bending of a circular plate, and other problems.

CHAPTER 1

THEORY OF PERFECTLY PLASTIC SOLIDS

1.Generalized variables

If more than one component of stress is acting simultaneously, the description of material behavior becomes much more complex. In order to keep the ensuing discussion as general as possible, we shall follow Prager [1.1] and introduce generalized stress and strain variables. The stress variables will be denoted by Q₁, Q₂, …, Q_n, where n is the number of variables needed to specify the stress state. For example, in a beam subjected to bending, the only variable would be Q₁ = M. In the torsion of a cylindrical rod we could set Q₁ = τ_xz, Q₂ = τ_yz. Alternatively, it might be convenient to use dimensionless variables Q₁ = τ_xz/k, Q₂ = τ_yz/k, where k is the yield stress in pure shear. The general three-dimensional stress state is included as a special case by setting Q₁ = σ_x, Q₂ = σ_y, Q₃ = σ_z, Q₄ = τ_yz, Q₅ = τ_zx, Q₆ = τ_xy.

The choice of stress variables above is, to some extent, arbitrary. Thus in the torsion problem one might prefer to set Q₁ = τ_rz, Q₂ = τ_θz. Or, more profoundly, one might wish a more exact solution to the beam problem and set Q₁ = σ_x, Q₂ = σ_y, Q₃ = τ_xy. However, once the stress variables have been chosen, the strain variables q_i are determined to within a single multiplicative constant by a requirement that the internal energy be of the form *

Thus, in the bending problem if Q₁ = M, then q₁ = κ. In the bending of circular plates which will be considered in more detail in Chapter 5, it is convenient to define

where Y is the tensile yield stress, 2h the plate thickness, and a the radius. In this case all stresses and strains are dimensionless and the internal energy per unit area is

in agreement with (1.1).

A second example is provided by the radially symmetric deformation of axially symmetric shells. Let M_θ and Μ_ϕ be the moments per unit length, and N_θ and Ν_ϕ the corresponding direct stresses. We then define

as the stress variables, and

Here again it is readily verified that

In view of the defining relationship between stresses and strains, only quantities which contribute to the internal energy can be taken as generalized variables. Thus, in the usual beam theory the shear force S must be treated as a reaction, not as a stress, since shear deformations are neglected.

For linear elasticity, the generalized stresses and strains will be linear functions of the component stresses and strains respectively. It follows that they will be linearly related to each other, so that

The values of B_ij can always be computed in terms of Young’s modulus and Poisson’s ratio (or more generally in terms of the anisotropic elastic constants). Thus, for example, in the symmetric bending of circular plates, the elastic moments and curvatures are related by *

Solution of these equations for the curvatures, and substitution of the values from (1.2) leads to

Therefore, in this case

In this example it is obvious that B_ij is symmetric. Also, since

, it is positive definite. It can be shown that these two properties are generally true, and use will be made of this fact in Chapter 4.

A basic assumption of plasticity theory is that the total strain rate can always be decomposed into an elastic part and a plastic part. Thus, if

denotes the plastic strain rate, it follows from (1.4) that

where dots indicate differentiation with respect to time. In much of the theory to follow we shall consider only the effect of the plastic strains p_i. If this is done, the results will be immediately meaningful only for a rigid-plastic material (Fig. 1e and f). However, by means of (1.8), any such results can be generalized to include elastic-plastic materials.

2.Yield condition and flow law

In the Introduction we showed that two current yield stress values were necessary to specify the elastic range of a one-dimensional stress system at any instant. This was because we had a choice of two directions of loa...