Mecanica De Suelos

2
Stresses, strains, elasticity, and plasticity
2.1

Introduction

In many engineering problems we consider the behaviour of an initially unstressed
body to which we apply some first load-increment. We attempt to predict the consequent
distribution of stress and strain in key zones of the body. Very often we assume that the
material is perfectly elastic, and because of the assumedlinearity of the relation between
stress-increment and strain-increment the application of a second load-increment can be
considered as a separate problem. Hence, we solve problems by applying each loadincrement to the unstressed body and superposing the solutions. Often, as engineers, we
speak loosely of the relationship between stress-increment and strain-increment as a ‘stress
– strain’relationship, and when we come to study the behaviour of an inelastic material we
may be handicapped by this imprecision. It becomes necessary in soil mechanics for us to
consider the application of a stress-increment to a body that is initially stressed, and to
consider the actual sequence of load-increments, dividing the loading sequence into a
series of small but discrete steps. We shall beconcerned with the changes of configuration
of the body: each strain-increment will be dependent on the stress within the body at that
particular stage of the loading sequence, and will also be dependent on the particular
stress-increment then occurring.
In this chapter we assume that our readers have an engineer’s working
understanding of elastic stress analysis but we supplement this chapter withan appendix A
(see page 293). We introduce briefly our notation for stress and stress-increment, but care
will be needed in §2.4 when we consider strain-increment. We explain the concept of a
tensor being divided into spherical and deviatoric parts, and show this in relation to the
elastic constants: the axial compression or extension test gives engineers two elastic
constants, which we relateto the more fundamental bulk and shear moduli. For elastic
material the properties are independent of stress, but the first step in our understanding of
inelastic material is to consider the representation of possible states of stress (other than the
unstressed state) in principal stress space. We assume that our readers have an engineer’s
working understanding of the concept of ‘yieldfunctions’, which are functions that define
the combinations of stress at which the material yields plastically according to one or other
theory of the strength of materials. Having sketched two yield functions in principal stress
space we will consider an aspect of the theory of plasticity that is less familiar to engineers:
the association of a plastic strain-increment with yield at a certaincombination of stresses.
Underlying this associated ‘flow’ rule is a stability criterion, which we will need to
understand and use, particularly in chapter 5.

2.2

Stress

We have defined the effective stress component normal to any plane of cleavage in
a soil body in eq. (1.7). In this equation the pore-pressure uw, measured above atmospheric
pressure, is subtracted from the (total) normalcomponent of stress σ acting on the cleavage
plane, but the tangential components of stress are unaltered. In Fig. 2.1 we see the total
stress components familiar in engineering stress analysis, and in the following Fig. 2.2 we
see the effective stress components written with tensor-suffix notation.

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Fig. 2.1 Stresses on Small Cube: Engineering Notation

The equivalence betweenthese notations is as follows:
σ x = σ '11 +u w
τ xy = σ '12

τ xz = σ '13

τ yx = σ '21

σ y = σ '22 +u w

τ yz = σ '23

τ zx = σ '31

τ zy = σ '32

σ z = σ '33 +u w .

We use matrix notation to present these equations in the form
⎡σ x τ xy τ xz ⎤ ⎡σ '11 σ '12 σ '13 ⎤ ⎡u w 0
⎥⎢
⎢
⎥⎢
⎢τ yx σ y τ yz ⎥ = ⎢σ ' 21 σ ' 22 σ '23 ⎥ + ⎢ 0 u w
⎢τ zx τ zy σ z ⎥ ⎢σ '31 σ '32 σ '33 ⎥ ⎢ 0...