LECTURE
STRESS

- STRAIN

9
RELATIONS

Stress
Strain Relations: The Hook's
law, states that within the elastic limits the
stress is proportional to the strain since for most materials it is impossible to describe
the entire stress
strain curve with simple mathematical expression, in any given
problem the behavior of the materials is represented by an idealized stress
strain
curve, which emphasizes those aspects of the behaviors which are most important is
that particular problem.
(i) Linear elastic material:
A linear elastic material is one in which the strain is proportional
shown below:

to stress as

There are also other types of idealized models of material behavior.
(ii) Rigid Materials:
it is the one which
applied stress.

(iii) Perfectly

donot experience

plastic(non-strain

any strain

regardless

of the

hardening):

A perfectly plastic i.e non-strain hardening material is shown below:

(iv) Rigid Plastic material(strain
A rigid plastic material
below:

(v) Elastic Perfectly

hardening):
i.e strain hardening

is depicted

in the figure

Plastic material:

The elastic perfectly
shown below:

plastic material is having the characteristics

as

(vi) Elastic - Plastic material:
The elastic plastic material
depicted in the figure below:

exhibits

a stress Vs strain diagram

as

Elastic Stress - strain Relations

:

Previously stress strain relations were considered for the special case of a
uniaxial loading i.e. only one component of stress i.e. the axial or normal component
of stress was coming into picture. in this section we shall generalize the elastic
behavior, so as to arrive at the relations which connect all the six components of
stress with the six components of elastic stress. Futher, we would restrict overselves
to linearly elastic material.
Before writing down the relations let us introduce a term lSOTROPY
ISOTROPIC: if the response of the material is independent of the orientation of the
load axis of the sample, then we say that the material is isotropic or in other words we
can say that isotropy of a material in a characteristics, which gives us the information
that the properties are the same in the three orthogonal directions x y z, on the other
hand if the response is dependent on orientation it is known as anisotropic.
Examples of anisotropic
directions are

materials,

whose

properties

are

different

in different

(i) Wood
(ii) Fibre reinforced plastic
(iii) Reinforced concrete
HOMOGENIUS: A material is homogenous if it has the same composition through our
body. Hence the elastic properties are the same at every point in the body. However,
the properties need not to be the same in all the direction for the material to be
homogenous.
isotropic materials have the same elastic properties in all the
directions. Therefore, the material must be both homogenous and isotropic in order to
have the lateral strains to be same at every point in a particular component.
Generalized
Hook's
Law:
proportional limit.

We

know

that

for stresses

not

greater

than

the

These equation expresses the relationship between stress and strain (Hook's
law) for
uniaxial state of stress only when the stress is not greater than the proportional limit.
in order to analyze the deformational effects produced by all the stresses, we shall
consider the effects of one axial stress at a time. Since we presumably are dealing
with strains of the order of one percent or less. These effects can be superimposed
arbitrarily. The figure below shows the general triaxial state of stress.

Let us consider a case when ox alone is acting. it will cause an increase in dimension
in X-direction whereas the dimensions

in y and z direction will be decreased.

Therefore

the

resulting

strains

in

three

directions

are

Similarlylet us considerthatnormalstresscy aloneis actingandthe resultingstrains
are

Now let us consider the stress 62 acting alone, thus the strains produced are

in the following analysis shear stresses were not considered. it can be shown that for
an isotropic material's
a shear stress will produce only its corresponding shear strain
and will not inuence the axial strain. Thus, we can write Hook's
law forthe individual

Sill-.?1F...
...5U'3lll.5...,...
.300..... hear

Stresses

in

The

f00Wl"Q

manner-

The Equations (1) through (6) are known as Generalized Hook's
law and are the
constitutive equations for the linear elastic isotropic materials. When these equations
isotropic materials. When these equations are used as written, the strains can be
completely determined from known values of the stresses. To engineers the plane

stress situation is of much relevance (i.e. 02 = rxz = 1yz = O), Thus then the above set
of equations reduces to

Hook's
law is probably the most well known and widely used constitutive equations
for an engineering materials. However, we can not say that all the engineering
materials are linear elastic isotropic ones. Because now in the present times, the new
materials are being developed every day. Many useful materials exhibit nonlinear
response and are not elastic too.
Plane Stress: in many instances the stress situation is less complicated for example
if we pull one long thin wire of uniform section and examine
small parallepiped
where x axis coincides with the axis of the wire

Soif we takethexy planethenox , cry, rxywill betheonlystresscomponents
acting
on the parrallepiped.
situation

This combination of stress components

is called the plane stress

A plane stress may be defined as a stress condition in which all components
associated with a given direction ( i.e the z direction in this example) are zero

Plane strain: if we focus our attention on a body whose particles all lie in the same
plane and which deforms only in this plane. This deforms only in this plane. This type
of deformation is called as the plane strain, so for such a situation.

eZ=yzx= yzy= 0 andthenon zerotermswouldbeex,e y& yxy
i.e. if strain componentsex, ey and yxyand angle 6 are specified,the strain
components
ex,rayandyXy'
withrespectto someotheraxescanbedetermined.
ELASTIC CONSTANTS
in considering the elastic behavior of an isotropic materials under, normal, shear and
hydrostatic loading, we introduce a total of four elastic constants namely E, G, K, and
'Y.
it turns out that not all of these are independent to the others. in fact, given any two of
them, the other two can be foundout . Let us define these elastic constants

(i)

E = Young's
Modulus of Rigidity
= Stress / strain

(ii) G = Shear Modulus or Modulus of rigidity
= Shear stress / Shear strain
(iii) y= Possion's
ratio
=

lateral strain / longitudinal strain

(iv) K = Bulk Modulus of elasticity
= Volumetric stress / Volumetric strain

Where
Volumetric
strain
=sum
of
Iinear
stress
in
x,yand
zdirection.
Volumetric stress = stress which cause the change in volume.
Let us find the relations between them
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