ME231: Solid Mechanics-I

Stress and Strain

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 Example 7
For a slender beam, the equilibrium equations were derived as,

We will prove these equations again using general equilibrium equations derived in this
chapter, i.e.,
··········(7a)

··········(7b)

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Integrate (7b) in the thickness direction (i.e., y-direction) as,

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Now, multiply (7a) with y and integrate in the thickness direction (i.e., y-direction) as,

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 ME231: Solid Mechanics-I

Stress, Strain and Temperature relationship

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 Introduction
●
In the previous chapter, we derived three equations (for 3D) of equilibrium for the six
components of stress.
●
In addition to that there are three components of displacements in the six equations
relating strain to displacement.
●
Thus, to determine the distributions of stress and strain in a body more equations are
required.
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The distribution of stress and strain will depend on the material behavior of the body.
●
In this chapter we shall discuss the relations between stress and strain (constitutive law).
●
Different materials follow different relationship between stress and strain, and development
of constitutive model for materials is an active field of research.

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 Tensile test
Tensile test specimen Universal Testing Machine (UTM)

Gage length

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 Tensile test
https://www.youtube.com/watch?v=D8U4G5kcpcM

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 Stress-strain curve

Engineering Stress , Engineering Strain .

1 ksi = 6.89476 MPa

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Low carbon steel Aluminum alloy
Features of stress-strain curve
●
The very first region is where stress
is linearly proportional to the
strain. The proportional limit is
defined as the maximum stress upto
which this proportionality exists.
●
The elastic limit is defined as the
maximum stress up to which
material behaves elastically, i.e.,
there is no permanent strain on
release of stress.
●
However, neither the proportional
nor the elastic limits can be
determined precisely. They deal
with the limiting cases of zero
deviation from linearity and of no
permanent set. 6
 ●
Since plastic deformations of the
order of the elastic strains are often
unimportant, instead of reporting
the elastic limit it has become
standard practice to report a
quantity called the yield strength,
which is the stress required to
produce a certain arbitrary plastic
deformation.
●
For many of the common steels the
plastic deformation begins abruptly,
resulting in an increase of strain
with no increase, or perhaps even a
decrease, in stress.

●
For such materials, the stress at which plastic deformation first begins is called the upper
yield point; subsequent plastic deformation may occur at a lower stress, called the lower
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yield point.
●
As plastic deformation is continued,
the load required for further plastic
flow increases. This phenomenon is
called strain hardening.
●
Finally a point is reached where the
load required to cause further
elongation begins to decrease. At
this point the load has passed a
maximum and, consequently, so
also has the engineering stress. This
maximum value of the engineering
stress is termed the ultimate tensile
stress (or tensile strength).

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●
After the ultimate tensile strength, it is
observed that at a certain cross-section
reduction in the cross-sectional area is
higher than that in the other places. This
non-uniform deformation is called necking,
●
The tensile test reaches its conclusion when
a small crack develops at the center of the
neck and spreads outward to complete the
fracture.
●
The stress at which complete fracture
occurs is called the breaking stress.