Stress :
Mechanical stress can be defined as the force acting on a body per unit area.
In above figure, left figure shows a prismatic bar applied with a load 'P', which develops a total internal resisting force (R) on cross-section, and figure on right shows the free body diagram of left fig. Then the Stess can be define as the magnitude or intensity of internal resisting force developed at a point in a member under given load.
Stress Vs Pressure :
Stress |
Pressure |
stress magnitude can be different in different direction |
Same in every direction |
Stress can't be measured |
Pressure gauges can be use to measure pressure |
Acts normal or along to the applied load |
Acts normal to the surface |
Stress may be tensile, compressive and shear |
Pressure is always compressive |
Stress is a Second order tensor |
Pressure is a scalar quantity |
Zero order tensor :
Zero order tensor are basically Scalar quantity means it has only magnitude.
First order tensor :
First order tensor are basically Vector quantity means it has magnitude and one direction.
Second order tensor :
Second order tensor have both magnitude and 2 direction. Example :- Stress , Strain
Engineering stress : It is the ratio of applied load at any instant to average original cross-sectional area. Engineering Stress is denoted S or .
Where P is applied load and is Avg. Cross-sectional area
True stress : It is the ratio of applied load at any instant to instantaneous cross-sectional area. True Stress is denoted as .
Where P is applied load and is instantaneous Cross-sectional area
Units of Stress : Pa ( ) , MPa ( = 1MPa ) ,
Relation between True stress and Engineering stress:
= Engg. Strain
= Engg. Stress
= True Stress
Engineering strain : It is the ratio of change in dimension to original dimension om application of load. Engineering Strain is denoted e.
Where is change in dimesion and is original length
Longitudinal Strain : It is the normal strain in the direction of applied load
Lateral Strain : It is the normal strain in the direction perpendicular of applied load
Note :
(1) Longitudinal Strain and Lateral Strain are always unlike in nature
(1) Every longitudinal strain is associated with 2 lateral strain
True strain : True strain equals the natural log of the quotient of current length over the original length. True Strain is denoted as .
Here,
L
0 and A
0 is intial length and cross-section area respectively
L and A is final length and cross-section area respectively
q = percentage reduction in cross-sectional area
Units of Strain : Dimensionless
Relation between True stran and Engineering strain :
= Engg. Strain
= Engg. Stress
= True Stress
Volume strain :
Volumetric strain is defined as the total change in volume divided by the original volume, i.e., ΔV/V. Let assume a block is strained as in x, y, and z direction respectively. Then volumeric strain will be-
On neglecting square and Cube term
For Rectengular block :
For Spherical block :
Shear strain :
Shear strain is defined as the change in initial right angle between two lines elements which are parallel to x and y axes. The above figure shows the simple shear strain is a combination of pure shear and rotation. Shear strain can be calculated as
For homogeneous and isotropic material the number of elastic constants are 4. and the number of independent elastic constant are 2 (i.e., )
Elastic Modulus (E) :
Modulus of Rigidity (G) :
Bulk Modulus (K) :
Poission ratio ( ) :
Material |
Independent Elastic Constant |
Isotropic |
2 |
Orthotropic |
9 |
Anisotropic |
21 |
Relationship between elastic constant :
For metals
Value of Elastic Constants
E, K, G > 0 and Ex.
From 2nd equation
Volumetric strain under triaxial state of stress :

As mentioned earlier , Every longitudinal strain is associated with 2 lateral strain. For above case the longitudinal strain in x-direction is associated with 2 lateral strain . So for all direction :
|
X-Direction |
Y-Direction |
Z-Direction |
Px |
|
|
|
Py |
|
|
|
Pz |
|
|
|
Hence,
Total strain in x-direction
Total strain in x-direction
Total strain in x-direction
As for a rectangular block :