UNIT :4 state of stress in 3 dimension
1) What are the types of failure?
The 2 types of failure are;
I. Brittle failure
II. Ductile failure
2) Define Brittle failure
Failure of a material represents direct separation of particles from each other, accompanied
by considerable deformation is known as Brittle failure.
3) Define ductile failure
Slipping of a material accompanied, by considerable plastic deformations is known as ductile
failure.
4) Define tensor
State of a stress at a point is defined by three components on each of the three mutually
perpendicular axis in mathematical terminology is called tensor.
5) List out the theories of failure
Maximum principal stress theory(Rankine’s theory)
Maximum principal strain theory(St.Venant’s theory)
Maximum shear stress theory(Tresca’s & guest theory)
Maximum shear strain energy theory(Von-Mises-Hencky theory)
Maximum strain energy theory(Haigh’s theory)
6) Define Maximum principal stress theory (Rankine’s theory)
According to this theory, the failure of a material will occur when the maximum principal
tensile stress (σ1) in the complex system reaches the value of the maximum stress (σ t*) at the elastic
limit in simple tension or the minimum principal stress (i.e., the maximum principal compressive stress)
reaches the value of the maximum stress at the elastic limit in simple compression..
σ1= σ t*
7) Define Maximum principal strain theory (St.Venant’s theory)
According to this theory, the failure of a material will occur when the maximum principal strain
(e1) reaches the strain due to the yield stress in simple tension (σ t*/E).
e1= σ t*/E
In 3D e1= [σ1-μ (σ2+σ3)] = σ t*/E
In 2D e1= [σ1-μσ2] = σ t*/E
8) Define Maximum shear stress theory (Tresca’s & guest theory)
According to this theory, the failure of a material will occur when the maximum shear stress in
the body will reaches the value of Maximum shear stress in simple tension at the elastic limit.
In 3D (σ1- σ3) = σ t*
In 2D σ1 = σ t*
9) Define Maximum shear strain energy theory (Von-Mises-Hencky theory)
According to this theory, the failure of a material will occur when the total shear strain energy
per unit volume in the stressed material reaches a value equal to the shear strain energy per unit
volume at the elastic limit in simple tensile test.
In 3D shear strain energy due to distortion U = (1/12C) [(σ1- σ2) ^2+ (σ2- σ3) ^2+ (σ3- σ1) ^2]
In 3D shear strain energy due to simple tension U = σ t*^2\6C
U= (1/6C) [(σ1- σ2) ^2+ (σ2- σ3) ^2+ (σ3- σ1) ^2] = σ t*^2\6C
In 2D shear strain energy due to simple tension
U= (1/6C) [(σ1- σ2) ^2+ (σ2) ^2+ (σ1) ^2] = σ t*^2\6C
10) Define Maximum strain energy theory (Haigh’s theory)
According to this theory, the failure of a material will occur when the total strain energy per
unit volume in the stressed material reaches the strain energy per unit volume of the material at the
elastic limit in simple tensile test.
In 3D shear strain energy due to deformation
U= (1/2E) [σ1^2+ σ2^2+ σ3^2+-2μ (σ1σ2 +σ2σ3 +σ3 σ1)]
In 3D shear strain energy due to simple tension U= σ t*^2\2E
U= [σ1^2+ σ2^2+ σ3^2+-2μ (σ1σ2 +σ2σ3 +σ3 σ1)] = σ t*^2
In 2D U= [σ1^2+ σ2^2 -2μ (σ1σ2)] = σ t*^2
11) Write the limitations of Maximum principal stress theory (Rankine’s theory)
• This theory disregards the effect of other principal stresses & effect of shearing
stresses on other planes through the planes through the element.
• Material in tension test piece along 45º to the axis of the test piece, where normal
stress is neither maximum nor minimum, but the shear stress is maximum.
• Failure is not brittle but cleavage failure.
12) Write the limitations of shear stress theory (Tresca’s & guest theory)
This theory does not give accurate results for the state of stress of pure shear in which the
maximum amount of shear is developed (in torsion test).
13) Write the limitations of Maximum shear strain energy theory (Von-Mises-Hencky theory)
This theory cannot be applied to materials under hydro static pressure
14) Write the limitations Maximum strain energy theory (Haigh’s theory)
This theory does not apply to brittle materials for which elastic limit in tension and
compression is quite different.
15) Define strain Rosette
Linear strains are measured in all direction by strain gauges known as strain Rosette.
16) Define Plasticity ellipse
The graphical surface of a maximum shear strain energy theory (Von-Mises-Hencky theory) is
a straight circular cylinder.
[σ1^2+ σ2^2 -σ1σ2] = σ t*^2, which is called Plasticity ellipse.
17) Define Octahedral Plane
The plane which is equally inclined to the three axes of reference is called Octahedral Plane.
18) Define Octahedral Stresses
The normal and shear stress acting on the Octahedral Plane (The plane which is equally
inclined to the three axes of reference) is known as Octahedral Stresses.
19) Define volumetric strain per unit volume
Volumetric strain per unit volume is defined as the ratio of change in volume of a material to
its unit volume.
ev = ex +ey +ez
20) What are residual Stresses?
Maximum principal stress and Maximum shear stress are collectively known as residual
Stresses.
