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# ConstitutiveRelationship

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```Stress-Strain Relationship | Constitutive Relationship
Ilaksh Adlakha
Department of Applied Mechanics, IIT Madras
Content
❑ Stress-stress relation
❑ Material properties: Young’s modulus; Shear modulus; Bulk modulus; Poisson ratio
❑ Hooke’s Law &amp; Generalized Hooke’s Law
❑ Plane stress and Plane strain
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Flow in Mechanics Problem
FBD/equilibrium
External
load
Internal
forces
Strain-deformation
Material models/
relations
Constitutive relations
Stress
Strain
Deformation
Differential equation of equilibrium / governing equations of equilibrium
Deformation
Strain
AM2200 (July-Nov. 2022), Ilaksh Adlakha
Stress
Internal
forces
External
loads
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Stress-Strain in 1D:
Normal stress and strain
Shear stress and strain
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain in 1D: Elastic Constant
Young’s Modulus or Modulus of elasticity, E
Below the yield stress or within proportionality limit
Hooke’s Law
Strength is affected by alloying, heat treating, and
manufacturing process, but stiffness (Modulus of
Elasticity) is not.
Stress-strain diagram
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Stress-Strain in 1D: Elastic Constant. How?
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Tensile test of materials
True stress-strain diagram
-- Elastic Behavior
-- Yielding
-- Strain Hardening
Nominal/Engg.
stress-strain diagram
-- Necking
Necking
Source: Wikipedia
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain diagram: Elastic Behavior
Elastic Behavior: The material is said to show elastic behavior if the materials returns to its original shape and
size once the load is withdrawn.
Hooke’s Law
Young’s Modulus or Modulus of elasticity, E
Below the yield stress or within proportionality limit. Material follows Hooke’s law. Beyond this is plastic behaviour.
Proportionality limit: The point in stress-strain graph till which stress is proportional to strain. The
material shows linear elastic behavior and follows Hooke’s law.
Elastic limit: The point in stress-strain graph till which the material shows elastic behavior.
Yielding
A slight increase in stress above the elastic limit will result in a breakdown of the material and cause it to deform
permanently. This behavior is called yielding.
The stress that causes yielding is called the yield stress or yield point..
The upper yield point occurs first, followed by a sudden decrease in load-carrying capacity to a lower yield point.
Strength is affected by alloying, heat treating, and manufacturing process but stiffness (Modulus of Elasticity) is not.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain diagram: Plastic behavior
Plastic Behavior: The material is said to show plastic behavior if the material does not return to its original shape and
size, once the load is withdrawn and leaves a residual strain in the material.
Hooke’s law is no more valid for the materials under plastic deformation. The stress-strain; force-deformation relations
are nonlinear.
Notice that once the yield point is reached, the specimen will continue to elongate (strain) without any increase in load.
When the material is in this state, it is often referred to as being perfectly plastic.
Strain Hardening: When yielding has ended, an increase in load can be supported by the specimen, resulting in a
curve that rises continuously but becomes flatter until it reaches a maximum stress referred to as the ultimate stress.
The rise in the curve in this manner is called strain hardening.
Necking: Up to the ultimate stress, as the specimen elongates, its cross-sectional area will decrease. This decrease is
fairly uniform over the specimen’s entire gauge length; however, just after, at the ultimate stress, the cross-sectional
area will begin to decrease in a localized region of the specimen. As a result, a constriction or “neck” tends to form in
this region as the specimen elongates further.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
True-stress-strain diagram
Engineering stress-strain diagram: based on the original
geometrical configuration, like cross-sectional area, A0
and gauge length, L0
True stress-strain diagram: based on the actual
/deformed geometrical configuration, like instantaneous
cross-sectional area, Ai and instantaneous length, Li.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain diagram: 0.2% proof stress
Any material that can be subjected to large strains before it fractures is called a ductile material
Aluminum, wood, steel
It shows cup-cone formation and necking
For some materials like, Aluminum, there is NO
proper yield point.
We calculate yield point based on 0.2% proof stress.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Stress-Strain Diagrams:
Al alloy
Low-carbon steel
Brittle materials
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Materials types based on stress-strain diagram
Ductile materials: Any material that can be subjected to large strains before it fractures is called a
ductile material. Example: Mild steel, Al-, and Mg- alloys.
One way to specify the ductility of a material is to look at its percent elongation or percent reduction in
area at the time of fracture. The percent elongation is the specimen’s fracture strain expressed as a
percent.
Engineers often choose ductile materials for design because these materials are capable of
absorbing shock or energy, and if they become overloaded, they will usually exhibit large deformation
before failing.
Brittle material: Materials that exhibit little or no yielding before failure are referred to as brittle
materials. Gray cast iron, concrete are examples.
Since the appearance of initial cracks in a specimen is quite random, brittle materials do not have a
well-defined tensile fracture stress.
Brittle materials are good in compression than in tension.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain diagram: Compression
Materials that exhibit little or no yielding before failure are referred to as brittle materials.
Cast iron, Concrete.
Not suited for tensile stresses but are very good COMPRESSION materials
Stress-strain diagram of concrete
J. of Advanced Research in Applied Mechanics: 2289-7895 | Vol. 22, No. 1. Pages 13-48, 2016
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Strain Energy
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Modulus of Resilience and Toughness
Modulus of Resilience: The strain energy when the stress reaches the
proportional limit.
Physically a material’s resilience represents the ability of the material to
absorb energy without any permanent damage to the material.
Modulus of Toughness: This quantity represents the entire area
under the stress–strain.
It indicates the strain-energy density of the material just before it fractures. This
property becomes important when designing members that may be accidentally
overloaded.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Problem 01
The stress-strain diagram for an Al alloy that is used for making aircraft parts in Figure.
If the material was stressed to 600 MPa. Determine the permanent strain that remains
in the specimen when the stress is released. Find the modulus of resilience both at
points A and B.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Problem 01
The stress-strain diagram for an Al alloy that is used for making aircraft parts in Figure.
If the material was stressed to 600 MPa. Determine the permanent strain that remains
in the specimen when the stress is released. Find the modulus of resilience both at
points A and B.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Stress-Strain diagram: strain hardening
Stress-strain diagrams portray the behavior of engineering materials when the materials are loaded in tension or
compression, as described in the preceding section. To go one step further, let us now consider what happens
when the load is removed, and the material is unloaded.
The property of a material, by which it returns
to its original dimensions during unloading, is
called elasticity. Note that the stress-strain
curve from O to A need not be linear in order
for the material to be elastic.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
The unloading line is parallel to the tangent to the stressstrain curve at the origin. A permanent set is observed as a
residual strain, or permanent strain, remains in the
material. Thus, during unloading the bar returns partially to
its original shape, and so the material is said to be partially
elastic.
ilaksh.adlakha@iitm.ac.in
problem.
Rigid
Materials:
A(ii)
rigid
material i.e strain hardening is depicted in the figure below:
A perfectly plastic i.e non-strain hardening material
is plastic
shown
below:
Material Classification:It isStress-Strain
the one which donot experience anydiagrams
strain regardless of the applied stress.
(i) Linear elastic material:
A linear elastic material is one in which the strain is proportional to stress as
shown below:
A perfectly plastic i.e non-strain hardening material is shown below:
(v) Elastic Perfectly Plastic material:
(iv) Rigid Plastic material(strain hardening):
Linear elastic model
Rigid material model
(iii)elastic
Perfectly
plastic(non-strain
hardening):
There are also other types of idealized models of material behavior.
The
perfectly
plastic material
is having the
A rigid plastic material i.e strain hardening is depicted in the figure below:
(ii) Rigid Materials:
It is the one which donot experience any strain regardless of the applied stress.
Rigid plastic model
(Strain hardening)
(iii) Perfectly plastic(non-strain hardening):
(v)
Elastic Perfectly Plastic material:
Plastic
model
(iv) Rigid
Plasticmaterial
material(strain
hardening):
characteristics as shown below:
A rigid plastic material i.e strain hardening is depicted in the figure belo
Elastic-perfectly plastic
Elastic plastic model
(vi) Elastic
model– Plastic material:
(v) Elastic Perfectly Plastic material:
Elastic Stress – strain Relations :
The elastic perfectly plastic material is having the characteristics as shown below:
The elastic plastic material exhibits a stress
Vs strain
diagram
as material
depictedis in
the figure
The elastic
perfectly
plastic
having
the characteristics as show
below:
Previously stress – strain relations were consider
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Shear stress-strain in 1D:
Shear Modulus/ Modulus of Rigidity
Material under pure shear shows a linear relation between shear strain and shear stress. The constant
of proportionality is called Shear Modulus or Modulus of Rigidity
𝐸
2(1 + 𝜈)
For uniform shear stress/strain distribution, the material is assumed homogeneous and isotropic.
𝐺=
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Poisson Ratio:
It has been observed that for an elastic materials, the lateral strain is proportional to the longitudinal strain. The
ratio of the lateral strain to longitudinal strain is known as the Poisson's ratio.
Poisson’s Ratio
Poisson’s ratio is a dimensionless quantity that has a value between 0 and 0.5 for most materials, although some
composite 1 materials can have negative values for 𝜈. The theoretical range for Poisson’s ration is –1 ≤ 𝝂 ≤ 0.5.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Generalized Hooke’s Law:
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Generalized Hooke’s Law:
The use of Poisson’s ratio to relate strains in perpendicular directions is valid not only for Cartesian coordinates but
for any orthogonal coordinate system. Thus the generalized Hooke’s law may be written for any orthogonal
coordinate system, such as spherical and polar coordinate systems.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Bulk modulus | Dilatation
The constant k is known as the bulk modulus of the material.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
Problem 02
A circle of diameter, d = 9 in. is scribed on an unstressed Al plate of thickness, t = &frac34;
in. The plate is subjected to normal stress along X and Z directions of 12 and 20 ksi,
respectively. E for Al = 10 x 106 psi and 𝜈= 1/3. Determine the change:
a) AB
b) CD
c) Thickness of plate
d) Volume of plate
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Plane Strain
AM2200 (July-Nov. 2022), Ilaksh Adlakha
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Plane Stress
State of stress
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State of strain
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Problem 03
The stresses at a point on steel were found to be σxx = 15 MPa (T), σyy = 30 MPa (C), and 𝜏𝑥𝑦 = 25 MPa. Using E =
210 GPA and 𝜈 = 0.25, determine the state of strains εxx , εyy, γxy, εzz and the stress σzz assuming (a) the point is in a
state of plane stress. (b) the point is in a state of plane strain.
AM2200 (July-Nov. 2022), Ilaksh Adlakha
ilaksh.adlakha@iitm.ac.in
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