mechanics of solid DR salahuddin

1. Dr. Salah Uddin,
PhD (Geotech), (Univ of Nottingham, UK)
ME (Structure), (NED UET Karachi, Pakistan)
BE (Civil), (Balochistan UET Khuzdar, Pakistan)
Associate Professor
Email: Salahuddin@buetk.edu.pk
WhatsApp: +923337950583
Relationship between Elastic
Constants

2. Topics
Contents
1. Stress, Strain and Mechanical Properties of
Materials
• Uniaxial state of stress and strain,
• Relationships between elastic Constants
• Response of materials under different sets of
monotonic loading (including impact),
• Normal and shearing stress and strains,
Distribution of direct stresses on uniform and
non-uniform members,
• Thermal stresses and strains
Serial No. of lectures: 01-08 (Total Classes: 08)

3. Goal of the lecture
• To understand and learn elastic constants and their relationship

4. Elasticity
CLO-1 (PLO-1)
Elasticity, ability of a deformed material body to return to its original shape and
size when the forces causing the deformation are removed. A body with this
ability is said to behave (or respond) elastically
There is a limit to the magnitude of the force and the
accompanying deformation within which elastic
recovery is possible for any given material. This
limit, called the elastic limit, is the
maximum stress or force per unit area within a solid
material that can arise before the onset of
permanent deformation.

6. Elastic Constants
Elastic constants are the constants describing mechanical response of a material
when it is elastic.
Elastic constants measure the proportionality between strain and stress
If this mechanical response is linear , one
can define a set of constants that relate any
applied stress to the corresponding strain

7. Young’s Modulus
For a simple tension or compression test, the easiest elastic constant to define
is Young's modulus, E. Young's modulus is the elastic constant defined as
the proportionality constant between stress and strain:
The Young's modulus is the slope of the linear
elastic response for a number of materials
𝐸 =
𝜎
𝜖
Young’s Modulus is the ability of any material to
resist the change along its length
Modulus of elasticity is often called Young’s
modulus, after another English scientist, Thomas
Young (1773–1829).

9. Bulk Modulus
When a body is subjected to mutually perpendicular direct stresses which
are alike and equal, within its elastic limits, the ratio of direct stress to the
corresponding volumetric strain is found to be constant. This ratio is called
bulk modulus and is represented by letter “K”. Unit of Bulk modulus is MPa.
Bulk Modulus K =
Direct Stress
Volumetric Strain

10. Shear Modulus
A type of stress called a shear stress, 𝜏, can also be defined which can produce a
shear strain, 𝛾.
Just as for normal stresses, shear loading can
result in a mechanical response that is elastic and
nearly linear, but the shape change is called a
shear strain.

11. The elastic constant that describes the linear relation between 𝜏 and 𝛾 is called
the shear modulus, G.
𝐺 =
𝜏
𝛾
Shear Modulus
G is the shear modulus of elasticity (also called the modulus
of rigidity).

12. Poisson’s Ratio
Another elastic constant that can be obtained in a tension or compression test
wherein the strain along the loading direction and orthogonal to it is
called Poisson's ratio, ν.
When a prismatic bar is loaded in tension, the axial
elongation is accompanied by lateral contraction
(that is, contraction normal to the direction of the
applied load).

13. Poisson’s Ratio
Lateral contraction is easily seen by stretching a rubber band, but in metals the
changes in lateral dimensions (in the linearly elastic region) are usually too small
to be visible. However, they can be detected with sensitive measuring devices.
https://extrudesign.com/poisson-ratio-definition/

14. Poisson’s Ratio
The lateral strain e at any point in a bar is proportional to the axial strain 𝜖 at
that same point if the material is linearly elastic. The ratio of these strains is a
property of the material known as Poisson’s ratio. This dimensionless ratio,
usually denoted by the Greek letter 𝜈 (nu), can be expressed by the equation
𝜈 =
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛
𝐴𝑥𝑖𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛
= −
𝜖′
𝜖
The minus sign is inserted in the equation
to compensate for the fact that the lateral
and axial strains normally have opposite
signs.

15. Relationship between Elastic Constants
𝐺 =
𝐸
2(1 + 𝜈)
This relationship, will be derived later in our discussion of Torsion, shows that
E, G, and 𝜈 are not independent elastic properties
of the material.
The relationship between Young’s modulus (E), rigidity modulus (G) and
Poisson’s ratio (𝜈) is expressed as

16. Relationship between Elastic Constants
 The relationship between Young’s modulus (E), bulk modulus (K) and
Poisson’s ratio (𝜈) is expressed as :
𝐾 =
𝐸
3(1 − 2𝜈)
 Young’s modulus can be expressed in terms of bulk modulus (K) and rigidity
modulus (G) as :
𝐸 =
9𝐾𝐺
(3𝐾 + 𝐺)

17.  Poisson’s ratio can be expressed in terms of bulk modulus (K) and rigidity
modulus (G) as :
Relationship between Elastic Constants
𝜈 =
3𝐾 − 2𝐺
6𝐾 + 2𝐺

18. Thanks