UNIT 4
Conditions for Static Equilibrium
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Mechanical Properties of Solids
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6

 

Mechanical Properties of Solids

Elasticity

All solids

— have a three dimensional structure of atoms held together by equilibrum of strong attactive and repulsive sorces forces.
— Some solids are rigid because atoms are difficult to displace,
— but in other 'elastic' materials the atoms may be arranged in long flexible molecules.
— Even in 'rigid' materials here is some level of elasticity as when atoms are displaced these forces tend to return them to their original positions.
— When stretched they may return to their original positions, become permanently stretched or break depending on the force applied (Hooke's Law).

Stress (deforming force per unit area) causes strain (deformation) and may be of the following types:
stress extension (ΔL) is caused by tensile stress
compression (ΔL) is caused by compressive stress
Fig 1
 
stress shearing (ΔL) is caused by a shear stress
Fig 2
 
stress hydraulic (ΔV) is caused by hydraulic force
Fig 3
 

Stress = Modulus of Elasticity X Strain.

Modulus of Elasticity = Stress / Strain.

Strain = ΔL/L (no unit - it is a pure number or dimensionless )

Stress = F/A (UNIT - Nm-2 or Pa)

stress

Typical stress-strain curve for a solid metal rod.
B - Limit of Proportionality. Hooke's law obeyed (F=kx).
C - Yield Strength. Beyond the yield strength increased stress will cause permanent deformation.
F - Fracture Point

A - B Extension is proportional to the applied force. F=kx. The material is elastic and returns to its original size when the stress is relieved.
B - C (elastic limit) Extension is NOT proportional to the applied force. F≠kx. The material is elastic and returns to its original size when the stress is relieved.
C - D Extension is NOT proportional to the applied force. F≠kx. The material is NOT elastic and DOES NOT return to its original size when the stress is relieved. When the stress is relieved the material follows the line DE.
F - Wire breaks.

Fig 4
 
stress

Difference in shapes of Stress-strain curves for different materials.

A - Brittle material like High Carbon Steel. The material is strong because there is little strain for large stress. There is little or no plastic deformation. and the material fractures suddenly.
B - Strong materials but not ductile eg. Steel. Very little stretch and breaks suddenly.
C - Ductile material e.g. Copper. Large plastic region where necking occurs.
D - Plastic Material e.g. rubber.

Fig 5
 

Tension and Compression

Recall Stress = Modulus of Elasticity X Strain and

Strain = ΔL/L

Also Stress = F/A

so F/A = E ΔL/L where E is called Young's Modulus and has the units Nm-2 or Pa.

Note for some materials like steel the compressive Young's Modulus may be similar to the tensile Young's Modulus but for others like concrete they are very different and this may nbe very important in engineering construction.

Shearing

Although shearing stress is also force per unit area, the strain lines in a different plane. In the case of tensile or copressive stress the force is perpendicular to the crosssectioanal area, here it is in the same plane. This is a twisting or cutting kind of action rather than an extending or compressing force as in tensile or copressive stress.

F/A = E ΔL/L where ΔL is shown in Fig 2.

Hydraulic Stress

P = B ΔV/V

We used this is when we were calculating speed of waves in liquids like water in PHYS2005.

 

Summary

Stress is the restoring force per unit area which causes strain (fractional change in dimension).
Types of Stress

— tensile stress
— compressive stress (caused by compression)
— longitudinal stress (caused by streching)
— shearing stress (caused by shear forces)
— hydraulic stress (caused by hydraulic forces)

Hooke's law

— for small deformations, stress is directly proportional to the strain for many materials (linear part of stress-strain curve).
— this modulus of elasticity is the constant of propotionallity.
— Young's modulus is used for stress / stain caused by tensile and compressive forces.
— shear modulus is used for stress / stain caused by shear forces.
— bulk modulus are used is used for stress / stain caused by hydraulic stress.
— describe the elastic behaviour of objects as they respond to deforming forces.

A class of solids called elastomers does not obey Hooke's law. Some polymers which are elastomers include polyisoprene or natural rubber, polybutadiene, polyisobutylene, and polyurethanes. Elastomers can be stretched to many times their original length without permanent deformation.

When an object is under tension or compression Hooke's law takes the form: Y =  [(F/A) /(ΔL/L)] where Y is the Young's modulus for the object, F is the magnitude of the applied force causing the strain, A is the cross-sectional area over which F is applied (perpendicular to A), ΔL is the extension or compression and L is the original length of the material. ΔL/L is the tensile or compressive strain of the object and the stress is F/A.

Shear Stress

— a pair of forces (F) when applied parallel to the upper and lower faces of a solid deforms the solid so that the upper face moves sideways with respect to the lower.
— the horizontal displacement ΔX of the upper face is perpendicular to the vertical height L.
— the corresponding stress [(F)/(A)] is the shearing stress and causes a strain [(ΔX)/(L)].
— this type of stress is possible only in solids.
— Hooke's law takes the form [(F)/(A)] = G [(ΔX)/(L)] where G is the Shear Modulus of the material.

Hydraulic Stress

— hydraulic compression due to a stress (P) exerted by a surrounding fluid where p is the pressure(hydraulic stress) on the object due to the fluid.
— Hooke's law takes the form P = B [(ΔV)/(V)] where B is the Bulk Modulus of the material.
— the volume strain [(ΔV)/(V)] is the absolute fractional change in the object's volume due to that pressure .

Young's Modulus

— Metals have larger values of than alloys and elastomers.
— large value of Young's modulus requires a large force to produce small changes in its length (strength)
— only holds for solids

Bulk modulus

— for solids, liquid and gases and is
— the change in volume when the entire body is under the uniform stress and the shape of the body remains unchanged.

Shear Modulus

— only holds for solids

Poisson ratio

— a deforming force in one direction can produce strains in other directions.
— the proportionality between stress and strain in such situations cannot be described by just one elastic constant.
— wire under longitudinal strain, the lateral dimensions (radius of cross section) will undergo a small change, which is described by another elastic constant of the material.
— Poisson ratio μ = - εt / εl . where. μ = Poisson's ratio. εt = transverse strain. εl = longitudinal or axial strain

Physicist vs Layman
Layman — thinks that a material which stretches more is more elastic
Physicist — material which stretches to a lesser extent for a given load is considered to be more elastic.

   
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Concept by Kishore Lal. Programmed by Kishore Lal... Copyright © 2015 Kishore Lal. All rights reserved.