Elements of Solid MechanicsBy: Md.Mohit-ul Alam

Solid mechanics is the branch of mechanics that studies the behavior of solid materials

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Solid Mechanics Deals with

•Motion and deformation of material under action of • Force• Temperature change• Phase change• Other external or internal agentsThese changes lead us to some properties that are called Mechanical properties

Mechanical properties

• Some of the Mechanical Properties• Ductility• Hardness• Impact resistance• Fracture toughness• Elasticity• Fatigue strength• Endurance limit• Creep resistance• Strength of material

Mechanical Properties• Ductility: ductility is a solid

material's ability to deform under tensile stress

Hardness of a material may refer to resistance to bending, scratching, abrasion or cutting. Impact resistance is the ability of a material to withstand a high force or shock applied to it over a short period of timePlasticity: ability of a material to deform permanently by the application of force

Mechanical Properties• fracture toughness is a property which describes the ability of a

material containing a crack to resist fracture• Elasticity is the tendency of solid materials to return to their

original shape after being deformed• Endurance strength/ Fatigue strength: The highest stress that a

material can withstand for a given number of cycles without breaking —called also endurance strength

• Endurance limit: In fatigue testing, the maximum stress which can be applied to a material for an infinite number of stress cycles without resulting in failure of the material is called Endurance limit

• Creep Resistance: It’s the ability of a material not to deform permanently or slowly under the influence of Mechanical Stress.

Creep means Deformation

Creep(Slow Deformation)• Reasons Behind Creep• Long term stress below yield strength• Material subjected to long term heat when in operation• Working condition near to the melting point of the material• Total time of useCreep is a time dependent phenomenonStages of Creep:

Hardness Test

• Hardness measurement scales• Rockwell • Brinell• Vickers• Knoop• Shore• Nano Indentation

Brinell: Brinell hardness is determined by forcing a hard steel or carbide sphere of a specified diameter under a specified load into the surface of a material and measuring the diameter of the indentation left after the test. Load applied is much high (0.5 - 3 ton) for about 10 - 30 sec..

Vickers: Hardness of ceramics and metals with extremely hard surfaces are measured by this method. The surface is subjected to a standard pressure (120 kgf) for a standard length of time by means of a pyramid-shaped diamond (1360). Here also indentation diameter is measured.

Stress and Strength

• Stress in a material: In solid mechanics, stress is a physical quantity that express the internal force per unit area that neighboring particles of a continuous material exert on each other.• Strength of material: it is the measurement in

engineering of the capacity of metal, wood, concrete, and other materials to withstand stress and strain.• Strain: It is the deformation of material due to

stress• Strain = del l/L

Strength of Material

•Different strengths are• Yield strength/ Tensile strength•Ultimate Tensile strength•Rupture strength•Compressive strength• Impact strength

Stress strain Curve of Mild Iron

Stress Concentration• Stress concentration is defined as Localized

Stress considerably Higher than average due to abrupt changes in geometry or localized loading

• Causes of Stress concentration• Sharp corner• Geometric Discontinuities• Cracks, Hole in the member• Abrupt changes is cross section PreventionRemove sharp corner and edge by filleting or chamferingAvoiding abrupt changes in cross section

Wear• In materials science, wear is erosion or sideways

displacement of material from its "derivative" and original position on a solid surface performed by the action of another surface

• Stages of wear• Primary stage where surfaces adapt to each other and the

wear-rate might vary between high and low.• Secondary stage, where a steady rate of ageing is in motion.

Most of the components operational life is comprised in this stage.

• Tertiary stage, where the components are subjected to rapid failure due to a high rate of ageing.

Fatigue

Deflection• In engineering, deflection is the degree to which a structural

element is displaced under a load. It may refer to an angle or a distance.

Different type of Beam Deflection

• End Loaded Cantilever Beams:• Formulas

• F= Force acting on the tip of the beam• L= Length of the beam (span)• E= Modulus of elasticity• I= Area moment of inertia• ∂= Deflection• ⱷ = angle of deflection

Uniformly loaded cantilever beam

• The deflection, at the free end B, of a cantilevered beam under a uniform load is given by

• q= Uniform load on the beam (force per unit length

Center loaded simply supported beam

• The elastic deflection (at the midpoint C) of a beam, loaded at its center, supported by two simple supports is given by:

Intermediately loaded beam• The maximum elastic deflection on a beam supported by two

simple supports, loaded at a distance from the closest support, is given by:

• a = Distance from the load to the closest support

Torsion• Torsion :It is the twisting of an object due to an applied torque

(twisting moment)is expressed in newton metres (N·m)

• T = is the applied torque or moment of torsion in Nm• is the maximum shear stress at the outer surface• J= Polar moment of Inertia• r= is the distance between the rotational axis and the farthest point

in the section (at the outer surface).• l = is the length of the object • φ= is the angle of twist in radians.• G= Modulous of Rigidity

• The angular frequency can be calculated with the following formula:

• The torque carried by the shaft is related to the power by the following equation:

SPRINGS• A spring is an elastic body, which deflects under

load and recover to its original shape upon release of the load. • It is also resilient member which stores energy

once deflected and releases the same as it recovers to its original shape.

APPLICATIONS OF SPRINGS1. Applying forces and controlling motions, as

found in brakes and clutches.2. Measuring force, as in the case of spring

balance. Ex weighing machine (Analogue).3. Storing energy, as in the case of clock springs &

springs used in toys.4. Reduce the effect of shock loading, as in the

case of vehicle suspension ring.5. Changing the vibration characteristics of

machine mounted on foundation beds.

CLASSIFICATION OF SPRINGS1. Helical tension and compression spring:

• The helical springs are made up of a wire coiled in the form of a helix and are primarily intended for compressive or tensile loads.

• The cross-section of the wire from which the spring is made may be circular, square or rectangular.

• Helical compression springs have applications to resist applied compression forces

CLASSIFICATION OF SPRINGS• The major stresses produced in helical springs are

shear stresses due to twisting. The load applied is parallel to or along the axis of the spring.

CLASSIFICATION OF SPRINGS

Helical compression spring

CLASSIFICATION OF SPRINGS

2. Helical torsion springs:

• The principal stress induced are tensile and compressive due to bending.• These are similar to the helical

tension and compression springs.• In these springs, the load is

subjected to torsion about its axis.

CLASSIFICATION OF SPRINGS

Helical torsion springs

CLASSIFICATION OF SPRINGS3. Spiral Springs: • The principal stress induced are

tensile and compressive due to bending.

• These are made of flat strip, wound in the form of spiral.

• This is subjected to torsion about its axis.

CLASSIFICATION OF SPRINGS

Spiral Spring

CLASSIFICATION OF SPRINGS4. Leaf or laminated Springs :• The principal stresses are tensile and compressive de to

bending.• These are made of flat strips of varying lengths , clamped

together. • These may be cantilever, semi-elliptic or full elliptic in

form.

CLASSIFICATION OF SPRINGS

Leaf Springs

CLASSIFICATION OF SPRINGS5. Belleville springs:

• The principal stress are tensile and compressive de to bending.• These are made in the form

of coned discs which may be stacked so as to give the required spring load-deflection characteristics.

CLASSIFICATION OF SPRINGS

Belleville springs

MATERIALS OF SPRINGS• Commonly from alloy steels, High carbon steel (0.7 – 1 % C)

or carbon alloy steel.• The most common spring steels are music wire, oil tempered

wire, silicon, Chrome vanadium.• Stainless steel, Spring brass, Phosphor bronze, monel &

titanium are used for corrosion resistance spring.

TERMINOLOGY IN SPRINGS

TERMINOLOGY IN SPRINGS• Solid Length :When the compression spring is compressed

until the coils come in contact with each other, then the spring is said to be solid. The solid length of a spring is the product of total number of coils and the diameter of the wire.

Solid length, L s = n x dWhere, n = number of coils

• Free Length (Lo) : The free length of a compression spring is the length of the spring in the free or unloaded condition.

Free length, Lo = Solid Length + Maximum Compression deflection + Clearance between adjacent coils (1mm).

TERMINOLOGY IN SPRINGS• Spring Index (C): The ratio of mean coil diameter to wire

diameter. A low index indicates a tightly wound spring (a relatively large wire size wound around a relatively small diameter mandrel giving a high rate).

C=d/D• Spring rate(K): The Spring rate is defined as the force required

to produce unit deflection of the spring. It can also be said as stiffness or spring constant.

K =F/ ᵟ Where F is the load applied,

ᵟ is the deflection of the spring.

TERMINOLOGY IN SPRINGS• Pitch (P) : The distance from center to center of the wire in

adjacent active coils. The pitch of the coil is defined as the axial distance between adjacent coils in uncompressed state.

P = Free length / (n-1)

SPRING COMBINATIONS• Parallel arrangement: In parallel the spring are arranged side by

side. The deflection in spring combination is equal to individual spring.

Ke = K1 + K2 + ...... + Kn

SPRING COMBINATIONS• Series Arrangement: When the spring are arranged in series, the

total deflection of the spring combination is equal to sum of the deflection of individual springs.

1/ Ke = 1/ K1 + 1/ K2 +... + 1/ Kn