Chapter 3

Elasticity and Strength of Materials

Elasticity and Strength of Materials*When a force is applied to a body, the shape and size

of the body change .*Depending on how the force is applied, the body:

may be stretched, compressed, bent, or twisted.Elasticity: is the property of a body that tends to return the body to its original shape after the force is removed.

If the applied force is sufficiently large, however, the body is distorted beyond its elastic limit, and the original shape is not restored after removal of the force.

Elasticity*It is possible to change the shape or size of an object

or both by applying external forces.*The material that is deformed always resists the

deformation by internal reaction forces.*The internal reaction forces that resist the

deformation are due to short-range forces between atoms or between molecules.

ElasticityElasticity is the ability of a substance to restore its original shape and size after deformation.

Types of materials according to its Elasticity:1-Elastic materials

Materials that restore its original shape and size after deformation are called elastic materials.

2-Inelastic materialsMaterials that cannot restore its original shape and size after deformation are called inelastic materials.

*The deformation of materials is described in terms of the concept of stress and strain.

stressThe stress is the internal reaction (resistance force) inside a material to an applied external force.

The internal resistance force is equal in magnitude and opposite in direction to the external force causing the deformation and they are both distributed over a given area of the body. See Fig. (3.1)

The stress is defined as the force per unit area.Stress = Force /area N/m2 (Pa)

The strainThe strain is a measure of the degree of deformation in an object as a result of stress.Hook's LawIt is found that, for small values of stresses, the stress proportional to the strain,This proportionality between stress and strain is called Hook's law.

The constant of proportionality between the stress and strain depends on:

1 .The type of material being deformed and2 .The nature of deformation.

Elastic modulusThe proportionality constant between the stress and strain is called the elastic modulus. The elastic modulus is the ratio of stress to strain:

Elastic modulus = stress / strain

Types of deformations and an elastic modulus

1-Young’s moduluswhich measure the resistance of a solid to a change in its length.

(Elastic modulus)a. Tensile b. Compressive

2-Shear moduluswhich measure the resistance to motion of the planes of a solid sliding past each other

3.Bulk moduluswhich measures the resistance that solids or liquids offer to changes in their volumeTypes of stress and strain

1 .Tensile and compressive stress and strain.A. Tensile stress

Tension results in a body when it is acted by two forces equal in magnitude directed away from each other and acting along the same lineThis stress is called tensile stress. The tensile stress increases the body length as shown in figure (3.2).

Fig. (3.2) tensile stress

Tensile stress = Fn / A

Where Fn is the force acting perpendicular to the area A.

B. Compressive stressCompression results when a body is acted by tow forces that are equal in magnitude, directed toward each other, and acting along the same line.This stress is called compressive stress. The compressive stress decreases the body length as shown in figure (3.3).

Fig. (3.3) compressive stress

Compression stress = F/A

c. Tensile and compression strainFigure 3.5 shows a bar whose natural length is lo and which elongates to a length l when equal and opposite pulls are exerted at its ends .

The tensile strain in the bar is defined as the ratio of the increase in the length to the original length:

Tensile strain = l - lo / lo = Δl / lo

Fig 3.4

The compressive strain of a bar in compression is defined :as the ratio of the decrease in length to the original length.

Compressive strain = - Δl / lo

Young’s modulus YThe ratio of tensile stress to tensile strain for a given material equals the ratio of the compressive stress to compressive strain. This ratio is called Young’s modulus.

Hook's law for tensile and compressive stress and strain is given as

i-Shear stressA body is said to be in shear or

under tangential stress if it is not moving and is acted by two forces

of equal magnitude directed towards each others and not acting

along the same line.

3.2 Shear stress (tangential stress) and strain

Fig. (3.5) a block subjected to shearing stress

Shear stress = F / A

ii-Shear strain is defined as the ratio of the displacement x of corner b to the transverse dimension h :

iii-Shear modulus S

2.3Volume stress (the pressure) and strainIn fluids, the force per unit area is called pressure. Since fluids can flow (slips) no shearing (tangential force) can occurs in fluid. The force exerted by a fluid is always normal to the surface acted by this force.

Volume strainThe strain produced by a hydrostatic pressure, called a volume strain, is defined as the ratio of the change in volume, ΔV, to the original volume V. It also has no units:

Volume strain = ΔV / V

The volume stress (pressure) P = F/A .

The bulk modulusThe modulus relating the hydrostatic pressure to the corresponding volume strain is called the bulk modulus, denoted as B .

The bulk modulus is defined as the negative of the ratio of the change in pressure and the change in volume.

B = - ΔP / (ΔV / V)Hook's law for volume stress (pressure) and strain is written as

Elasticity and plasticityStress-strain graph

Figure (3.7): Stress strain curve and the hysteresis loop .

*When any stress is plotted against the appropriate strain, the resulting stress-strain diagram is found to have several different shapes, depending on the kind of material.

*A typical stress-strain diagram for a ductile metal is shown in Fig.3.6. We note from this figure that :

*During the first portion of the curve (up to a strain of less than 1 percent), the stress and strain are proportional to the point a, the proportional limits, is reached. The region in which stress and strain are proportional is called Hook’s law region

The proportional limit The maximum stress up to which the stress proportional to strain i.e. the material obeys Hook’s law.

*From a to b, stress and strain are not proportional, but if the load is removed at any point between O and b, the curve will be retraced and the material will return to its original length.

*In the region Ob, the material is said to be elastic, and the point b is called the elastic limit, or the yield point. Up to this point, the forces exerted returns to its original shape; work done in producing the deformation is recovered .

The elastic limit:The maximum stress, up to which if the stress is removed the material will return to its original length.

* If the material is loaded further, the strain increases rapidly, but when the load is removed at some point beyond b, say c, the material does not come back to its original length i.e.: permanent strain.

*Further increase load beyond c produces a large increase in strain until a point d is reached at which fracture takes place.

* From b to d, the metal is said to undergo plastic flow or plastic deformation .

Ductile materials If large plastic deformation takes place between the elastic limit and the fracture point, the metal is said to be ductile.

Brittle materials If, however, fracture occurs soon after the elastic limit is passed, the metal is said to

be brittle .

Elastic hysteresis.The lack of coincidence of the curve for increasing and decreasing stress is known as elastic hysteresis. When the stress-strain relation exhibits this behavior the work done by the material in returning to its original shape is less than the work required to

deform .

Breaking stress The stress required to cause actual fracture of a material is called the breaking stress or the ultimate

strength .Hook’s law can be stated as: Within the proportional limit where the stress proportional to the strain, the elastic modulus of a given material is constant, it depends only on the nature of the material.

K is the force constant, From the relation ,

Example 1 A human biceps may exert a force of the order of 600 N on the bones to which it is attached. If the muscle has a cross-sectional area at its center of 50 cm2 = 0.005 m2, and the tendon attaching its lower end to the bones below the elbow joint has a cross-section of 0.5 cm2 = 5 x10 -5 m2, find the tensile

stress in each of the muscle and tendon .