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**Elasticity and Strength of Materials**

Chapter 3 Elasticity and Strength of Materials

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**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.

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

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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 materials Materials that cannot restore its original shape and size after deformation are called inelastic materials.

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

The 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)

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**The stress is defined as the force per unit area**

The stress is defined as the force per unit area. Stress = Force /area N/m2 (Pa) The strain The strain is a measure of the degree of deformation in an object as a result of stress. Hook's Law It 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.

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The constant of proportionality between the stress and strain depends on: 1. The type of material being deformed and 2. The nature of deformation. Elastic modulus The 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

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**Types of deformations and an elastic modulus**

1-Young’s modulus which measure the resistance of a solid to a change in its length. (Elastic modulus) a. Tensile b. Compressive 2-Shear modulus which measure the resistance to motion of the planes of a solid sliding past each other

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3.Bulk modulus which measures the resistance that solids or liquids offer to changes in their volume Types 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 line This stress is called tensile stress. The tensile stress increases the body length as shown in figure (3.2).

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Fig. (3.2) tensile stress Tensile stress = Fn / A Where Fn is the force acting perpendicular to the area A.

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B. Compressive stress Compression 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

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

c. Tensile and compression strain Figure 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

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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 Y The 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.

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

i-Shear stress A 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

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**ii-Shear strain Shear stress = F / A**

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:

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iii-Shear modulus S

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**2.3Volume stress (the pressure) and strain**

In 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 strain The 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.

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The bulk modulus The 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

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**Elasticity and plasticity**

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

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*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

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**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.

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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.

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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.

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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.

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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,

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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 = 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.

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