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Chapter 4 Elasticity by Ibrhim AlMohimeed BMTS /11/2013

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Video BMTS /11/2013

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BMTS 3533 Deformation When a object crash a car, the car may not move but it will noticeably change shape. A change in the shape due to the application of a force is a deformation. Even very small forces are known to cause some deformation. 19/11/2013

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BMTS 3534 Cont. Deformation 19/11/2013

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BMTS 3535 Hooke’s law 19/11/2013

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BMTS 3536 Cont. Hooke’s law The proportionality constant k depends upon a number of factors for the material. For example, a guitar string made of nylon stretches when it is tightened, and the elongation ΔL is proportional to the force applied. Thicker nylon strings and ones made of different material (steel) stretch less for the same applied force, implying they have a larger k. 19/11/2013

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BMTS 3537 Hooke’s law Examples Example 4.1 Example 4.1: A force of 600 N will compress a spring 0.5 meters. What is the spring constant of the spring? Example 4.2: Example 4.2: A spring has spring constant 0.1 N/m. What force is necessary to stretch the spring by 2 meters? 19/11/2013

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BMTS 3538 Elastic Modulus ΔL depends on the material of the subject. ΔL is proportional to the force F. ΔL is proportional to the original length L0. ΔL is inversely proportional to the cross-sectional area of the subject 19/11/2013

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BMTS 3539 Cont. Elastic Modulus 19/11/2013

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BMTS Stress and Strain 19/11/2013

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BMTS Cont. Stress and Strain 19/11/2013

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BMTS Stress and Strain Example Example 4.3: Example 4.3: A steel wire 10 m long and 2 mm in diameter is attached to the ceiling and a 200-N weight is attached to the end. What is the applied stress? If the wire stretches 3.08 mm. What is the longitudinal strain? 19/11/2013 L LL A = π × r 2

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BMTS Stress and Strain Curve 19/11/2013

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BMTS Cont. Stress and Strain Curve 19/11/2013 Flash

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BMTS Cont. Stress and Strain Curve 19/11/2013 the object will returns to its original length (or shape) when the stress acting on it is removed. Elastic behavior: the object will returns to its original length (or shape) when the stress acting on it is removed. the object will NOT returns to its original length (or shape) when the stress acting on it is removed. Plastic behavior: the object will NOT returns to its original length (or shape) when the stress acting on it is removed. the stress above is not longer proportional to strain. Proportionality limit: the stress above is not longer proportional to strain.

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BMTS Cont. Stress and Strain Curve 19/11/2013 the object will returns to its original length (or shape) when the stress acting on it is removed. Elastic behavior: the object will returns to its original length (or shape) when the stress acting on it is removed. The maximum stress that can be applied without resulting permanent deformation. Elastic Limit: The maximum stress that can be applied without resulting permanent deformation. at which there are large increases in strain with little or no increase in stress. steel exhibits this type of response. Yield Stress: at which there are large increases in strain with little or no increase in stress. steel exhibits this type of response.

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BMTS Cont. Stress and Strain Curve 19/11/2013 when yielding has ended, a further load can be applied to the object. Strain Hardening: when yielding has ended, a further load can be applied to the object. the maximum stress the object can withstand. Ultimate Strength: the maximum stress the object can withstand. after the ultimate stress, the cross-sectional area begins to decrease in a localized region of the object. Necking: after the ultimate stress, the cross-sectional area begins to decrease in a localized region of the object.

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BMTS Stress & Strain Curve Example 19/11/2013 Example 4.4: Example 4.4: The elastic limit for steel of 3.14 x m 2 is 2.48 x 10 8 Pa and The ultimate strength is 4.89 x 10 8 Pa. a)What is the maximum weight that can be supported without exceeding the elastic limit? b) What is the maximum weight that can be supported without breaking the wire?

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BMTS Stress & Strain Curve Example 19/11/2013 Example 4.5: a stress applied to an object was 6.37 x 10 7 Pa and its strain was 3.08 x Find the modulus of elasticity for object?

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BMTS Stress & Strain Curve Example 19/11/2013 Example 4.6: Young’s modulus for brass is 8.96 x Pa. A 120-N weight is attached to an 8-m length of brass wire; find the increase in length. The diameter is 1.5 mm. 8 m LL 120 N

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BMTS Strain Hardening 19/11/2013

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BMTS Cont. Strain Hardening 19/11/2013 Strain Hardening - If the material is loaded again from the beginning stage of strain hardening, the curve will be the same Elastic Modulus (slope). - The material will has a higher yield strength.

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BMTS Types of Material 19/11/2013 Isotropic materials: have elastic properties that are independent of direction. Anisotropic materials: whose properties depend upon direction.

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BMTS Material Behavior 19/11/2013 Behavior of materials can be broadly classified into two categories: Brittle (Example: glass, ceramics) Ductile (Example: Metals; Gold, silver, copper, iron )

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BMTS Brittle Behavior 19/11/2013 Brittle \fracture at much lower strains. yielding region is nearly nonexistent. often have relatively large Young's moduli and ultimate stresses.

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BMTS Ductile Behavior 19/11/2013 Ductile \withstand large strains before failure. yielding region often takes up the majority of the stress-strain curve capable of absorbing much larger quantities of energy before failure.

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BMTS Resilience & Toughness 19/11/2013 Resilience: Amount of energy stored in material up to elastic limit up to elastic limit per unit volume. \ Amount of energy stored in material up to elastic limit up to elastic limit per unit volume. Toughness: Amount of energy stored in material up to fracture per unit volume

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BMTS Cont. Resilience & Toughness 19/11/2013

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BMTS Shear modulus 19/11/2013 The shear modulus (S) is the elastic modulus we use for the deformation which takes place when a force is applied parallel to one face of the object while the opposite face is held fixed by another equal force.

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BMTS Cont. Shear modulus 19/11/2013

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BMTS Bulk Modulus 19/11/2013 An object will be compressed in all directions if inward forces are applied evenly on all its surfaces. It is relatively easy to compress gases and extremely difficult to compress liquids and solids.

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BMTS Bulk Modulus 19/11/2013

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BMTS Bulk Modulus Example 19/11/2013 Example 4.6: A hydrostatic press contains 5 liters of oil. Find the decrease in volume of the oil if it is subjected to a pressure of 3000 kPa. (Assume that B = 1700 MPa.)

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End of the Chapter 19/11/2013BMTS 35334

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