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2E4: SOLIDS & STRUCTURES Lecture 9 Dr. Bidisha Ghosh Notes: lids & Structures

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Hookes Law A material which regains its shape when the external load is removed is considered as perfectly elastic. From tensile tests, it can be seen within the range of elastic behaviour of a material the elongation is proportional to both the external load and the length of the bar. For linearly elastic materials, this Stress is proportional to strain. The factor of proportionality between stress and strain is called, Modulus of Elasticity or Youngs modulus. E has the dimension of stress

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We already know Hookes law, but what does it tell us?

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Tensile Test

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Tensile Test Check this link for tensile test movie:

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Extensometer

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Tensile Test 1. Linear elastic region. Slope of this linear part is the youngs modulus. 2. The proportional limit is the stress when stress-strain relationship is starts to become nonlinear. (Beyond this limit the material is not elastic) 3. Yielding (strain hardening) 4. Ultimate strength 5. necking 6. Fracture Stress 7. Unload-reloading creates strain hardening/work hardening Permanent deformation

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Stress-Strain Diagram The Load-deformation plot does not provide material properties. But, when converted to stress-strain plot it provides all the information needed. Notice elastic limit and proportionality limits are different! Some materials are still elastic beyond the linear (proportional) section of the curve. But in all practical cases they are same. Notice ultimate stress is higher than fracture stress. This is because this graph do not plot the true stress accounting for the reduction in area due to necking. This is called engineering stress. The true stress actually is higher at fracture.

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Glossary Proportionality Limit: The point till which the stress-strain curve is linear. Elastic Limit: The point beyond which the material will no longer go back to its original shape when the load is removed. Yield Point: It is the point at which the material will have an appreciable elongation or yielding without any increase in load. Ultimate Strength: The maximum ordinate in the stress-strain diagram is the ultimate strength or tensile strength. Fracture Strength: It is the strength of the material at rupture. This is also known as the breaking strength. Residual Strain: In the plastic region, after unloading the material does not go back to its original shape and the remaining strain in the material is called residual strain and the elongation is called permanent set. Work Hardening: Also known as strain hardening, after yielding occurs the material can withstand increase amount of stress, showing increase in strength. True stress-strain & engineering stress-strain: The engineering strain is calculated using the initial cross-sectional area of the specimen. Creep: A solid material deforms permanently under the influence of continuous loading below yield stress.

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Stress-Strain Diagram Ductile materials are those which can yield and undergo significant deformation in normal temperature. Brittle materials rupture with little deformation.

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Concrete Concrete is very weak in tension (10% of its compressive strength) and very strong in compression. Concrete behaves like a brittle material when assumed homogenous. compression testing of concrete

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Properties of Typical Materials Material Young's Modulus (Modulus of Elasticity) (GPa) Ultimate Strength (MPa) Yield Strength (Mpa) Aluminum Bone (compression) Concrete (high strength)(compression) 3040 Diamond (C) 1220 Wood (compression) Glass Steel

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Hookes Law: Shear Modulus shear modulus or modulus of rigidity, G Elasticity can be measured for shear loading. Generally a direct shear tests or torsion test can be used. Using Hookes law for the linear elastic part of the stress-strain diagram, Direct shear test on soil!

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Poissons Ratio In elastic range, the ratio of lateral strain to elastic strain is constant. The lateral strain caused due to Poisson's ratio do not result/create any stress in lateral direction. dx dy dz

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Values of Values of The concept is only valid for uniaxial strain and isotropic material. In case of perfectly incompressible material, is 0.5. For all practical cases, 0< Generally, between For steel, assumed to be 0.3 For concrete, assumed to be 0.1 For incompressible material, 0.5 (may be, water) Relation between elastic moduli:

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Strain Energy The external work done on an elastic body in causing it to distort/deform from its original state is stored in the body as strain energy. For perfectly elastic body no dissipation of energy occurs and this energy is recoverable on unloading. Strain energy is the area under the linear part of stress strain curve

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Strain Analysis What happens when we apply 1-D stress? What happens when we apply 2-D stress?

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What happens when we apply 3-D stress? stress and strain are not proportional any more!! Strain Analysis

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How much does the volume change? 3-D case Lets assume initial volume, abc Final volume = Change in volume = Hence, strain or volumetric strain, a b c

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