Strain_energy - Strain energy MECH320-Spring 2008...

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Unformatted text preview: Strain energy MECH320-Spring 2008 Nasser-Eddin M., PhD 1 Outlines 2 3 Car’s Example One component of the suspension system is the springs, which provide a sort of energy reservoir. To be useful, springs must be able to absorb and release energy repetitively without sustaining damage. A material's ability to absorb energy elastically is an important property for such applications. 4 Car’s Example There is another important way in which materials must deal with energy. In this case the issue is not routine elastic handling of nuisance energy, but rather the handling of destructive energy during a catastrophic event. 5 Car’s Example The more energy the material absorbs, the less damage is done to the vehicle's occupants. The ability of a material to absorb energy inelastically is also important. 6 Strain energy To examine how we can quantify the storage of energy in a material, consider the simple catapult shown. By cranking the handle we are able to store energy in the cantilever beam… If we cut the rope, the projectile will be launched into the air as the stored energy in the beam is released. Let's consider a small block of material from the beam in order to examine how the energy is stored… 7 Strain energy (cont.) 8 Strain energy (cont.) 9 Strain energy (cont.) 10 Strain energy (cont.) 11 Strain energy (cont.) The increment in the stored (strain) energy can be calculated by multiplying the average stress by its area to obtain force, and then multiplying by the length change, given by the strain increment times the length. 12 Formulas Formulas To obtain the change in stored energy for a finite change in state, we simply integrate from one strain state to the other. 13 Strain energy (cont.) 14 Strain energy (cont.) 15 Relation to Stress-Strain Curve 16 Linear Elastic Case 17 Linear Elastic Case 18 Modulus of resilience 19 Modulus of toughness 20 21 More on strain energy 22 General case 23 General case 24 Stress decomposition 25 Stress decomposition (cont.) 26 Shear Strain 27 Shear Strain (cont.) 28 Shear Strain (cont.) 29 Shear Strain (cont.) 30 Shear Strain (cont.) 31 Strain (cont.) 32 We thus have these simple relations between deviatoric and total stress components. We now consider the strain energy. 33 Strain Energy 34 Strain Energy 35 Strain Energy 36 Strain Energy 37 Strain Energy 38 Strain Energy 39 Strain Energy 40 Strain Energy 41 Strain energy 42 Deviatoric Stress-Strain 43 Deviatoric stress-strain 44 Equations 45 Prove that: 46 Equations (cont.) 47 Equations (cont.) 48 Equations (cont.) 49 Equations (cont.) 50 Equations (cont.) 51 Equations (cont.) 52 Strain Energy 53 Strain Energy 54 Strain Energy 55 conclusion 56 ...
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