derive the generalized hook's law for an isotropic homogeneous elastic solid in terms of lame's constats
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derive the generalized hook's law for an isotropic homogeneous elastic solid in terms of lame's constats
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- Problem 1 (6-1 , Eringen) Prove that in a nonlinear, isotropic, elastic solid the only possible initial stress is a state of hydrostatic stress. In this regard, what can be said about an anisotropic elastic solid?Consider a 50 mm × 50 mm × 50 mm element of graphite-reinforced material that is heated 50°C above some reference state and is restrained in the 2 direction. (a) What are the changes in dimensions of the heated element in the 1 and 3 directions? (b) What stress 02 is required to restrain the element against deformation in the 2 direction? Assume the material properties are independent of temperature. (c) How do the changes in dimensions of the heated element for this partially restrained case compare numerically with the changes in dimensions of the heated element for the case of free thermal strain, equation (2.101)? Why are they different? (d) What are the mechanical strains for this case? Numerically compare the mechanical strains with the fully constrained case, that is, equation (2.119). Hint: To solve the problem use either equation (2.107) or (2.108), assume &₁# 0 and 83 ‡ 0, and solve for 81, 83, and 02.The compressive strength of alumina quartz (Al2O3) is 2.1 GPa. A quartz tube is being used for a vacuum furnace. Both ends of the tube are constrained and cannot move. To what temperature can the tube be heated up from room temperature such that the tube won’t fracture due to the thermal stress?
- 5. Derive expressions for a simple Voigt element in the following experiments: (a) Creep experiment: time-dependent compliance, J(t) (b) Stress-relaxation experiment: time-dependent modulus, E(t) (c) Constant extension rate experiment: change in stress, o(t)(7) A copper rod is 8 m long at 20°C and has a coefficient of linear expansion a = 17 x 10-6 (C°)-!. What is the increase in the rod's length when it is heated to 40 °C?A cylindrical block of concrete is 30 cm long and has a circular cross-section of 10 cm diameter, figure 2-7. It carries a total compressive load of 70 kN, under this load contracted 0.02 cm. Estimates the average compressive stress over a normal cross- section and the compressive strain. 2:47 PM
- Consider a metal with shear modulus G = 40 GPa and atomic radiusr0 = 0.15 nm. Suppose the metal has a solute that results in a misfit of ε =(R − r0)/r0 = 0.14. Compute the elastic misfit energy per mole of solute.Consider dislocations in nickel. If the plastic flow stress is controlled by the stress necessary to operate a Frank-Read source, compute the dislocation density p in the crystal when it is deformed to a point where the resolved shear stress on the slip plane is 100 MPa.Determine the strain energy per unit volume (units of psi) and the strain energy per unit weight (units of in ) that can be stored in each or the materials listed in the accompanying table, assuming that the material is stressed to the proportional limit. DATA FOR PROBLEM 2.7-5 Material Weight Density (lb/in3) Modulus of Elasticity (ksi) Proportional Limit (psi) Mild sleel 0.284 30,000 36,000 Tool steel 0.284 30,000 75,000 Aluminum 0.0984 10,500 60,000 Rubber (soft) 0.0405 0.300 300
- Explain the following microstructures: -Lamellar -Bimodal -EquiaxedConsider dislocations blocked with an average spacing of in a coppercrystal. If the flow stress is controlled by the stress necessary to operate aFrank--Read source, compute the dislocation density ρ in this crystal when itis deformed to a point where the resolved shear stress in the slip plane is 42MPa. Take G = 50 GPa.(6) For an aluminum alloy under constant load, the creep rate increased by a factor of 2 when the temperature was increased from 71°C to 77°C. Find the activation energy and express it in Joule/mole.