By L. I. Sedov, J. R. M. Radok
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Extra resources for A course in continuum mechanics Vol. I Basic equations and analytical techniques
A copper sleeve loosely surrounds a steel bar, they are of the same length and stress free at temperature Γ 0 , and their ends are rigidly fixed together. Determine the stress in the copper and steel at temperature T. Superscripts c and s will be used to denote the two materials ; A = cross-sectional area; E = modulus of elasticity; a = co efficient of expansion. Equilibrium The change in temperature will cause a change in length of the assembly, but because of the different values for a, each material will put a restraint on the other.
Calculate the nominal stresses and strains from the data and plot on one sheet the nominal stress-strain curves (a) for all the values, (b) for strains below 0 005 to a larger scale. Also plot a true stress-true strain curve for the results beyond a strain of 0-025. e. Λ AQLQ From the curves determine the following: (1) Limit of proportionality stress; (2) Upper yield stress; (3) Lower yield stress; (4) 0-2 per cent proof stress; (5) 0-5 per cent proof stress; (6) Tensile strength ; (7) Nominal stress at fracture; (8) Actual stress at fracture; (9) Elongation on 2 in.
1 it is clear that each of the bars will stretch a small amount under the action of the load and the joint O will move to O' where 00' is δ12. For the three members to remain joined at 0', a condition of geo metrical compatibility, there will be a simple relation for the geometry of deformation between O'B, O'C and O'A, as shown by the dashed lines. Thus OB= OC=lx = i/2 x l2 and the extension of OB and OC δΐ^ = δ12 cos 45° = (5/ 2 /Λ/2. This step appears to have complicated the solution rather than assisted it, since there is now a further unknown quantity (5/2.