Choosing between the two classical descriptions of deformation arising from a combined stress state will depend upon the manner in which stress is distributed. When the latter is uniform, there is little difference between predictions from the Hencky deformation theory and the Prandtl-Reuss flow theory. In fact, experiment appears to agree better with the simpler, closed-form Hencky predictions to deformation under non-radial loading paths. In the analysis of the deformation behaviour of structures under non-uniform stress states, experiment shows that the effects of history are insignificant with proportional loadings from zero stress. The Hencky total-strain deformation theory will represent, within closed solutions of acceptable accuracy, the distribution of stress and the load-deformation response with the transition from elastic to fully plastic behaviour. Deviations arise between the total and incremental theories under non-proportional loadings, including stepped-stress paths or paths in which one component of strain is constrained. Although the Hencky theory appears to remain conservative in its prediction of strain, the generally held view, partly substantiated here, is that the incremental Prandtl-Reuss theory (with elastic compressibility) provides more realistic predictions whenever history effects are present. However, it is known that errors can arise from Prandtl-Reuss. We have seen in Chapter 3 that it does not account for plasticity in anisotropic material. Other deficiencies include the neglect of softening resulting from stress reversal and cold creep, i.e. low-temperature, time-dependent strain.
References
1. Hencky H. Zeit Angew Math Mech, 1924,4,323.
2. Ilyushin A. A. Prik Matem Mekh, AN SSSR, 1946,10, 347.
3. Nadai A. Plasticity: A Mechanics of the Plastic State of Matter, 1931, McGraw-Hill, New York.
4. Prandtl L. Proc. 1st Int Congr Appl Mech. Delft, 1924, p.43 (also Zeit Angew Math Mech 1928,8,85.)
5. Reuss A. Zeit Angew Math Mech, 1930,10,266.
6. ZyczkowsM M. Combined Loadings in the Theory of Plasticity, 1981, PWN, Warsaw.
7. Massonnet C. Plasticity in Structural Engineering, Fundamentals and Applications, 1979, Ch. 1, Springer Verlag.
8. Hohensemer K. Zeit Angew Math Mech, 1951,11,515.
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10. Hohensemer K. and Prager W. Zeit Angew Math Mech, 1932,12,1.
11. Prager W. Proc: 5th Int Congr Appl Mech, Cambridge, 1938, S.234.
12. Neuber H. Kerbspannungslehre, 2 Aufl, Berlin Gottingen, Heidelberg 1958.
13. BettenJ. Zeit Angew Math Mech, 1975,55,119.
14. Gill S. S. /. Appl Mech, 1956,23,497.
15. Schlafer J. L. and Sidebottom O. M. Expl Mech, 1969,9,500.
16. Shammamy M. R. and Sidebottom O. M. Expl Mech, 1967,7,497.
17. Zyczkowski M. Rozpr Inz, Polska Akademia, 1955,3,285.
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19. Sved G. and Brooks D. S. Acta Techn. Hung, 1965,50, 337.
20. Smith J. O. and Sidebottom O. M. Inelastic Behaviour of Load-Carrying Members, 1965, J. Wiley and Son.
21. Prager W. and Hodge P. Theory of Perfectly Plastic Solids, 1951, J. Wiley and Son, New York.
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and Spence J.) , Elsevier.
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25. Mii H. J. Japan Soc Appl Mech, Tokyo, 1950,3,195.
Exercises
Uniform Stress States
4.1 A thin-walled cylinder is stressed to the point of yield under an axial-shear stress ratio rfr= 2.
The shear strain is then held constant while the cylinder is subjected to increasing tension. Derive expressions for the subsequent variation in axial strain with the axial and shear stresses (a, r) according to the Hencky and Prandtl-Reuss theories. Compare, graphically showing the normalised stress asymptotes a/Y and tlY. Take G = 77 GPa, v= 0.3 and F = 230 MPa.
4.2 A thin-walled cylinder is stressed to the point of yield under an axial-shear stress ratio ofr= 2.
The axial strain is then held constant while the cylinder is subjected to increasing torsion. Derive expressions for the subsequent variation in shear strain with the axial and shear stresses (a, r) according to the Hencky and Prandtl-Reuss theories. Compare, graphically showing the normalised stress asymptotes alY and tfY. Take G = 77 GPa, v= 0.3 and Y = 230 MPa.
NON-HARDENING PLASTICITY 12S 4.3 The internal pressure in a closed-end, thin tube is increased to produce yielding of the tube material under a ratio erg/a, = 2. The circumferential strain is then held constant while the tube is progressively twisted. Show, according to the Prandtl-Reuss theory, that r varies with total shear strain y*as
Y = — — In, 2Gfl + v} \ 1 - T
1 - 2v)T"
(I + v)
where T' = ttk. Why is axial plastic strain absent when the radial stress is ignored?
4.4 In a thin tube of non-hardening material the axial strain at yield is held constant. Show, with progressively increasing circumferential tension, that the Prandtl-Reuss prediction to the ensuing hoop strain is given by
AE in
Y(2v- 1)
•J3S
e/2
1-3 5 ,
where Sg= aJE. Hence, show that the normalised stress asymptotes are og/F=2A/3 and aJY=> 1A/3.
4.S A thin-walled tube is twisted undeT torsion to the point of yield. The total shear strain is then held constant while the tube is subjected to an increasing internal pressure for which ag = 2av Show, for the ensuing deformation, that the respective total circumferential and axial strains are given by:
ea = (1 (1 -
- 2 F ) SZ
2v)S,
+ (1 + v) 2^2 - In
i
+n$\
I - ns
t)
where S, = erJY. Plot Sz versus ea'EIY and e^ElY showing the St asymptotes.
4.6 A non-hardening, thin-walled cylinder, with inner and outer radii 7.1 and 7.6 mm respectively, is strained to the tensile yield point. Thereafter the cylinder is subjected to an increasing torque whilst the simultaneous tension is adjusted to maintain the tensile yield strain F/E constant. Compare predictions to the deformation that ensues from the Hencky incompressible theory ( v = Vi) and the Prandtl-Reuss theory with v= %. Take E = 207 GPa and F = 283 MPa.
Non-Uniform Stress States
4.7 A solid cylindrical bar, 14.75 mm diameter, is made from a non-hardening steel with a yield stress of 283 MPa. Given that the torque C and the axial load J" increase proportionately in the ratio P/C
= 0.375 m m " ' , find P and C: (i) at the limit of elastic deformation, (ii) at the elastic-plastic mean radius and (iii) for a fully plastic condition. Assume elastic ^compressibility with E = 207 GPa.
[Answer: (i) 80.5 Nm, 30.1 fcN, (ii) 94.12 Nm, 35.3 kN, (iii) 95.7 Nm, 35.8 kN]
4M Plot for the cylinder in exercise 4.7, the distribution of tensile and shear stress with radius for each of the conditions (i), (ii) and (iii). Assuming elastic unloading, plot the distribution of residual stress corresponding to (ii) and (iii). Take E = 207 GPa and assume elastic incompressibility.
4.9 A solid cylindrical bar, 12.8 ram diameter, is made from a non-hardening steel with a yield stress of 305 MPa. The bar is subjected to a constant torque of 60.7 Nm with a steadily increasing axial force. Calculate the value of the force corresponding to: (i) the limit of elasticity, (ii) an elastic-plastic interface coincident with the mean radius and (iii) full plasticity. Take E = 207 GPa and v=¥t.
[Answer: 21.9 kN, 31.05 kN» 31.6 fcN]
4.10 Plot the distributions of axial and shear stress with radius for the cylinder of exercise 4.9 in each of (i), (ii) and (iii). Determine which condition (ii) and (iii) has the greatest extent of the residual compression following unloading.
4.11 Apply the Prandtl-Reuss solution, with compressible elasticity, to the deformation in a solid bar subjected to torque and axial load which increase proportionately within the plastic range.
4.12 Derive expressions for the normalised stress components S = c?zfY and T = ttk from the incompressible Prandtl-Reuss theory when the shear strain in a solid circular bar is increased while the elastic axial strain is held constant.
4.13 A solid circular bar supports a constant, initially elastic-plastic torque while being subjected to increasing tension. Determine the Hencky stress distributions for suitable stages of the ensuing deformation to M l plasticity.
4.14 Derive the Prandtl-Reuss solutions, for v= 'A, to the stress distributions in a solid circular bar under increasing axial force combined with a constant, initially elastic torque.
4.15 Establish the stress distributions from exercise 4.14 for m = 0.83, when the force has produced a fully elastic bar (p^ = 1) and an elastic-plastic bar with pep = 0.6. Compare graphically with the Hencky solution (see Fig. 4.11) for the same conditions.
4.15 Establish the governing equations from the Prandtl-Reuss theory, with compressible elasticity, for the deformation which ensues when an initially elastic torque Cv is held constant under an increasing force P in a solid bar.
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