St Venant theory

These solutions, derived by St. Venant55, apply only to small values of 6 and to isotropic materials. A basic assumption of the St. Venant theory is that the displacement u is given by... 

Torsional work is normally based on St. Venant s theory," with a correction being made for the axial load." Ladizesky and Ward have shown that the St. Venant theory for small strain elasticity is not adequate in the situation where two shear moduli contribute to the elastic behaviour and have demonstrated that experimental extrapolation to zero axial stress is also required in this case. 

St. Venant theory. Time dependence, and the anisotropy of the strain dependence of the compliances were discussed. 

Also in 1963 Raumann reported measurements on specimens prepared from amorphous sheets stretched below the glass transition temperature. The experimental method was identical with that reported earlier for polyethylene. Major features were that as the result of orientation q increased to over five times the isotropic value, and the shear modulus at 90° dropped to about half the isotropic shear modulus other moduli showed only small variations from the isotropic values (Ladizesky and Ward have commented that Raumann s torsional moduli are probably in error, owing to her assumptions of the validity of the St. Venant theory, and an inadequate correction for tensile stress). 

The St Venant theory (see Reference 15, p. 283) gives the torque gz required to produce the twist r in a specimen of length I, thickness a and width b ... 

The double extrapolation procedure invoked involves firstly extrapolation of measurements of torque on specimens with various known axial tensions to zero axial tension. These extrapolated values were then used in the St. Venant expression for torsion of non-circular cylinders to obtain apparent values of 566 for rectangular prisms of various aspect ratios (width rthickness). If the theory was rigorous these values of 566 would be independent of aspect ratio. However, this was not the case and a second extrapolation was made to obtain a value for 5166 at the limiting aspect ratio at which the theory could be regarded as most rigorous. 

Now it cannot escape the casual reader that the various theories of necking in cold-drawing all describe the comportment of fibers and thin strips. This is not accidental indeed, to the untutored eye the specimens used by Zapas and Crissman were about 15 cm. long, only several centimeters wide, and of negligable thickness. How Is this related to the necking Can a full three dimensional theory based on appropriate principles also predict this instability or Is the geometry of the specimen also very important The mechanical phenomenon is not restricted in this way it occurs in tubes for example. Some work of Spector [23] may be of use here. From a different perspective, it is possible to ask if there are families of time dependent St. Venant-type solutions for this sort of material which display the appropriate behavior. 

Finally the problem of bending of a circular cylinder with cylindrical cavities is solved according to St. Venant s theory ... 

The maximum elastic strain theory (St. Venant s theory) states that inception of failure is due if the largest local strain, 3, within the material exceeds somewhere a critical value e. The failure criterion, therefore, is derived as... 

The term in parentheses is an orientation-dependent and minor kinetic contribution derived from the assumption of successive failure of many elements (in the technically relevant range of values from 40 to 200 kJ/mole the term in parentheses assumes values between 1.17 and 1.0). If the nature of the breaking elements is not changed during sample treatment or fracture development, then P can be considered to be constant. The major effect on 7 is derived from the local stress concentration 0/ 0 which here is equal to the stiffness ratio E/E. This theory essentially predicts, therefore, that the increase in strength is equal to the increase in stiffness it is based on the presumption that the elements break indeed at a critical local strain (kinetic version of St. Venant s criterion of maximum strain). A different interpretation of the strength of oriented samples which is based on the rotation of flaws has been presented in an earlier section of this chapter. 

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