Stress-strain relations shell

Reissner E. Stress-strain relations in the theory of thin elastic shells, J. Math. Physics, Vol. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

Principle. The quantity, E (s), in transform space is analogous to the usual Young s modulus for a Unear elastic materials. Here, the Unear differential relation between stress and strain for a viscoelastic polymer has been transformed into a linear elastic relation between stress and strain in the transform space. It will be shown in the next chapter that the same result can be obtained from integral expressions of viscoelasticity without recourse to mechanical models, so that the result is general and not limited to use of a particular mechanical model. Therefore, the simple transform operation allows the solution of many viscoelastic boundary value problems using results from elementary solid mechanics and from more advanced elasticity approaches to solids such as two and three dimensional problems as well as plates, shells, etc. See Chapters 8 and 9 for more details on solving problems in the transform domain. 

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