Abstract: | Summary Finite elastic straining is analysed with all quantities referred consistently to the deformed body taken as the function domain. The straining-displacement of a typical point is relative to a set of axes imbedded in the body at one arbitrary point and rotating in fixed space with that neighborhood if necessary in a particular problem. The resulting |plane stress' equations have precisely the same form as in the classical theory but relate to |true' quantities in the deformed body.The solution of a circular hole in a deformed sheet under simple tension is given and checks closely with experiment on rubber. Cauchy strains of order 65% and local rotation of order 30° are found to occur at the hole boundary.The solution of a deformed quadrantal cantilever is given. Cauchy strains of several hundred percent and local rotation of order 90° occur.Any boundary value problem already solved for the classical infinitesimal strains theory can be applied directly as a finite strains solution for the deformed body.Notation
x, y, z, r, , z
Cartesian and polar co-ordinates respectively
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,
Normal and shear true stresses respectively
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,
Normal and shear true strains respectively
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r
Position vector
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Airy stress function
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S
Simple tensile stress applied to sheet
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a
Radius of circular hole in deformed sheet
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a, b
Inner and outer radii of quadrantal cantilever
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u
Straining-displacement vector
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u, v
Straining-displacement scalar components
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E,
True Young's modulus and Poisson's ratio respectively
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c
1, c
2
Local unit vectors in principal normal strains directions
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i, j
Cartesian axes constant unit vectors
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Stress dyadic or tensor
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First stress invariant
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I
Idemfactor or spherical tensor
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P
Shear load per unit thickness applied to quadrantal cantilever
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A, B, D, N, H, K, L
Arbitrary constants of integration |