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Title: Solvability conditions in simple inverse/design torsion problems
Authors: Ioakimidis, Nikolaos
Keywords: Computer algebra
Elastic bars
Gröbner bases
Inequality constraints
Inverse/design problems
Milne volume function
Quantifier elimination
Solvability conditions
Sturm–Habicht sequences
Source: Recent advances in mechanics and related fields, special volume in honour of Professor Constantine L. Goudas
Abstract: The inverse/design problem of torsion of a straight isotropic elastic bar of an elliptical cross-section is revisited. Under the assumption that the shear modulus, the applied torque, the angle of twist per unit length and the maximum shear stress are known in advance, the related values for the semi-axes of the elliptical cross-section are the unknown quantities. This problem is easily reduced to a simple system of two polynomial equations in two unknowns (the semi-axes) under appropriate inequality constraints. Here necessary conditions for the solvability of this inverse/design torsion problem are derived. The derivation of these conditions is mainly based on the use of Sturm–Habicht negated polynomial remainder sequences and their generalizations. Gröbner bases and Milne’s volume function method are also employed. The present approach can be generalized to a variety of equally significant and even more complex mechanics problems.
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