Three-dimensional elasticity based on quaternion-valued potentials

Abstract

International audienceOne of the most fruitful and elegant approach (known as Kolosov-Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starting point is the classical harmonic potential representation due to Papkovich and Neuber. Alike the classical complex formulation, two monogenic functions very similar to holomorphic functions in 2D and conserving many of interesting properties, are used in this contribution. The completeness of the potential formulation is demonstrated rigorously. Moreover, body forces, residual stress and thermal strain are taken into account as a left side term. The obtained monogenic representation is compact and a straightforward calculation shows that classical complex representation for plane problems is embedded in the presented extended formulas. Finally the classical uniqueness problem of the Papkovich-Neuber solutions is overcome for polynomial solutions by fixing explicitly linear dependencies

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“…A three dimension equivalent to a Westergaard stress function would have to be based on a hypercomplex number system. Hamilton's quaternions are a hypercomplex number system [16] that is sometimes used in elasticity [17]. In a quaternion it is assumed that i 2 = j 2 = k 2 = -1.…”

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“…In this section we study a recent approach [24] to a spatial generalization of the Kolosov-Muskhelishvili formulae [21] in the framework of hypercomplex function theory. A survey to this topic and references to related works can be found in [6,24].…”

“…In this section we study a recent approach [24] to a spatial generalization of the Kolosov-Muskhelishvili formulae [21] in the framework of hypercomplex function theory. A survey to this topic and references to related works can be found in [6,24]. Based on the general solution of PapkovicNeuber [22,23] we construct an H-valued representation of the displacement field in terms of a monogenic and an anti-monogenic function.…”

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