Unacknowledged fatal error in expressions for shear strains in 3-D theory of elasticity in curvilinear co-ordinates and the consequences

Abstract

An undetected fatal error has existed in the 3-D theory of elasticity in curvilinear co-ordinates since its formulation in the 19th century, whereby the linear expressions for shear strains as functions of the three orthogonal displacements fail to satisfy St. Venant’s compatibility equations. This error has, consequently, invalidated all thin-shell theories and finite-elements (other than those in Cartesian co-ordinates). The error is shown to have been caused by defining all six strain components mathematically to fit a single strain tensor as if they were independent quantities not constrained by any need for compatibility, instead of calculating what they really were. With six strain components (three direct and three shear) and six compatibility equations, this procedure was always scientifically unacceptable. It is further shown that the three real shear-strain components satisfying compatibility are fully defined derivative functions of the three direct strains. The correct shear-strain characterizations are established here by two totally independent methods, the first by analysis, and the second by revealing that St. Venant’s compatibility equations actually specify mathematically what the three shear strains must be. The article includes the correct specific strain components for thin cylindrical shells and exposes the bogus compatibility equations that were derived decades ago from the incompatible classical strain components, which concealed the existence of this error for cylindrical and spherical shells. This “fix,” universally accepted, whereby the compatibility equations were made to fit a set of independent strain components, defeats the purpose of compatibility requirements constraining the options for possible strains that cannot all be independent if they are to be compatible. It is recommended that the several related malpractices in the analysis and design of thin-shell structures be discontinued—and that new shell theories and new finite-elements and structural analysis computer codes be developed to replace them.

Keywords

errors in classical theory shear strains thin-shell buckling structural analysis curvilinear co-ordinates

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