Continuous analytical elasticity solutions for arbitrary cross-sections and combined loads

Abstract

A geometrically exact formulation of three-dimensional elasticity is presented in which coupled translational and rotational deformation is represented within a special Euclidean group SE(3) configuration field defined over the material body. Each material point is represented as an element of SE(3), and the continuum is described by a smooth configuration map whose local strain is obtained via the left-trivialized Maurer–Cartan form. This yields a Lie-algebra-valued strain field capable of describing coupled rotational and translational deformation modes in a unified manner. The configuration is defined by a first-order SE(3) evolution equation whose solution is equivalently represented by a path-ordered exponential of the strain field, providing exact nonlinear reconstruction of the deformed geometry. Constitutive response is formulated directly on the SE(3) strain space using only the material parameters Young’s modulus E and shear modulus G. All classical sectional stiffness measures, such as axial stiffness EA, bending stiffness EJ, and torsional stiffness GJ, arise as geometric moment invariants of the cross-section under the SE(3)-induced strain field rather than by being introduced as constitutive inputs. A unified force–moment resultant (wrench) is obtained as the dual of an affine strain measure, providing a direct analytical relation between stress fields without explicit use of moment arms. Classical beam and rod theories, including torsion, bending, axial deformation, and Saint-Venant warping, are recovered as special cases of the general SE(3) formulation. For noncircular cross-sections, warping arises naturally from compatibility conditions expressed in differential form on the SE(3) strain field. The formulation identifies the centroid as the natural reference point for the affine strain measure, eliminating first-moment coupling terms and providing a canonical gauge for the cross-sectional representation.