One-dimensional model for the combined bending, stretching, shearing and torsion of rods derived from three-dimensional elasticity

The displacement field in rods can be approximated by using a third-order Taylor-Young expansion in the transverse dimension of the rod. These involve that the highest-order term of shear is of first order in transverse dimension of the rod. Then we are motivated to consider a simplified theory based on the transverse-directions expansion of the potential energy truncated at fourth order in the transverse dimension of the rod. In the same way as Pruchnicki [1], the Euler-Lagrange equations are modified so as to be compatible with equilibrium equations. These lead to an analytical expression for one-dimensional potential energy in terms of the zeroth-order displacement field and its derivatives that includes non-standard transverse shearing energy and torsion energy. As a consequence this potential energy satisfies the stability condition of Legendre-Hadamard which is necessary for the existence of a minimizer. When the lateral surface of the rod is free of charge and body force is equal to zero, the minimization of the potential energy leads to a boundary value problem which can be integrated analytically.