Computational rod models have emerged as efficient tools to simulate the bending and twisting deformations of a variety of slender structures in engineering and biological applications. The dynamics of such deformations, however, strongly depends on the constitutive law in bending and torsion that, in general, may be nonlinear, and vary from material to material. Jacobian-based computational rod models require users to change the Jacobian if the functional form of the constitutive law is changed, and hence are not user-friendly. This paper presents a scheme that automatically modifies the Jacobian based on any user-defined constitutive law without requiring symbolic differentiation. The scheme is then used to simulate force-extension behavior of a coiled spring with a softening constitutive law.