Extensible elastica solutions on the large deflection of fiber cantilever with circular wavy crimp. II. Classification of equilibrium configurations

Abstract

The types of equilibrium shapes in the extensible elastica solutions of planar deflection of two-element crimped fiber cantilever with circular wavy crimp were classified for one end clamped boundary under concentrated, inclined and dead tip load. Material property of the fiber was also regarded linear elastic, while crimp was described as a combination of semicircular arcs smoothly connected with each other having constant curvature of all the same magnitude and alternative sign. Investigating the mathematical features of the previously obtained solutions shows that the possibility and variety of multiple equilibrium shapes originates from the following two reasons: the existence of Case 1/Case 2 type transformation, and the mathematical features of Case 1 type transformation. In two-element crimped fiber, there exist four different choices for Case1/Case2 type transformation. Regarding Case 2-Case 2 type equilibrium shape as the default and most probable one, other three types of equilibrium shape can be regarded as multiple equilibrium shapes. Moreover, Case 1 type transformation itself may lead to multiple configurations due to the possibility of the contra-flexure point after bending while Case 2 type transformation doesn’t allow such characteristics. Case 1-Case 1 type equilibrium shape shows maximum varieties in shape, while Case 2-Case 1 and Case 1-Case 2 type shapes occasionally leads to the identical results with different choice of the sign. Tables and shapes in each configuration were also presented.