Large Deflection Analysis of Fibers with Nonlinear Elastic Properties

Abstract

This paper provides useful insights into the nonlinear buckling phenomenon of fibrous materials. In the analysis, a fiber is considered to be a rectangular, prismatic, inextensible column with no shear effect, whose constitutive equation corresponds to a Ludwick or modified Ludwick type. Governing equations of the buckling of a fiber as a nonlinear elastic column are established and solved. These equations cover not only the buckling situation of the column but also three other loading conditions: a horizontal direction of point load and a vertical direction of the distributed load, a vertical direction of point load and horizontal direction of distributed load, and a vertical direction of both point and distributed loads. As a result, in a linear material, the shape of the deflected fiber is fully described by the geometric boundary conditions of the path. In a nonlinear material, however, the shape depends on many other factors, including the cross-section, stress-strain response, and loading. Moreover, the deflected shape of a fiber with specified angles at both ends is almost the same for point or combined loading, regardless of the choice of constitutive relationship. What makes difference between the choices of constitutive equations is the load level. In a nonlinear fiber, the same deflected shape can be achieved at different load levels from the linear case. Another distinctive feature in nonlinear fiber is that the solution has double roots of the load parameter β at the same value of tip slope angle, whereas it has only one root for the linear case. Finally, the accuracy of the solution is estimated by comparing the results with well-known elliptic integral solutions for a linear elastic case under point load. Selected examples of the deflected shape are also provided.