An analytical fiber bundle model for pullout mechanics of root bundles

Roots in soil contribute to the mechanical stability of slopes. Estimation of root reinforcement is challenging because roots form complex biological networks whose geometrical and mechanical characteristics are difficult to characterize. Here we describe an analytical model that builds on simple root descriptors to estimate root reinforcement. Root bundles are modeled as bundles of heterogeneous fibers pulled along their long axes neglecting root-soil friction. Analytical expressions for the pullout force as a function of displacement are derived. The maximum pullout force and corresponding critical displacement are either derived analytically or computed numerically. Key model inputs are a root diameter distribution (uniform, Weibull, or lognormal) and three empirical power law relations describing tensile strength, elastic modulus, and length of roots as functions of root diameter. When a root bundle with root tips anchored in the soil matrix is pulled by a rigid plate, a unique parameter, kappa, that depends only on the exponents of the power law relations, dictates the order in which roots of different diameters break. If kappa < 1, small roots break first; if kappa > 1, large roots break first. When kappa = 1, all fibers break simultaneously, and the maximum tensile force is simply the roots' mean force times the number of roots in the bundle. Based on measurements of root geometry and mechanical properties, the value of kappa is less than 1, usually ranging between 0 and 0.7. Thus, small roots always fail first. The model shows how geometrical and mechanical characteristics of roots and root diameter distribution affect the pullout force, its maximum and corresponding displacement. Comparing bundles of roots that have similar mean diameters, a bundle with a narrow variance in root diameter will result in a larger maximum force and a smaller displacement at maximum force than a bundle with a wide diameter distribution. Increasing the mean root diameter of a bundle without changing the distribution's shape increases both the maximum force and corresponding displacement. Estimates of the maximum pullout forces for bundles of 100 roots with identical diameter distribution for different species range from less than 1 kN for barley (Hordeum vulgare) to almost 16 kN for pistachio (Pistacia lentiscus). The model explains why a commonly used assumption that all roots break simultaneously overpredicts the maximum pullout force by a factor of about 1.6-2. This ratio may exceed 3 for diameter distributions that have a large number of small roots like the exponential distribution.