Modeling root reinforcement using a root-failure Weibull survival function

. Root networks contribute to slope stability through complex interactions with soil that include mechanical compression and tension. Due to the spatial heterogeneity of root distribution and the dynamics of root turnover, the quantiﬁcation of root reinforcement on steep slopes is challenging and consequently the calculation of slope stability also. Although considerable progress has been made, some important aspects of root mechanics remain neglected. In this study we address speciﬁcally the role of root-strength variability on the mechanical behavior of a root bundle. Many factors contribute to the variability of root mechanical properties even within a single class of diameter. This work presents a new approach for quantifying root reinforcement that considers the variability of mechanical properties of each root diameter class. Using the data of laboratory tensile tests and ﬁeld pullout tests, we calibrate the parameters of the Weibull survival function to implement the variability of root strength in a numerical model for the calculation of root reinforcement (RBMw). The results show that, for both laboratory and ﬁeld data sets, the parameters of the Weibull distribution may be considered constant with the exponent equal to 2 and the normalized failure displacement equal to 1. Moreover, the results show that the variability of root strength in each root diameter class has a major inﬂuence on the behavior of a root bundle with important implications when considering different approaches in slope stability calculation. Sensitivity analysis shows that the calibration of the equations of the tensile force, the elasticity of the roots, and the root distribution are the most important steps. The new model allows the characterization of root reinforcement in terms of maximum pullout force, stiffness, and energy. Moreover, it simpliﬁes the implementation of root reinforcement in slope stability models. The realistic quantiﬁcation of root reinforcement for tensile, shear and compression behavior allows for the consideration of the stabilization effects of root networks on steep slopes and the inﬂuence that this has on the triggering of shallow landslides.


Introduction
Root reinforcement, the strength roots impart to soil, is recognized to be an important factor affecting directly and indirectly several hydro-mechanical processes in hydrology and earth surface systems. For example, the roots of riparian vegetation may consistently influence the morphodynamic of rivers by anchoring sediments (Edmaier et al., 2011) or stabilizing river banks (Pollen and Simon, 2005;Petrone and Preti, 2008) and represent an important factor in river restoration projects. Moreover, vegetation contributes to the mitigation of erosion and shallow landslides at the catchment scale, regulating the yield and transport of sediments (Schmidt et al., 2001;Bathurst et al., 2007). In mountain catchments, root reinforcement is one of the most important contributions of vegetation to slope stability (Phillips and Watson, 1994;Sidle, 1992;Rickli and Graf, 2009). In the last 30 yr, three distinct methods have been employed to quantify root reinforcement. The approach of Wu et al. (1979) has been, and still is, used because of its simplicity (it requires only minimal information about root critical tensile strength and the cross-section area of roots crossing the failure surface). However, recent studies (Pollen and Simon, 2005; have demonstrated that Wu et al. (1979) hypothesis that all roots break simultaneously can lead to an order-of-magnitude error in the estimation of root reinforcement and is thus untenable. More recently, Pollen and Simon (2005) used the fiber bundle model with a stress-step loading to estimate root reinforcement. The advantage of this model is that roots of different dimensions do not all break at the same load. This approach, however, does not easily permit calculation of root elongation for realistic root bundles (e.g., roots with different apparent elasticities). To overcome this problem Schwarz et al. (2010c) implemented the strain-step loading approach in the Root Bundle Model (RBM). The main advantages of the RBM are (1) calculation of the complete force-displacement curve of a bundle of roots, and (2) redistribution of forces on each single root based on their geometrical and mechanical properties (and not statistically imposed). In a further simplification of the RBM, Cohen et al. (2011) proposed an analytical solution implementing only the most relevant parameters (rootsize distribution, root tensile force, secant Young's modulus, length, and tortuosity).
Since Wu et al. (1979) model, the main improvements in modeling root reinforcement are (1) roots do not all break at the same time (Waldron and Dakessian, 1981;Pollen and Simon, 2005); (2) roots have different failure mechanisms, break or slip out (Waldron and Dakessian, 1981); (3) root geometry (length) and secant Young's modulus are functions of root diameter (Waldron and Dakessian, 1981;Schwarz et al., 2010b); (4) root tortuosity affects the apparent elasticity of roots and the failure mechanism .
Based on empirical observations ) and on biomechanics studies of roots (Loades et al., 2010), root mechanical properties are highly variable. Despite this evidence, all numerical and analytical models thus far implement the mechanical variability of roots only as a function of their diameters, usually given as a distribution, assuming that roots within a diameter class are homogeneous. A more realistic assumption is that, for a given diameter or small diameter range, there is a variability due to the presence of "weak spots" related to the anatomy and geometry of roots (Loades et al., 2010). Root age, root constituents, and environmental conditions in which roots grow are important factors that influence root biomechanics (Loades, 2007). All of these factors contribute to the variability of root mechanical properties. Thus, it is important to implement this variability in root reinforcement models and analyze how this variability affects the mechanical behavior of root bundles.
The objective of this work is to present a new approach for quantifying root reinforcement that considers the intrinsic variability of mechanical properties of roots of similar diameters. The new model is presented in Sect. 2. In Sect. 3 we present new field and laboratory root strength data used for the calibration and validation of the model (Sect. 4). A discussion of the model and comparisons with others is given in Sect. 5.

Root geometry and mechanics
We assume that each root is a linear-elastic fiber that breaks at a threshold displacement. Although roots stretched in tensile tests in the laboratory show a decrease in the slope of the stress-strain curve associated with plastic behavior (Waldron and Dakessian, 1981;Loades et al., 2013), cyclic laboratory pullout tests of Czarnes et al. (1999) of roots in soil show little irreversible deformation (less than 5 %), supporting the use of a linear model for roots. Then, estimating the tensile force in a root using the fundamental equation of linear elasticity requires knowledge of its geometry (diameter, length, tortuosity) and mechanical properties (maximum tensile force, Young's modulus).
Data on roots (Operstein and Frydman, 2000;Schmidt et al., 2001;Ammann et al., 2009;Giadrossich et al., 2013) provide support for modeling the average root length, L, the average maximum tensile force, F max , and the average Young's modulus, E, as power-law functions of root diameter (φ): where φ 0 , L 0 , F 0 , and E 0 are scaling factors and γ , ξ , and β power-law exponents, and where φ 0 is assumed to equal 1 and will not be explicitly written in the following equations: For Young's modulus, we use the secant Young's modulus, the ratio of root strength over strain at failure as done by Waldron and Dakessian (1981) for barley. This value, lower than the initial value of the Young's modulus estimated from tensile laboratory (e.g., Loades et al., 2013), is more appropriate to estimate the maximum stress at failure. In addition, in natural soils, roots are not straight but tortuous and the force necessary to pull a root is small until the root is fully stretched out. The effect of root tortuosity when using laboratory tensile tests to estimate root reinforcement in natural soils on slopes is considered by using a coefficient that reduces the Young's modulus (e.g., Schwarz et al., 2010b): where the coefficient r ranges between 0.3 and 0.5. This coefficient does not affect the estimation of the maximum tensile force of a root, only the stretching (displacement) at which the maximum force is observed. In this study we backcalculate the apparent value of the secant Young's modulus from field pullout tests using only measured displacement and tensile force (see Schwarz et al., 2010b). Using Eqs. (1) and (4) together with the equation of elasticity, the tensile force, F , in a single root as a function of displacement, x, is and the displacement, x fit max , at which that root fails is

Weibull survival function for roots
A survival function, also known as a complementary cumulative distribution function, is a probability function used in a broad range of applications that captures the failure probability of a complex system beyond a threshold. The Weibull distribution originates from the study of fatigue (Weibull, 1939) and is used in engineering as the time to failure or in biological systems as a survivorship curve (Pinder et al., 1978). The Weibull is adaptable to many scientific applications and particularly to the study of fiber failure (e.g., Curtin and Takeda, 1989) and roots (Pollen and Simon, 2005;. We hypothesize that the probability of a root to survive is a function of a normalized displacement, x * , and is given by the two-parameter Weibull survival function where ω is the Weibull exponent (shape factor) and λ * the scaling factor. The normalized displacement is given by This normalization, which eliminates the effect of root diameter on maximum displacement, is needed to construct a survival function where ω expresses the relative variability of root strength independently of root diameter.

Root bundle reinforcement
The tensile force (root reinforcement) of a bundle of roots is obtained by summing the force contributions from each root multiplied by the survival function S where N is the number of roots. This new extension of the RBM is called RBMw. Equation (9) can be rewritten considering the number of roots (n) in a given root diameter classes ( ) of a bundle in the form where φ is the mean root diameter of each root diameter class, max is the maximum root diameter class considered, and x * is the normalized displacement of each root diameter class . The RBMw was implemented in a R code and can be downloaded at the following link: www.ecorisq.org/ openFTP/Schwarz.zip.

Calibration of the survival function
Fitting of the Weibull exponent for a data set of field pullout or laboratory tensile experiments is the novelty of this new RBMw approach. To illustrate the method, we use a small hypothetical data set of five pullout experiments with root diameter ranging from 1 to 4 mm. Figure 1a shows the measured (F meas ) and the fitted (F fit ) values of maximum pullout/tensile force as a function of root diameter. Each measurement is represented by a red dot and labeled with a number ranging from 1 to 5. Using Eq. (5), we compute the force-displacement for each root (Fig. 1b), assuming either the measured (red dashed lines) or fitted (green lines) values of maximum pullout force (see Table 1, column 3 and 4). To obtain the Weibull exponent ω and the scaling factor λ * of the Weibull survival function (Eq. 7), we first rank roots in ascending order ( where and we compute their survival probability (Table 1, column 9) of each root using the following equation

Data collection for the calibration of the RBMw
The use of field experiments for calibration is considered the most appropriated for model calibration. Furthermore, by using field measurements, one could forgo root length data and secant Young's modulus calculations by using Hooke's law for elasticity which directly relates force to displacement through a spring constant: where H (φ) corresponds to a spring constant as a function of root diameter φ, which summarizes the mechanical properties of the root-soil system under specific conditions (root diameter, tree species, stand, soil type, and moisture conditions). The application of the Hooke's law would simplify the calculation and reduce the number of parameter considered in the calculation with the consequence to reduce the source of errors. This approach is possible only using time consuming and complicated field pullout experiments. This is probably one of the reasons why there is a big lack of data for this type of experiments and future works should focus on providing such data set for different tree species, especially testing large root diameters.

Implications of root reinforcement quantification in hydrology and earth surface systems
The presented RBMw may find application in several types of data analysis and process modeling. The prediction of the pullout of riparian plants due to drag forces of water flow may be characterized with the RBMw using data of pullout experiments on single roots for the calibration, as shown in Edmaier et al. (2012), and upscaled to entire root systems or root networks considering the distribution of roots and the variability of the root-soil mechanical properties. Consequently, the RBMw could allow the implementation of root reinforcement in models for the simulation of long-term fluvial morpho-dynamics. In an analogous way, the quantification of root reinforcement distribution within root system could be applied in models for the study of tree stability during wind storms or rock fall impacts. Overall, the advantages quantifying root reinforcement in term of forcedisplacement behavior with a minimal computational effort makes the RBMw attractive for its implementation in slope stability model at large temporal and spatial scales. Recent studies (Schwarz et al., 2010a; have shown the importance of the dynamic of root reinforcement during the triggering of shallow landslides remarking the importance of changes in stiffness and total mobilized energy of rooted soil volumes loaded under tension and compression. In particular Schwarz et al. (2010b) propose an upscaling framework of root reinforcement at the hillslope scale considering the structure and the type of forest cover. Such heterogeneous spatial characterization of root reinforcement is implemented in a 3-D slope stability model called SOSlope Schwarz and Thormann, 2012). For the characterization of root reinforcement at such a large-scale, site-specific calibration of parameters and the implementation of parameter variability is needed to predict realistic values and a simplified model such as the RBMw may help in reduce parametrization and thus reducing/optimizing the efforts of field investigation needed for the calibration. However, more data sets of field pullout experiment are needed for the further validation of the RBMw under different combination of factors (soil type, soil moisture, and tree species). The same approach of the RBMw