Aurélien Doitrand
email: aurelien.doitrand@gmail.com

Dominique Leguillon

Numerical modeling of the nucleation of facets ahead of a primary crack under mode I+III loading

Keywords: Mode I+III loading, Crack front fragmentation, Finite Fracture Mechanics, Coupled criterion

A numerical study of crack front segmentation under mode I+III loading is proposed. Facets initiation ahead of a parent crack is predicted through a tridimensional application of the coupled criterion. Crack initiation shape, orientation and spacing are determined for any mode mixity ratio by coupling a stress and an energy criterion using matched asymptotic expansions. The stress and the energy conditions are computed through a 3D finite element modeling of a periodic network of facets ahead of the parent crack. Facets initiation shape, loading and spacing depend on the blunted parent crack tip radius. A good estimate of facet orientations is obtained based on the direction of maximum tensile stress. The facet shapes, determined using the stress isocontours, are qualitatively similar to those observed experimentally. The order of magnitude of numerical predictions of facets spacing is very close to experimental measurements.

Introduction

Crack growth under combined mode I+III loading has been widely studied over the past decades. Under such loading combination, the crack tends to rotate around the direction of propagation in order to reduce mode III and reach a pure mode I situation. This is achieved by a fragmentation of the initial crack into multiple daughter cracks (also called facets). Once initiated, some of these facets grow and coalesce so as to form a stepped fractured surface that becomes coarser as the crack grows [START_REF] Pons | Helical crack-front instability in mixed-mode fracture[END_REF]. Mode I+III crack patterns have been investigated experimentally for some years. Since the pioneering works of Sommer [START_REF] Sommer | Formation of fracture lances in glass[END_REF] and Knauss [START_REF] Knauss | An observation of crack propagation in antiplane shear[END_REF] who studied facet nucleation respectively in glass and in polymer, mode I+III crack patterns have been observed in various studies such as fault formation [START_REF] Cox | On the formation and growth of faults: an experimental study[END_REF], cracks in rocks [START_REF] Younes | Fringe cracks: key structures for the interpretation of the progressive Alleghanian deformation of the Appalachian plateau[END_REF], in PMMA [START_REF] Buchholz | Fracture analyses and experimental results of crack growth under general mixed mode loading conditions[END_REF][START_REF] Lazarus | Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments[END_REF] or in plexiglass [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF].

The geometrical description of facets is generally performed by observing post-mortem specimens in which the initial crack has segmented into daughter cracks that have already propagated and possibly coalesced. Cambonie and Lazarus [START_REF] Cambonie | Quantification of the crack fragmentation resulting from mode I+III loading[END_REF] proposed a systematic method to quantify the geometry of the facets from post-mortem specimens by profilometry. They determined the distance between the facets and their twisting angles as a function of the distance to the parent crack front. Eberlein et al. [11] performed Arcan tests with CTSRspecimen under mode I+III fatigue loading so as to study the crack front segmentation profile using postmortem specimens. They determined the influence of the amount of mode III on some geometrical features of the segmented crack front such as facet angle, projected facet length, facet distance and bridging width. In order to quantify more in depth facet nucleation, Pham and Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] designed a test so as to load a specimen in mode I+III through the combination of a wedge load and of a confining load resulting in a global compressive stress field ahead of the crack front. Multiple facet nucleation was observed by stop-ping the test just after crack initiation, which particularly highligthed that the facet coalescence does not happen during initiation, but at a later stage of crack growth. Mittelman and Yosibash [START_REF] Mittelman | Energy release rate cannot predict crack initiation orientation in domains with a sharp V-notch under mode III loading[END_REF][START_REF] Yosibash | A 3-D failure initiation criterion from a sharp V-notch edge in elastic brittle structures[END_REF] carried out experiments on inclined V-notched specimens of PMMA, Graphite and ceramics under mode I+III loading, and concluded that facet initiation orientation can be determined from stress considerations.

Leblond et al. [START_REF] Leblond | Theoretical analysis of crack front instability in mode I+III[END_REF][START_REF] Leblond | Multiscale cohesive zone model for propagation of segmented crack fronts in mode I+III fracture[END_REF] proposed a theoretical analysis of the instability of the crack front and the growth of the facets under mode I+III loading, which was also studied numerically by several authors, mainly from a macroscopic point of view. Using the modified virtual crack closure integral method, Dhondt et al. [START_REF] Dhondt | Computational fracture analysis of different specimens regarding 3D and mode coupling effects[END_REF], Buchholz et al. [START_REF] Buchholz | Fracture analyses and experimental results of crack growth under general mixed mode loading conditions[END_REF] and Lazarus et al. [START_REF] Lazarus | Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments[END_REF] proposed finite element (FE) simulations of crack propagation in different configurations such as three point bending of specimens with a tilted initial crack . They qualitatively well reproduced the progressive crack rotation with respect to its initial direction until it reaches a pure mode I situation. Citarella and Buchholz [START_REF] Citarella | Comparison of crack growth simulation by DBEM and FEM for SEN specimens undergoing torsion or bending loading[END_REF] showed the consistency of the dual boundary element modeling for modeling crack propagation under mode I+III loading compared to FE modeling. However, these approachs involve a progressive twist of the crack, and are unable to take into account the initial crack front segmentation into facets. Pons and Karma [START_REF] Pons | Helical crack-front instability in mixed-mode fracture[END_REF] and Chen et al. [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF] simulated crack growth based on a continuum phase field method, which allows reproducing the facet coarsening under mode I+III loading.

Facet nucleation was studied numerically by Pham and Ravi-Chandar [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF] using the boundary element simulation. The stress intensity factors K I and K III along the parent crack front for which a first daughter crack has already initiated were computed, allowing the determination of a range for the spacing between facets corresponding to the locations maximizing K I and minimizing K III . Pham and Ravi-Chandar concluded that the distance between facets was dictated either by the fracture process zone for a natural crack, or by the local radius of curvature for a machined crack. The same authors [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] also highlited the shielding of some daughter cracks resulting from the propagation of neighboring cracks, hence leading to facet coarsening as the crack grows. These models allow the study of daughter crack initiation, after the nucleation of the first facet which has not been investigated until now.

The objective of the work is the prediction of daughter crack initiation ahead of a primary crack under mode I+III loading. In section 2, some experimental data and observations about facet nucleation are presented in order to justify the modeling hypotheses. The crack initiation prediction method, namely the coupled criterion, is presented in Section 3. In Section 4, daughter crack initiation is modeled by FE, and numerical prediction of facet nucleation are presented and confronted to some experimental data in Section 5. Throughout this paper, crack and facet are synonymous. When the main crack is under consideration, the denomination parent crack is used.

Experimental observations and modeling hypotheses

Crack rotation aiming at reaching a pure mode I situation during crack propagation under mode I+III loading usually occurs due to the parent crack front segmentation into multiples facets [1-3, 23, 24]. However, Eberlein et al. [11] showed experimentally that in some cases, below a certain K III /K I ratio threshold, no facet formation are observed and the crack grows continuously. This phenomenon was not observed by Pham and Ravi-Chandar [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF][START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF], who highlighted facet formation for ratios as small as K III /K I = 0.001. Herein, only crack front segmentation into facets is adressed, whereas the following continuous crack growth is beyond the scope of this work.

Experimental observations under mode I+III loading show simultaneous initiation of several facets ahead of the parent crack. These cracks initiate forming a "nice nearly periodic pattern" [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF], all cracks being (i) oriented with similar angles with respect to the parent crack [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF][START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF][START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF], (ii) separated by almost the same distance [START_REF] Cambonie | Quantification of the crack fragmentation resulting from mode I+III loading[END_REF]11,[START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] and (iii) having almost the same size [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF][START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]. Moreover, the nucleated facets are nearly planar [START_REF] Cambonie | Quantification of the crack fragmentation resulting from mode I+III loading[END_REF][START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF], except the crack tip that is gently curved [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF]. In this plane, the initiation crack exhibits a nearly elliptic crack shape [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]. Several approaches exist in order to determine the crack orientation such as, e.g., analytical formulae [START_REF] Lazarus | Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments[END_REF][START_REF] Pollard | Formation and interpretation of dilatant echelon cracks[END_REF]. In this work, we will assume that the normal to the crack plane is determined by the direction of the maximum tensile stress at the parent crack tip. Experimentally, mode I+III loading also includes a portion of mode II, especially near the free edges [11], which will be ignored in the proposed modeling.

Given the experimental observations described previously, the modeling hypotheses assumed in this work are the following : o The imposed loading is a combination of mode I and mode III, without any mode II. It is independent of the curvilinear abscissa along the crack front. o Aa a consequence, an infinite network of periodic cracks is modeled, which is representative of facet initiation far from free edges.

o Facets extend in a plane during the nucleation phase, hence the gently curved crack tip is not taken into account. o The crack angle with respect to the parent crack is determined as the one maximizing the tensile stress ahead of the parent crack tip. o In the crack plane, no a priori hypothesis is made on the crack shape, which is directly determined based on the normal stress field, as explained in Section 3.4, following the approach proposed by Doitrand and Leguillon [START_REF] Doitrand | 3D application of the coupled criterion to crack initiation prediction in epoxy/aluminum specimens under four point bending[END_REF].

3 The coupled criterion

Stress and energy conditions

According to the coupled criterion (CC) [START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF], crack initiation requires two separate conditions that must be fulfilled. On the one hand, the stress state prior to crack initiation must be high enough, and on the other hand, sufficient energy must be released by the crack opening. Both conditions can be gathered to predict crack initiation.

The energy condition of the coupled criterion is obtained through an energy balance between an elastic state prior to crack initiation and after the nucleation of a crack of surface S (Eq. 1).

δW p + δW k + G c S = 0 (1)
where δW p is the change in potential energy, δW k the change in kinetic energy and G c the material toughness. A quasi-static initial state leads to a production of kinetic energy (δW k ≥ 0), which results in the energy condition for crack initiation (Eq. 2).

G inc = - δW p S ≥ G c (2)
The stress condition of the CC compares the tensile stress acting in the direction orthogonal to the crack to the tensile strength σ c (Eq 3).

σ ≥ σ c on the presupposed whole crack surface S (3) Combining both conditions (2) and (3) allows the determination of the initiation crack surface and loading level. As mentionned in Section 2, the shape of the facets is determined by the isocontours of the tensile stress, ensuring the stress condition to be fulfilled on the whole surface. When S is small the isocontours are roughly homothetic curves, each one being defined by its extension along a choosen direction.

Crack initiation ahead of a parent crack

Experimental investigations of crack initiation and growth under mode I+III usually involve specimens with a blunted parent crack tip [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]. Crack initiation ahead of this parent crack tip depends on the crack tip radius, which can be considered as a characteristic length of the problem. The coupled criterion has been successfully applied to crack initiation prediction at blunted notches under mode I loading [21, [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF][START_REF] Picard | A method to estimate the influence of the notch-root radius on the fracture toughness measurement of ceramics[END_REF], using matched asymptotics expansions in a 2D framework. This method consists in determining the displacement field for two problems, one far from the crack (the outer expansion) and one near it (the inner expansion), and then determine matching conditions in an intermediate zone (close to the crack in the outer problem and far from it for the inner problem) where both solutions are valid. The application of matched asymptotic expansions to crack initiation prediction at blunted notches under pure mode I loading using the coupled criterion is detailed in [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF]. In the following, the choice of the inner domain and the method employed to solve the coupled criterion in pure mode I or in mixed mode I+III loading is described.

Pure mode I loading

Regardless of the blunting, in the outer domain spanned by the space variables x i (i = 1, 3), the crack is sharp (see the Appendix). The inner domain is obtained by dilatation of the space variables compared to a characteristic length of the problem. Classically [START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF], in the case of a sharp parent crack, the dilatation is carried out with respect to the crack extension l, whereas in the case of a blunted parent crack tip it can be performed either with respect to the crack extension l or the crack tip radius R, provided that both lengths are of the same order of magnitude, property which is checked afterwards. For the sake of ease of FE mesh generation, we choose to perfom the dilatation with respect to the crack tip radius to obtain the inner domain. Therefore, the inner domain is an unbounded media spanned by the space variables y i = x i /R, containing a semi-infinite crack which root radius is equal to 1, and a crack extension of dimensionless length λ = l/R. Leguillon et al. [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF] determined the change in potential energy from calculations in the inner domain (Eq. 4) in a problem under plane strain assumption, e being the thickness of the specimen:

   -δW p = K I 2 R(B(λ) -B(0))e G inc = - δWp S = - δWp le = K I 2 B(λ)-B(0) λ (4)
where K I is the stress intensity factor at the crack tip in the outer domain (keep in mind that in this domain, the crack is sharp) and B(λ) can be computed by FE modeling calculations on the inner domain with various values of λ (cf. Section 4.1). Of course, the FE calculations of B(λ) cannot be performed on an infinite domain. Therefore, the inner domain is artificially bounded at a large distance from the crack. In the case of pure mode I loading, the crack initiates and grows in the plane of the parent crack. Therefore, the domain corresponding to the inner problem for FE calculations is defined as a cylinder of any thickness and outer radius Π embedding a parent crack with a blunted tip of radius 1 (cf. Fig. 1a).

Using Matched asymptotic expansions in the 2D case, Leguillon et al. [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF] determined the problem to solve in order to obtain the loading level K app Ic and the length l c of a crack initiating ahead of a blunted crack tip of radius R (Eq. 5 and 6).

1 σ(λ c ) 2 B(λ c ) -B(0) λ c = 1 R G c σ 2 c ( 5
)
where λ c = l c /R is the non-dimensional initiation crack length, σ(λ) stands for the non-dimensional tensile stress in the inner domain at a distance λ from the blunted tip prior to crack initiation , and B(λ)-B(0) λ is computed by FE for several values of λ (refer to [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF] and to the Appendix for more precise definitions of these terms).

K app Ic = λ c G c B(λ c ) -B(0) (6)
The same approach can be applied to the 3D case treated herein, assuming that the initiation crack front is parallel to the parent crack front, the 3D problem being reduced to a 2D problem in the plane orthogonal to y 3 .

Mixed mode I+III

Under mode I+III loading, facets that initiate ahead of the parent crack form an angle φ (see Figure 1) with this parent crack. In the following, the mode mixity ratio is denoted β (Eq. 7), where K I and K III are the mode I and III stress intensity factors in the outer domain (see the Appendix).

β = K III K I + K III (7)
Following the hypotheses described in Section 2, the choosen inner domain is a cylinder of outer radius Π and thickness containing a semi-infinite parent crack with a blunted tip of radius unity in which a single crack initiates. Periodic boundary conditions with period = e/R are prescribed along y 3 -direction in order to model the simultaneous initiation of several cracks of similar size and orientation, regularly spaced ahead of the parent crack, being the dimensionless spacing (unknown up to now) between facets. The possible initiation cracks, whose modeling is described in Section 4.1, exhibit a complex 3D shape, which may no longer be described by a length, but rather by some parameters (and possibly an infinite number, as evoked in [START_REF] Doitrand | Damage onset modeling in woven composites based on a coupled stress and energy criterion[END_REF][START_REF] Weißgraeber | A review of Finite Fracture Mechanics: crack initiation at singular and non-singular stress raisers[END_REF]). In order to overcome this difficulty, Doitrand and Leguillon [START_REF] Doitrand | 3D application of the coupled criterion to crack initiation prediction in epoxy/aluminum specimens under four point bending[END_REF] proposed to determine the possible crack shapes based on the stress isocontours. The main advantage of this method is that the possible crack shapes can be described by a single parameter, e.g., their surface S or their extension l along (Oy 1 ) axis, assuming that the crack surface can be written S = αl 2 (where α is a scaling coefficient). Once the dilatation of space variables (detailed in the Appendix) has been performed, equation ( 4) rewrites (powers differ slightly from 2D to 3D, refer to [START_REF] Leguillon | An attempt to extend the 2D coupled criterion for crack nucleation in brittle materials to the 3D case[END_REF] and to the Appendix):

   -δW P = K I 2 R 2 (B(λ) -B(0)) G inc = K I 2 B(λ)-B(0) αλ 2 (8)
Thus, the equation to solve in order to obtain the initiation crack extension length λ c is :

1 σ(λ c ) 2 B(λ c ) -B(0) αλ 2 c = 1 R G c σ 2 c ( 9
)
Leading to

     K app Ic = αλ 2 c Gc B(λc)-B(0) K app IIIc = β 1-β K app Ic (10)
Solving Eqs. 6 or 10 requires the computation of the normalized incremental energy release rate B(λ)-B(0) αλ 2 , which is performed by FE and explained in next Section.

FE modeling of daughter crack nucleation

FE model

Solving the coupled criterion requires some FE element calculations in order to determine the stress fields prior to crack initiation and the potential energy change between undamaged and damaged states for various facet size. The mesh generation of the undamaged inner domain, whose geometry is depicted in Fig. 1a, is quite straightforward. A 2D mesh in the y 1 y 2 plane is generated and extruded along the y 3 direction, the mesh being refined in the vicinity of the parent crack tip. Classical 15 nodes quadratic elements are used in order to compute the stress fields. While periodic boundary conditions are prescribed on the inner domain faces whose normal lies in the y 3 -direction, displacements corresponding to a mode I+III far field loading around the sharp crack tip [START_REF] Westergaard | Stresses at a crack, size of the crack and the bending of reinforced concrete[END_REF] of the outer domain are imposed on the fictitious boundary of the inner domain (cf. Eq. 11). It is nothing more than the implementation of the matching conditions in the matched asymptotic expansions method (see the two first assumptions at the end of Section 2 and the Appendix).

             u 1 = KI 2µ ρ 2π cos θ 2 (3 -4ν -cosθ) u 2 = KI 2µ ρ 2π sin θ 2 (3 -4ν -cosθ) u 3 = 2KIII µ ρ 2π sin θ 2 (11)
where (ρ, θ) (ρ = r/R) are the polar coordinates in y 1 y 2 plane of the inner domain, K I and K III the stress intensity factors independent of y 3 as a main assumption (Section 2), and µ the second Lamé coefficient (shear modulus). The FE calculations have been performed using Zset FE solver [34]. The material properties used for the calculations are E = 2.1 GPa, ν = 0.395, G c = 88 10 -3 MPa mm and σ c = 35 MPa, which are repre-sentative of a polymer such as Homalite-100 studied by Pham and Ravi-Chandar [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF][START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF].

The calculation of the potential energy change between undamaged and damaged states requires the generation of meshes containing cracks. Therefore, it is necessary to determine the possible initiation crack shape. If no assumptions are made on the 3D crack shape, it can possibly be described by an infinite number of parameters [START_REF] Doitrand | Damage onset modeling in woven composites based on a coupled stress and energy criterion[END_REF][START_REF] Weißgraeber | A review of Finite Fracture Mechanics: crack initiation at singular and non-singular stress raisers[END_REF], which complexifies the initiation crack shape determination. Doitrand and Leguillon [START_REF] Doitrand | 3D application of the coupled criterion to crack initiation prediction in epoxy/aluminum specimens under four point bending[END_REF][START_REF] Doitrand | Comparison between 2D and 3D applications of the coupled criterion to crack initiation prediction in scarf adhesive joints[END_REF] recently overcome this difficulty by determining the possible crack shapes based on the stress isocontours, so that the 3D crack can be described by a single parameter. The same approach is used herein in order to determine the possible crack shapes in the plane maximizing the tensile stress at the blunted crack tip. Fig. 2a-b shows the normal stress field and isocontours in this plane for a ratio β = 0.37. From a qualitative point of view, the stress isocontours exhibit a shape that is similar to that observed experimentally by Pham and Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] (cf. Fig. 2c-d). Therefore, it seems reasonable to select the possible crack shapes based on the stress isocontours.

Once the possible crack shapes are determined, it is necessary to generate 3D meshes containing such cracks in order to compute the potential energy difference for various crack surfaces. First, a 2D mesh of the cross section of the inner domain containing the projection of the crack outline in the plane (Oy 1 y 2 ) is generated (Fig. 3a-b). Then, the mesh is extruded so as to obtain a 3D mesh which is geometrically sheared in the (Oy 2 y 3 ) plane in order to get the crack back in the plane maximizing the tensile stress (Fig. 3c-e). The uncracked case is obtained from the cracked one by merging the double nodes lying on the opposite crack faces. Thus, the mesh topology is the same reducing the errors in the calculation of the difference in potential energy. It can be noted that the representative volume element faces in the y 3 -direction are inclined with an angle corresponding to the crack angle. This is really convenient since it allows modeling an overlap between cracks, which would not be possible with vertical faces and a single inclined crack. Besides, the inner domain thickness in y 3 -direction, which has no influence under pure mode I loading, now corresponds to the dimensionless spacing between the cracks in the modeled periodic network along y 3 -direction. The determination of the initiation crack spacing is adressed in Section 5.

Inner domain size

Although the inner domain is theoretically unbounded, to the aim of FE calculations it is artificially bounded by a fictitious boundary on which displacements are prescribed to represent the behavior at infinity. Therefore, the inner domain for the FE calculations must be large enough so that the blunted crack tip radius is small compared to the inner domain radius. In addition, it must also be large enough so that the area corresponding to the perturbation induced by the daugh-ter crack nucleation in the displacement field does not interact with the fictitious inner domain boundary.

In order to calibrate the inner domain size, the incremental energy release rate in pure mode I has been calculated for different inner domain radii Π = 50, 100, 200, 400 (keep in mind the dimensionless crack tip radius is 1) and compared to the incremental energy release rate obtained using the method described by Leguillon et al. [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF] (cf. Fig. 4). The incremental energy release rate associated to a crack in an unbounded domain is a monotonic increasing function of the crack surface. For each inner domain radius, it is clear that if the crack extension length (in y 1 or y 2 direction, cf. Fig. 1) is larger than Π/20, a non-monotonic incremental energy release rate is obtained, consequence of an interaction between the crack (that is too large compared to the inner domain size) and the outer fictitious boundary of the inner domain. However, if the crack extension is lower than Π/25, the difference on the incremental release rate is lower than 1% compared to the reference obtained in [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF]. Therefore, in the following, all the calculations have been performed ensuring an inner domain radius corresponding to at least 25 crack extensions along y 1 and y 2 direction.

Pure mode I solution

FE calculations using the inner domain presented in Fig 1a have been performed under pure mode I loading. The inner domain thickness (which has no influence on the results for pure mode I since the crack extends in the 0y 1 y 3 plane), was choosen equal to 50 dimensionless crack tip radius. The incremental energy release rate has been calculated for several normalized crack length λ (crack depicted in Fig. 1b). Then, by solving Eq. 5 for 4 Incremental energy release rate as a function of the dimensionless crack length under pure mode I loading obtained for different inner domain radius (the blunted crack tip radius is 1) and using [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF].

different values of blunted crack tip radius R, the corresponding initiation normalized crack length has been determined, as well as the initiation load (given by Eq. 6). The initiation normalized length and loading as a function of the crack tip radius are presented in Fig. 5a andb. For instance, a crack tip radius R = 89 10 -3 mm ( [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]) leads to an initiation crack length 56.2 10 -3 mm at a loading level K app Ic = 0.58 MPa m 1/2 . As a matter of comparison, using [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF], an initiation crack length of 55. 3 10 -3 mm and a loading level K app Ic = 0.59 MPa m 1/2 are obtained for pure mode I loading, which shows the consistency of the proposed 3D FE modeling.

Mixed mode I+III solution FE calculations using the inner domain presented in Fig 1c have been performed

under mode I+III loading. The incremental energy release rate has been calculated for several crack spacing (e.g. the crack depicted in Fig. 1c and3). The initiation normalized crack surface and loadings were then obtained by solving Eq. 9 and using Eqs. 10 and 7. Fig. 6 shows the initiation normalized crack surface and loading as a function of the blunted crack tip radius for an inner domain thickness corresponding to 10 times the dimensionless crack tip radius and a ratio β = 0.44. A crack tip radius R = 89 10 -3 mm ( [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]) leads to an initiation crack surface 3.04 mm 2 at a loading level K app Ic = 0.71 MPa m 1/2 .

5 Numerical results

Comparison with experiments by Pham and

Ravi-Chandar [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF] and Chen et al. [START_REF] Chen | Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture[END_REF] Fig. 7 shows the crack initiation angle as a function of the mode mixity ratio β obtained experimentally by Pham and Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] and numerically by FE. It seems that determining the crack orientation based on the maximum tensile stress ahead of the parent crack tip provides a good estimate of the crack initiation angle compared to experimental results. The proposed model allows determining the initiation crack surface and loading for a given crack spacing. Therefore, the initiation loading was determined as a function of the crack spacing to crack tip radius ratio = e/R, which is presented in Fig. 8 for a mode mixity ratio β = 0.37. It can be higlighted that the loading necessary to initiate a crack decreases with increasing crack spacing until a critical crack spacing above which it reaches a constant value. Indeed, there is a shielding effect that prevent the initiation of two cracks that are too close to each other. However, as soon as the crack spacing is sufficiently high, the cracks no longer interact so that the same loading level is obtained. Leguillon et al. [START_REF] Leguillon | Multi-cracking in brittle thin layers and coatings using a FFM model[END_REF] encountered the same situation in the case of multi-cracking in brittle thin layers and showed that the crack spacing at initiation may be determined as the mimimum spacing among those corresponding to the constant initiation loading. Repeating the same procedure for other mode mixity ratio β between 0.17 and 1, the initiation loadings K app Ic and K app IIIc can be determined (cf. Fig. 9). The large gap between β = 0 and β > 0 may be explained by two different mechanisms, continuous crack growth in the former case and fragmentation of the crack front in the latter.

In addition, the initiation crack spacing were found to lie between 0.44 mm and 0.51 mm for a blunted crack tip radius R = 89 10 -3 mm, corresponding roughly to 5 crack tip radii. Therefore, it seems that the initiation crack spacing only depends on the parent crack tip dimensions, which is in agreement with Pham and Ravi-Chandar conclusions [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF]. Moreover, the crack spacing determined numerically is of the same order of magnitude as the one found experimentally by Pham and Ravi-Chandar [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF] for Homalite-100, who measured an initiation crack spacing corresponding to around 4 parent crack tip radius. Similar order of magnitude was also obtained by Chen et al., who measured crack spacings in PMMA between 0.13 and 0.48 mm, however, no indication was given about the crack tip radius.

Comparison with experiments by Pham and

Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] Pham and Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] designed a test so as to load a specimen in mode I+III through the combination of a wedge load and of a confining load resulting in a global compressive stress field ahead of the crack front. This particular set up promoted the sequential initiation of several cracks (at nearly similar loading levels) rather than a simultaneous initiation of these cracks. Moreover, an unstable propagation of the cracks just Fig. 7 Crack initiation angle as a function of the mode mixity ratio β obtained experimentally by Pham and Ravi-Chandar [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF] and numerically by FE.

after initiation was observed. Therefore, the first crack likely reached a bigger size than in the case of simultaneous crack initiations [START_REF] Pham | Further examination of the criterion for crack initiation under mixed mode I+III loading[END_REF] . This phenomenon highlights the difficulty to precisely measure an initiation length in certain experiments. Indeed, in the present case, the initiation length l c is not an arrest length since an unstable growth is observed rigth after crack initiation. This can be explained considering the relation between the differential (G) and incremental (G inc ) release rates (cf. Eq. 12) [START_REF] Doitrand | Mesoscale analysis of damage growth in woven composites energy criterion[END_REF].

G(S) = dG inc (S) dS S + G inc (S) (12)
At crack initiation, G inc = G c , therefore it is clear that for a strictly monotonic increasing evolution of G inc , ) Fig. 10 Normalized incremental energy release rate as a function of the crack spacing obtained for a mode mixity ratio β = 0.37 under the assumption that the first crack has propagated in an unstable maneer just after initiation. G(S) > G inc (S) = G c , resulting in unstable crack propagation immediatly following the initiation. A spacing between cracks ranging from 4.9 to 6.6 mm was measured experimentally, both cracks exhibiting nearly the same size (crack major and minor axes were about 6.3 and 3.7 mm assuming a nearly elliptic shape) and angles (between 29 and 36 deg.). We apply herein the model presented in Section 4.1 assuming that the crack shape is fixed with dimensions close to those measured in [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF], with a tilt angle φ = 33 deg. for various crack spacings. Fig. 10 shows the normalized incremental energy release rate as a function of the crack spacing. Similarly to the results obtained previously, the initiation crack spacing can be determined as the minimum spacing among those corresponding to a constant normalized incremental energy release rate (or a constant initiation loading). A initiation crack spacing of 5.3 mm is obtained, which is in good agreement with the spacings measured experimentally [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF].

Conclusion

Adressing crack propagation under mode I+III loading requires studying the parent crack front segmentation in multiple facets that grow and later on coalesce. The crack front segmentation thus looks like a crack initiation problem rather than a problem of crack growth. The proposed approach, which is a new application of the coupled criterion to the tridimensional case, allows the nucleation of facets ahead of a parent crack under mode I+III loading to be simulated. Crack initiation shape, orientation and spacing is determined for any mode mixity ratio by coupling a stress and an en-ergy criterion in the framework of matched asymptotic expansions. Crack orientation is determined based on the maximum tensile stress ahead of the parent crack tip. The obtained crack angles for several mode mixity ratios show a good agreement with experimental results. Crack initiation size, loading and spacing are dependent on the blunted crack tip radius. Similar crack spacings are obtained whatever the mode mixity ratio, which are of the same order or magnitude as crack spacings measured experimentally. The possible crack shapes based on the stress isocontours are qualitatively close to those observed experimentally, which indicates the consistency of determining the possible cracks using these isocontours. It is worth highlighting the convenience of such a method, which allows parameterizing the cracks by a single variable. The only assumptions made on the crack shape is that it initiates in a plane, which is representative of the cracks observed experimentally except for their tips, which are gently curved. A possible improvement of the proposed method would be the determination of the 3D crack shape without making the assumption of a plane extension.

Appendix

The matched asymptotic expansions

The actual domain embeds the parent blunted crack (root radius R) and a group of small slanted crack regularly spaced. The outer domain is obtained for R → 0, and as a consequence l → 0 since it is assumed that l is smaller or of the same order of magnitude than R (to be checked afterwards). In the outer domain, the blunted crack becomes a slit with no thickness. Both actual and outer domains are spanned by the Cartesian space variables x i (i = 1, 3) (r, θ, x 3 , in cylindrical coordinates).The outer expansion with respect to the small parameter R can be written (∇ x is the gradient operator with respect to the x i 's)

U R (x 1 , x 2 , x 3 , l) = U 0 (x 1 , x 2 , x 3 ) + ... σ(U R ) = C : ∇ x U R = C : ∇ x U 0 + ... (13)
The dots denote a remainder that is small and decreases with R. The behavior of the leading term of Eq. ( 13) near the crack tip is described by the Williams series (polar and Cartesian coordinates are mixed without risk of confusion)

U 0 (x 1 , x 2 , x 3 ) = C + K I √ ru I (θ) + K III √ ru III (θ) + ... U 0 (x 1 , x 2 , x 3 ) =C + K I √ r[u I (θ) + mu III (θ)] + ... with m = K III K I = ( 1 β -1) -1 (14)
The leading term is an irrelevant constant (rigid translation), K I and K III are the modes I and III stress intensity factors and u I and u III are the associated opening and shear modes. The intensity factors K I and K III are independant of x 3 as a consequence of the first assumption stated at the end of Section 2.

The inner domain is obtained by zooming in the actual domain by 1/R and considering again the limit as R → 0. It is an unbounded domain spanned by the dimensionless space variables y i = x i /R (i = 1, 3); it embeds the semi-infinite parent crack and an infinite number of regularly spaced slanted cracks with length λ = l/R along y 1 axis. In the y 3 direction, a periodicity assumption is done reducing the domain to a single period of length = e/R in this direction (see Figures 1 to 3).

After having fulfilled the matching conditions [START_REF] Leguillon | Computation of singular solutions in elliptic problems and elasticity[END_REF], the inner expansion can be written

Where W = V I +mV III behaves at infinity like √ ρu I (θ)+ m √ ρu III (θ). According to [START_REF] Leguillon | Prediction of crack initiation at blunt notches and cavitiessize effects[END_REF], the change in potential energy takes the form -δW p = Ψ x (U R (x 1 , x 2 , x 3 , l), U R (x 1 , x 2 , x 3 , 0)) -δW p = Ψ x (U R (x 1 , x 2 , x 3 , l), U 0 (x 1 , x 2 , x 3 )) -Ψ x (U R (x 1 , x 2 , x 3 , 0), U 0 (x 1 , x 2 , x 3 )) [START_REF] Leguillon | An attempt to extend the 2D coupled criterion for crack nucleation in brittle materials to the 3D case[END_REF] where

Ψ x (F , G) = 1 2 Γx [σ(F ).n.G -σ(G).n.F ]ds (18)
The contour Γ x starts from the stress free edges of the crack and embeds the blunted notch and the crack extension. The integral [START_REF] Leguillon | Multi-cracking in brittle thin layers and coatings using a FFM model[END_REF] is path indepedent for any two functions F and G fulfilling the equilibrium equations. The index x in Eqs. ( 17) and [START_REF] Leguillon | Multi-cracking in brittle thin layers and coatings using a FFM model[END_REF] recalls that the calculations are a priori carried out in the actual (using the actual solution) or the outer domain (using the outer expansion but this requires calculating the remainder in 13 [START_REF] Leguillon | Determination of the length of a short crack at a v-notch from a full field measurement[END_REF]).

Taking into account the change of variables defined by the dilatation in the inner domain, [START_REF] Leguillon | Multi-cracking in brittle thin layers and coatings using a FFM model[END_REF] rewrites

Ψ x (F , G) = R
2 Γy [σ(F ).n.G -σ(G).n.F ]dS Ψ x (F , G) = RΨ y (F , G)

Because dS = R 2 × ds (while dS = R × ds in 2D leading to two similar relationships expressed either with respect to the x i 's or the y i 's). Then

Note that B(λ) -B(0) is nothing but the change in potential energy computed in the y i variables when moving from W (y 1 , y 2 , y 3 , 0) to W (y 1 , y 2 , y 3 , λ). With a newly created crack surface S = αl 2 (α is a scaling coefficient) it comes the energy condition

G inc = - δW P S = K 2 I B(λ) -B(0) αλ 2 ≥ G c (21)
Denoting σ and σ the tensile component of the corresponding stress tensors, the stress condition can be written

σ(U R ) = K I √ R σ(W ) ≥ σ c (22)
here σ(W ) is nothing but the tensile stress computed in the y i variables. Combining (21) and [START_REF] Mittelman | Energy release rate cannot predict crack initiation orientation in domains with a sharp V-notch under mode III loading[END_REF] gives the equation for the dimensionless crack extension length λ c 1 σ(λ c )

B(λ) -B(0) αλ 2 c = 1 R G c σ 2 c (23)
In this equation σ(λ c ) denotes the tensile stress associated with W , expressed in the y i variables and computed at a dimensionless distance λ c of the notch root along the y 1 axis. Then the load at failure can be derived from (21)

K app Ic = αλ 2 c G c B(λ c ) -B(0)
; K app IIIc = mK app Ic [START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]

3 ΠFig. 1 (

 31 Fig. 1 (a) Inner domain of radius Π and thickness considered around the parent crack blunted tip of radius 1. Crack of length l initiating ahead of the parent crack tip under (b) pure mode I and (c) mixed mode I+III loading.

Fig. 2

 2 Fig. 2 Normal stress (a) field and (b) isocontours in the plane maximizing the tensile stress at the blunted crack tip. (c) Possible crack shapes based on the stress isocontours and (d) crack shape observed experimentally (Reprinted from[START_REF] Pham | On the growth of cracks under mixed-mode I+III loading[END_REF]).

Fig. 3

 3 Fig. 3 (a) 2D mesh containing the 3D crack projection in the (Oxy) plane and (b) focus around the crack. (c) 3D and (d-e) lateral views of the 3D mesh obtained by extrusion and shearing of the 2D mesh.

Fig. 5

 5 Fig. 5 Initiation (a) Normalized crack length and (b) loading as a function of blunted crack tip radius under pure mode I loading.

Fig. 6

 6 Fig. 6 Initiation (a) Normalized crack surface and (b) loading as a function of blunted crack tip radius under mode I+III loading with a mode mixity ratio β = 0.44.

Fig. 8 Fig. 9

 89 Fig.8Crack initiation loading as a function of the crack spacing obtained for a mode mixity ratio β = 0.37 .

U

  R (x 1 , x 2 , x 3 , l) = U R (Ry 1 , Ry 2 , Ry 3 , Rλ) U R (x 1 , x 2 , x 3 , l) = C + K I √ RV I (y 1 , y 2 , y 3 , λ) + K III √ RV III (y 1 , y 2 , y 3 , λ) + ... U R (x 1 , x 2 , x 3 , l) = C + K I √ R[V I (y 1 , y 2 , y 3 , λ) + mV III (y 1 , y 2 , y 3 , λ)] + ...

( 15 )

 15 where V I (respectively V III ) behaves like√ ρu I (θ) (respectively √ ρu III (θ)) at infinity (ρ = r/R → ∞). Relationship (15) can be rewritten for the purpose of full FE calculationsU R (x 1 , x 2 , x 3 , l) = C + K I √ RW (y 1 , y 2 , y 3 , λ) + ... σ(U R ) = 1 R C : ∇ y U R = KI √ R C : ∇ y W = KI √ R σ(W )

-δW p = K 2 I R 2 [

 22 Ψ y (W (y 1 , y 2 , y 3 , λ), √ ρ[u I (θ) + mu III (θ)]) -Ψ y (W (y 1 , y 2 , y 3 , 0), √ ρ[u I (θ) + mu III (θ)]) -δW p = K 2 I R 2 (B(λ) -B(0)) + ...


Ignacio Avellino
email: avellino@isir.upmc.fr

Carman Neustaedter
email: carman@sfu.ca

Putting the Coach in the Loop: Improving Practice in Outdoor Activities Through Technology

Keywords: Outdoors, Sports, Sports Tracking, Wearables, Expert Feedback, Coaching, Performance Coaching ACM Classification H.5.m [Information interfaces and presentation (e.g., HCI)]: Miscellaneous

Outdoor activities provide many benefits, including physical and mental health, skill acquisition and social development. Wearable devices that measure activity data, such as GPS position, speed or heart rate, can potentially help users improve their performance. For this purpose, current systems either present data to the user, or have an expert system evaluate it and provide recommendations. But both these approaches leave out a key player: the expert coach. Expert mentoring has an important and irreplaceable role when it comes to making decisions to improve performance. We argue that in order to effectively improve performance, systems that use wearables to capture data during outdoor activities should provide useful visualizations to experts when these are available.

Introduction

It goes without saying that outdoor activities have a myriad of benefits for physical and mental health. Physical inac-tivity, defined as less than 30 minutes of brisk walking per day, increases several causes of mortality: cardiovascular diseases including hypertension or stroke; metabolic diseases including diabetes or obesity; several types of cancer including breast, prostate and pancreatic; pulmonary diseases such as asthma; immune dysfunction; musculoskeletal and neurological disorders (see [START_REF] Booth | Waging war on physical inactivity: using modern molecular ammunition against an ancient enemy[END_REF] for a review).

One of the goals of performing outdoor activities, e.g. running, hiking, cycling, skating, skiing or open-water swimming, is training to improve performance. For this purpose, wearable devices that can measure a wide variety of data bring new opportunities. Current systems though, follow one of two approaches: they either burden the user, making him interpret data on his own and take action, or use an algorithm to make decision for the user. But these approaches leave out a key player for improving performance: the expert coach.

Imagine having to interpret the results of a medical test, such as an electrocardiogram, to make recommendations for improving your own health. Or having to interpret the results of a personality test to determine the changes you should make to reach your goals. When we are left to interpret our own data, and we do not have the necessary knowledge to do so, we might make inaccurate decisions or have a biased perspective, falling for confirmation bias [START_REF] Nickerson | Confirmation bias: A ubiquitous phenomenon in many guises[END_REF]. Expertise becomes crucial when making not only a decision, but the right decision. So why is it then that research and commercial system haven't given experts a center role when it comes to improving performance in outdoor activities? We argue that systems that use data from wearable devices and aim at improving performance should include experts that can access and interpret activity information.

Coaching and Its Benefits

Lyle [START_REF] Lyle | Sports coaching concepts: A framework for coaches' behaviour[END_REF] distinguishes between two types of coaching: participation and performance. Participation coaching is less intensive, usually not systematic, and not all of the performance elements (e.g. strength training or technique) are attended to. Performance coaching has competition goals, longer-term horizons, significant athlete commitment and an attempted control of contributory variables. In this position paper we refer to performance coaching, although the ideas could be applied to less intensive coaching for amateur athletes, as long as their goal is improving performance.

McPherson and colleagues have long studied expert and novices in sports, mainly through tennis [START_REF] Mcpherson | Relation of knowledge and performance in boys' tennis: Age and expertise[END_REF][START_REF] Mcpherson | Expert-Novice Differences in Performance Skills and Problem Representations of Youth and Adults during Tennis Competition[END_REF][START_REF] Mcpherson | Changes in Cognitive Strategies and Motor Skill in Tennis[END_REF]. They find that experts learn to solve problems in a more sophisticated manner, having developed cognitive structures for this purpose. Experts generate more plans for action than novices, which are more varied and with more sophisticated goals. In contrast, novices show minimal planning strategies and poor interpretations of events, focusing on the moment and developing weak solutions to current problems [START_REF] Mcpherson | Expert-novice differences in planning strategies during collegiate singles tennis competition[END_REF].

Performance coaches develop and improve the individuals' performance. Their expertise is domain-specific, they recognize patterns faster than novices, their knowledge is structured and easy to recall, they are more flexible, more able to adapt to situations and take deeper meanings from cues than novices [START_REF] Nash | Tacit Knowledge in Expert Coaching[END_REF]. They have both procedural and tacit knowledge that allows them to make correct decisions.

Supporting Outdoor Activities with Technology

Wearable devices that can measure outdoor activity data are becoming more available and affordable. They include sensor such as pedometer, accelerometer/gyroscope, GPS, heart rate monitor and temperature [START_REF] Li | Wearable Performance Devices in Sports Medicine[END_REF]. Research has looked into using the data from these devices to improve performance mainly in two ways: by having the user make decisions based on his own data, or by having a system provide recommendations based on automatic analysis. Some systems record user data during outdoor activities and show it to them in real time. RunRight [START_REF] Nylander | Runright: Real-time Visual and Audio Feedback on Running[END_REF] shows a live visualization of a runner's body movement so that they can correct their posture. Stienstra et al. [START_REF] Stienstra | Embodying Complexity Through Movement Sonification: Case Study on Empowering the Speed-skater[END_REF] propose a system for professional speed skaters that measures speedskating strokes and provides auditory feedback, as a way to present complex information that the skater can use to improve his technique. The sonic golf club 1 is a system that provides auditory feedback to improve the swing during golf. These systems rely on the user having the sufficient knowledge for decision making, which is not always the case.

The second use of the data is to have it analyzed by a system that acts as a virtual mentor and provides recommendations for improvement. FootStriker [START_REF] Wiehr | FootStriker: An EMS-based Assistance System for Real-time Running Style Correction[END_REF] senses the contact point of the feet during running and gives feedback as electrical muscle stimulation, so that runners avoid striking with their heels. Around Me [START_REF] Tominaga | Around Me: A System with an Escort Robot Providing a Sports Player's Self-images[END_REF] is a robot that follows a person while it jogs (from the front), and provides a realtime visualization of the ideal jogging posture so users can adapt their posture. Spelmezan & Borchers [START_REF] Spelmezan | Real-time Snowboard Training System[END_REF] proposed a system that measures forces applied by the feet during snowboard and provides feedback on the posture.

Sensor data can also be used by a coach as part of the decision process for improving performance [START_REF] Li | Wearable Performance Devices in Sports Medicine[END_REF]. But this has only been recently introduced in HCI. Wakefield et al. [START_REF] Wakefield | The Informatics Needs of Amateur Endurance Athletic Coaches[END_REF] interviewed coaches to determine which data they collect from athletes and how they use it. They identify that recording data is tedious for athletes, and that it can be difficult 1 www.sonicgolf.com for coaches to read, parse and understand it. Sharing data is cumbersome as many times this happens face-to-face. Athletes sometimes forget to present some of the relevant information, and even if they use a journal, they may forget to record data or do not know exactly what data are relevant to record in the first place. This qualitative study shows that there is a need for data to be exchanged between athletes and coaches. Nevertheless, specific data needs for coaches to provide useful feedback towards improving athletes' performance remain unstudied, as well as how can technology support this process.

The only published work in HCI, to my knowledge, of a system that aims at coaches providing feedback to athletes, is in the context of snowboarding practice. Spelmezan et al. [START_REF] Daniel Spelmezan | Tactile Motion Instructions for Physical Activities[END_REF] designed a system for improving snowboarding, where a trainer can provide live feedback to a novice by giving him electric stimulation. This study though, aimed at determining how accurately athletes can interpret feedback, and did not study how coaches could visualize and analyze data during decision making. Their prototype was one-way as the expert could provide feedback, but there was no sensory data that an expert could see. Some companies in industry are starting to produce hardware for professional rugby and american football teams, that can measure in real time fatigue index, collision load and distance covered, as recently reported in the news 2 . Decrease in injury and improved performance are reported, but research that can validate these claims is still missing.

Discussion

With the increased availability of wearable devices, various types of data can be recorded automatically. This opens the possibility for both live and offline coaching. Live coaching is a challenge for outdoor activities as these usually involve moving through long distances. Following every athlete is physically demanding, thus systems could take advantage of real-time data from wearable devices and enable remote live coach feedback to improve athletes' performance.

In running, coaches could provide real-time feedback on stepping (as FootStriker [START_REF] Wiehr | FootStriker: An EMS-based Assistance System for Real-time Running Style Correction[END_REF]) or posture (as Around Me [START_REF] Tominaga | Around Me: A System with an Escort Robot Providing a Sports Player's Self-images[END_REF]). During swimming, feedback could include breathing, arm movement errors and upper and lower body coordination, which are some of the problems in the crawl technique [START_REF] Mandzák | Correction of errors front crawl swimming technique of students of physical education and sport[END_REF]. Data logging though, does not need to be restricted to an automatic process, as technology can assist manual logging. For instance, smart glasses could be used to log data such as Rated Perceived Exertion (RPE) [START_REF] Wiehr | Interfaces for Assessing the Rated Perceived Exertion (RPE) During High-intensity Activities[END_REF].

Outside of training sessions, other important data can also be automatically recorded such as average heart rate, step count and sleeping time. These data can later be used along with that of the activity when developing long-term training plans. Data sharing with current technologies can happen transparently for both the athlete and the coach. But in order for coaches to make correct choices, systems need to present all the gathered information in a way that is easy to understand. Systems need to create useful data visualizations that satisfy coaching goals.

Lastly, besides improving practice, having data to support decisions can bring additional benefits for coaches, for instance in understanding the rationale behind their decisions. Coaches are many times seen as "masters of the instantaneous response" [START_REF] Launder | Coach education for the twenty first century[END_REF], making decisions based on intuition. Research has recognized that there is value in studying the why and how of coach behavior, and not only the what [START_REF] Potrac | The invisible ingredient in coaching knowledge: A case for recognising and researching the social component[END_REF]. Many existing courses for coaching however develop "parrots" that mimic the instructor, and evaluate candidates on their ability to do so. But a pedagogic ap-proach based on data that makes transparent the rationale behind decision making would be more successful than the current approach that relies largely on luck [START_REF] Nash | Tacit Knowledge in Expert Coaching[END_REF].

Conclusion

Digital devices that collect data during outdoor activities have a great potential for improving performance. But so far, systems have largely neglected performance coaches. Research has focused on providing the user with visualizations for making decisions on how to improve, or on automatic systems that make these decisions for the user. We argue that we should aim for systems that provide useful visualizations of athletes' data to coaches. Experts of outdoor activities can provide valuable and useful feedback to athletes that want to improve their practice if they have access to their data. This feedback can be in real-time while the activity is being performed, or after the activity in a retrospective analysis.

Because outdoor activities, such as running, cycling or skiing involve traveling, technology can help to bring the expert closer to the person performing the activity. Video feeds, position data, speed, heart rate and other measures can all be sent to the coach in real time for live feedback, or be recorded for later analysis.

Given the findings presented in this paper and the open design space, we plan to pursue qualitative studies to identify the data needs of performance coaches for improving performance. Then, we plan to explore the design space of future systems that gather and present this data to coaches, with the goal of prototyping and testing one or more prototypes. Ultimately, we plan to create technology that helps people become better athletes. we look forward to the possibility of discussing these topics with others as part of workshop activities.

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