Ibrahim Bitar
email: ibrahim.bitar@ec-nantes.fr

Nathan Benkemoun
email: nathan.benkemoun@univ-nantes.fr

Panagiotis Kotronis
email: panagiotis.kotronis@ec-nantes.fr

Stéphane Grange
email: stephane.grange@ujf-grenoble.fr

A novel

Keywords: Timoshenko, Multi-fiber, Discontinuity

A NOVEL MULTI-FIBER TIMOSHENKO BEAM FINITE ELEMENT FORMULATION WITH EMBEDDED DISCONTINUITIES TO DESCRIBE REINFORCED CONCRETE FAILURE UNDER STATIC LOADINGS

Ibrahim Bitar * , Nathan Benkemoun † , Panagiotis Kotronis * and Stéphane Grange ‡

Introduction

Different kinematic assumptions are used in structural analysis in order to simplify the global equilibrium equations and to reduce the required number of degrees of freedom. The Timoshenko beam theory considers that plane sections remain plane after deformation but not necessary normal to the deformed axis. The advantage of this theory is that it takes into account the influence of shear strains (contrary to Euler-Bernoulli (EB) assumption.

In a multi-fiber beam context, the section of the beam is divided into several fibers with specific stress/strain relations [START_REF] Owen | Finite elements in plasticity[END_REF]. Finite elements of this type are efficient for various applications in civil engineering: nonlinear analysis of beam type or bearing wall structures with nonhomogenous sections (eg. reinforced concrete) [START_REF] Kotronis | A simplified modelling strategy for R/C walls satisfying PS92 and EC8 design[END_REF], [START_REF] Grange | Numerical modelling of the seismic behaviour of a 7-story building: NEES benchmark[END_REF], arbitrarily geometrical plane or hollow shape sections [START_REF] Grange | The effects of Soil-Structure Interaction on a reinforced concrete viaduct[END_REF], [START_REF] Desprez | Seismic vulnerability assessment of a RC structure before and after FRP retrofitting[END_REF] submitted to bending, shear or torsion [START_REF] Mazars | Using multifiber beams to account for shear and torsion[END_REF], Soil Structure Interaction problems [START_REF] Grange | The effects of Soil-Structure Interaction on a reinforced concrete viaduct[END_REF], vulnerability assessment cases [START_REF] Desprez | Seismic vulnerability assessment of a RC structure before and after FRP retrofitting[END_REF] and Fiber-Reinforced Polymer retrofitting [START_REF] Desprez | Damage model for FRPconfined concrete columns under cyclic loading[END_REF].

In this paper we present a novel multifiber beam finite element formulation based on the Timoshenko model to simulate failure of reinforced concrete structural elements subjected to static monotonic loadings. The new displacement-based Finite Element formulation proposed by [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] is used. This formulation uses shape functions of order three (3) for the transverse displacements, and two (2) for the rotations and an additional internal node. This results to a finite element free of shear locking. We refer to [START_REF] Bitar | A review on Timoshenko multifiber beams Finite Element formulations and elasto-plastic applications[END_REF] where we show the performance of this formulation with respect to other formulations in the literature [START_REF] Pegon | A Timoshenko simple beam element in Castem[END_REF], [START_REF] Guedes | A Fibre/Timoshenko beam element in Castem[END_REF].

The multi-fiber section consists of concrete and steel fibers. The concrete fibers behave according to a damage law and the steel fibers are modeled with an elasto-plastic behavior.

The lack of an internal length parameter in the behavior laws leads to mesh dependency. To solve this, different approaches can be found in the literature: local approaches [START_REF] Hillerborg | Application of fracture mechanics to concrete: summary of a series of lectures[END_REF], [START_REF] Needleman | Material rate dependence and mesh sensitivity in localization problems[END_REF]; non local approaches [START_REF] Bažant | Wave propagation in a strain-softening bar: exact solution[END_REF], [START_REF] Bažant | Pijaudier?Cabot, Softening in Reinforced Concrete Beams and Frames[END_REF], [START_REF] Chambon | Etude de la localisation unidimensionnelle à l'aide d'un modèle de second gradient[END_REF]; and the enhanced approaches [START_REF] Ortiz | A finite element method for localized failure analysis[END_REF], [START_REF] Simo | An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids[END_REF]. In this work, the focus is set on the enhanced approach [START_REF] Simo | An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids[END_REF], and more specifically on the embedded discontinuity approach (EDA) [START_REF] Wells | Application of embedded discontinuities for softening solids[END_REF]. EDA represents an alternative approach to the smeared and discrete crack representations avoiding some of their drawbacks as described in [START_REF] Loffi | Embedded Representation of Fracture in Concrete With Mixed Finite Elements[END_REF].

The EDA based on the strong discontinuity approach [START_REF] Simo | An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids[END_REF] is adopted, hence localized failure is incorporated into the standard disablement-based Finite Element using discontinuities variables. The fibers are enhanced in order to describe concrete crack openings and the development of plastic hinges for steel [START_REF] Jukić | Failure analysis of reinforced concrete frames by beam finite element that combines damage, plasticity and embedded discontinuity[END_REF]. The strong discontinuity is introduced by adding a jump in the displacement field. Accordingly, additional shape functions are added to interpolate the displacement jump within the enhanced finite element.

Furthermore, the materials behavior at the discontinuity is characterized by a cohesive law linking the axial stress and the displacement jump by a linearly decreasing relation, which allows capturing the released fracture energy.

The variational formulation is presented in the context of the incompatible modes method. The additional modes are statically condensed at the element level in order to maintain the same architecture of the global resolution.

The outline of the paper is as follows. In the next section the governing equations of the multi-fiber Timoshenko beam with the corresponding interpolation functions describing the new based-displacement Finite Element [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] are briefly recalled. In section 3, we present shortly the behavior laws for continuum and cohesive materials. In section 4, we focus on the variational formulation. A numerical illustration of the proposed model is presented in section 5. The article ends with some concluding remarks.

Governing equations

Consider a beam of length Ł discretized into n elements e = [x i ; x j ] of length L = x jx i and external nodes i and j. The generalized displacement vector is approximated by an equation of the form U (x) = N U e , where U e is a vector containing the external nodal displacements of the element e and N is the matrix of the shape functions depending on x. For simplicity reasons presentation is made hereafter in 2D.

U (x) = U x (x) U y (x) Θ z (x) T (1)
U x (x) being the longitudinal displacement, U y (x) the transverse displacement and Θ z (x) the rotation of the section at the neutral axis.

The continuous displacements u x (x, y), u y (x, y) of another point of the section (or of a "fiber" f (x, y)) can be evaluated using the displacements of the section as follows:

u x (x, y) =U x (x) -yΘ z (x), u y (x, y) =U y (x) (2)
The strain field becomes:

ε x (x, y) = ∂u x ∂x = U ′ x (x) -yΘ ′ z (x), γ xy (x, y) = ∂u x ∂y + ∂u y ∂x = U ′ y (x) -Θ z (x) (3)
with ε x (x, y) the axial strain and γ xy (x, y) the transverse shear strain of the fiber f (x, y). The line over the variables indicates that they are continuous.

The theoretical formulation, based on equilibrium equations and the kinematic assumption and using the virtual work principle, provides the beam element stiffness matrix and the internal nodal forces expressions as follows

K element = L 0 B T K S Bdx F int,element = L 0 B T F S dx (4)
where B is a matrix containing the derivatives with respect to x of the shape functions, K S is the multi-fiber section stiffness matrix and F S is the generalized force vector of the section.

In order to enhance the element kinematics, the fiber axial displacement field ( 2) is written with an additional term ( [START_REF] Armero | On the characterization of localized solutions in inelastic solids: an analysis of wave propagation in a softening bar[END_REF], [START_REF] Ehrlich | Finite element methods for the analysis of softening plastic hinges in beams and frames[END_REF], [START_REF] Pham | Stress-resultant models for ultimate load design of reinforced concrete frames and multi-scale parameter estimates[END_REF] and [START_REF] Jukić | Failure analysis of reinforced concrete frames by beam finite element that combines damage, plasticity and embedded discontinuity[END_REF]), as follows:

u x (x, y) = u x (x, y) + αM α (x), (5)
where u x represents the continuum expression of the axial displacement (2) and

• M α (x) = H α (x) -N (x), (6)
• H α (x) = {1 if x > x α or 0 if x ≤ x α }, (7)
•

N (x) = x L , ( 8
)
and α is the discontinuity variable, x α is the position of the discontinuity within the element. Therefore the enhanced axial strain field becomes:

ε x (x, y) = ∂ ∂x (u x (x, y)) + ∂ ∂x (αM α (x)) = ε x (x, y) + α ∂ ∂x (-N (x)) + α ∂ ∂x (H α (x)) = ε x (x, y) + α G(x) ε + α δ α (x) ε (9)
where δ α (x) is the Dirac function at x α . Thus, the latter expression of the axial strain is singular (presence of ε). To solve this singularity, [START_REF] Ibrahimbegović | Combined hardening and softening constitutive model of plasticity: Precursor to shear slip line failure[END_REF] proposed to introduce a cohesive law, that models the material discontinuity and thus eliminates the singular term.

2.1 Displacement-based Finite Element formulation In this paragraph, the new displacementbased Full Cubic Quadratic Finite Element FCQ is presented [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF]. Cubic functions are used to interpolate the transverse displacements and quadratic for the rotations. The element is free of shear locking and uses an additional internal node. Caillerie & al. [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] proved that one FCQ element is able to predict the exact tip displacements for any complex loading (shear/flexion) submitted to an homogeneous elastic beam (see also [START_REF] Bitar | A review on Timoshenko multifiber beams Finite Element formulations and elasto-plastic applications[END_REF]). The nodal displacement field takes the following form,

U T e = [U xi , U yi , Θ zi , ∆U 1 yk , ∆Θ zk , ∆U 2 yk , U xj , U yj , Θ zj ],
where ∆U 1 yk , ∆Θ zk and ∆U 2 yk are the degrees of freedom of the internal node (with no specific physical meaning). The generalized displacement field is U (x) = N (x)U e , and the interpolation functions matrix N (x) can be found in the original paper [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] and also in the appendix.

Material Constitutive laws

The constitutive laws for the continuum and cohesive materials are briefly elaborated. We notice that shear components for both continuous models are linear elastic. Therefore the fiber shear stress is:

τ = kGγ, ( 10
)
where k is the shear correction coefficient, G is the shear modulus and γ is the fiber elastic shear strain.

Ψ d = σ d T ε d - 1 2 σ d T Dσ d + 1 2 ξ d K d h ξ d (11)
The strain-strain relation can be deduced as:

σ d = D -1 ε d with D ∈ [E -1 c , ∞) (12)
where E c is the concrete young modulus. The hardening law takes a linear form

q d = K d h ξ d
where K d h is the hardening modulus which is different in compression and traction. The damage criteria is defined by the following damage surface

φ d = |σ d | -(σ d i -q d ) ≤ 0 ( 13
)
σ d i = σ d c for compression σ d t for traction

Bulk steel behavior

The steel fibers are modeled with an elastoplastic behavior with isotropic hardening. The main ingredients of the model are given in the equations below. We start with the classical decomposition of the continuous strain into elastic and plastic terms:

ε = ε e + ε p (14)
The thermodynamic free energy is expressed in terms of the internal variables: the elastic deformation ε e and the strain-like hardening variable ξ s .

Ψ s (ε e , ξ s ) = 1 2 ε e E s ε e + 1 2 ξ s K s h ξ s ( 15
)
where E s is the steel elastic modulus and K s h is the steel hardening modulus. The stress/strain relation reads

σ s = E s (ε -ε e ) with        εp = γ ∂φ s ∂σ ξs = γ ∂φ s ∂q (16)
where q s is the stress-like variable defining the linear hardening law q s = K s h ξ and φ s is the elastic yield surface:

φ s = |σ s | -(σ s e -q s ) ≤ 0. ( 17
)
3.3 Cohesive material behavior at the discontinuity Linear cohesive laws are used. The thermodynamic free energies relative to concrete and steel:

Ψ d = 1 2 K d 2 coh ξ d , Ψ s = 1 2 K s 2 coh ξ s ( 18
)
K coh is the softening modulus (< 0) of the cohesive material. The cohesive laws are expressed as a relation between the traction at the discontinuity and the discontinuity variable:

t d = K d coh α d , t s = K s coh α s (19)
In order to pass from the continuum model to the discrete one, a failure criteria should be verified. This criteria is defined by the function φ such that:

φ i = |t(α i )| -(σ i u -q i ) ≤ 0 (20)
where

q i = -∂Ψ i ∂ξ i = -K i coh ξ i and i = {d, s}.

Variational formulation

The objective of this work is to embed discontinuity into the fiber axial displacement field. In the spirit of the Hu-Washizu variational formulation [START_REF] Washizu | Variational methods in elasticity and plasticity[END_REF], [START_REF] Belytschko | A finite element with embedded localization zones[END_REF], the variational formulation can be written within the framework of a finite element problem and leads to the following set of non linear equations,

n elem e=1 n act f ib f =1 f int e,f (U e , α e,f ) -f ext e,f = 0 (21) h e,f (U e , α e,f ) = 0, ∀e ∈ [1, n elm ], ∀f ∈ [1, n act f ib ] (22)
where denotes the standard assembly operator, n act f ib indicates the fibers with active discontinuity. The first equation above concerns the global equilibrium, and the second one is relative to the local equilibrium corresponding to the active embedded discontinuities in the fibers (n act f ib ).

The linearized form of the system of equations [START_REF] Loffi | Embedded Representation of Fracture in Concrete With Mixed Finite Elements[END_REF] at incremental pseudo-time n + 1 and iteration k + 1 reads

n elem e=1 n act f ib f =1 K BB K BG K GB K GG + K coh k n+1 ∆U e,f ∆α e,f k+1 n+1 = f int e,f -f ext e,f h e,f k n+1 (23)
where,

K BB = ∂f int e,f ∂U e,f = V f B T f K f B f dV f ; (24)
K BG = ∂f int e,f ∂α e,f = V f B T f K f G f dV f ; (25)
K GB = ∂h e,f ∂U e,f = V f G T f K f B f dV f ; (26)
K GG = ∂h e,f ∂α e,f = V f G T f K f G f dV f ; (27)
and B f represents the interpolation function of the strain field at the fiber level, G f is the enhanced interpolation function, U e,f is the nodal displacements of the fiber and K f is the material tangent modulus of the fiber. The local equilibrium equation represented by the second equation of the linearized system ( 23) is solved locally for the active discontinuities, therefore, we find the increment of the jump ∆α e,f . Using the static condensation technique, the following condensed fiber stiffness is found

K cond = K BB -K BG K -1 GG K GB ( 28
)
5 Numerical applications To illustrate our model, two numerical applications are presented hereafter.

Application 1

In order to prove the efficiency of the enhancement method to solve the mesh dependence problem, we consider a cantilever beam of length Ł = 1m subjected to transverse displacement at the right end. Firstly, the fibers of the beam are modeled with an elasto-plastic material with continuum softening. The beam is tested with several mesh sizes. Secondly, the fibers are modeled with an elasto plastic behavior for the continuum part coupled to a cohesive law for the softening part. The results of both applications are showed in the figures below.

-0.02 -0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0 - The difference between the figures (3) and ( 4) is clear. The elasto-plastic behavior with continuum softening is not objective relative to the mesh size as we can see in figure [START_REF] Grange | Numerical modelling of the seismic behaviour of a 7-story building: NEES benchmark[END_REF]. Indeed, the quantity of dissipated energy decreases with fine discretization. On the other hand, the enhanced multi-fiber beam improved the softening response with respect to mesh dependency since all the curves in figure (4) coincide at the end of the softening phase. Nevertheless, the total response is not yet free of mesh dependence. Furthermore, the dissipated energy in the beam softening response for the enhanced beam is captured by the cohesive law according to the following expression D = t α-Ψ; therefore, since discontinuities appear only in the first element whatever the mesh size, the captured energy in the softening process is the same.

The strain localization is also examined at local level. The axial strain distribution in the extreme fiber (located at section's edge) along the beam length is showed with both models (with and without enhancement) in figures [START_REF] Desprez | Seismic vulnerability assessment of a RC structure before and after FRP retrofitting[END_REF], [START_REF] Mazars | Using multifiber beams to account for shear and torsion[END_REF]. The curves are obtained by joining the strain values at the integration points of the elements. We remark that the strain localization is regularized once the fiber is enhanced. Figure [START_REF] Desprez | Seismic vulnerability assessment of a RC structure before and after FRP retrofitting[END_REF] clearly shows the strain localization within the first element of the fiber by refining the mesh of the beam. Otherwise, one can notice the presence of active discontinuity in the first enhanced element of the fiber by the decreased strain values within the element, (6). Furthermore, figure [START_REF] Desprez | Damage model for FRPconfined concrete columns under cyclic loading[END_REF] shows that the number of the fibers in the section doesn't have a big influence on the global response of the beam discretized into two elements.

Application 2

We consider a cantilever reinforced concrete beam with fixed support at the left end, and submitted to transversal displacement at the free end. The beam has a length of Ł = 1m and the section (0.2 × 0.5) is divided into 62 fibers (figure [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF]). 2 bars of diameter 32mm are added to the top of the section as reinforcement (fiber 4 in figure [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF]). The materials properties are the following: The global response of the reinforced beam, the cohesive law at an active discontinuity and the behavior of the extreme compressed fiber are showed in the figures below: Figure [START_REF] Guedes | A Fibre/Timoshenko beam element in Castem[END_REF] shows that the extreme compressed concrete fiber (number (62) in fig. (8)) is in the hardening phase. Figure [START_REF] Pegon | A Timoshenko simple beam element in Castem[END_REF] shows that the extreme tight concrete fiber (1) is totally cracked.

Concrete properties

E c = 40 GPa ν c = 0.2 K d h = 2 GPa

Conclusion

In this paper, a new multi-fiber displacement based Timoshenko Finite Element is proposed. Higher order shape functions with additional internal degrees of freedom are used to interpolate the element displacement field. The embedded discontinuity approach is used to enhance the kinematic of the fibers. Therefore, the objectiveness of the softening response with respect to the mesh size is improved. The section is divided into fibers of concrete and steel. Each material is modeled with two coupled behaviors, a continuous model to describe the bulk behavior and a cohesive model to illustrate the localized zones. The variational formulation is briefly elaborated. A numerical illustration of the proposed model is presented and it showed that the strain localization is limited. We are currently working on improving the discontinuity kinematic of the enhanced fibers, and we look to extend the current model to make cyclic applications in the forthcoming work.

Appendix A FCQ element

The interpolation functions of the new displacement based Full Cubic Quadratic Finite Element FCQ [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] are

N (x) =   N 1 0 0 0 0 0 N 7 0 0 0 N 11 0 N 13 0 N 15 0 N 17 0 0 0 N 21 0 N 23 0 0 0 N 27   (29)
where

                                                     N 1 =1 - x L N 7 = x L N 11 =(1 - x L ) 2 (1 + 2 x L ) N 13 =2(1 - x L ) 2 ( x L ) N 15 = -2( x L ) 2 (1 - x L ) N 17 =( x L ) 2 (3 -2 x L ) N 21 =(1 - x L )(1 -3 x L ) N 23 =1 -(1 -2 x L ) 2 N 27 = -( x L )(2 -3 x L ) (30)
The three internal degrees of freedom can be treated locally (inside the element subroutine) using static condensation method (see [START_REF] Caillerie | A Timoshenko finite element straight beam with internam degrees of freedom[END_REF] for more details and the analytical expressions of the condensed matrices and vectors).

Figure 1 :

 1 Figure 1: FCQ element

Figure 2 :

 2 Figure 2: (a) Continuous model ; (b) Discontinuous model

Figure 3 :Figure 4 :

 34 Figure 3: Elasto-plastic model with continuous softening

Figure 5 :Figure 6 :

 56 Figure 5: Strain at fiber level : Elasto-plastic model with continuous softening

Figure 7 :

 7 Figure 7: Influence of the fibers number on the global response

K d coh = - 5 . 2

 52 GPa σ comp elastic = 40 MPa σ comp crack = 45 MPa σ trac crack = 7 MPa Steel properties E s = 200 GPa ν s = 0.3 K s h = 30 GPa K s coh = -0.5 GPa σ s elastic = 80 MPa σ s ultimate = 90 MPa

Figure 8 :

 8 Figure 8: Multi-fiber section

Figure 9 :

 9 Figure 9: Global response of the enhanced multi-fiber reinforced concrete beam

Figure 10 :Figure 11 :

 1011 Figure10: Cohesive law at the active discontinuity within extreme concrete fiber[START_REF] Owen | Finite elements in plasticity[END_REF]

Acknowledgments

The authors would like to thank the PIA-SINAPS@ project (Sésisme et Installation Nucléaire: Améliorer et Pérenniser la Sureté), approved and funded by the National Agency of Research (ANR) following the RSNR 2012 call for projects on future investments post-


François Hild

Didier Marquis

A Statistical Approach to the Rupture of Brittle Materials

published or not. The documents may come   L'archive ouverte pluridisciplinaire


André Zaoui

On the Average Pull-Out length of Fiber-Reinforced Composites 1 Mécanique des solides/Mechanics of Solids On the Average Pull-Out Length of Fiber-Reinforced Composites

In this Note, an expression of the average pull-out length is established when the length of a representative element varies with the external load. This expression is derived from a study of fracture locations generalized to varying lengths, and can be used to determine ultimate strengths of fiber-reinforced composites.

Sur la longueur moyenne d'extraction pour des composites à fibres longues

Résumé -Dans cette Note, une expression de la longueur moyenne d'extraction est obtenue lorsque la longueur d'un élément représentatif varie avec le chargement. Cette

expression correspond à une généralisation des lieux de rupture dans le cas d'une longueur variable. Le résultat peut être utilisé dans le calcul de résistance de composites à fibres longues.

Version française abrégée -Le phénomène d'extraction, ou de pull-out [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF], qui accompagne souvent la rupture de composites à fibres longues différencie un composite à matrice céramique (CMC) d'un faisceau de fibres parallèles [START_REF] Daniels | The Statistical Theory of the Strength of Bundles of Threads[END_REF]. L'extraction est produite par le glissement d'une fibre rompue par rapport à la matrice, et est caractérisé par une contrainte critique de glissement τ à l'interface. Le calcul de la résistance des CMC nécessite la connaissance de la longueur moyenne d'extraction et une étude dans laquelle la longueur de l'élément représentatif varie avec la charge appliquée.

On considère un élément de composite de longueur 2L R (Fig. 1) , où L R est appelée longueur de transfert et correspond à la longueur maximale d'une fibre qui peut être arrachée et conduire à une diminution de charge par rapport au cas où elle ne serait pas rompue. La contrainte de référence T, contrainte des fibres non rompues dans un plan de référence Π R d'une fissure matricielle (z=0), est la contrainte longitudinale maximale dans le composite. probabilité cumulée de rupture P F (T,L R (T)) d'un élément de fibre de longueur 2L R <2L est écrite en (4). La quantité P F0 (T,z) représente la probabilité cumulée de rupture d'un maillon de longueur L 0 dont le plan médian est situé à la cote z. A cet élément, on peut associer la densité de probabilité de rupture (5.1) et récrire la probabilité cumulée de rupture (5.2). La fonction Φ(T,z) correspond à la densité de probabilité de rupture d'un élément à la cote z lorsque la contrainte de référence varie de T à T + dT (6.1), et la fonction Ψ(T) à la densité de probabilité de rupture liée à l'augmentation de longueur de transfert de L R (T) à L R (T) + L' R (T) dT (6.2). Lorsque L R est indépendante de T, ou lorsque P F0 (T,L R (T)) = 0, les résultats de [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF][START_REF] Oh | On the Location of Fracture in Brittle Solids-I Due to Static Loading[END_REF] s'appliquent. Par analyse dimensionnelle et en utilisant (5.2), la moyenne < > h(T) des lieux de rupture h(T) d'un composite de longueur 2L R (T) est donnée par (7), où α est une constante. En tension, à l'aide de (2), ( 5) et ( 7), il n'est pas nécessaire d'expliciter la distribution des lieux de rupture, et on trouve α = 1/2. Le premier terme de (7) est identique à celui donné dans [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF][START_REF] Sutcu | Weibull Statistics Applied to Fiber Failure in Ceramic Composites and Work of Fracture[END_REF]. Le second terme est dû à la dépendance de la longueur de transfert avec T. L'expression (7) ne prend pas en compte les fibres qui ont cassé hors de la zone de transfert et qui lors d'une augmentation de la charge se trouvent à l'intérieur de la zone de transfert. Cette approximation est la plus importante dans le cas d'un champ de contrainte constant sur 2L R (tension). Dans ce cas, il a été montré que l'approximation n'est pas très pénalisante [START_REF] Hild | Tensile and Flexural Ultimate Strength of Fiber Reinforced Ceramic-Matrix Composites[END_REF].

Les résultats précédents peuvent être utilisés pour calculer la LME. On suppose que la probabilité de rupture est modélisée par une loi de Weibull à deux paramètres m et S 0 [START_REF] Weibull | A Statistical Theory of the Strength of Materials[END_REF] Les contraintes ultimes en flexion sont les contraintes apparentes élastiques.

1. Introduction -A feature that distinguishes Ceramic-Matrix Composites (CMC's) from a classical dry fiber bundle [START_REF] Daniels | The Statistical Theory of the Strength of Bundles of Threads[END_REF] is that fiber breakage is usually accompanied with fiber pull-out [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF] characterized by an interfacial sliding resistance τ. This latter is due to frictional sliding that occurs as the broken ends of the fiber pull out of the matrix.

To determine the ultimate strength of a CMC, one needs to determine the average pull-out length as a function of the load level in the fiber. This Note is devoted to establishing an expression of the average pull-out length.

σ(T,z)

T 0 z 2L R 1 2τ/R
Fiber Failure Rupture de fibre

σ p 2L R 2L Π R z Figure 1.
The cell length is denoted by 2L R (Fig. 1) , where L R is called recovery length, and refers to the longest fiber that can be pulled out and cause a reduction in load carrying capacity. The reference stress T, which corresponds to the stress in the unbroken fibers at a reference plane Π R (z=0) of one of the matrix cracks, is the maximum longitudinal fiber stress. The matrix cracking process is supposed to be saturated. It is assumed that if a fiber breaks, it equally shares the load among all the unbroken neighboring fibers (global load sharing), i.e. the matrix cracks traverse the whole composite. Within the recovery length it is assumed that a fiber breaks at most one time.

Throughout this Note, the longitudinal stress field, σ(T,z), is assumed to be symmetric about z=0. When the interfacial sliding resistance is assumed constant, the recovery length, L R (T) , is given by

(1) L R (T) = RT 2τ if L R (T) < L
where R is the fiber radius and 2L the total composite length (Fig. 1). The contribution of the broken fibers within the recovery length or pull-out stress, σ _ po , at z=0 is written as

(2) σ _ po (T) = f ⌡  ⌠ 0 T σ _ p (t) dP F (t,L R (t))
dt dt where f is the fiber volume fraction, σ _ p (t) the average stress at z=0 when a fiber breaks at a location z, and at a reference stress level t, and P F (t,L R (t)) is the failure probability that a fiber break within the recovery length L R at or below a reference stress t (Fig. 1). In tension, the probability density of fracture locations is uniform, therefore σ _ p (t) = t / 2. On the other hand, when the stress field along the fiber direction is not homogeneous, the previous result does not apply, and one needs to determine the average fracture location to estimate the pull-out stress. The average stress applied to the composite at the plane z=0, σ _ , is given by

(3) σ _ f = T { } 1 -P F (T,L R (T)) + 2τ R < > h(T) P F (T,L R (T))
The first term of the right hand side of Eqn.

(3) corresponds to the contribution of the unbroken fibers, and the second term is the pull-out stress σ, _ po (T) . The quantity < > h(T)

represents the average pull-out length, or average fracture location in 2L R (T) when the reference stress is less than or equal to T.

Average

Fracture Locations -Under the weakest link assumption and the independent events hypothesis, the cumulative failure probability P F (T,L R (T)) of a piece of fiber of length 2L R (T) is given by

(4) P F (T,L R (T)) = 1 -exp       2 L 0 ⌡ ⌠ 0 L R (T) ln { } 1 -P F0 (T,z) dz
where P F0 (T,z) denotes the cumulative failure probability of a single link of length L 0 whose mid-section is located at z. The failure probability density is obtained by differentiation of Eqn. ( 4) with respect to T

(5.1) dP F (T,L R (T))

dT = 2 ⌡ ⌠ 0 L R (T) Φ(T,z) dz + 2 Ψ(T)
and therefore

(5.2)

P F (T,L R (T)) = ⌡  ⌠ 0 T 2       ⌡ ⌠ 0 L R (t) Φ(t,z) dz + Ψ(t) dt
where Φ(T,z) corresponds to the failure probability density for a reference stress varying between T and T + dT, of an element of length dz centered at z, and Ψ(T) is the failure probability density associated with the length increase from L R (T) to L R (T) + L' R (T) dT.

The first term is identical to that given in [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF][START_REF] Oh | On the Location of Fracture in Brittle Solids-I Due to Static Loading[END_REF] when using a Weibull law, the second term is due to the load dependence of the recovery length L R . The expressions of the functions Φ and Ψ are respectively given by (6.1)

Φ(T,z) = -{ } 1 -P F (T,L R (T)) 1 L 0 ∂ ∂T ( ) ln { } 1 -P F0 (T,z) (6.2) Ψ(T) = -{ } 1 -P F (T,L R (T)) L'(T) L 0 ln { } 1 -P F0 (T,L R (T))
When L R is independent of T, or P F0 (T,L R (T)) vanishes, Ψ(T) vanishes as well, and the results given in [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF] still apply in the case of a Weibull law .

By dimensional analysis and by inspection of Eqn. (5.2), the average < > h(T) of the fracture location h(T) for a piece of composite of length 2L R (T) at a reference stress T is

(7) < > h(T) = 1 P F (T;L R (T)) ⌡  ⌠ 0 T 2       ⌡ ⌠ 0 L R (t) z Φ(t,z) dz + α L R (t) Ψ(t) dt
where α is a constant. The first term is identical to that given in [START_REF] Thouless | Effects of Pull-Out on the Mechanical Properties of Ceramic-Matrix Composites[END_REF][START_REF] Sutcu | Weibull Statistics Applied to Fiber Failure in Ceramic Composites and Work of Fracture[END_REF]. The second term is due to the load dependence of the recovery length. In tension, by using Eqns. ( 2), ( 5) and ( 7), there is no need to know the exact expression of the fracture location distribution, and one obtains α = 1/2. This expression of the average fracture location does not consider the fibers that originally broke outside the recovery length but were brought into it as the load level increased. The approximation made here is the worst when the stress field is constant over the whole length (i.e. pure tension). In that case however, it has been shown that this hypothesis is not very strong [START_REF] Hild | Tensile and Flexural Ultimate Strength of Fiber Reinforced Ceramic-Matrix Composites[END_REF].

3

. Average Pull-Out Length -In the following, we assume that the cumulative failure probability is modeled by a two-parameter Weibull law [START_REF] Weibull | A Statistical Theory of the Strength of Materials[END_REF]: m denotes the shape parameter and S 0 the scale parameter. In pure tension, the pull-out stress becomes

(8) σ _ po (T) = fS c 2 γ       m+2 m+1 ,       T S c m+1
where S c denotes the characteristic strength (S c m+1 = S 0 m L 0 τ/R) , and γ{.,.} the incomplete Gamma function.

4. Application -To determine the ultimate strength of a fiber-reinforced composite, the following condition has to be satisfied at the most loaded point(s) in tension ( 9) ∂σ _ ∂T = 0 since the average strain is proportional to the reference stress T. It is worth noting that the ultimate strength is scaled by fS c provided L R <L at the ultimate. In the following we compare the predictions given by the previous results to experimental data on CMC's subjected to tension, four-point flexure, and three-point flexure. In flexure it is assumed that the behavior in compression is elastic and the Young's modulus is equal to the Young's modulus of the composite, E c . In tension, the behavior is non-linear (fiber breakage and fiber pull-out) and the Young's modulus is that of the fiber volume fraction, fE f because of the presence of matrix cracks. The comparison concerns a Lithium Alumino-Silicate (LAS) matrix reinforced by unidirectional SiC fibers [START_REF] Jansson | The Mechanics of Failure of Silicon Carbide Fiber-Reinforced Glass-Matrix Composites[END_REF]. The predictions are summarized in Table 1, in flexure the stresses correspond to the apparent elastic stresses. Table 1.

The agreement between the predictions and the experiments is quite good. One source of discrepancy is given by the hypothesis that the matrix cracking stress is equal to zero.

Figure caption

Strength

Figure 1 :

 1 Figure 1: Elementary cell and stress field in broken and unbroken fibers.

Figure 1 :

 1 Figure 1: Cellule élémentaire et contraintes dans les fibres rompues et non rompues.

  On suppose que la fissuration matricielle est saturée. Dans le cas où les fissures matricielles sont traversantes et lorsqu'une fibre rompt, la charge qu'elle supportait est également distribuée sur toutes les fibres non rompues (partage global d'effort). Enfin, une fibre ne peut rompre qu'une seule fois à l'intérieur de la zone de transfert. Dans toute cette Note, le champ de contrainte le long de la fibre, σ(T,z), est supposé symétrique par rapport à z=0. Lorsque τ est constante, la longueur de transfert est donnée par (1), où R est le rayon des fibres et 2L la longueur totale des fibres. Sous l'hypothèse de partage global d'effort, la contribution des fibres rompues à z=0 en terme de contrainte d'extraction σ _

	rupture P F (t,L R (t)) d'un élément de longueur 2L R (t) pour une contrainte de référence
	inférieure ou égale à t. En tension, la densité de probabilité de lieux de rupture est uniforme,
	donc σ _	p (t) = t / 2. Par contre, lorsque le champ de contrainte n'est pas uniforme le long de la
	fibre, le résultat précédent ne s'applique pas, et il faut déterminer la distance moyenne de
	rupture par rapport à z=0 pour estimer la valeur de la contrainte d'extraction. En effet, la
	contrainte appliquée au composite en z=0 est donnée par (3). Le premier terme correspond à
	la contribution des fibres non rompues, le second terme à la contrainte d'extraction σ _	po (T)
	où <	h(T)	>	représente la longueur moyenne d'extraction (LME) pour une contrainte de
	référence égale à T.
			Sous l'hypothèse du maillon le plus faible et d'indépendance des événements, la

po (T) est exprimée en (2), où f est la fraction volumique de fibres, σ _ p (t) est la contrainte moyenne à z=0 de toutes les fibres rompues à une cote |z| ≤ L R (t) pour une contrainte de référence t, et une probabilité de

Table 1 :

 1 Comparison between predictions and experimental data on a LAS matrix reinforced by unidirectional SiC fibers (f=0.5, S c =2500 MPa, R=7.5 µm, τ=5 MPa, fE f /E c = .62, m=4).

	Tableau 1: Comparaison entre prévisions et expériences sur une matrice LAS renforcée par
	des fibres unidirectionnelles en SiC.
	Rédacteur : François HILD
	Laboratoire de Mécanique et Technologie
	E.N.S. Cachan / C.N.R.S. / Université Paris 6
	61, avenue du Président Wilson
	F-94235 CACHAN Cedex.


Alain Burr

Pascal Feillard

François Hild

Identification of a Potential to Model the Behavior of Fiber-Reinforced Composites

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Christophe Denoual

Gilles Barbier

François Hild

A Probabilistic Approach for Fragmentation of Brittle Materials under Dynamic Loading

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François Hild

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Ultimate strength properties of fiber-reinforced composites


James Venning

Samuel Smith

Paul Brandner

Dean Giosio

Bryce Pearce

S Y Mpo

O Tatin

The influence of nuclei content on cloud cavitation about a hydrofoil

Keywords:

e dynamics of cloud cavitation about a D hydrofoil are investigated experimentally in a cavitation tunnel with both an abundance and dearth of freestream nuclei. e rectangular-planform, NACA hydrofoil was tested at a Reynolds number of 1.4 × 10 6 , a cavitation number of . and an incidence of 6°. High-speed photography of cavitation shedding phenomena was acquired simultaneously with unsteady force measurement to enable identification of cavity shedding modes corresponding with force spectral peaks. Two shedding modes are evident for both the nuclei deplete and abundant cases, although each are driven by different flow phenomena. e high-frequency mode for the nuclei deplete case is driven primarily by large-scale re-entrant jet formation during the growth phase, but shockwave propagation for the collapse phase of the cycle.

e weaker low-frequency mode occurs because the strength of shedding at the hydrofoil tip varies at half the fundamental frequency. e dominant mode for the abundant case is the low-frequency mode which is some . times slower than the nuclei deplete case. e high-frequency mode for the nuclei abundant case is due to the propagation of two shockwaves; the passage of the first only partially condenses the cavity, while the second condenses a much larger region of the cavity.

. INTRODUCTION e random or periodic formation, separation and collapse of sheet cavities (i.e. partial cavitation) is generally described as cloud cavitation. For sufficiently thick cavities the re-entrant jet, stemming from the cavity closure, has sufficient intensity to grow upstream and impinge upon the cavity interface, detaching a portion that condenses to a bubble cloud and is entrained downstream by the overlying flow [ ]. is phenomenon has been widely studied in various contexts including hydrofoils [ , , , ], rotating machinery [ , ], nozzles/venturies [ , , , ], wall-mounted cavitators [ ] and bluff bodies [ , , ]. Recently it has been shown that the formation of condensation shockwaves produce another instability mechanism for partial cavitation, either singularly or coupled with a re-entrant jet [ , ]. ree-dimensional flow features have been shown to significantly alter the reentrant flow topology and the resulting shedding physics [ , , ]. e present work has also observed that the threedimensional nature of a cavitating flow alters the shockwave behaviour if present.

Inception of hydrodynamic cavitation in practical flows is invariably due to heterogeneous nucleation. ese discrete sites of weakness, termed nuclei, are typically microbubbles either free in the liquid volume, or volumes of gas trapped in or on adjacent surfaces. Nuclei have been shown to control not only inception [ ] but also the dynamics of cavitation [ ]. e water tunnel at the Australian Maritime College has been specifically designed to control the free microbubble content serving as free nuclei in the test water and has the capacity to investigate the effect of varying nuclei content on cavitating flows. In the present work a preliminary study has been undertaken on the effect of nuclei content on cloud cavitation about a finite span hydrofoil and results are presented for two cases: nuclei deplete water and water seeded with a high nuclei concentration (referred to as the abundant case), maintaining all other flow conditions. e model hydrofoil of anodised aluminium has a rectangular planform of . m span and . m chord with constant NACA section and a faired tip. e model is mounted vertically from the test section ceiling on a -component force balance for dynamic force measurement.

. EXPERIMENTAL OVERVIEW

Measurements were made at a cavitation number of . and a Reynolds number (based on chord length) of 1.4 × 10 6 with the hydrofoil set at a fixed incidence of 6°. e cavitation number is defined as σ = 2(p ∞p v )/ρU 2 ∞ and Reynolds number as Re = U ∞ c/ν, where p ∞ is the static pressure at the test section centreline, p v is the vapour pressure, ρ is the water density, U ∞ is the test section velocity, c is the chord and ν is the kinematic viscosity of the water. e freestream velocity was approximately . m/s. Two seeding conditions were investigated where the freestream is either deplete or seeded with an abundance of nuclei. For the deplete case no nuclei are injected such that only background nuclei are present in the tunnel water which do not provide active nuclei in the freestream for this flow condition [ ]. For the abundant case poly-disperse nuclei with a dominant size of to µm are injected at a concentration of about /cm 3 [ ]. is volumetric concentration results in an area concentration of activated nuclei of approximately nuclei /cm 2 on the hydrofoil. In both cases the tunnel water was maintained at a dissolved O 2 level of ppm.

Simultaneous measurements were made of hydrofoil normal force and high-speed photography of cavitation taken from the side of the test section normal to the flow direction.

e high-speed photography was recorded using a LaVision HighSpeedStar at a resolution of 1024 × 1024 pixels using a Nikkor f/ . mm lens. Simultaneous forces and highspeed images were recorded at kHz for s. Long-time series measurements of force for obtaining high-resolution spectra were recorded at kHz for s giving about , cycles of the dominant frequency. Power Spectral Density (PSD) were derived using the Welch estimate of the PSD [ ] with a window size of points ( . s) and overlap between windows of points ( . s). Time series of measured side force and local pixel intensity from high-speed imaging are compared to gain insight into the phasing of shed structures and the forces produced.

. RESULTS AND DISCUSSION

A sequence of extracted images through one shedding cycle from the high-speed photography for the nuclei deplete and nuclei abundant cases are shown in figure . For both cases the shedding was highly periodic as can be seen from the time series of the measured li (figure ) and PSD of the side force (figure ). e third PSD shown in figure is for the noncavitating case to distinguish the balance/hydrofoil response to turbulent excitation from that due to cavitation dynamics. Additional frequency analysis beyond the PSD was carried out using continuous wavelet transforms of forces and fast Fourier transform (FFT) analysis of every pixel from the high speed photography. ese analyses enabled discrimination of the peaks of the PSD between those actually associated with cavitating flow physics and those being artefacts of the FFT process. at is, wavelet transforms correlate all variations over the range of time and frequency scales investigated avoiding harmonic peaks. Amplitudes of all peak frequencies can also be mapped from FFTs of the high-speed photography to highlight regions of maximum variation or energy for each frequency to reveal which mechanisms are associated with which frequencies. Full details of these analyses are provided in a companion full length journal article in preparation.

For the nuclei deplete case, the high-speed imaging and unsteady force measurements show the dominant frequency to be St = 0.28 which is the second peak evident on the PSD. e analysis described above show this frequency to be associated with shedding involving large-scale cavity growth and collapse/condensation. Large-scale in this case refers to shedding involving cavity growth and collapse ranging over almost the full chord and through the midspan extending some one-half to two-thirds of the span. e first frequency peak at St = 0.14 can be shown to be associated with local shedding from the cavity end near the hydrofoil tip. e first frequency peak is half that of the fundamental and from the above described analyses and chordwise/spanwise spacetime plots it can be shown that the strength of the local tip shedding varies at half the frequency of the bulk cavity shedding. at is, there are two separate but coupled shedding regions through the midspan and near the tip.

e third peak on the PSD at St = 0.55 is also present at the tip when the cavity sheds sometimes at twice the frequency of the fundamental.

As a result of the above described shedding behaviour, the cavity physics and topology are complex and highly threedimensional. Since the large-scale shedding occurs over the mid-half to two-thirds of the span it is only in this region that the cavity growth and collapse oscillations occur from the hydrofoil leading to trailing edges. As a result of this topol-

(a) τ = 0 (b) τ = 1/4 (c) τ = 1/2 (d) τ = 3/4 (e) τ = 0 (f) τ = 1/4 (g) τ = 1/2 (h) τ = 3/4
Figure .

Photographs at one-quarter increments of the shedding cycle for the nuclei deplete (top) and abundant (bo om) conditions. Note that the shedding cycle for the deplete case is . times faster than with abundant nuclei.

ogy the cavity has a crescent shaped appearance during the growth phase (figure a) while the cavity regrows midspan from the leading edge. e cavity growth at midspan is much faster than that near the root and tip, quickly reaching the trailing edge (figure b,c) triggering a shockwave that propagates upstream. is shockwave is focused as it travels upstream due to shockwaves that form simultaneously at the root and tip travelling in the spanwise direction toward midspan.

is focusing condenses the cavity through the mid span region (figure d) such that the cavity has to grow anew leading to the crescent shape described above. During the growth phase the cavity forms a re-entrant jet, the leading edge of which remains almost stationary as the trailing edge of the growing cavity reaches the hydrofoil trailing edge. Although the re-entrant jet may occasionally break through the surface of the growing cavity it does not initiate a shockwave as has been observed for flows about bodies with much greater adverse pressure gradients [ ]. In addition to the strength of the local tip shedding varying at half the fundamental frequency, it can also be shown that the bulk shedding cavity growth rates and inter-cavity time between growth phases also vary at this lower frequency.

With an abundance of nuclei, the shedding physics were significantly altered due to the high concentration of continuously activated nuclei. e fundamental frequency of the bulk shedding was St = 0.15, corresponding to the first peak in the force PSD (figure ). is shedding was approximately two-dimensional, occurring over the full hydrofoil span, and was . times slower than the large-scale shedding for the nuclei deplete case. e second peak in the PSD is twice that of the fundamental (St = 0.31) and can be shown to be associated with the propagation of the first of two shockwaves each cycle.

e first shockwave forms once the growing cavity reaches the trailing edge. is shockwave propagates upstream (figure g) but loses strength and speed, stalling before reaching the cavity leading edge. Shortly a er this another shockwave forms that travels upstream with greater strength and velocity (figure h) causing large scale condensation with subsequent near-D re-growth along the hydrofoil span. Due to the continuous supply of activated nuclei there is no clear cavity detachment line but a discontinuous cavity leading edge. Due to the discontinuous nature of the cavity leading edge and the continuous supply of activated nuclei the cavity is not condensed completely each shedding cycle as was the case for the nuclei deplete case. Nuclei activated at the hydrofoil leading edge grow as they are advected downstream merging to form a contiguous cavity volume. During the growth phase, advected bubbles reach the cavity trailing edge and condense simultaneous with the formation of reentrant flow as evidenced by the lighter regions along the cavity trailing edge (figure e). Cavity growth is due to the volume of activated nuclei being greater than the condensing volume at the cavity trailing edge. As described above, once the slow growing cavity reaches the trailing edge, the first shockwave is initiated. As with the nuclei-deplete case the re-entrant jet apparently has no direct influence on the shedding. e peaks at higher frequencies can be shown to be harmonic artefacts of the FFT process.

With the addition of seeding, the li coefficient dropped by % from . to . . e drag coefficient remained for the deplete case, decreasing to . with the addition of nuclei.

. CONCLUSIONS e dynamics of cloud cavitation about a hydrofoil have been shown to be significantly affected by free steam nuclei content for otherwise identical flow conditions. e two bounding cases considered: nuclei deplete where there are essentially no active free freestream nuclei, and nuclei abundant where there is a large concentration of active nuclei, develop different mechanisms that lead to instability and shedding. For both cases there are two shedding modes of different frequencies but in each case the high mode frequency is a multiple of the low mode. For the nuclei deplete case the dominant mode corresponding to large-scale shedding is the high-frequency mode. e lower frequency mode being due to the variation on every alternate high mode cycle of the strength of local shedding at the cavity end near the hydrofoil tip. For the nuclei abundant case the dominant mode corresponding to large-scale shedding is the low frequency mode which is . times slower than the dominant mode for the nuclei deplete case.

e high frequency mode for the seeded case is due to the propagation of the first of two shockwaves, the passage of which only partially condenses the cavity. e second shockwave condenses a much larger region of the cavity a er which there is re-growth to begin another shedding cycle. e use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE transactions on Audio and Electroacoustics, ( ): -, .

Figure .

 . Figure . Tunnel schematic showing the experimental layout including the microbubble nuclei injection, contraction, test section, and diffuser.

Figure .

 . Figure .Time series of the li coefficient (blue and red) and image intensity from the high-speed movies in the shedding zone (gray) for the deplete (top) and abundant (bo om) cases.

Figure .

 . Figure .Power spectra of the normal force for the hydrofoil under two nuclei conditions: deplete (blue) and abundant (red). Freestream conditions are Re = 1.4 × 10 6 and σ = 0.55. e foil is at a °angle of a ack. e single-phase (non-cavitating) spectra is given in gray.

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[

ACKNOWLEDGEMENTS

is project was supported by the University of Tasmania, the Defence Science and Technology Group and the US Office of Naval Research (Dr. Ki-Han Kim, Program Officer) and ONR Global (Dr. Woei-Min Lin) through NICOP S&T Grant no. N ---.