J Duriez

R Wan

A micromechanical µUNSAT effective stress expression for stress-strain behaviour of wet granular materials

Keywords: effective stress, Discrete Element Method (DEM), granular, pendular, µUNSAT

The definition of an effective stress variable for idealized triphasic granular media as the contact stress arising from interparticle forces is examined through discrete element modelling computations in concert with appropriately derived analytical stress expressions based on homogenization. The latter take a more practical importance, in that they also circumvent the need for direct measurements of interparticle contact forces. Considering dry or pendular-regime conditions for slightly polydisperse dense and loose packings, the contact stressstrain behaviours in dry or wet conditions are compared along a variety of loading paths, depending on the level of plastic dissipation involved. The contact stress indeed emerges as an effective stress variable with a remarkable stressstrain character along contractant loading paths and for dense solid packings where the behaviour is close to be non-dissipative. Along more dilatant loading paths or for looser packings, plastic dissipation increases and the coincidence of the constitutive behaviour in both dry and triphasic conditions is restricted to the initial stages of the loading paths, limiting the applicability of the contact stress to the estimation of initial stiffnesses. The stark constitutive differences between the proposed effective stress and Bishop's stress are also highlighted.

Introduction

Energy and Environmental Engineering operations require a precise mechanical description of multiphase porous media such as in oil and gas reservoirs, or partially water-saturated (unsaturated) soils. Obviously, the presence of various fluids within the pore space network of these multiphasic materials greatly modifies their mechanical behaviour with respect to the dry (air-saturated) state which is well understood. For biphasic (or two-phase) porous media penetrated by a single pore fluid, a unified mechanical description by way of dry state constitutive models actually exists whatever the pore fluid and its pressure are following Terzaghi's and Biot's effective stress equations [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF][START_REF] Biot | General theory of three-dimensional consolidation[END_REF]. Turning to triphasic (or three-phase) conditions with an immiscible fluid mixture, it would also be highly beneficial to extend this effective stress concept so that triphasic materials can be described by just inserting an adequate single effective stress into models designed for dry materials. In other words, the objective is to obtain a stress-strain-strength effective stress that would allow failure criterion and constitutive relations from dry conditions, all together, to be applied to triphasic conditions in order to easily predict limit stress states and strains. However, the exact determination of such a single effective stress has revealed to be highly challenging ever since the early Bishop's attempts [START_REF] Bishop | Some aspects of effective stress in saturated and partly saturated soils[END_REF].

When describing failure in triphasic media, it has been often possible to resort to an effective stress framework based on limit stress states according to dry shear strength criteria [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF][START_REF] Hicher | A microstructural elastoplastic model for unsaturated granular materials[END_REF][START_REF] Nuth | Effective stress concept in unsaturated soils: Clarification and validation of a unified framework[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF]. In these works, the effective stress is either phenomenologically deduced from Bishop's equation with appropriate expressions for χ coefficient [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Nuth | Effective stress concept in unsaturated soils: Clarification and validation of a unified framework[END_REF], or as the contact forces stress contribution which can be indirectly accessed by subtracting a so-called suction stress from the total stress [START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF][START_REF] Hicher | A microstructural elastoplastic model for unsaturated granular materials[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF]. The suction stress [START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF] encompasses all relevant stress contributions in triphasic materials, excluding the contact stress stemming from the contact forces between the solid grains of the skeleton, and can be identified for granular materials with the capillary stress since other physico-chemical forces are then negligible. As for the contact stress, the latter appears as a logical stress-strength effective stress since pore water is reported to have little or no influence onto the contact-scale local frictional threshold [START_REF] Cavarretta | A new interparticle friction apparatus for granular materials[END_REF][START_REF] Senetakis | The inter-particle coefficient of friction at the contacts of leighton buzzard sand quartz minerals[END_REF], which thus leads to unique limit states for the macroscopic contact stress and a unified failure description irrespective of saturation.

Whereas in the biphasic case unique constitutive relations are related to strains via Biot's effective stress, no effective stress has yet been found that would relate to strains in triphasic conditions through dry constitutive relations [START_REF] Nuth | Effective stress concept in unsaturated soils: Clarification and validation of a unified framework[END_REF]. As such, the application of an effective stress principle to triphasic materials has often been challenged since Bishop himself [START_REF] Bishop | Some aspects of effective stress in saturated and partly saturated soils[END_REF], with alternate approaches relying on total stress and another stress variable such as matric suction to describe unsaturated conditions using additional constitutive equations that are specific to these triphasic conditions [START_REF] Fredlund | The shear strength of unsaturated soils[END_REF][START_REF] Alonso | A constitutive model for partially saturated soils[END_REF]. A compromise may, nevertheless, be found in that quasi-elastic behaviour, e.g. volume change along overconsolidated oedometric loadings, may still be described in triphasic conditions using an effective stress framework [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF][START_REF] Baille | Effective stress in clays of various mineralogy[END_REF]. Recent micromechanical results by the authors [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Partially saturated granular materials: Insights from micro-mechanical modelling[END_REF] suggest that the same contact stress also enjoys such a quasi-elastic strain-stress effective nature, in addition to its stress-strength character. This seems to be indeed coherent because granular material strains derive from solid particle relative displacements that relate to contact forces through a unique contact law, irrespective of saturation.

The present work aims to provide new evidences that demonstrate the stressstrain effective character of the contact stress in any regime showing small plastic dissipation by considering a wide range of loading paths which were absent from previous studies [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Partially saturated granular materials: Insights from micro-mechanical modelling[END_REF].

It is further noted that direct measurements of contact forces and contact stresses are still beyond reach in triphasic materials in spite of a current research in the field [START_REF] Hurley | Extracting inter-particle forces in opaque granular materials: Beyond photoelasticity[END_REF]. To address this shortcoming, the present paper first recalls in Section 2 the µUNSAT expressions that enable one to indirectly access the contact stress through capillary stresses and the fluid phase microstuctures, whose measurements appear to be more feasible thanks to e.g. computed tomography [START_REF] Culligan | Interfacial area measurements for unsaturated flow through a porous medium[END_REF][START_REF] Bruchon | X-ray microtomography characterisation of the changes in statistical homogeneity of an unsaturated sand during imbibition[END_REF]. For the purpose of the paper, the choice is actually made to con-sider a Discrete Element Method (DEM) model for wet granular materials [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF], presented in Section 3, which allows both direct and indirect measurements of the contact stress from a comprehensive description of the microstructure of all solid and fluid phases, including interfaces. Numerical comparisons of dry-wet behaviours for slightly polydisperse materials along various loading paths are then presented in Section 4, validating the stress-strain effective nature of the contact stress in regimes with small dissipation, and hence the present µUNSAT effective stress expression. Section 5 finally evidences the fundamental differences in the pendular regime between the present µUNSAT effective stress and Bishop's stress, whose inapplicability to wet conditions is once again demonstrated.

Micromechanical µUNSAT effective stress expression

Total stress decomposition from micromechanics

The µUNSAT approach expresses the contact stress by isolating from the macroscopic total stresses Σ, all microscopic stress contributions due to the various phases and interfaces that encompass triphasic granular media. First, the solid s and fluid pn, wq bulk phases are considered with their corresponding volumes V s , V n and V w . Here, n stands for the non-wetting fluid, e.g. air, whereas w refers to the wetting one, e.g. water. Additionally, all three interfaces S αβ , α, β " n, s, w are as well included in the approach since the corresponding surface tensions γ αβ necessarily belong to the internal forces of the triphasic material. As such, the total stresses are given as follows:

Σ " 1 V ˜ÿ α"n,s,w ż Vα σ α dV `ÿ α,β"n,s,w ż S αβ π αβ dS ¸(1)
In Eq. ( 1), the fluid stresses σ α , α " n, w, are directly given by the corresponding pressures u α δ with δ as the identity tensor, whereas the solid stress σ s can be classically expressed in terms of the tractions acting along the solid particles surfaces S p [START_REF] Duriez | The micromechanical nature of stresses in triphasic granular media with interfaces[END_REF]. The surface stress tensors π αβ " γ αβ pñ b ñ ´δq, with ñ as the normal to S αβ , finally describe the surface tension fields acting as internal forces within any interface S αβ [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF][START_REF] Chateau | Homogenization of a non-saturated porous medium: Hill's lemma and applications[END_REF][START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF][START_REF] Duriez | Stress in wet granular media with interfaces via homogenization and discrete element approaches[END_REF].

Algebraic manipulations applied to Eq. ( 1) lead to the following µUNSAT decomposition of Σ into the contact stress σ cont and the capillary stress σ cap , to which the solid fluid surface tensions γ sα eventually do not contribute in the case of rigid solid particles [START_REF] Duriez | The micromechanical nature of stresses in triphasic granular media with interfaces[END_REF][START_REF] Chateau | Homogenization of a non-saturated porous medium: Hill's lemma and applications[END_REF][START_REF] Duriez | Stress in wet granular media with interfaces via homogenization and discrete element approaches[END_REF]:

Σ ´un δ " σ cont `σcap (2)
σ cont " 1 V ÿ c f c b l (3) σ cap " ´1 V " u c ˆVw δ `żSsw ñ b x dS γnw ˆżSnw pδ ´ñ b ñq dS `żΓ ν b x dl ˙ " ´1 V " u c pµ V w `µSsw q `γnw pµ Snw `µΓ q  (4)
Eq. ( 3) refers to the standard definition of the contact stress tensor σ cont from the contact forces f c between solid particles, and the corresponding branch vector l connecting the centroids of contacting particles.

In Eq. ( 4), the capillary stress σ cap encompasses the remainder of the internal forces that arise both from the fluid-fluid surface tension γ nw and the capillary pressure u c " u n ´uw that is equal to suction in granular materials.

Four microstructure tensors µ X reflect the tensorial nature of these internal forces, with µ V w the only one being always spherical since it corresponds to the isotropic action of suction within V w . Other microstructure tensors have a general, non-spherical, tensorial expression with e.g. µ Ssw depending on the orientation of the wetted surface S sw since the latter sustains suction along its normal only. Also, µ Snw , µ Γ reflect the orientations of the interface S nw and of the three-phase contact lines Γ that are subjected to fluid-fluid surface tension, where ν in Eq. ( 4) is the tangent to S nw being normal to Γ [START_REF] Duriez | The micromechanical nature of stresses in triphasic granular media with interfaces[END_REF].

µUNSAT contact stress expression

Isolating σ cont in Eq. ( 2)-(4), the following µUNSAT expression for the contact stress is eventually obtained:

σ cont " Σ ´un δ `1 V " u c pµ V w `µSsw q `γnw pµ Snw `µΓ q  (5)
Such an expression is believed to provide an easier access to σ cont during experiments than the direct definition of Eq. ( 3). Imaging techniques such as computed tomography indeed are very promising in allowing precise measurements of the fluid phase microstructures [START_REF] Culligan | Interfacial area measurements for unsaturated flow through a porous medium[END_REF][START_REF] Bruchon | X-ray microtomography characterisation of the changes in statistical homogeneity of an unsaturated sand during imbibition[END_REF]. Then, one would be able to compute the microstructure tensors µ X in Eq. ( 5), as opposed to requiring more complex contact force measurements [START_REF] Hurley | Extracting inter-particle forces in opaque granular materials: Beyond photoelasticity[END_REF] that enter the direct contact stress expression, Eq. ( 3).

Leaving any experimental endeavour aside for now, Eq. ( 5) is herein used within the framework of DEM to investigate the very role of the contact stress in connection with strains, and whether it may serve as a strain-stress-strength effective stress expression.

DEM multiscale modelling of triphasic granular materials

Numerical DEM models indeed provide a convenient multiscale description of triphasic granular materials whereby the microstructure of all solid and fluid phases is easily measured. As such, the contact stress can be readily estimated using either Eq. ( 3) or ( 5) along various loading paths and for different packings.

Model formulation

Previously proposed in [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF] and implemented in the Yade code [START_REF] Šmilauer | The Yade Project[END_REF], the considered DEM model addresses the case of spherical solid particles and a wetting fluid distribution in terms of distinct capillary bridges, i.e. menisci between particles pairs (Fig. 1), in accordance with the so-called pendular regime, also referred to as wet conditions. Such conditions correspond to a low wetting saturation, S w , only, where S w is defined as:

S w " V w V w `Vn (6)
In line with another DEM approach [START_REF] Scholtès | On the capillary stress tensor in wet granular materials[END_REF], the model does not compute any biphasic pore flow, but instead assumes thermodynamic equilibrium and uniform capillary pressure conditions. As such, the menisci properties are computed

R 2 ≥ R 1 R 1 θ θ d s s w n Figure 1: A capillary bridge
from the following Laplace-Young equation ( 7) which relates capillary pressure to surface tension and the interface curvature div pñq, with ñ being the normal to S nw pointing from n to w:

u c " γ nw div pñq (7)
According to the thermodynamic equilibrium assumption, Eq. ( 7) is solved for uniform values of u c throughout the sample, also considering a uniform and constant contact angle θ (Fig. 1) [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF]. As discussed in greater depth in [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF],

Eq. ( 7) is actually solved depending on the particles radii and their distance, in addition to imposed u c and θ. Based on the S nw interface profile, the numerical solution procedure finally outputs the resultant capillary force f cap and other relevant microstructural quantities such as µ Snw " ş Snw pδ ´ñ b ñq dS for instance, thus enabling one to apply the µUNSAT Eq. ( 5) and measure contact stresses during DEM simulations [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF].

The model actually inserts capillary bridges at each new contact between solid particles, with properties given by the Laplace-Young equation. In the instance the particles subsequently separate, the capillary bridge remains until some limit distance beyond which the Laplace-Young equation does not yield any solution. This hysteretic mechanism of bridge formation and rupture is consistent with experimental observations [START_REF] Herminghaus | Dynamics of wet granular matter[END_REF] and occurs in the model in a fully drained manner. The presence of a capillary bridge between two particles corresponds to a capillary interaction with the corresponding attractive capillary force f cap due to the capillary pressure and surface tension loadings along the wetted surfaces S sw and the contact lines Γ.

In addition, the interaction between touching particles is governed by classical elastic-plastic frictional (cohesionless) contact laws whereby normal and tangential relative displacements between particles are restricted by normal and tangential contact forces f c n and f c t [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF]. A simplified linear elastic behaviour of the contact interaction is first defined by the normal and tangential local stiffnesses, k n and k t , which are directly obtained from the average diameter D of any pair of contacting particles and the model parameters k n { D and k t {k n :

k n " k n D D (8) k t " k t k n k n (9)
with the consideration of an elastic shear stiffness being eventually restricted by the following Coulombian friction threshold, with µ the friction coefficient:

|f c t | ď µ f c n ( 10
)
Table 1 lists all parameters of the DEM model. The retained values for the contact interaction parameters, k n { D and k t {k n , are similar to previous studies [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Stress in wet granular media with interfaces via homogenization and discrete element approaches[END_REF][START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF][START_REF] Duriez | The micromechanical nature of stresses in triphasic granular media with interfaces[END_REF]. As for the capillary interaction, the latter is fully determined by the fluid-fluid surface tension γ nw , the contact angle θ and the particle size distribution. Air-water surface tension at ambient temperature is considered for γ nw while a low though not zero θ-value corresponds to the wetting of glass beads by water, having at the same time the convenience of enhancing the mechanical effects of triphasic conditions [START_REF] Duriez | Contact angle mechanical influence for wet granular soils[END_REF]. Finally, silt-like particle diameters are chosen with a narrow particle size distribution being uniform in number between D min and D max .

Contact Capillarity Packing k n { D k t {k n µ γ nw θ D min D max (MPa) (-) (-) (N/m) ( 0 ) (µm) (µm) 10 0.5 0.5 0.073 10 21 25
Table 1: DEM model parameters

While the dynamic scheme of the DEM is classically carried out using some numerical damping, in order to obtain more easily a quasi-static evolution of the system, the damping coefficient [START_REF] Šmilauer | The Yade Project[END_REF] is taken as low as 0.05 to avoid numerical artifacts. Quasi-staticity is systematically verified through low values of the particle unbalanced forces [START_REF] Šmilauer | The Yade Project[END_REF] which were below 1% during all loading cases.

As such, particle inertia (mass) does not play any role in the computations, which allows the use of density scaling with an artificially high density value (ρ " 20,000 kg/m 3 instead of 2,600 kg/m 3 ) for the particles. Without any consequence on the quasi-static rheology, density scaling reduces computational costs through larger time steps, even though loading rates have to be marginally reduced for quasi-staticity to still hold.

The
n " V n `Vw V s `Vn `Vw " V v V ( 11
)
z c " 2 N c ´N1 N p ´N1 ´N0 (12)
with N c the total number of contacts, and N 0 `N1 the number of excluded rattlers, i.e. particles with zero or only one contact, and N p the total number of DE. All considered packings are initially isotropic and prepared such that dry vs wet comparisons can be conducted in an unbiaised manner.

Care is indeed taken in considering dry or wet solid packings to be comparable in terms of n and z c as these parameters obviously affect the dry constitutive relation which should also be described in wet conditions by an effective stress.

As such, an appropriate particle packing generation procedure is followed which first involves a classical isotropic compression until a desired confining pressure is reached to obtain dense or loose dry packings [START_REF] Scholtès | On the capillary stress tensor in wet granular materials[END_REF], without the consideration of any capillary interaction that is only introduced in a second stage for the wet samples. In this manner, wet packings show initial states which are barely affected by the presence of capillary forces, and thus compare well with dry packings.

Dry and wet strain proportional loading paths

In order to investigate the stress-strain effective nature of the contact stress, the DEM model is used to explore the existence of a unique constitutive relation F between σ cont and the strains ε in both dry (biphasic) and wet (triphasic) conditions. In the case of such an existence, F by definition would relate the current contact stress σ cont ptq to the strain loading path rεpτ q; ´8 ă τ ď ts such that:

σ cont ptq " F `rεpτ q; ´8 ă τ ď ts ˘(13)
Extending our previous work [START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Partially saturated granular materials: Insights from micro-mechanical modelling[END_REF] where classical triaxial paths were examined, the validity of the same Eq. ( 13) in dry and wet conditions with a unique F is herein probed considering axisymmetric strain proportional loading paths, with '1' as the axis of symmetry and '2, 3' the two other spatial directions:

dε 1 " cst ě 0 ; dε 2 " dε 3 ; dε 1 `2 R dε 3 " 0 (14)
The loading parameter R ě 0 is constant along a given path, R " 1 corresponds to an isochoric loading path, whereas dilatancy (resp. contractancy) is imposed for 0 ă R ă 1 (resp. R ą 1). The limiting case R " 0 is defined as another contractant path pdε 1 " 0 ; dε 2 " dε 3 " cst ě 0q.

Let's denote the volumetric strain as ε V :

ε V " ε 1 `2 ε 3 (15)
and choose ||ε|| (the Euclidean norm of ε) as a monotonously increasing loading parameter, i.e.

||ε|| "

a ε 1 2 `2 ε 3 2 (16)
Accordingly, Fig. 2 illustrates the corresponding dilatancy rates of four loading paths with R P t0; 0.2; 1; 5u that will be considered in the following analysis. In all cases, imposed strain rates are low enough to ensure quasi-static simulations

ε v (%, comp. >0) R=0 R=0.2 R=1 R=5
Figure 2: Dilatancy rates of imposed loading paths for R P t0; 0.2; 1; 5u as verified from the corresponding low values of the particle unbalanced forces [START_REF] Šmilauer | The Yade Project[END_REF] which were below 1%.

The main objective of imposing such strain loading paths on both dry and wet packings is to verify the hypothesis, σ 1 " σ cont , upon which is founded the µUNSAT framework. If Eq. ( 13) holds, then the resulting contact stress paths should coincide in both dry and wet conditions. Recognizing the axial symmetry of the problem, it is convenient to associate the tensor σ cont with the two classical scalars (invariants) p cont and q cont , i.e.

p cont " σ cont 1 `2 σ cont 3 3 (17)
q cont " σ cont 1 ´σcont 3 (18)
while similarly, p and q variables pertain to the total stress tensor Σ:

p " Σ 1 `2 Σ 3 3 (19) q " Σ 1 ´Σ3 (20)
4. Validity of the µUNSAT effective stress

Dense packings

Proportional strain loading paths are first imposed to dense solid packings.

In essence, the same strain loading path is imposed to similarly dense packings in both dry and wet conditions with the resulting contact stress paths recorded.

The contact stress obviously coincides with the total stress in dry conditions, whereas Eq. ( 3) or ( 5) can be indistinctly used to access the contact stress under wet conditions [START_REF] Duriez | Subtleties in discrete-element modelling of wet granular soils[END_REF].

Four strain loading paths are considered with R P t0; 0. Comparing the contact stress responses in dry and wet conditions (Fig. 3 and 4), it is indeed verified that virtually unique behaviours are observed along the contractant or isochoric loading paths R P t0; 1; 5u. As such, this establishes the validity of using the dry constitutive relation F together with σ cont as an effective stress along these loading paths.

As for the dilatant loading path R " 0.2, a unique behaviour is observed in the early stage of the test. Subsequently, the imposed dilatancy rate requires the stresses to eventually vanish in the dry case, whereas when wet the same packing 2) along contractant (R " 0) and isochoric (R " 1) strain loading paths (color version online) 2) along contractant (R " 5) and dilatant (R " 0.2) strain loading paths (color version online) gains a propensity to dilate originating from liquid bridges, thus avoiding a loss in bearing capacity. Hence, for such a dilatant loading path, the role of the contact stress as an effective stress is limited to the initial part of the loading path, and could still be used to predict initial stiffnesses in triphasic conditions.

0
It is noted that all contact stress responses obtained in both dry and triphasic conditions are found to be bounded by the same cohesionless Mohr-Coulomb failure criterion (with c " 0 Pa ; ϕ " 31 0 ) that was previously measured in dry conditions for such packings [START_REF] Duriez | Partially saturated granular materials: Insights from micro-mechanical modelling[END_REF]. The universal property of this limit stress line, irrespective of the phase condition, is another illustration of the previously mentioned stress-strength effective nature of the contact stress.

Loose packing

The same analysis procedure is now applied to loose packings (Table 3). The numerical results displayed in Figs. 5 and In particular, the existence of a unique constitutive relation F between σ cont and ε is identified for the most contractant loading path R " 0 only, and is somewhat approximated along the less contractant path R " 5.

As for the isochoric path R " 1, and at variance with the dense packing (Fig. 3), a dry-wet agreement is only obtained for the very initial phase during which p cont remains constant, consistent with a possible relationship between σ cont and ε through Hooke's elastic law. Then, the dry packing presents a 'liquefaction' behaviour with vanishing stresses, consistent with the constant volume condition and the underlying material contractant flow rule. In contrast, 3) along contractant (R " 0) and isochoric (R " 1) strain loading paths (color version online) 3) along contractant (R " 5) and dilatant (R " 0.2) strain loading paths (color version online) similar packings under triphasic conditions gain a propensity to dilate due to liquid bridges, thus leading the mean contact stress to increase. Some influence of the mean contact stress on the dry-wet comparison is finally observed along the dilatant loading path R " 0.2. Along such a path, dry conditions induce the stresses in this loose material to vanish as expected, while triphasic conditions enhance again the material dilatant nature, leading to non zero stress states. The exact behaviour of the wet material nevertheless depends on the confining pressure.

For a low confining pressure, the wet loose packing behaves similarly to the dense one because capillary stress plays an important part of the total stresses: |p cap |{p « 3.0 and |p cap |{p cont « 0.75 at the initial stage of this test. The behaviour then changes into one with a greater propensity to dilate as a result of triphasic conditions.

On the other hand, upon increasing the confining pressure, the contact stress also increases while the capillary stress remains fairly constant, leading to |p cap |{p cont « 0.42 and |p cap |{p « 0.72, initially. The changes in the relative contributions from contact and capillary stresses diminish the liquid bridge induced dilatancy in the loose packing.

Finally, and as previously noted for the dense packing, the plastic limit criterion (c " 0 Pa ; ϕ " 21 0 ) corresponding to this loose material under dry conditions [START_REF] Duriez | Partially saturated granular materials: Insights from micro-mechanical modelling[END_REF] also applies to the contact stress under triphasic conditions, once again confirming the stress-strength effective nature of σ cont .

Discussion: effective stress vs dissipation and elasticity

The influences of the loading path and the solid packing on the ability of σ cont to act as a stress-strain effective stress are now interpreted in relation to the variable dissipation associated with the observed behaviours. From f c n and f c t the normal and tangential components of the contact forces f c , the specific elastic energy e el of the material is:

e el " 1 2 V ÿ Nc ˆpf c n q 2 k n `pf c t q 2 k t ˙ (21)
which is easily measured from the DEM model, together with its increment δe el between successive states. Comparing this increment with the incremental external work Σ : dε, the following quantity λ quantifies the deviation of the mechanical behaviour of any dry test from a non-dissipative behaviour:

λ " ˇˇˇΣ : dε ´δe el Σ : dε ˇˇˇ ( 22)
The quasi-static nature of the loading paths ensures that both the kinetic energy and the energy term associated to numerical damping are negligible, and thus do not contribute to the energy balance quantified by the quantity λ. As such, during a non-dissipative, e.g. elastic, dry loading path, λ is equal to 0 throughout since all external work is completely stored as elastic energy. On the other hand, increasing values of λ reveal an increasing dissipation. It is noted that λ may be greater than 1 in the instance of elastic energy loss (δe el ă 0).

However, it is to stress out that a non-dissipative behaviour with negligible values of λ does not necessarily correspond to a stress-strain path being reversible, even though the opposite is true. These intricacies are specific to granular materials whose elastic energy is contained in a contact network whose properties (for instance, N c in Eq. ( 21)) fundamentally depend on irreversible strains. Also, granular materials are prone to irreversible yet non-dissipative local mechanisms such as contact losses. These very specific features can be interpreted in terms of a locked elastic energy concept [START_REF] Collins | The concept of stored plastic work or frozen elastic energy in soil mechanics[END_REF] and have also been widely commented, e.g. in [START_REF] Puebla | Irreversible processes without energy dissipation in an isolated Lipkin-Meshkov-Glick model[END_REF][START_REF] Pouragha | Non-dissipative structural evolutions in granular materials within the small strain range[END_REF], motivating here the distinction between the "non-dissipative" and "elastic" terminologies.

From the evolutions of λ shown in Figs. 7, and comparing with previous Figs. [START_REF] Bishop | Some aspects of effective stress in saturated and partly saturated soils[END_REF], it clearly appears that it is along the less dissipative loading paths that the stress-strain effective nature of σ cont is the most marked, e.g. R " 0. Such a result includes and extends the finding previously reported in [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF] that an effective stress approach is possible in the elastic regime.

Another illustration of the role of dissipation in the effective stress discussion is proposed in Fig. 8 which considers an extreme case of the DEM model, referred to as µ " 8. In this extreme case of the DEM model, an ideal contact Figure 7: Dissipative nature of the behaviour during various dry loading paths imposed on the dense packing (Tests A). Raw data have been smoothed without altering the average trends and λ-values for R=0.2 may be disregarded beyond ||ε|| « 1% since stresses vanish at this point description is adopted in which no frictional threshold [START_REF] Senetakis | The inter-particle coefficient of friction at the contacts of leighton buzzard sand quartz minerals[END_REF] limit is imposed on the shear contact force: f c t just evolves linearly with the tangential relative displacement according to the shear stiffness k t as long as a contact exists. The dry-wet comparisons shown in Fig. 8 then reveal an enhanced stress-strain effective character of σ cont with this model, as opposed to the already discussed dry-wet comparisons with the classical friction coefficient µ " 0.5 for the loose packing along the R " 5 loading path.

The present discussion finally sheds some light on the generality of thermomechanical approaches to effective stress, as proposed e.g. in [START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF][START_REF] Madeo | A continuum model for deformable, second gradient porous media partially saturated with compressible fluids[END_REF][START_REF] Nikooee | Effective stress in unsaturated soils: A thermodynamic approach based on the interfacial energy and hydromechanical coupling[END_REF].

Because the effective stress discussion herein appears to be widely different depending on the elastic or dissipative nature of the behaviour, and since such thermomechanical approaches usually rely on the assumption of an elastic behaviour for the solid phase with an appropriate potential, it is probably impossible to generalize the conclusions obtained through thermomechanics outside the assumed framework.

Comparison with Bishop's stress

Qualitative discussion

The present µUNSAT approach to effective stress is now compared with the commonly used effective stress expression as Bishop's stress σ Bish :

σ 1 ? " σ Bish " Σ ´un δ `uc χ δ (23)
Formally, significant differences appear between Bishop's equation [START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF] and the proposed µUNSAT equation [START_REF] Lu | Suction stress characteristic curve for unsaturated soil[END_REF]. It is herein recalled that the µUNSAT equation includes a surface tension dependency in addition to that of capillary pressure, as well as the possibility of a non-spherical, microstructure-dependent, capillary stress contribution. Such features that characterize the microstructural details in triphasic materials go beyond the scope of Bishop's stress that relies on an average fluid pressure, being proportionnal to u c and necessarily spherical. Since triphasic conditions by nature involve fluid-fluid interfaces and corresponding surface tension, the lack of γ nw in Bishop's approach is questionable, as is the hypothesis of an averaged (isotropic) fluid pressure, while the wetted solid surfaces for instance experience capillary pressure along their normals only. Actually, the deviatoric nature of the capillary stresses has been repeatedly demonstrated in the pendular regime from modelling approaches [START_REF] Hicher | A microstructural elastoplastic model for unsaturated granular materials[END_REF][START_REF] Scholtès | On the capillary stress tensor in wet granular materials[END_REF][START_REF] Wan | A tensorial description of stresses in triphasic granular materials with interfaces[END_REF][START_REF] Duriez | Stress in wet granular media with interfaces via homogenization and discrete element approaches[END_REF][START_REF] Duriez | The micromechanical nature of stresses in triphasic granular media with interfaces[END_REF], with possible variations in q cont or q cap independently of q during microstructural changes of the fluid phases, contrary to Bishop's general equality between q and q Bish .

Quantitative discussion

In order to illustrate the inadequacy of Bishop's stress to serve as an effective stress in the pendular regime, other dry-wet comparisons are presented for the dense packing along the two loading paths R P t1; 5u, from initial states presented in Such loading paths R P t1; 5u revealed a very good agreement of the σ cont -ε behaviours in both dry and wet conditions (Figs. 3 and4). However, the wet behaviour in terms of Bishop's stress here significantly deviates from the dry behaviour (Fig. 9), clearly showing that Bishop's stress is a much less appropriate effective stress measure along these loading paths than the contact stress. In addition to this degraded stress-strain effective character of Bishop's stress, it is also clear from the present comparison that Bishop's stress is also inadequate to describe the strength of wet materials from the dry plastic limit criterion, which σ Bish violates. The consideration of Bishop's stress in an effective stress approach with some dry shear strength criterion in practice would thus lead to an overestimation of the risk of material failure in the pendular regime, in 20

Conclusion

Micromechanical modelling approaches showed evidence of a single effective stress variable that can be used together with dry constitutive relations to describe the strains of idealized slightly polydisperse granular media in the pendular regime. The appropriate variable is the contact stress whose direct evaluation requires the prior knowledge of the contact force network. Using the proposed µUNSAT expression, contact stresses can be alternatively calculated through measurements or estimations of the fluid phase microstructure (interconnects), which are easier to determine than forces.

Unique contact stress-strain relationships have indeed been numerically demonstrated along a variety of loading paths for both dry and wet (pendular regime) conditions. This lends strong support to applying the effective stress designation to contact stress. However, the considered loading paths have to induce small dissipation for the stress-strain effective stress approach to hold. Namely, contractant loading paths and dense packings are well suited for the application of the present µUNSAT approach to hold.

Previous experimental and phenomenological works by [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF] had restricted the existence of a single stress-strain effective stress in triphasic media to the elastic regime. This work based on microstructural analysis suggests a less restrictive condition by extending it to even the irreversible regime but with small dissipation.

There remain open questions as to whether new definitions of strain quantities could cast the presented results into an even more general framework.

The present approach actually relies on observable strains or particle relative displacements that are due to both contact and capillary forces, through the classical DEM workflow (second Newton's law). Alternatively, it could be attempted to define distinct, non-observable, strain components: first, a contact strain uniquely caused by contact forces, and second, a capillary strain uniquely due to capillary forces, with the combination of the two leading to the observable strains herein considered. Such an alternative framework may lead to strictly unique constitutive relations between the contact stress and such a contact strain, and shed some light on the dissipation role through the yet to be defined link between contact strain and observable strain.

Figure 3 :

 3 Figure 3: Contact stress response of a dense packing (Table2) along contractant (R " 0) and isochoric (R " 1) strain loading paths (color version online)

Figure 4 :

 4 Figure 4: Contact stress response of a dense packing (Table2) along contractant (R " 5) and dilatant (R " 0.2) strain loading paths (color version online)

Figure 5 :

 5 Figure 5: Contact stress response of a loose packing (Table3) along contractant (R " 0) and isochoric (R " 1) strain loading paths (color version online)

Figure 6 :

 6 Figure 6: Contact stress response of a loose packing (Table3) along contractant (R " 5) and dilatant (R " 0.2) strain loading paths (color version online)

Figure 8 :

 8 Figure 8: Influence of local friction µ on the contact stress' effective nature for a loose packing along a contractant loading path (R " 5)

Figure 9 :

 9 Figure 9: Bishop's or total stress responses along isochoric (R " 1) and contractant (R " 5) loading paths imposed on a dense packing

Table 2 :

 2 Tests data for the dry and wet loading paths imposed on dense packings

		Tests A	Tests B
		Dry	Wet	Dry	Wet
	Initial p (kPa)	11.1	2	20.6	10
	Imposed u c (kPa)	-	25	-	125
	Initial S w (%)	-	7.18	-	0.85
	Initial n (-)	0.375 0.380 0.370 0.373
	Initial z c (-)	6.41	6.21	6.62	6.52

2; 1; 5u, under two different confining pressures referred to as "A" and "B". For each confining pressure A or B, and each R value, one dry and one wet test are considered, leading to 16 tests on dense packings in total. Table

2

gives details of the initial conditions for the dry-wet test pairs A, B pertaining to dense packings. These initial conditions are such that dry and wet tests of a given pair A or B show initial states (σ cont , n, z c ) being as close as possible.

  6 reveal less of an agreement between dry and wet behaviours in comparison to the previous dense case study.

		Tests A	Tests B
		Dry	Wet	Dry	Wet
	Initial p (kPa)	8.1	2	17.3	10
	Imposed u c (kPa)	-	25	-	125
	Initial S w (%)	-	4.17	-	0.50
	Initial n (-)	0.442 0.441 0.436 0.438
	Initial z c (-)	4.96	4.84	5.17	5.12

Table 3 :

 3 Tests data for the dry and wet loading paths imposed on loose packings

Table 4 .

 4 The present discussion actually narrows down to the classical choice χ " S w for Bishop's parameter, which can be continuously measured during DEM simulations thanks to the complete description of the microstructure. As such, the stress response observed in dry conditions is now compared to the Bishop's stress response measured during triphasic conditions along the same strain paths.

		Dry	Wet
	Initial p (kPa)	11.5	10
	Imposed u c (kPa)	-	125
	Initial S w (%)	-	0.85
	Initial n (-)	0.375 0.373
	Initial z c (-)	6.41	6.52

Table 4 :

 4 Tests data for the dry-wet comparison in terms of Bishop's stress

Acknowledgements

This work has been funded by the Natural Science and Engineering Research

Council of Canada and Foundation Computer Modelling Group.


D M Allen

N Duclos

I Garbutt

M Saumer

Ch Dhum

M Schmitt

J E Hoffmann

THE EFFECTS OF ADDITIVES ON THE PHYSICAL PROPERTIES OF ELECTROFORMED NICKEL AND ON THE STRETCH OF PHOTOELECTROFORMED NICKEL COMPONENTS

The process of nickel electroforming is becoming increasingly important in the manufacture of MST products, as it has the potential to replicate complex geometries with extremely high fidelity. Electroforming of nickel uses multi-component electrolyte formulations in order to maximise desirable product properties. In addition to nickel sulphamate (the major electrolyte component), formulation additives can also comprise nickel chloride (to increase nickel anode dissolution), sulphamic acid (to control pH), boric acid (to act as a pH buffer), hardening/levelling agents (to increase deposit hardness and lustre) and wetting agents (to aid surface wetting and thus prevent gas bubbles and void formation). This paper investigates the effects of some of these variables on internal stress and stretch as a function of applied current density.

INTRODUCTION

The process of nickel electroforming is becoming increasingly important in the manufacture of MST products, as it has the potential to replicate complex geometries with extremely high fidelity. In order to reduce the costs per piece of polymeric components, replication from electroformed precision moulds by micro-injection moulding or hot-embossing is attracting the attention of manufacturers of microfluidic devices and diagnostic/analysis micro-systems as it appears to be one of the few economically-viable processes available to industry when product demand exceeds millions of components per annum [START_REF] Uriarte | A comparison between microfabrication technologies for metal tooling[END_REF].

Nickel is the most commonly-used electroformed material in such applications as it may be deposited with minimal internal (residual) stress. It is a primary requirement that the electroformed metal replicates the mould geometry and dimensions precisely and does not distort after separation from the mould mandrel. However, this is not easily done, requiring strict temperature and humidity control especially if the mould is made by imaging and developing photoresist; a process known as photoelectroforming (PEF). A well-known phenomenon in PEF is the inadvertent production of parts where an expansion of dimensions occurs after release from the mould, commonly known as "stretch".

Electroforming of nickel uses multi-component electrolyte formulations in order to maximise desirable product properties. In addition to nickel sulphamate (the major electrolyte component), formulation additives can also comprise nickel chloride (to increase nickel anode dissolution), sulphamic acid (to control pH), boric acid (to act as a pH buffer), hardening/levelling agents (to increase deposit hardness and lustre) and wetting agents (to aid surface wetting and thus prevent gas bubbles and void formation).

Several distinct types of nickel can therefore be deposited from such a range of nickel electroplating compositions, including:

• Soft nickel, based on a nickel sulphamate electrolyte • Hard nickel, based on a nickel sulphamate electrolyte with an additional proprietary hardener and • Ni-speed, a high-speed, high-concentration, nickel sulphamate formulation, developed originally by International Nickel. The quantity of metal deposited follows Faraday's Laws of Electrolysis and is proportional to the applied current and time. To reduce deposition time, it is therefore advantageous to electro-deposit with a high current. Nickel electroforms can thus be fabricated to provide a range of characteristic properties including variable grain size, Young's modulus, yield strength,

D.M. Allen et al

The effects of additives on the physical properties of electroformed nickel and.... ultimate tensile strength, hardness and wear rate over a wide range of current densities [START_REF] Allen | Comparisons of various physical characteristics of electroformed nickels and nickelcobalt alloys as an aid for the selection of micro-mould insert materials[END_REF].

STRESS THEORY

Residual stress is a tension or compression that exists in the bulk of a material without application of an external load. It can exist in a macro form over a few grains (σ I ), over one particular grain (σ II ), or even within a grain across several interatomic distances (σ III ). The last two stresses may be termed microstresses [3] [4].

In electroformed metals such as nickel, the stress may be either tensile or compressive. In the case of tensile (+) stress, the deposit can be likened to an expanded spring that is trying to contract relative to the substrate (mandrel) that it is attached to. In the case of compressive (-) stress, the deposit, likened to a compressed spring, is trying to expand relative to its mandrel [START_REF] Stein | A practical guide to understanding, measuring and controlling stress in electroformed metals[END_REF].

As mentioned in the Introduction, electroforming of nickel uses multi-component electrolyte formulations and each component will affect the degree of stress in the deposit produced. In addition, internal stress will be affected by current density, temperature, agitation, plating cell geometry, the composition and condition of the anodes, the anode/cathode surface area ratio, the quality of the DC power supplied and the nature of the mandrel cathode. Indeed, it is hard to find a process variable that does not influence the deposit internal stress as pointed out by Stein [START_REF] Stein | A practical guide to understanding, measuring and controlling stress in electroformed metals[END_REF].

In industrial electroforming and PEF, the tolerances of the parts are often very tight and accurate dimensions are difficult to achieve even in a rigorous temperaturecontrolled environment. In considering the release of a compressively-stressed deposit from its mandrel, the electroformed part will expand and change its dimensions such that the part may be out of tolerance and a costly reject. It is therefore vital that stress is minimized towards zero in order to obtain separated parts within dimensional tolerance. The stress cannot be controlled unless it can be measured accurately.

Two methods for stress measurement have therefore been evaluated so that the results may be compared.

STRESS MEASUREMENT METHODS

Deposit stress analyzer

The deposit stress analyzer (Figure 1) is a popular instrument for measuring stress in a production environment as it offers significant cost-and timesavings in comparison with other methods [START_REF] Stein | A practical guide to understanding, measuring and controlling stress in electroformed metals[END_REF]6]. The method comprises fixing a strip of metal, slit into two "legs" that are coated on opposite sides to each other with an insulating lacquer, in a special test cell within the plating tank. As electroplating of the mandrel is carried out, metal also deposits on opposite sides of the metal legs. With the production of stressed deposits, the legs bend and the resulting separation between them can be measured directly as shown in Figure 1. As the stress in the deposit increases, the separation will become greater. A calibration chart allows conversion from the separation distance to the internal stress value.

X-ray diffraction (XRD)

The strain (ε) in the direction normal to the diffraction planes is related to the corresponding lattice spacing (d) that is linked directly to the diffraction angle (θ) by Bragg's Law: nλ = 2d.sinθ. For each inclination of the sample, defined by two angles, ψ and φ, the strain ε is thus related to the corresponding lattice spacing d.

The relationship between the strain ε and the stress tensor components σ is then derived, under specific conditions, through the sin 2 ψ law [START_REF] Macherauch | Das sin²ψ-Verfahren der röntgenographischen Spannungsmessung[END_REF] such that:

ε = ½S 2{hkl} σ φ sin 2 ψ + s 1 (σ 1 + σ 2 )
Thus a plot of ε vs. sin 2 ψ gives a straight line, the slope of which allows the calculation of the stress component σ φ , in the direction φ of the surface if the elastic constant (½S 2{hkl} ) is known [START_REF] Macherauch | Zur Zweckmäßigen Definition von Eigenspannungen[END_REF].

The diffractometer used to measure the stress in the electrodeposited material is shown in Figure 2.

D.M. Allen et al

The effects of additives on the physical properties of electroformed nickel and....

Comparisons between the two methods of measuring stress

There are many differences between the two aforementioned methods used for stress measurement. The small-area deposit stress test strip is quick to use (<20 minutes) but can never be positioned in the exact location of the cathode and may not receive the average cell current density as it can vary considerably across a widearea mandrel.

The test strip is also orientated perpendicular to the mandrel to prevent shielding of one side of the strip. This results in a macro-measurement that averages the overall effects of the different current densities and micro-stresses in the deposit.

On the other hand, the XRD equipment only measures to an average penetration depth of 3.5µm over a very small area of 0.126mm 2 . The analysis time is long and takes 3.5 hours.

RESULTS OF STRESS MEASUREMENTS

Samples

Nickel deposits, approximately 20µm thick, were electroplated onto large brass mandrels (48.8cm x 80.4cm) at current densities of ~1Adm -2 and ~5Adm -2 .

Smaller samples (2cm x 5cm) were cut out from the middle of these plated mandrels and used for the XRD investigations, firstly whilst still attached to the brass mandrel and secondly when manually stripped from the mandrel. These samples were compared with the results obtained from the deposit stress analyzer samples electroplated under the same electrolytic and temperature conditions. The results obtained are shown in Table 1 The results for the XRD analyses are plotted in Figure 3 and show some variation according to the angle of measurement and general movement towards zero internal stress, when released from the mandrel.

The highest amount of compressive internal stress, i.e. -120 MPa, is given by the nickel layer on a brass mandrel electroplated at the higher current density of 5 Adm -2 (sample St 57; see Figure 3 and Table 1).

The scatter band at the same measuring point of sample St 56 is between -50 to -100 MPa. The variation in sample St 56 could originate from a particular orientation of the nickel grains. Therefore the results may deviate with the variation in the angle of measurement.

Releasing the nickel layer from the brass mandrel, results in a decrease of the internal compressive stress. This is due to the lack of reinforcement of the thin layer by the much thicker mandrel. The internal stress must therefore relieve towards zero as confirmed by the measurements. Thus, releasing of any compressivelystressed thin electroform should result in stretching of the nickel.

In contradistinction, the results obtained by the deposit stress analyzer show another trend (Table 1). Here the measured compressive stress is higher in sample St 56 (-21 MPa) and nearly zero (+ 1 MPa) in sample St 57. However, for the same equipment, Stein [START_REF] Stein | Fast and accurate deposit internal stress determination[END_REF] has shown the compressive internal stress for nickel reducing to approximately zero as current density increases from 1 Adm -2 to about 4.5 Adm -2 when using a nickel sulphamate electrolyte. This result compares very favourably with our results for sample St 57, although the precise chemistry of Stein's nickel sulphamate electrolyte was not given.

Further experimental investigations are necessary to understand the relationships between results obtained under different plating conditions (e.g. flow of

D.M. Allen et al

The effects of additives on the physical properties of electroformed nickel and.... electrolyte, size and orientation of anode/cathode, substrate material and size) analysed by different stress measurement methods. An improved understanding of the stress characterisation will then allow the relationships between internal stress, stretch and electrolyte additives to be investigated in a more comprehensive manner.

Relationship of internal stress to stretch

The relationship between stress and stretch (measured in parts per million) is illustrated in Figure 4. The internal stress value has been derived from the deposit stress analyzer measurement and therefore gives a first approximation to the actual stress likely to be experienced by the electroformed part. The high correlation coefficient (R 2 = 0.8167) indicates a very strong correlation between internal compressive stress and the resulting stretch of an electroformed component despite the fact that the absolute value of the internal stress is in dispute.

Effects of additives on internal stress

Notwithstanding the difficulties of measuring stress accurately, some preliminary analysis on the general effects of additives on internal stress has been carried out [START_REF] Duclos | The effects of additives on the physical properties of electroformed nickel[END_REF]. In summary, additions of nickel chloride appear to reduce compressive stress. However, hardener additions not only increase the hardness of the deposit but in so doing also reduce grain size, increase the visual lustre and increase compressive stress as shown in Figure 5. Increasing the current density also increases the electrodeposit lustre. It is therefore vital in industrial PEF to measure and control not only the current density but also the electrolyte chemistry and additive concentrations.

CONCLUSIONS

Previous research has been carried out in comparing the values of internal stresses within electroformed nickel, obtained via a deposit stress analyzer, a spiral contractometer and an IS meter [9]. In this work, XRD results have been compared with deposit stress analyzer results.

Our preliminary investigations suggest that the results obtained from a deposit stress analyzer are self-consistent and allow a rapid evaluation of stress that can be used for electroplating control. The results obtained appear to give an average value for the coating relating to the particular geometrical arrangement within the electroplating cell that can be related to the perpendicular mandrel onto which a part is plated.

XRD is an extremely accurate measurement over such a very small area that, again, the stress value may not be typical of the overall stress across a much larger component. Indeed, the analysis may be so localized, and relevant only to a maximum depth of 3.5µm, that the results may not reflect the overall stress even in an electroformed micro-product.

The XRD method appears very sensitive to grain orientation that may be influenced by the mandrel crystallographic orientation and/or the current density. Even stresses induced by pulling the electroform from the mandrel may be detected. XRD also has particular value when the electroform thickness is extremely small -a condition that results in higher errors for a deposit stress analyzer [9].

However, XRD clearly confirms that on removal from a mandrel, the compressive internal stress is relieved and consequently becomes apparent in component stretch.

ACKNOWLEDGEMENTS

Prof. Allen, Prof. Saumer and Mr. Duclos wish to acknowledge the cooperation of Tecan Ltd, its staff and its Chairman and Owner, Mr. Paul Cane, and to express their thanks for access to the industrial manufacturing facilities and an insight into the complex world of commercial PEF.

D.M. Allen et al

The effects of additives on the physical properties of electroformed nickel and....

Figure 1 .

 1 Figure 1. Deposit stress analyzer

Figure 2 .

 2 Figure 2. Seifert ψ-Diffractometer XRD 3000 PTS

Figure 3 .

 3 Figure 3. Stress measured by XRD for St 56 on a brass mandrel (■) and free-standing (□) and St 57 on a brass mandrel (▲) and free-standing (∆).

Figure 4 Figure 5 .

 45 Figure 4 Graph showing direct relationship between internal stress and stretch

Table 1 .

 1 . Results of stress analysis by the two methods

		Internal Stress (MPa)
	Sample	(On brass)	(On brass)	(Free-
	and	from deposit	from XRD	standing)
	current	stress		from
	density	analyzer		XRD
	St 56 ~1Adm -2	-21 ± 10	-50 to -100 ± 15	-37 ±13
	St 57	+ 1± 10	-120 ± 20	-21 ±50
	~5Adm -2

D.M. Allen et al

The effects of additives on the physical properties of electroformed nickel and....


Samuel Deslauriers-Gauthier

Drew Parker

François Rheault

Rachid Deriche

Steven Brem

Maxime Descoteaux

Ragini Verma

Edema-informed anatomically constrained particle filter tractography

In this work, we propose an edema-informed anatomically constrained tractography paradigm that enables reconstructing larger spatial extent of white matter bundles as well as increased cortical coverage in the presence of edema. These improvements will help surgeons maximize the extent of the resection while minimizing the risk of cognitive deficits. The new paradigm is based on a segmentation of the brain into gray matter, white matter, corticospinal fluid, edema and tumor regions which utilizes a tumor growth model. Using this segmentation, a valid tracking domain is generated and, in combination with anatomically constrained particle filter tractography, allows streamlines to cross the edema region and reach the cortex. Using subjects with brain tumors, we show that our edema-informed anatomically constrained tractography paradigm increases the cortico-cortical connections that cross edemacontaminated regions when compared to traditional fractional anisotropy thresholded tracking.

Co-senior authors. They have contributed equally.

Introduction

Diffusion magnetic resonance imaging (MRI) allows to non-invasively reconstruct white matter tracts. In the context of pre-operative surgical planning, tractography is used to identify and locate fiber bundles in the vicinity of the lesion. Equipped with this knowledge, the surgeon attempts to maximize tumor removal while minimizing damage to functional white matter networks [START_REF] Mcgirt | Independent association of extent of resection with survival in patients with malignant brain astrocytoma[END_REF]. While the development efforts on every step of the tractography pipeline are extensive, most algorithms assume healthy brain and are tested on healthy white matter models. Diffusion tensor imaging tractography has been investigated in the context of neurosurgery [START_REF] Mandelli | MandeliQuantifying accuracy and precision of diffusion MR tractography of the corticospinal tract in brain tumors[END_REF][START_REF] Pujol | The DTI Challenge: Towards Standardized Evaluation of Diffusion Tensor Imaging Tractography for Neurosurgery[END_REF], but is subject to the well known limitation of the diffusion tensor. There is thus very little data on the performance of modern tractography algorithms in the presence of edematous or tumoral tissue. One notable exception is the use of multi-tensor models in the context of unscented Kalman filter (UKF) tractography, which has been investigated on brain tumor patients [START_REF] Chen | Reconstruction of the arcuate fasciculus for surgical planning in the setting of peritumoral edema using two-tensor unscented Kalman filter tractography[END_REF][START_REF] Chen | Corticospinal tract modeling for neurosurgical planning by tracking through regions of peritumoral edema and crossing fibers using two-tensor unscented Kalman filter tractography[END_REF][START_REF] Liao | Performance of unscented Kalman filter tractography in edema: Analysis of the two-tensor model[END_REF]. The authors show that two-tensor UKF increases the volume of the arcuate fasciculus and corticospinal tract and their coverage of the functional MRI Fig. 1. Segmentation of the edema obtained from GLISTR (first) and the tracking mask obtained by thresholding the FA at 0.2 (second). The edema lowers the FA and isolates the temporal lobe. The include map (third) and exclude map (fourth) show that in EI-PFT the edema can be traversed as it is in black in both masks. The tumor is visible in the exclude map indicating that streamlines cannot end in this region.

activation sites, especially when used with whole brain seeding. These results highlight the need for further investigation into the use of modern tractography strategies for pre-operative surgical planning.

Tractography algorithms reconstruct streamlines in the white matter of the brain by following the local diffusion directions obtained through diffusion MRI. While several criteria can be used to stop the tracking, the most common is to place a threshold on the fractional anisotropy (FA) map [START_REF] Jeurissen | Diffusion MRI fiber tractography of the brain[END_REF]. The rational is that the FA is high in regions with coherent fiber directions and therefore acts as a proxy for the white matter. However, it is well known that many phenomena can lower the measured FA, such as fiber crossings, free water contamination, and pathologies like edema in the presence of tumor [START_REF] Duffau | The dangers of magnetic resonance imaging diffusion tensor tractography in brain surgery[END_REF]. In these scenarios, the tracking may be prematurely terminated yielding a streamline that does not reach the cortex. Furthermore, because the FA does not encode anatomical information, algorithms that rely on it will produce physically impossible streamlines, such as those that start or end in the ventricles. Anatomically constrained tractography (ACT) [START_REF] Smith | Anatomically-constrained tractography: Improved diffusion MRI streamlines tractography through effective use of anatomical information[END_REF] alleviates this problem by defining a valid tracking domain based on the anatomy of the subject defined by the tissue segmentation. In combination with particle filter tractography, which backtracks and attempts to find alternate routes when it prematurely hits a region outside of the valid tracking domain, ACT was shown to reduce the number of false positive streamlines [START_REF] Girard | Towards quantitative connectivity analysis: reducing tractography biases[END_REF].

Because edema and tumor change the local tissue properties of the brain, tractography based on FA thresholding may not provide an accurate representation of fiber near the lesion. In this paper, we provide a paradigm that integrates anatomical information of both healthy and unhealthy tissues with particle filter tracking to obtain tractography through edema. This has been demonstrated on 4 tumor subjects.

Methods

Edema-informed anatomically constrained particle filter tractography

Tracking is rendered edema-informed by defining a valid tracking domain which integrates the knowledge of the various tissue types: white matter, gray matter, corticospinal fluid, edema, and tumor segmentation. More specifically, the tracking domain is defined by 3 probabilistic maps. The first map, referred to as the include map, determines valid start and end points for streamlines. That is, the value of a voxel of the include map gives the probability that a streamline will terminate within this voxel and be included in the final tractogram. An example of an include map is illustrated in the third image of Figure 1 where white regions indicate valid streamline start or end points. The second map, referred to as the exclude map, determines the probability that a streamline will terminate within a region and be excluded from the final tractogram. An example of an exclude map is illustrated in the fourth image of Figure 1 where white regions indicates invalid streamlines start or end points. Note that for any given voxel, the sum of the two maps must be between zero and one. Together, the include and exclude maps define a third implicit map which ensure the probabilities sum to one. This third map contains the regions that streamlines are free to cross, but where they cannot terminate. This separation of the tracking domain into 3 maps was used in [START_REF] Girard | Towards quantitative connectivity analysis: reducing tractography biases[END_REF], but their definition of the maps was based solely on the segmentation of healthy tissues, whereas we include edema and tumor information. Specifically, we define the include map to be the gray matter segmentation and the exclude map as P ex = P tumor +P csf -P tumor P csf where P tumor and P csf are the tumor and CSF segmentations, respectively. Note that the edema segmentation map is not used explicitly in the computation of the tracking domain, but it nonetheless affects it implicitly. Indeed, just like the white matter, the edema is considered as a region where the streamlines can traverse but not end. These changes to the tracking domain have two important effects on the final tractogram. First, streamlines entering the tumor will now be excluded, as the tumor is part of the exclude map. Second, streamlines displaced by the tumor are more likely to be identified because the particle filter will backtrack and attempt to find a way around the tumor in the edema. We refer to this method as edema-informed anatomically constrained particle filter tractography (EI-PFT).

It should be noted that common segmentation tools, e.g. FAST (FSL), assume a healthy brain and will therefore misclassify edematous or tumoral tissue as white matter, gray matter, or corticospinal fluid. These misclassifications are of critical importance in our paradigm because they directly influence the tracking domain and thus the final tractogram. For example, considering edematous tissue as gray matter will cause streamlines to prematurely terminate inside the lesion and yield an erroneous connection. Likewise, misclassifying a tumor may allow streamlines to traverse the tumor instead of being displaced by it. To produce the final segmentation, we therefore opted for the fusion of two segmentations. First, the white matter, gray matter, and corticospinal fluid segmentations are based on the partial volumes obtained with FAST (FSL) as it is very reliable in healthy regions away from the lesion. Second, the edema and tumor regions are based on the glioma image segmentation and registration (GLISTR) [START_REF] Gooya | GLISTR: Glioma image segmentation and registration[END_REF] tool which provides accurate tumor and edema segmentation based on tumor growth modeling. To ensure the maps of the 5 labels sum to one, the corrected maps were computed as P * = P * -P edema P * -P tumor P * where P edema and P tumor are GLISTR outputs and P * is the white matter, gray matter, or corticospinal fluid map obtained from FAST.

Data acquisition and preprocessing

For 4 subjects having a tumor near the temporal stem, multi-shell multi-band DWI [TR/TE = 5,216/100 ms, resolution = 2 mm isotropic, 64 diffusion directions at b = 2,000 s/mm 2 , 30 diffusion directions at b = 800 s/mm 2 , 15 diffusion directions at b = 300 s/mm 2 , and 7 b = 0 images] were acquired on an Siemens 3T TrioTim scanner using a 32-channel head coil. In a separate session on a Siemens 3T TrioTim, T1 MPRAGE, T1 contrast-enhanced (0.977 x 0.977 x 1 mm), T2 and FLAIR images were acquired using a 12-channel head coil. The diffusion MRI volumes were corrected for distortion correction and motion correction using eddy (FSL). Structural images were bias corrected with N3 (ITK), smoothed with susan (FSL) and then coregistered to T1 with flirt (FSL), and then all were brought to FA and b0 space with ANTs. The fiber orientation distribution functions were then computed using constrained spherical deconvolution using a spherical harmonic order of 8. FSL FAST was used to segment the T1 image into white matter, gray matter, and corticospinal fluid partial volume maps. GLISTR was used on the T1, T2, T1 contract-enhanced, and FLAIR to segment the lesion into edema and tumor regions. The output of FAST and GLISTR were used to define the include and exclude maps of the anatomically constrained tractography, as described in the previous section. An example of the include and exclude maps obtained for subject 1 are illustrated in Figure 1. Probabilistic particle filter tractography was then used to generate the streamlines by using 1 seed per voxel over the white matter mask. Finally, the streamlines were automatically segmented into bundles using recobundles [START_REF] Garyfallidis | Recognition of white matter bundles using local and global streamline-based registration and clustering[END_REF], which makes use of streamline geometry instead of anatomical landmarks. This streamlines segmentation strategy is critical as brain tumor subjects may have severe deformations which prevent typical landmarks from being identified.

Because our subjects all have a tumor near the temporal stem, we report the results for the inferior longitudinal fasciculus (ILF), uncinate fasciculus (UF), and the inferior fronto-occipital fasciculus (IFOF). These 3 bundles all funnel near the temporal stem and were the most affected by the lesion. To evaluate if EI-PFT is able to traverse edema regions, we also report result obtained by using FA thresholded tractography pipeline. The same fiber orientation distribution functions were used as input to probabilistic tractography where the tracking was stopped when the FA dropped below 0.2.

Results

To quantify our results, we computed the volume of the recovered bundles that intersect the edema mask. To obtain normalized values, we divide this volume by the total volume of the bundle, yielding a bundle fraction in edema. Since the volume of a bundle can be affected by shorter streamlines, we also compute the cortical surface area overlapping with the streamlines that traverse through the edema and reach the cortical surface. This surface area was estimated by selecting the streamlines that intersect the edema and by counting the number of start and end voxels of each streamline that intersect with the cortical surface.

For both the bundle fraction in edema and the cortical surface area of streamlines intersecting the edema, we expect larger values to be obtained by tractography pipelines are able to penetrate edematous regions. Figure 2 illustrates the bundle fraction in edema for each of our bundles of interest. Out of 9 bundles with a non-negligible bundle fraction in edema (above 5%), 8 had a larger fraction using EI-PFT than FA thresholding tractography. These results are notable because the complete paradigm includes an automatic streamline bundling step that removes streamlines that are not detected in bundles. In other words, EI-PFT explores the edematous region more than FA thresholding while still producing streamlines that are classified as part of a bundle. This is further illustrated in Figure 3, where the uncinate fasciculus of subject 1 is illustrated. It is clear that both EI-PFT and FA thresholding tractography produce 'bundle like' outputs, but EI-PFT has a greater intersection with the edema. Figure 4 compares the cortical surface area of streamlines intersecting the edema obtained using EI-PFT and FA thresholding tractography. Out of 10 bundles with a non-negligible area (above 500 mm 2 ), 8 have a larger area when obtained using EI-PFT. This highlights thats EI-PFT explores the edematous tissue while producing cortico-cortical streamlines. Examples of these surfaces are presented in Figure 5. The images were generated by coloring the cortical endpoints of the streamlines by their overlap with the edema mask. It can be observed that EI-PFT yields broader and more intense cortical surface area indicating regions where streamlines overlapping the edema terminate.

Conclusion

The surgical treatment of any intrinsic brain pathology necessitates careful planning to develop a safe trajectory from the cortical surface, through sulci and white matter to reach the tumor boundary while minimizing injury to the white matter tracts. Our evaluation demonstrates that EI-PFT is able to better achieve this in comparison to the FA thresholded tractography that is available in the surgical planning tools, potentially leading to safer resections. Additionally, larger tract volume in the edema, which our paradigm is able to achieve due to tracking through edema, is a precursor to increased margins of resection and hence better patient outcome.

Fig. 2 .

 2 Fig. 2. Comparison of the bundle fraction in edema obtained using EI-PFT or FA threshold tractography. Larger values indicates that a greater proportion of the bundle volume intersects with the edema.

Fig. 3 .

 3 Fig.3. Comparison of the UF of subject 1 (top) and the IFOF of subject 7 (bottom) reconstructed using EI-PFT (left) and FA thresholded tractography (right). For subject 1, the streamlines recovered by EI-PFT penetrate further into the temporal lobe and wrap the tumor. This is consistent with the maps of Figure1where the temporal lobe was isolated by the lowered FA of the edema. For subject 4, the streamlines projecting to the inferior portion of the occipital lobe have a greater intersection with the edema when reconstructed with IE-PFT.

Fig. 4 .

 4 Fig. 4. Comparison of the cortical surface area where streamlines intersecting the edema terminate recovered by EI-PFT and FA thresholding tractography. Larger values indicate that more streamlines traverse the edema and reach the cortex.

Fig. 5 .

 5 Fig.5. The cortical surface area of streamlines intersecting the area obtained using EI-PFT (left) and FA thresholded tractography (right) for the UF of subject 1 (top) and the IFOF of subject 4 (bottom). Warmer color indicates that more streamlines that intersect the edema terminate in that region of the cortex. EI-PFT has more intense and broader areas in the the temporal and frontal poles, indicating areas that may still be functionally active even if they are affected by the lesion.

Acknowledgements

This work was supported by NIH grant R01NS096606 (PI: Verma, Brem) and has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665: CoBCoM).