S Ryan

M Wicklein

A Mouritz

W Riedel

F Schäfer

K Thoma

Theoretical Prediction of Dynamic Composite Material Properties for Hypervelocity Impact Simulations

Keywords: Hypervelocity impact, Orbital debris, Hydrocode, CFRP, Numerical simulation 1

Recent advances in the description of fibre-reinforced polymer composite material behaviour under extreme loading rates provide a significant extension in capabilities for numerical simulation of hypervelocity impact on composite satellite structures. Given the complexity of the material model, extensive material characterisation is required, however, as the properties of composite materials are commonly tailored for a specific application, experimental characterisation is not efficient, particularly in preliminary design phases. As such, a procedure is outlined in this paper that applies a number of commonly accepted composite mechanics and shock physics theories in conjunction with generalised material properties which allows for the theoretical derivation of a complete material data set for utilisation of the new modelling capabilities. The derivation procedure has been applied to a carbon fibre/epoxy laminate, and is validated through a comparison of derived material properties with experimentally-characterised values and numerical simulation of damage induced by hypervelocity impact on a representative space debris shielding configuration employing the CFRP laminate. For the specific structures and impact conditions considered, application of the material property derivation procedure in place of experimental characterisation provided comparable accuracy in the prediction of damage induced by particles impacting at hypervelocity.

T P I R C S U N A M D E T P E C C A Introduction
Since 1957 over 4,000 successful launches by the international space community have deployed in excess of 17,000 objects in orbit about the Earth [START_REF] Klinkrad | Space Debris -Models and Risk Analysis[END_REF]. Ranging from derelict spacecraft and launch vehicle upper stages to paint flakes and solid rocket motor effluents,

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these launches have generated a current man-made debris population consisting of approximately 10,000 objects larger than 10 cm, 100,000 objects between 1 and 10 cm, and tens of millions of particles smaller than 1 cm. Although efforts are being undertaken to mitigate the generation of additional space debris (e.g. [START_REF]Inter-Agency Space Debris Coordination Committee. IADC Space Debris Mitigation Guidelines[END_REF]), the pollution of key altitudes has already (in some cases) exceeded a critical density, i.e. even if no new launches were conducted, the number of debris objects will continue to uncontrollably increase as a result of collisions between existing debris bodies [START_REF] Liou | Instability of the Present LEO Satellite Populations[END_REF]. As such, the threat to operational spacecraft posed by the impact of space debris particles at hypervelocity continues to increase. Due to their potentially high speeds (e.g. from 11 to 72 km/s for meteoroids) and collision energies, the impact of space debris can be critical to mission safety. However, assessing the risk is difficult as only approximately 40% of impact conditions relevant for low-Earth orbiting (LEO) spacecraft can be experimentally reproduced. Numerical hydrocodes are commonly used in the support of experimental risk analysis programs as they are ideal for simulation of these highly dynamic, shockwave-inducing events (e.g. [START_REF] Hayhurst | Ballistic Limit Evaluation of Advanced Shielding Using Numerical Simulations[END_REF] [START_REF] Ryan | An Excitation Function for Hypervelocity Impact-Induced Wave Propagation in Satellite Structures, 57 th IAC[END_REF]). Furthermore, recent advances [START_REF] Clegg | Hypervelocity impact damage prediction in composites Part I -Material model and characterisation[END_REF] [START_REF] Riedel | Hypervelocity impact damage in composites Part II -Experimental investigation and simulations[END_REF] in the modelling of composite materials provide an improvement over existing capabilities by allowing the macromechanical description of orthotropic constitutive behaviour, non-linear equation of state, orthotropic non-linear irreversible strain, and individual material plane interactive failure initiation criteria. Application of these new capabilities in modelling the hypervelocity impact of aluminium fragments on Aramid/epoxy plates [START_REF] Riedel | Hypervelocity impact damage in composites Part II -Experimental investigation and simulations[END_REF] and aluminium projectiles on carbon fibre reinforced polymer (CFRP)/Aluminum honeycomb (HC) sandwich panels (SP) [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF] has shown the ability to reproduce many of the damage phenomena observed in experiments.

In previous applications of the advanced orthotropic composite material model for analysing hypervelocity impact events (e.g. [START_REF] Riedel | Hypervelocity impact damage in composites Part II -Experimental investigation and simulations[END_REF], [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]), material data has been determined through experimental characterisation programs. As the directional performance of a fibre-reinforced polymer can be altered through composition, orientation of the constituents, and fibre packing geometry, full experimental characterisation is required for each new material configuration.

Subsequently, the time and expense involved in determination of accurate material data sets for application of the advanced composite material model can be prohibitive.

In this paper an analytical procedure is outlined that combines a number of shock physics and composite mechanics theories to derive a full set of material data for a space-representative

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CFRP laminate. This procedure enables application of the advanced composite model in the simulation of hypervelocity impact on composite materials when experimental material characterisation is not viable. Although not intended to fully replace the need for experimental characterisation, the analysis procedure provides a means of performing material sensitivity studies in the first phase of satellite structural design that allows the quick and cost-effective comparison of the debris shielding performance of competing material configurations.

Advanced Orthotropic Composite Continuum Material Model

A new material model for composite materials was derived by Riedel and Clegg et al. [START_REF] Clegg | Hypervelocity impact damage prediction in composites Part I -Material model and characterisation[END_REF] [7] to address material anisotropy, shock response and anisotropic strength degradation (damage) under impact loading. Key aspects of the model are reviewed herein.

Modelling basic orthotropic stiffness and shock response

A shock wave can be defined as a discontinuity in pressure, P, temperature (or internal energy, E) and density, ρ. Therefore, in the analysis of shock wave propagation within a continuum, and its effect on the continuum properties, equations are developed by considering regions immediately ahead (reference condition indicated by the subscript 0) and behind an infinitely thin shock. The conservation laws across a shock front were originally defined by Rankine and Hugoniot for fluids, and are defined as:

Conservation of mass:

( ) p s s u U U - = ρ ρ 0 (1)
Conservation of momentum:

( ) p s u U p p 0 0 ρ = - (2)
Conservation of energy:

( )( ) V V p p E E - + = - 0 0 0 2 1 (3)
where:

ρ 1 = V U s -shock velocity u p -particle velocity
The Equation of State (EOS) is required to solve the conservation equations. For isotropic solid continua, the EOS is commonly expressed in the Mie-Grüneisen form:

( ) ( ) ( ) [ ] v E E v v v p p r r - Γ + = (4) T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 4
where Γ is the Grüneisen gamma:

( ) v E p v v ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = Γ
The functions p r (v) and E r (v) refer to the internal pressure and energy of the compressed material in terms of volume, and are generally known functions on some reference curve. For example, with the Hugoniot as reference:

( )

H H E E p p - Γ = - ρ (5)
The Mie-Grüneisen formulation of the hydrodynamic EOS describes the thermodynamic response of a material under shock loading (i.e. the material behaves like a gas or fluid, having no strength). During hypervelocity impact, pressures are generated that can exceed the strengths of impacting materials (projectile, target) by orders of magnitude [START_REF] Lambert | The Validation of Hydrocodes for Orbital Debris Impact Simulation[END_REF], thus the materials are effectively behaving hydrodynamically. However, there exist two regimes -far field and late time -where strength effects may be important. As the impact-induced stresses decrease with distance from the impact location, there may be far-field regions in the target where local stress levels are of the order of material strength or lower. Furthermore, as the projectile is slowed within the target the impact-induced stresses will also decrease. If the target is sufficiently thick, the stresses may decrease to the level of magnitude (or lower) of the material strength.

Description of the strength effects under shock loading is referred to as constitutive modelling.

Historically, the thermodynamic (EOS) response of a material and its ability to carry shear loads (strength) were dealt with separately by partitioning the strain into volumetric and deviatoric components. However, for anisotropic materials this approach is unsuitable, as (from [START_REF] Anderson | A Constitutive Formulation for Anisotropic Materials Suitable for Wave Propagation Computer Programs -II[END_REF]):

• Strain is not uniform in all three (principal) directions under hydrostatic pressure, and;

• Deviatoric strain will produce volumetric dilation since the coefficients in the stress/strain relation are not all equal.

A new formulation of the EOS for composite materials that provides a coupled deviatoric and volumetric response was derived by Anderson et al. [START_REF] Anderson | A Constitutive Formulation for Anisotropic Materials Suitable for Wave Propagation Computer Programs -II[END_REF] and takes the form:

( ) [ ] [ ] [ ] d d d H H C C C C C C C C C E E p p
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The volumetric portion of Eq. ( 6) can be expressed by a polynomial formulation of the Mie-Grüneisen EOS:

( )

2 3 1 2 3 0 1 0 p K K K B B E μ μ μ μ ρ = + + + + (7)
where µ -volumetric strain = 1 0 ρ ρ K i and B i are material constants which can be defined separately for compression and expansion, i.e.:

For μ > 0 (compression):

( ) E B B A A A p 0 1 0 3 3 2 2 1 ρ μ μ μ μ + + + + = (8)
For μ < 0 (tension):

E B T T p 0 0 2 2 1 ρ μ μ + + = (9)
The polynomial formulation of μ in Eq. ( 8)-( 9) is an alternate means of describing the Hugoniot pressure, p H . The material constants K i must then be correlated with the Hugoniot parameters in order to approximate the values of the shocked material parameters. In the case of a material with a linear shock velocity-particle velocity relationship (i.e. Hugoniot curve in the U s -u p plane), the parameters K i of the polynomial relationship can be calculated using:

2 1 0 B K c ρ = (10)
( )

2 2 0 1 1 B K c S S ρ = + - ⎡ ⎤ ⎣ ⎦ (11)
( ) ( )

2 2 3 0 2 1 3 1 B K c S S ρ ⎡ ⎤ = -+ - ⎣ ⎦ (12)
For an isotropic Hookean material, Eq. ( 10) is equivalent to a linear EOS, where K 1 represents the bulk modulus of the material. For an anisotropic material, to ensure consistency in the elastic regime, the bulk modulus is considered as an "effective" or "average" bulk modulus K 1 '

and is defined from the orthotropic stiffness matrix [C]:

( ) [ ] 23 13 12 33 22 11 1 2 9 1 C C C C C C A + + + + + = ′ (13)
The second part of the polynomial expression (Eq. ( 7)) is associated with shock conditions located off the Hugoniot curve and is formulated using two parameters to distinguish the density dependence of the Grüneisen gamma, Γ, where:

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 6 0 1 1 B Bμ μ + Γ = + (14)
and The nine a ij coefficients are referred to as plasticity coefficients, and are used to describe the amount of anisotropy in plasticity, while k is a state variable defining the instantaneous value of the limit surface. In the quadratic yield function a so-called master effective stress-effective strain curve is defined. The nine plasticity coefficients are then capable of collapsing specific loading conditions onto the master curve, and thus the plasticity of an orthotropic laminate in any 3D loading case can be determined.

Γ = constant for B 0 = B 1 Γρ = constant for B 1 = 0 Γ is linear in ρ for B 0 ≠ B 1 ≠ 0 2.

Modelling failure initiation

For impact loading on reinforced composite materials, the dominant modes of damage are: delamination caused by through thickness strain, shear strain causing matrix cracking, and fibre failure. These damage modes lead to a reduction in the load carrying capability of the material, which is referred to as softening. To account for the anisotropic nature of these damage modes, modified forms of the well-known Hashin criteria which include out-of-plane shear stresses for fibre failure and matrix cracking are used to evaluate failure initiation [START_REF] Hou | Prediction of impact damage in composite plates[END_REF]:

2 2 2 2 , , , 1 ij ii ki iif ii u ij u ki u e σ σ σ σ σ σ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = + + ≥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (16)
NOTE: Initiation criteria only applied in tension

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If one of the failure criteria are exceeded (

1 2 > iif e
), the orthotropic post-failure softening model is activated.

Modelling non-linear softening

When the state of stress is calculated to lie outside the limit surface defined by Eq. ( 16) an inelastic strain increment is accumulated as crack strain, ε cr . The maximum stress that can be sustained by the laminate is reduced as a function of crack strain (damage), according to:

( )

1 u f a i l Dam σ σ = - (17)
where

2 2 cr ij ij ij ij f ij ij L F Dam G F ε ⎛ ⎞⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ and 2 ij
F is the initial failure stress in the three material directions (kPa)

Composite Continuum Material Model

An experimental program (described in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]) was used to characterise the properties of a spacerepresentative CFRP laminate for implementation in the advanced orthotropic composite material model. Details of the laminate are given in Table 1. In the following section, the theoretical procedure for derivation of CFRP dynamic properties is applied to this laminate. Although no single elastic or strength theory receives universal application, the accuracy of

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8 micromechanics for predicting the elastic properties of composite lamina is widely accepted. To identify the elastic micromechanics theory best suited for predicting the properties of spacegrade CFRP composite laminates, six of the most common theories are subject to a critical review: mechanics of materials [START_REF] Chamis | Simplified Composite Micromechanics Equations for Strength, Fracture Toughness and Environmental Effects[END_REF]; semi-empirical approach [START_REF] Halpin | Effects of Environmental Factors on Composite Materials[END_REF]; self-consistent field [START_REF] Hermans | The Elastic Properties of Fibre Reinforced Materials When the Fibres are Aligned[END_REF]; equivalent inclusion method [START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF]; method of cells [START_REF] Aboudi | Mechanics of Composite materials. A Unified Micromechanical Approach[END_REF], and; bridging method [START_REF] Huang | A Bridging Model Prediction of the Ultimate Strength of Composite Laminates Subjected to Biaxial Loads[END_REF].

The critical review is performed by evaluating the predictions of the six theories for three freely available carbon fibre/epoxy composite material data sets: IM6/3501-6 (V f = 63.5%) [START_REF] Anon | Properties for Graphite/Epoxy (IM-6/3501-6)[END_REF], T300/BSL914C (V f = 60%) [START_REF] Soden | Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates[END_REF], and AS4/3501-6 (V f = 60%) [START_REF] Soden | Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates[END_REF]. The variance of the micromechanics predictions from the measured values is shown in Figure 1. For the longitudinal tensile modulus (E 11 ) and major Poisson's ratio (ν 12 ) the various methods (with the exception of Method of Cells) all predict an effectively identical result which is within 10% of the reported value for the three composites. It appears that the predictive accuracy of the six micromechanics theories is dependent on the material data sets. For example, the transverse tensile modulii (E 22 ) of AS4/3501-6 and T300/BSL914C are significantly under-predicted, however for the IM6/3501-6 material the predictions are reasonably accurate.

The accuracy of the six micromechanics theories for predicting the elastic properties of the three carbon/epoxy composites considered is assessed in two ways: 1) a direct calculation of the average error (%), and; 2) a Rule of Measure (ROM) which ranks the theories according to their prediction for each elastic property of each material. In each case, the theories are ranked from best to worst and assigned a score ranging from 1 (best) to 6 (worse). A summary of the accuracy evaluation is given in Table 2. With both evaluation methods Huang's bridging method provides the best overall agreement with the reported quasi-static composite material propeties.

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Once the elastic properties of the composite material are known, Classical Laminate Theory (CLT) provides the means to determine the elastic behaviour of composite laminates within the framework of the following assumptions:

• Each ply in the laminate is quasi-homogenous and orthotropic;

• The laminate is thin with lateral dimensions much larger than its thickness;

• The laminate is loaded in a state of plane stress (i.e. σ z = τ xz = τ yz = 0);

• There is no transverse shear.

CLT assumes a state of plane stress to enable derivation of the in-plane properties without the need for consideration of the through-thickness behaviour of the laminate. To derive the laminate out-of-plane properties, Hooke's law for an orthotropic composite can be used as it uses a transformation matrix which is a factor of the stress tensor (and therefore considers through-thickness effects), i.e.:

For plane stress (CLT),

[ ] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 6 2 1 σ σ σ σ or in body-coordinate system [ ] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = xy yy xx σ σ σ σ (18)
For two-dimensional stress-tensor (Hooke's law),

[ ] ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 6 5 4 3 2 1 σ σ σ σ σ σ σ or in body-coordinate system [ ] ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = xy xz yz zz yy xx σ σ σ σ σ σ σ (19)
For symmetric laminates no coupling exists between in-plane loading and out-of-plane deformation, i.e. no flexural/twisting moments are produced under in-plane loading. In this case the laminate stiffness matrix, C, can be determined simply by summing the contributions of each ply in terms of their respective thickness, i.e.:

∑ = × = n k lam k ply k ij ij t t C C 1 , , (20)
where k -individual ply index t ply,k -thickness of ply k n -total number of plies t lam -total thickness of the laminate For the UMS2526/Krempel BD laminate defined in Table 1, the orthotropic stiffness properties have been calculated using constituent properties listed in Table 3 with the micromechanical

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Shock response

The shock response of the CFRP laminate is defined by Eq. ( 6), incorporating a coupled deviatoric and volumetric strain response. The volumetric response (EOS) is expressed using a polynomial form of the Mie-Grüneisen EOS (Eq. ( 7)). In this case, it is assumed that shock response under compression and tension is the same, and therefore A 1 ' = T 1 and A 2 = T 2 .

Flyer plate impact tests on isotopic materials allow shock and particle velocity properties to be deduced, from which the reference curve (generally the Hugoniot) can be defined. However, as a result of the coupled strain response, these tests are not accurate for anisotropic materials and should only be used for validation of complete numerical models via a comparison of rear side velocity-time signals. A theory for the direct derivation of the coefficients in the polynomial

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS
12 equation EOS, coupled with orthotropic response, is currently not available. Nonetheless, flyer plate impact tests on orthotropic materials are commonly used for the definition of the Hugoniot reference curve (e.g. [START_REF] Hiermaier | Advanced Material Model for Hypervelocity Impact Simulations[END_REF], [START_REF] Riedel | Advanced Material Damage Models for Numerical Simulation[END_REF]).

Three different CFRP laminates subjected to flyer plate impact tests (IM7/8552 [START_REF] Riedel | Shock Properties of a HEXCEL IM7/8552 -Experimental Investigations and Material Modelling[END_REF],

T800H/B413M15 [START_REF] Machens | Shock Properties of a Carbon Fibre Composite for Space Applications[END_REF], and UMS2526/Krempel BD [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]) showed a high level of similarity when normalised in terms of the material primary sound speed (c p ) at shock inducing impact velocities, as shown in Figure 2a. For particle velocities less than ~500 m/s, a plastic wave is induced which is expected to be dispersive as a result of plastic damage such as matrix cracking, delamination, etc. Continuous stress measurements have been made during PPI tests on mortar by Kawai et al. [START_REF] Kawai | The Dynamic Behaviour of Mortar Under Impact-Loading, Shock Compression of Condensed Matter[END_REF] which demonstrate the relationship between plastic damage and reduced wavespeed (dispersive plastic phase). It is considered that a similar phenomenon is measured in the CFRP samples, resulting in decreasing shock velocity measurements for increasing particle velocity below the onset of shock wave generation.

The p-wave soundspeed of the three laminates was measured via ultrasound as 3100 m/s, 2900 m/s, and 2800 m/s respectively. A linear best fit is made to the normalised experimental data (considering particle velocities above 500 m/s), shown in Figure 2a. The linear curve is defined as:

( ) p p s c E u E U ⋅ - + ⋅ - = 01 707 . 6 04 242 . 6 (21)
Using the generalised U s -u p relationship (Eq. ( 21)) and a measured p-wave soundspeed of 2800 m/s, the polynomial equation of state parameters for the UMS2526/Krempel BD CFRP laminate are calculated from Eq. ( 10)- [START_REF] Chamis | Simplified Composite Micromechanics Equations for Strength, Fracture Toughness and Environmental Effects[END_REF]. Additionally, the Grüneisen gamma is calculated using the approximation proposed in [START_REF] Dugdale | The Thermal Expansion of Solids[END_REF]:

1 2 1 0 - ≅ Γ = = S B B (22)
A comparison with the experimentally-determined parameters is made in Table 5. The principle differences between the calculated and measured equation of state parameters lie in the quadratic and cubic compression modulii (A2, A3), and the Grüniesen gamma terms (B0, B1).

The experimental values of A2 and A3 were set to zero as it provided the best overall response with all types of planar plate impact tests performed in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]. Hugoniot reference curves defined by the parameters in Table 5 are shown in Figure 2b. The calculated curve shows a non-linear increase with compression, whereas the experimental curve shows a constant linear relationship.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 13

Characterising irreversible, non-linear damage

Similarly to the orthotropic elastic properties, prediction of laminate irreversible plastic damage begins on a micromechanics level. For the prediction of strength, there exist significantly more micromechanical theories than for elasticity; however they are less accurate in general and subsequently less widely accepted. Strength characteristics of composite materials are very sensitive to pre-existing material defects, which must be considered in any strength-based model. In the derivation of material stress, integrated averages are usually expressed for the material volume, smoothing the local stress and strain perturbations. However, in mechanical testing of composite materials, failure initiation often occurs in defective regions, thus their effect is very significant. Two simple strength models are assessed for their accuracy in predicting the strength properties of the three CFRP composites used previously: the strength equations of Chamis [START_REF] Chamis | Simplified Composite Micromechanics Equations for Strength, Fracture Toughness and Environmental Effects[END_REF] and a continuation of the bridging model [START_REF] Huang | A Bridging Model Prediction of the Ultimate Strength of Composite Laminates Subjected to Biaxial Loads[END_REF]. In the following X, Y and S xy refer to strength in the longitudinal, transverse, and in-plane shear directions respectively. The subscripts t and c indicate tensile and compressive strengths, respectively.

The average variance of the micromechanics strength predictions from the measured properties of the three CFRP composites is shown in Figure 3a. The strength of materials predictions provide better agreement with the measured properties that those of the bridging model, with the exception of longitudinal compression strength, X c . Similar to the elastic models, the accuracy of the methods is found to depend on the material data source. In the strength of materials theory, longitudinal compression strength is calculated as the minimum of three

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 14
different possible failure mechanisms: fibre compression, delamination, and microbuckling.

Figure 3b shows, that by ignoring the delamination failure mechanism, agreement of the prediction with the reported composite properties considerably increases. As such, for the prediction of CFRP compression strength this failure mechanism is not considered.

Laminate strength properties are calculated, in general, by the analysis of stress and strains in each individual ply which are checked against ply failure criteria. Once failure in an individual ply has occurred, degradation models are used to re-evaluate the contribution of failed plies to the continuing load performance of the intact laminate. The World Wide Failure Exercise (WWFE) [START_REF] Hinton | Failure criteria in fibre reinforced polymer composites: The World Wide Failure Exercise[END_REF] assessed the maturity and capability of nineteen leading theoretical approaches for predicting the deformation and failure response of polymer composite laminates when subject to complex states of stress. Puck's action plane failure theory [START_REF] Puck | Failure Analysis of FRP Laminates by Means of Physically Based Phenomenological Models[END_REF] was deemed the "winner" of the exercise, based on a systematic comparison with experimental measurements. Puck's failure theory defines two independent fracture criteria to determine lamina strength: fibre failure (FF)

and inter-fibre failure (IFF) within the matrix. The FF criterion is based on a modified version of the maximum stress criteria, in which the effect of stress magnification caused by the differing constituent modulii during biaxial loading is taken into account:

1 1 2 11 12 1 1 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + σ ν ε ε σf f f T m E for (…) ≥ 1 (23) ( ) 1 10 1 2 21 2 11 12 1 1 = + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + γ σ ν ε ε σf f f C m E for (…) < 0 ( 24
)
where m σf is the mean magnification factor of the transverse stress (fibre).

The IFF criterion is based on the Mohr failure criteria. Three IFF modes (A, B and C) were identified by Puck, and are used to generate a so-called master fracture curve, shown in Figure 4. Mode A corresponds to failure under applied transverse tension or in-plane shear; mode B corresponds to a longitudinal shear failure under high applied shear stress and moderate transverse compression stress; and mode C describes failure that occurs on an inclined fracture plane under a combination of high applied transverse compression and shear stresses. These modes are described mathematically as:

Mode A (σ 2 ≥ 0): (25)
T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 15 ( ) ( ) ( ) 1 1 | 2 || 2 21 2 2 || || || 0 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = + ⊥ + ⊥ + ⊥ ⊥ ⊥ °= σ τ σ σ p p R R R f fp E Mode B (σ 2 < 0): (26) ( ) ( ) ( ) 1 1 | 2 || 2 2 || 2 21 || 0 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = - ⊥ - ⊥ ⊥ °= σ σ τ θ p p R f fp E Mode C (σ 2 < 0): (27) ( ) ( ) ( ) ( ) ( ) ( ) 1 4
= - + - ⋅ + ⋅ = - ⊥ - ⊥ ⊥⊥ - ⊥ ⊥ °≠ R R R p R f A E fp σ σ τ θ where ( ) + ⊥ R is the transverse tensile strength of a unidirectional laminate ( ) - ⊥
R is the transverse compressive strength of a unidirectional laminate

( ) + ⊥|| p & ( ) - ⊥||
p are the fracture surface inclination parameters Puck's theory is used to calculate the in-plane stress-strain performance of the composite laminate, using the constants given in Table 6. The onset of non-linear, irreversible damage is defined in the advanced orthotropic composite material model by a quadratic yield function [START_REF] Chen | A Quadratic Yield Function for Fiber-Reinforced Composites[END_REF]. To describe the limit of reversible loading, the yield function requires a master effective stress-effective plastic strain curve and nine plasticity coefficients which define the degree of anisotropy of the hardening surface. For the UMS2526/Krempel BD laminate, the laminate 22-direction is used to define the master curve (thus, the corresponding plasticity parameter a 22 = 1.0). The remaining in-plane plasticity parameters are using Poisson's ratios calculated from the derived in-plane σ-ε curves:

The remaining in-plane and shear plasticity parameters can be defined by simplifying the elastic-plastic stress-strain performance of the laminate under the relative loading conditions to either perfectly elastic (

T P I R C S U N A M D E T P E C C A 2 , 3 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ULT ii L ii a σ σ in unidirectional loading (i = 1,2,3) (29) 2 , 3 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ULT ij L rr a τ σ in shear loading (i ≠ j, r = 4,5,6) (30)
The plasticity coefficients a 13 and a 23 are approximated using idealised assumptions, e.g. under loading in the 22-direction, the laminate Poisson's ratio is adopted for the plastic Poisson's ratio, effectively defining a rate of transverse plastic strain equal to the elastic transverse strain once the 1 st yield condition has been fulfilled. Under loading in the through-thickness direction, minimal plastic strain is expected in the 11-direction due to the high axial stiffness of the fibres, therefore 0 13 ≅ a . The calculated irreversible, non-linear damage properties of the UMS2526/Krempel BD laminate are listed in Table 7, compared to those determined experimentally in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]. The theoreticallyderived master curve shows significantly higher effective stress values than the experimentallydetermined values as a result of the higher predicted directional strength. These higher effective stress values result in higher out-of-plane shear stress plasticity parameters (a 44 and a 55 ) as well as a higher through-thickness plasticity parameter (a 33 ). The effect of these variations can be considered by visually comparing the defined failure surface. However, as the quadratic yield function is six-dimensional in stress space, it is impossible to visualise the described surface. By simplifying the yield function to consider only normal (Eq. ( 31)) or only shear stresses (Eq. ( 32)), a visualization can be made. Figure 5a shows the theoretically-and experimentally-derived limit surface defining the onset of non-linear irreversible strain in normal stress space. Although both surfaces form a closed ellipse, the experimentally-derived surface is significantly more elongated than the theoretical surface. This is due to an assumption in the experimentally-characterised data set of no plastic strain in the material 11-direction, which ensures directional failure occurs prior to the onset of inelastic strain. For the theoretical data set, a small degree of plasticity was predicted.

( )

k
T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 17
The curves shown on the failure surfaces in Figure 5a correspond to cross section plots, an example of which is given in Figure 5b for the theoretical data set at σ 33 = 0.

Characterising failure initiation

In [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF] uniaxial quasi-static tensile tests were used to determine laminate in-plane strength properties. However, for the through-thickness tensile strength, dynamic spall flyer plate impact tests were performed in which the loading rate (impact velocity) varied from 159 m/s to 1073 m/s and an average value, based on the three best spallation signals, was defined (shown in Figure 6).

This average dynamic tensile strength is significantly higher than what would be expected for a quasi-static though thickness tensile test. It is considered that the through-thickness tensile strength is limited by the allowable tensile strain in the matrix, which varies dependent on loading rate (see e.g. [START_REF] Gilat | Strain Rate Sensitivity of Epoxy Resin in Tensile and Shear Loading[END_REF]), resulting in the varying spallation strengths measured in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]. Unless dynamic loading conditions equivalent to those induced during hypervelocity impact can be generated, the definition of dynamic tensile strength in the through-thickness direction is somewhat arbitrary. For the theoretical derivation, the through-thickness tensile strength is estimated as a quasi-static property, based on the methodology for calculating the transverse

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 18
tensile strength of a unidirectional laminate in [START_REF] Chamis | Micromechanics Strength Theories[END_REF]. A maximum tensile strain of 2% is adopted for the Krempel BD epoxy resin.

33 max 33 E m ⋅ = ε σ (33)
In [START_REF] Chamis | Simplified Composite Micromechanics Equations for Strength, Fracture Toughness and Environmental Effects[END_REF] micromechanics equations for calculating the longitudinal and transverse interlaminar shear strengths of unidirectional laminates are presented:

Longitudinal: 3 1 ( ) m f m f f S G G V V S ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - - = 12 13 1 1 (34) Transverse: 3 2 ( ) (
)

m f m f f m f S G G V G G V S ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - - - - = 23 23 23 1 1 1 1 (35)
A simple scaling between the two limits is made, where a scaling parameter (χ) is defined based on the alignment of the individual layers in the composite laminate. The transverse shear strengths (S 23 , S 31 ) are then estimated using: The failure properties calculated for the UMS2526/Krempel BD laminate are given in Table 8, compared to the experimentally-measured values. Significant variation is noted for many of the failure properties, particularly the tensile failure stress in the 22-direction. To a degree this is to be expected as (noted previously) the theoretical calculation of composite strength properties is subject to large variation due to the dependence of composite material strength on the presence of manufacturing defects. As such, although the applied theories have been selected based on their accuracy for similar materials, a significant degree of variation from experimentallymeasured values is to be expected. With the exception of σ 22 (223%), and τ 23 (169%), all strength predictions are within ± 50% of the experimental values. In the case of hypervelocity impact, in which impact pressures exceed material strengths by orders of magnitude, this is not expected to have a significant effect on the overall result of an impact simulation (i.e. structural perforation). However, the lateral extension of damage is expected to be adversely effected.

( ) ij trans ij long ij S S S χ χ - + = 1 (36) where ∑
T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 19

Characterising non-linear softening

CFRP laminates usually fail in a brittle-type fracture mode, regardless of load direction, thus post failure "softening" is not applicable. However, in a macromechanic model the post-fracture softening must also incorporate the effects of geometric fracture. Under in-plane tensile loading CFRP laminates exhibit brittle failure, thus:

2 22 11 / 0 m J G G f f = = (37)
Tensile loading in the through-thickness direction is a matrix-dominated failure mode.

Experimentally, this fracture energy is determined in a double-cantilever beam (DCB) test. It has been shown in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF] that under this loading type, even for highly brittle epoxy resins, fracture is not instantaneous and therefore G f33 ≠ 0. For mode-I delamination of the laminate, the fracture energy of the epoxy resin is adopted, thus:

ii mf ii f G G , , 33 = (38)
Under in-plane shear loading, the fracture of a multi-directional CFRP composite is similarly brittle to that observed under uni-axial tensile loading. In Figure 7 the results of uni-axial tensile tests performed on a bias [±45°] CFRP laminate to simulate an in-plane shear load are shown.

It is seen that upon reaching the laminate failure stress, the load-carrying capability of the laminate diminishes immediately to zero. Thus, it is reasonable to assume:

2 12 / 0 m J G f = (39)
For transverse shear failure (mode II) a large databank of delamination fracture toughness values for laminates was compiled in [START_REF] O'brien | Composite Interlaminar Shear Fracture Toughness, G IIc : Shear Measurement of Sheer Myth?[END_REF]. The ratio of mode II to mode I fracture toughness for polymer composites is shown in Figure 8. Although there is a large degree of scatter in the

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 20
polynomial fit, it is considered to be a reasonable approximation for the transverse (i.e. loading in the through-thickness direction) mode II fracture toughness. The polynomial fit is defined as:

α - ⋅ ′ = Ic Ic IIc G A G G (40)
where A' = 59.244 α = 0.4811

The calculated post-failure, non-linear softening properties of the UMS2526/Krempel BD laminate are shown in Table 8, compared to those determined experimentally in [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]. Generally, a good degree of agreement is found. The experimentally-derived data set shows an increased fracture energy for out-of-plane shearing 23 than 31 (derived from ENF tests), probably as a result of the decreased fibre alignment in the 22-direction. The theoretical assumption, based on a relationship between mode I and mode II fracture energies of CFRP composites is unable to reproduce this difference.

Application of the Material Model for Simulation of Space Debris Impact

Two hypervelocity impact experiments have been performed on EMI's two-stage light-gas guns to assess the effect of material data on the accuracy of numerical simulation damage predictions and the applicability of the theoretical derivation procedure.

Target set-up

Two experimental set-ups using the UMS2526/Krempel BD laminate were considered: (1) a CFRP/Al honeycomb sandwich panel with 1.37 mm thick CFRP facesheets and a 25mm thick Al honeycomb core, and (2) two 1.37 mm thick CFRP laminates with 25mm spacing between them. Target 1 was subject to oblique (60°) impact of a 1.775 mm Al-sphere at 6.095 km/s (EMI no. 4848), and target 2 was subject to normal impact (perpendicular to the target surface) of a 1.179 mm Al-sphere at 4.935 km/s (EMI no. 236). The two target configurations are shown in Figure 9.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 21
In Figure 10 a comparison between the experimentally-induced (both visual and ultrasonic) and numerically-predicted damage in on Target 1 (EMI no. 4848) is shown. On the experimental target a roughly circular perforation is observable on the front facesheet, with a number of small cracks are barely visible on the rear facesheet. The ultrasonic scan reveals more significant internal damage (delamination). The honeycomb core shows a damage cone biased to one side due to the oblique impact velocity vector (to right of page). The numerical simulation applying the experimentally-characterised data set from [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF] shows a near-triangular shaped perforation hole in the front facesheet and two separate areas of damage are observable on the rear facesheet: one large perforation hole corresponding to debris ejected normally from the front facesheet, and a second smaller perforation biased in the direction of the impact velocity vector.

The damage to the honeycomb core is significantly less than that observed in the experiment.

The numerical simulation applying the data set derived from the theoretical procedure also For the second impact test (EMI no. 236), damage induced by the normal impact of a 1.179 mm Al-sphere at 4.935 km/s target 2 is shown in Figure 11 compared to the simulated damages using both the experimentally-and numerically-characterised data sets. The experimental result shows a clear perforation hole in the front facesheet and a small amount of cracking on the rear

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 22
facesheet. Ultrasonic scans of the rear facesheet show internal damage (delamination). Both numerical models show similar predictions, predicting a circular perforation hole in the front facesheet and no noticeable damage on the back of the rear facesheet. As with simulation of the oblique impact experiment, the numerical simulation using the experimentally-characterised data set shows significantly more delamination in the front facesheet than the theoreticallycharacterised model. This represents the major difference between the two models in predicting the damage induced by impact of a space debris particle at hypervelocity.

In Table 10 the damage measurements made from the experiment and numerical simulations are presented. As shown in Figure 10 and Figure 11, the agreement between the experimental damages and numerical predictions are quantitatively acceptable with the exception of predicted perforation of the sandwich panel rear facesheet in EMI experiment 4848 which is considered to be a result of excessively channelling of projectile fragments by the overly-strong honeycomb core cell walls. The material EOS is defined as a polynomial formulation of the Mie-Grüneisen EOS, which uses an effective bulk modulus calculated from the stiffness matrix and polynomial terms derived

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 23
from a general EOS defined for CFRP laminates. The post-failure performance of the laminate is described in terms of a damage algorithm that requires directional fracture energies determined from approximated properties and a mode I:mode II fracture energy relationship.

A carbon fibre/epoxy composite laminate representative of those used in spacecraft primary structures has been used for verification of the derivation procedure. Towards this end a comparison between theoretically-and experimentally-characterised properties has been made.

The laminate elastic properties have been satisfactorily predicted to within ± 30% of the experimentally-measured values, however the strength parameters (particularly in the material 22-direction) were, in general, over-predicted. For the equation of state, differing assumptions by the authors from those used in definition of the experimental data set (from [START_REF] Wicklein | Hypervelocity Impact on CFRP: Testing, Material Modelling, and Numerical Simulation[END_REF]) led to differences in the polynomial terms and the degree of anisotropy in the hardening surface.

Numerical simulations of hypervelocity experiments on spaced CFRP dual-wall targets and honeycomb sandwich panels with CFRP facesheets were performed with the experimentallyand theoretically-characterised material data sets. In a comparison with experimental measurements, it was found that both numerical models were qualitatively able to reproduce impact processes and damages. For normal impact on a spaced dual-wall configuration slight cracking and internal delamination of the rear wall was observed in the experiment. Both numerical models showed no damage on the back face of the rear wall. For oblique impact on a CFRP/Al honeycomb sandwich panel small cracks and internal delamination were observed on the rear CFRP facesheet in the experiment. Both numerical models predicted a clear perforation result, however this over-prediction of damage may be the effect of overly-capable honeycomb core foils, which act to channel the fragment cloud. Quantitatively, the numerical simulations using the theoretically-characterized material data set were able to reproduce front facesheet damage measurements to within ± 50% of the experimental results. Rear facesheet damage extension was generally underpredicted, while penetration was conservateively predicted (however, this is influenced by the honeycomb core model). The numerical model incorporating the experimentally-characterized data set showed a slight improvement in front facesheet damage prediction (agreement to within ±30%) while rear facesheet predictions provided comparable accuracy with the theoretically-characterized material data set. This suggests that for this particular case the theoretical characterisation procedure provided a very reasonable

  ε -deviatoric strain component C ij -terms of the orthotropic stiffness matrix

G

  is the fracture energy for each mode of failure (J) ij L is the characteristic cell dimension in the direction of failure 3 Derivation of Material Parameters for Application in the Advanced Orthotropic

  Angle of ply relative to load plane t k -Thickness of ply k t -Thickness of laminate

  shows a non-circular perforation hole in the front facesheet (although dimensions of the perforation hole are comparable with the experiment) and two separate perforation holes in the rear facesheet. The perforation hole dimensions and locations are similar to those in the experimentally-characterised numerical simulation.In general, a comparison of the predicted damage in the two numerical models shows little difference. The experimentally-characterised data set predicts larger delamination in the front CFRP facesheet, agreeing more closely to the damage seen in the ultrasound image.Conversely, the honeycomb core shows considerably more damage in the theoreticallycharacterised model than the experimentally-characterised numerical model, however it is still much less than that observed in the experiment. The reduced extension of damage in the numerical model honeycomb cores suggests the Al-foils of the honeycomb may have been modelled overly strong, restricting expansion of the fragment cloud and channelling the debris to a more localised area on the rear facesheet (which could justify the increased rear facesheet penetration damage prediction in the numerical model).

Figure 1

 1 Figure 1Accuracy of various micromechanics methods for predicting the elastic properties of AS4/3501-6 (Vf = 60%), T300/BSL914C (Vf = 60%), and IM6/3501-6 (Vf = 63.5%). Legend shown in E11 plot, in all cases predictions presented in order of (from left to right): Eshelby, Halpin-Tsai, Method of Cells, Bridging Model, Mechanics of Materials, Self-Consistent Field.

  Figure 1

Figure 11

 11 Figure 3a

2 Modelling irreversible, non-linear damage

	Chen et al. [11] derived a three-dimensional (3D) quadratic yield stress function which relaxes
	the constant pressure assumption of the Tsai-Hill anisotropic yield criterion, allowing a change
	in volume (and thus pressure) contribution from deviatoric strain. The yield function includes a
	hardening parameter to describe increasing strength with increasing plastic deformation. The
	quadratic yield function is expressed as:
	( ) f ij σ	=	a 11 σ	2 11	+	a 22 σ	a + 13 22 2 2 σ a 11 33 σ σ	33 2 33	+ +	a a 44 12 2 2 σ σ	a + 55 2 22 + 2 σ a σ 23 2 11	a 66 33 σ + 12 2 ... σ 22 + 2 σ 31 2 23	=	k	(15)

3.1 Characterising basic orthotropic stiffness and shock response 3.1.1 Basic orthotropic stiffness

	Table 1	Properties of the CFRP laminate subject to characterisation
	Property		Value / Description
	Fibre type		Tenax UMS2526
	Resin type		Krempel BD System 120°C epoxy resin
	Fibre volume content	52 %
	Lay-up		[0/45/-45/-45/45/0]
	Thickness		1.37 mm
	In the absence of manufacturer-supplied material data, constituent (fibre and matrix) product
	data sheets represent a starting point for derivation of composite laminate mechanical

properties. Micromechanical theories can generally be classified into elastic or strength theories.

Table 2

 2 Evaluating the predictive accuracy of six micromechanics theories for three CFRP materials.

		Ave. % error	ROM total
	Theory	Value Rank	Value	Rank
	Mech. of materials	11.5	3	66	5
	Semi-empirical	11.6	4	63	4
	Self-consistent field	10.9	2	75	6
	Equivalent incl.	11.8	5	61	2
	Method of cells	14.4	6	62	3
	Bridging method	10.2	1	51	1

  10 bridging model, CLT, and Hooke's law. The calculated properties (see Table4) show reasonable agreement with experimentally characterised values, particularly the in-plane parameters. However, given the uncertainties in estimating unknown constituent properties, the agreement is considered good.

Table 3

 3

	Constituent properties for micromechanics
	Property	Value	Source
	Fibre:
	Density (ρf)	1.79 g/cm³	(a)
	Long. tensile mod. (Ef11)	395 GPa	(a)
	Trans. tensile mod. (Ef22)	13.33 GPa	(b)
	Poisson's ratio (νf12)	0.2385	(c)
	Trans. Poisson's ratio (νf23)	0.2981	(c)
	Shear modulus (Gf12)	24.80 GPa	(b)
	Trans. shear modulus (Gf23)	5.80 GPa	(d)
	Long. tensile strength (Xft)	4560 MPa	(a)
	Long. comp. strength (Xfc)	2042 MPa	(c)
	Resin:
	Density (ρm)	1.22 g/cm³	(a)
	Tensile modulus (Em)	3.10 GPa	(c)
	Poisson's ratio (νm)	0.375	(c)
	Shear modulus (Gm)	1.127 GPa	(d)
	Tensile strength (Xmt)	73.82 MPa	(c)
	Compression strength (Xmc)	280.4 MPa	(e)
	Shear strength (Smxy)	86.63 MPa	(c)
	Fracture energy (Gfm)	240 J/m²	(c)
	(a) Manufacturer data sheet	(d) Isotropy / transverse isotropy
	(b) Constituent relationship [21]	(e) 10 % rule [22]
	(c) Common / average property

Table 4

 4 Theoretically and experimentally derived orthotropic stiffness properties of the UMS2526/Krempel BD laminate.

	Parameter	Theory	Exp.
	Reference density [g/cm 3 ]	1.516	1.563
	Young's Modulus 11 [GPa]	79.24	72.90
	Young's Modulus 22 [GPa]	30.25	22.89
	Young's Modulus 33 [GPa]	8.71	9.07
	Poisson's Ratio 12 []	0.84	0.77
	Poisons Ratio 23 []	0.33	0.55
	Poisson's Ratio 31 []	0.0071	0.0187
	Shear Modulus 12 [GPa]	36.32	48.35
	Shear Modulus 23 [GPa]	2.87	0.558
	Shear Modulus 31 [GPa]	3.36	0.873

Table 5

 5 Estimated and experimentally derived equation of state properties of the UMS2526/Krempel BD laminate.

	Parameter	Theory	Experiment
	Bulk Modulus A1 [GPa]	28.24	25.04
	Parameter A2 [GPa]	5.35	0
	Parameter A3 [GPa]	16.97	0
	Parameter B0 []	2.496	1.098
	Parameter B1 []	2.496	1.098
	Parameter T1 [GPa]	28.24	25.04
	Parameter T2 [GPa]	5.35	0

Table 6

 6

	Parameter values for Puck's theory
	Constant	Symbol		Value
	Inclination parameter	( 12 + p	)		0.35
	Inclination parameter	( 12 -p	)		0.3
	Inclination parameter	p	( 22 +	/	-	)	0.3
	Degradation constant		c				4
	Degradation constant		ξ				2
	Remaining stiffness		ηr			0.25

Table 7

 7 Derived strength model properties of the UMS2526/Krempel BD laminate.

	Parameter	Theory	Exp.	Parameter	Theory	Exp.
	A11 []	0.364	0.025	Eff. Stress #1 [MPa]	404.277	120.025
	A22 []	1	1	Eff. Stress #2 [MPa]	411.432	145.745
	A33 []	3.576	0.660	Eff. Stress #3 [MPa]	418.587	168.157
	A12 []	-0.341	-0.1285	Eff. Stress #4 [MPa]	427.416	186.774
	A13 []	0	0	Eff. Stress #5 [MPa]	438.265	199.633
	A23 []	-0.3345	-0.473	Eff. Stress #6 [MPa]	465.098	210.166
	A44 []	23.066	3.157	Eff. Stress #7 [MPa]	491.930	218.984
	A55 []	21.405	2.128	Eff. Stress #8 [MPa]	518.763	227.068
	A66 []	0.388	0.061	Eff. Stress #9 [MPa]	536.651	232.702
				Eff. Stress #10 [MPa]	549.173	238.825
				Eff. Plastic Strain #1 []	0	0
				Eff. Plastic Strain #2 []	2.701E-6	1.7236E-4
				Eff. Plastic Strain #3 []	8.295E-6	3.4473E-4
				Eff. Plastic Strain #4 []	1.697E-5	5.1709E-4
				Eff. Plastic Strain #5 []	3.997E-5	6.8945E-4
				Eff. Plastic Strain #6 []	1.670E-4	8.6140E-4
				Eff. Plastic Strain #7 []	3.676E-4	0.00103
				Eff. Plastic Strain #8 []	6.907E-4	0.00121
				Eff. Plastic Strain #9 []	9.088E-4	0.00138
				Eff. Plastic Strain #10 []	1.070E-3	0.00155

Table 8

 8 Theoretically and experimentally derived failure properties of the UMS2526/Krempel BD laminate.

	Parameter	Theory	Experimental
	Tens. Failure Stress 11 [MPa]	901.177	619
	Tens. Failure Stress 22 [MPa]	448.398	195
	Tens. Failure Stress 33 [MPa]	174.20	245.7
	Max. Shear Stress 12 [MPa]	374.639	280.5
	Max. Shear Stress 23 [MPa]	66.018	39.0
	Max. Shear Stress 31 [MPa]	68.524	47.5

Table 9

 9

	Theoretically and experimentally derived non-linear softening properties of the
	UMS2526/Krempel BD laminate.
	Parameter	Theory	Exp.
	Fracture Energy 11 [J/m 2 ] Fracture Energy 22 [J/m 2 ] Fracture Energy 33 [J/m 2 ] Fracture Energy 12 [J/m 2 ] Fracture Energy 23 [J/m 2 ] Fracture Energy 31 [J/m 2 ]	1E-6 1E-6 240 1E-6 1018 1018	1E-6 1E-6 333.5 1E-6 1378 747

material data set suitable for application in hypervelocity impact simulations used for damage assessment.


Giacomo Germani

Amar Andrew Kenneth Burroughs

Paul Dhillon

Amar P Dhillon
email: a.dhillon@medsch.ucl.ac.uk

Prof Paul Amar

Dhillon

The relationship between liver disease stage and liver fibrosis: a tangled web

Keywords: liver, fibrosis, stage, collagen proportionate area, image analysis

come

The relationship between liver disease stage and liver

fibrosis: a tangled web

subsequent histological "scoring" systems used to describe systematically chronic "hepatitis activity". Knodell et al., [START_REF] Knodell | Formulation and application of a numerical scoring system for assessing histological activity in asymptomatic chronic active hepatitis[END_REF] evaluated the histological features of chronic hepatitis as potential prognostic markers, based on the understanding current at that time of the pathophysiology of chronic aggressive hepatitis. In 1981 aetiologically the literature was mainly about autoimmune hepatitis and HBV infection. Occasionally drug toxicity was identified. Knodell et al. [START_REF] Knodell | Formulation and application of a numerical scoring system for assessing histological activity in asymptomatic chronic active hepatitis[END_REF] studied 14 liver biopsy specimens from five patients with asymptomatic chronic active hepatitis. One patient had HBV and the other four were "presumed to be cases of non-A, non-B chronic hepatitis". When Crawford studied "Citation Classics in Hepatopathology" in 2006, the paper by Knodell et al. [START_REF] Knodell | Formulation and application of a numerical scoring system for assessing histological activity in asymptomatic chronic active hepatitis[END_REF] achieved the highest "number of citations". [START_REF] Crawford | Evidence-based interpretation of liver biopsies[END_REF] Thus, the Knodell ''histological activity index'' system (HAI) was proposed and promulgated. The Knodell system evaluated separately, each of four histopathological axes (periportal ± bridging necrosis, intralobular degeneration/focal necrosis, portal inflammation, and "fibrosis"), and assigned a "numerical" score to each. To enhance the discriminant potential of the system, the difference between mild and serious disease was accentuated by eliminating the "number" (category) 2 from the scoring system for each histopathological axis. By omitting "2", the Knodell system allows a "simple" and practical everyday use of the system, by providing a gap between no/insignificant and appreciable/severe visual change for each axis of assessment i.e. 0/1 vs 3/4, (with some additional options for periportal ± bridging necrosis which was thereby given extra weight, because at that time, it was considered the most important determinant of chronic aggressive hepatitis). The score is therefore discontinuous, which has been considered a drawback by some. This objection is redundant when one considers that in any case the numerical labels are not measurements of a continuous variable. A more pernicious fault was the labelling as "fibrosis" of the axis of assessment which subsequently became known as the disease "stage" (which is a blend of features relating to fibrosis and architectural changes resulting in the effacement of the normal lobular structure of the liver). This confusion has persisted to the present day. In recent times use of categorical labels of scoring systems, as numerical measurements of a continuous variable, has been appreciated as being conceptually incorrect. This realisation will help to avoid misunderstandings between histopathologists and clinicians, as well as statistical errors in handling histopathological data. Similarly, the formal acknowledgement of the essential difference between liver fibrosis and liver disease stage, should result in better understanding, and benefits in framing research, and validation of "non invasive markers" of fibrosis. These considerations are relevant to several issues: Firstly the currently used histological hepatitis scoring systems, which include the Ishak 3 , the METAVIR scoring systems [START_REF]Intraobserver and interobserver variations in liver biopsy interpretation in patients with chronic hepatitis C. The French METAVIR Cooperative Study Group[END_REF] , and other hepatitis scoring systems, use the same principles to describe the "stage" of the liver disease. In all systems the ''stage'' or ''fibrosis'' score comprise a mixture of features. Notably, no individual feature is related specifically to the amount of fibrous tissue in a liver biopsy sample. For example, in the Ishak system the higher scores depend more on the degree of nodularity and on architectural changes, rather than the amount of fibrous tissue.

Secondly these histological staging systems have been misunderstood by many clinicians.

There has been inappropriate use statistically, of the histological stage "scores", which are documented in many publications, including randomized controlled trials. As noted above, histological stage scores describe features that depend on architectural changes and not only on the degree of fibrosis. In addition they are descriptive notations so that, at best, they are ordered categorical data and not numbers. The "number" assigned in the stage score is not a measurement. The "number" is a shorthand label, as explained above, for a morphological description. Thus the scores are categories and not a range of numbers with any known arithmetical relationship to each other ie. stage 2 is not half of stage 4. Therefore only statistical methods appropriate for ordered categorical assignments should be used to evaluate stage "scores". [START_REF] Lagging | Progression of fibrosis in untreated patients with hepatitis C virus infection[END_REF] This particular attribute of the nature of histological scoring has not been understood by many investigators. Virtually none of the extensive literature on the subject, particularly in clinical trials of antiviral therapeutic agents, has avoided this statistical error. What should be done is to convert categorical data into frequency data, in order to evaluate scores in an appropriate statistical manner.

A third issue is that a scoring system devised for one particular liver disease is not necessarily applicable to a different aetiology. Thus, a scheme designed for the assessment of steatohepatitis, for example by Brunt et al. [START_REF] Brunt | Nonalcoholic steatohepatitis: a proposal for grading and staging the histological lesions[END_REF] is not suitable for the assessment of chronic viral hepatitis, for which scoring systems such as the Knodell or Ishak are used, because fibrosis develops and progresses in different ways in these diseases.

The misuse of histopathological scoring systems confounds efforts to understand the natural history of different types of liver disease. Natural history studies, such as rate of accumulation of fibrosis, should be based on specific measurement of fibrosis with long term longitudinal follow up. Clinical outcomes of cohorts of patients with disease of defined aetiology should be studied. Generalisations, extrapolated from ideas derived from a historical and limited understanding of pathophysiology of undefined causes of liver disease, should not be used.

Histopathological measurement of liver fibrosis

One might argue about the definition of liver fibrosis, but one convenient and widely accepted proposal is: [START_REF] Anthony | The morphology of cirrhosis. Recommendations on definition, nomenclature, and classification by a working group sponsored by the World Health Organization[END_REF] Fibrosis is ''the presence of excess collagen due to new fiber formation.'' In order to validate surrogate markers of fibrosis such as serum markers [START_REF] Burroughs | Non-invasive tests for liver fibrosis: encouraging or discouraging results?[END_REF] or liver stiffness measurement (transient elastography), [START_REF] Castera | Non-invasive evaluation of liver fibrosis using transient elastography[END_REF] and to evaluate changes with treatment that affect fibrosis, a precise assessment of the amount of liver fibrosis is required. This can be performed only with a proper measurement of liver collagen.

The amount of collagen can be measured in tissue homogenates using biochemical methods, but this destroys the tissue, without the possibility of obtaining further information from the biopsy, which is unsatisfactory in the assessment of a diagnostic liver biopsy.

Histological evaluation of fibrosis is usually performed using trichrome or reticulin stains.

Each of these stains identify collagen to some extent, but the amount of trichrome or reticulin staining does not necessarily correspond quantitatively with the amount of hepatic collagen. [START_REF] Standish | An appraisal of the histopathological assessment of liver fibrosis[END_REF] The picroSirius red technique has been shown to be more suitable for quantitative assessment.

Sirius red has an affinity for most hepatic collagens, including types I and III [START_REF] Puchtler | Are picro-dye reactions for collagens quantitative? Chemical and histochemical considerations[END_REF][START_REF] Rojkind | The extracellular matrix of the liver[END_REF] ; and the binding correlates with biochemical hydroxyproline assay under standardized laboratory conditions. [START_REF] James | Sirius red histophotometry and spectrophotometry of sections in the assessment of the collagen content of liver tissue and its application in growing rat liver[END_REF] The quantity of bound stain also correlates well with biochemically determined collagen content and morphometrically determined hepatic fibrosis, and the method appears to be reliable and reproducible. [START_REF] Jimenez | Measurement of fibrosis in needle liver biopsies: evaluation of a colorimetric method[END_REF][START_REF] Lopez-De Leon | A simple micromethod for collagen and total protein determination in formalin-fixed paraffin-embedded sections[END_REF] Thus Sirius red staining is the preferred histochemical method when quantifying liver fibrosis, even though the staining may not be stoichiometric. [START_REF] Puchtler | Are picro-dye reactions for collagens quantitative? Chemical and histochemical considerations[END_REF] The routine histopathological analysis of liver biopsies is not compromised by computerassisted digital image analysis of Sirius red stained histological sections; this additional analysis can be performed simply, to measure liver biopsy fibrosis specifically. The evaluation is based on segmentation of digital images to measure the area of collagen, as well as the area of tissue, producing a ''fibrosis ratio'' or proportion of the area of the biopsy occupied by collagen (collagen proportionate area, CPA) [START_REF] Bedossa | Sampling variability of liver fibrosis in chronic hepatitis C[END_REF][START_REF] Calvaruso | Computer-assisted image analysis of liver collagen: relationship to Ishak scoring and hepatic venous pressure gradient[END_REF] . With this technique structural collagen (e.g. blood vessel wall, large septal portal tracts), which is irrelevant to disease processes and other areas that confound analyses (e.g. subcapsular fibrous tissue which often exaggerates the degree of fibrosis compared to deeper parts of a diseased liver) are edited from the measurement. [START_REF] Standish | An appraisal of the histopathological assessment of liver fibrosis[END_REF] Image analysis studies have evaluated fibrosis in many liver conditions [START_REF] Jimenez | Measurement of fibrosis in needle liver biopsies: evaluation of a colorimetric method[END_REF][START_REF] Bedossa | Sampling variability of liver fibrosis in chronic hepatitis C[END_REF][START_REF] Calvaruso | Computer-assisted image analysis of liver collagen: relationship to Ishak scoring and hepatic venous pressure gradient[END_REF][START_REF] Duchatelle | Changes in liver fibrosis at the end of alpha interferon therapy and 6 to 18 months later in patients with chronic hepatitis C: quantitative assessment by a morphometric method[END_REF][START_REF] Kage | Long-term evolution of fibrosis from chronic hepatitis to cirrhosis in patients with hepatitis C: morphometric analysis of repeated biopsies[END_REF][START_REF] Nohlgard | Liver fibrosis quantified by image analysis in methotrexate-treated patients with psoriasis[END_REF][START_REF] O'brien | An assessment of digital image analysis to measure fibrosis in liver biopsy specimens of patients with chronic hepatitis C[END_REF][START_REF] Pilette | Histopathological evaluation of liver fibrosis: quantitative image analysis vs semi-quantitative scores. Comparison with serum markers[END_REF] . Published range from 1-7% CPA in normal liver, to 12-36% in cirrhosis. However, the results of these studies are not directly comparable, due to the differences in the technology and methodology used (Table 1). Some of these studies [START_REF] Kage | Long-term evolution of fibrosis from chronic hepatitis to cirrhosis in patients with hepatitis C: morphometric analysis of repeated biopsies[END_REF][START_REF] Pilette | Histopathological evaluation of liver fibrosis: quantitative image analysis vs semi-quantitative scores. Comparison with serum markers[END_REF][START_REF] Manabe | Interferon-alpha 2b therapy reduces liver fibrosis in chronic non-A, non-B hepatitis: a quantitative histological evaluation[END_REF][START_REF] Fontana | Relationship of serum fibrosis markers with liver fibrosis stage and collagen content in patients with advanced chronic hepatitis C[END_REF][START_REF] Hui | Quantitative assessment of fibrosis in liver biopsies from patients with chronic hepatitis B[END_REF][START_REF] Lazzarini | Advances in digital quantification technique enhance discrimination between mild and advanced liver fibrosis in chronic hepatitis C[END_REF][START_REF] Levine | Assessment of fibrosis progression in untreated Irish women with chronic hepatitis C contracted from immunoglobulin anti-D[END_REF][START_REF] Maduli | Evaluation of liver fibrosis in chronic hepatitis C with a computer-assisted morphometric method[END_REF] tried to correlate the image analysis measurements (digital image analysis, DIA) of fibrosis with the categorical stage scores, with different results. Manabe et al. [START_REF] Manabe | Interferon-alpha 2b therapy reduces liver fibrosis in chronic non-A, non-B hepatitis: a quantitative histological evaluation[END_REF] found that CPA did not correlate well with the components of the Knodell scoring system. However, the same group later described a more complex scoring system, which correlated with Sirius red staining on the one hand, and Knodell stage scores on the other. [START_REF] Chevallier | A histological semiquantitative scoring system for evaluation of hepatic fibrosis in needle liver biopsy specimens: comparison with morphometric studies[END_REF] The correlation between semiquantitative scoring system and surface density of total collagen was 0.73 (p<0.00001). In the study by Pilette et al. [START_REF] Pilette | Histopathological evaluation of liver fibrosis: quantitative image analysis vs semi-quantitative scores. Comparison with serum markers[END_REF] there was a good correlation between the semiquantitative fibrosis score (Knodell and Metavir (modified) scoring systems) and the area of fibrosis (r=0.84, p<0.0001). Kage et al., [START_REF] Kage | Long-term evolution of fibrosis from chronic hepatitis to cirrhosis in patients with hepatitis C: morphometric analysis of repeated biopsies[END_REF] evaluating the correlation between CPA and a modified stage score, based on the Desmet and Scheuer systems, found that there was a significant correlation in both hepatitis C and B patients (HCV: r=0 .75, p<0.001; HBV: r=0 .67, p<0.001 on the Spearman rank test). In the study by O'Brien et al. [START_REF] O'brien | An assessment of digital image analysis to measure fibrosis in liver biopsy specimens of patients with chronic hepatitis C[END_REF] a correlation was found between image analysis measurements and Ishak stage score. However it was restricted to biopsy specimens with Ishak stage scores in the range 3 to 6 (Spearman correlation coefficient between stage score and fibrosis ratio was 0.66; p<0.001), whereas among biopsy specimens with Ishak stage scores in the range 0 to 3 it was only 0.14 (p = ns). Although later studies [START_REF] Fontana | Relationship of serum fibrosis markers with liver fibrosis stage and collagen content in patients with advanced chronic hepatitis C[END_REF][START_REF] Hui | Quantitative assessment of fibrosis in liver biopsies from patients with chronic hepatitis B[END_REF][START_REF] Lazzarini | Advances in digital quantification technique enhance discrimination between mild and advanced liver fibrosis in chronic hepatitis C[END_REF][START_REF] Levine | Assessment of fibrosis progression in untreated Irish women with chronic hepatitis C contracted from immunoglobulin anti-D[END_REF] showed good correlations between quantitative morphometric data and Ishak scores of fibrosis, in the study by Maduli et al. [START_REF] Maduli | Evaluation of liver fibrosis in chronic hepatitis C with a computer-assisted morphometric method[END_REF] score. The discrepancies between stage and fibrosis DIA are not surprising once the difference between fibrosis measurement and stage scoring is appreciated (Figure 1). The overlapping of the fibrosis measurements between stage scores should be recognised as a biological reality, that could be informative, representing, as it does, additional information.

Up to now in clinical settings, image analysis has been used infrequently. Four studies have used morphometry to study changes in hepatic fibrosis after alpha interferon therapy. [START_REF] Duchatelle | Changes in liver fibrosis at the end of alpha interferon therapy and 6 to 18 months later in patients with chronic hepatitis C: quantitative assessment by a morphometric method[END_REF][START_REF] Manabe | Interferon-alpha 2b therapy reduces liver fibrosis in chronic non-A, non-B hepatitis: a quantitative histological evaluation[END_REF][START_REF] Caballero | Liver fibrosis assessment with semiquantitative indexes and image analysis quantification in sustained-responder and nonresponder interferon-treated patients with chronic hepatitis C[END_REF][START_REF] Sakaida | Quantitative analysis of liver fibrosis and stellate cell changes in patients with chronic hepatitis C after interferon therapy[END_REF] One study has used morphometry to study disease progression in untreated patients with chronic hepatitis C, [START_REF] Kage | Long-term evolution of fibrosis from chronic hepatitis to cirrhosis in patients with hepatitis C: morphometric analysis of repeated biopsies[END_REF] and one evaluated the fibrosis progression in patients with treatmentrefractory chronic hepatitis C enrolled in a placebo-controlled clinical trial of interferon gamma-1b (IFN-g1b). [START_REF] Goodman | Progression of fibrosis in advanced chronic hepatitis C: evaluation by morphometric image analysis[END_REF] In the study by Goodman et al. [START_REF] Goodman | Progression of fibrosis in advanced chronic hepatitis C: evaluation by morphometric image analysis[END_REF] morphometry was found to be a more sensitive tool than histological staging to demonstrate fibrosis progression, because changes in collagen content between baseline and week 48 were more evident than changes in Ishak score. In the study by Fontana et al., [START_REF] Fontana | Relationship of serum fibrosis markers with liver fibrosis stage and collagen content in patients with advanced chronic hepatitis C[END_REF] the computerized morphometric assessment was evaluated in 386 patients with chronic hepatitis C with an Ishak stage score of at least 3. The relationship between serum markers and quantitative hepatic collagen content was also determined. A significant relationship between hepatic collagen content and the Ishak stage scores was shown (r=48, p<0.0001). There was a weaker relationship with cirrhosis compared to a panel of markers of fibrosis. These studies [START_REF] Fontana | Relationship of serum fibrosis markers with liver fibrosis stage and collagen content in patients with advanced chronic hepatitis C[END_REF][START_REF] Goodman | Progression of fibrosis in advanced chronic hepatitis C: evaluation by morphometric image analysis[END_REF] evaluated patients with either Ishak stage ≥2 32 or Ishak stage ≥3. [START_REF] Fontana | Relationship of serum fibrosis markers with liver fibrosis stage and collagen content in patients with advanced chronic hepatitis C[END_REF] In these studies the image analysis methodology did not express the amount of collagen as a simple proportionate area (the thickness of the histological section (for which a specialized microscope is needed) was used to calculate the quantity of collagen).

When methodology and study population are (more or less) aligned, the data are more coherent. For example, even though Bedossa et al. [START_REF] Bedossa | Sampling variability of liver fibrosis in chronic hepatitis C[END_REF] [START_REF] Calvaruso | Computer-assisted image analysis of liver collagen: relationship to Ishak scoring and hepatic venous pressure gradient[END_REF] used transjugular liver biopsies of post-transplant HCV patients, and Ishak staging, both groups used Sirius red staining and expressed their results as CPA. The range of CPA from normal liver (mean 2% vs. 2.7%) to HCV cirrhosis (25.1% vs.

22.6%) is similar, and as far as the intermediate METAVIR and Ishak stages can be compared, the CPAs appropriate to each is also similar.

Recently Gailhouste et al. [START_REF] Gailhouste | Fibrillar collagen scoring by second harmonic microscopy: a new tool in the assessment of liver fibrosis[END_REF] proposed a new approach to the analysis of liver fibrosis. This process is based on an application of multiphoton microscopy enabling the observation of unstained samples using endogenous sources of nonlinear signals such as Two-Photon Excitation Fluorescence (TPEF) and Second Harmonic Generation (SHG). They applied this method to 119 biopsies from patients with chronic liver disease showing that this approach allows a measurement of fibrosis that correlates with the "Fibrosis"-Metavir (stage) score (rho = 0.75, p<0.0001). They were able to distinguish patients with Metavir stage F0,1 vs. F2,3,4 (AUROC = 0.88, p<0.0001) and F0,1,2,3 vs. F4 (AUROC = 0.89, p<0.0001).

Relationship between liver disease stage, fibrosis and prognosis

Clinical outcome in chronic progressive liver disease has been related most often to hepatic venous pressure gradient (HVPG), which is a good predictor of survival and decompensation in cirrhotic patients. [START_REF] Triantos | Review article: the therapeutic and prognostic benefit of portal pressure reduction in cirrhosis[END_REF][START_REF] Burroughs | Assessment of therapeutic benefit of antiviral therapy in chronic hepatitis C: is hepatic venous pressure gradient a better end point?[END_REF][START_REF] Armonis | Hepatic venous pressure measurement: an old test as a new prognostic marker in cirrhosis?[END_REF][START_REF] Burroughs | HVPG in 2010: optimal measurement is key[END_REF] The clinical application of HVPG measurement has been demonstrated in patients transplanted for HCV liver disease. [START_REF] Samonakis | Hepatic venous pressure gradient to assess fibrosis and its progression after liver transplantation for HCV cirrhosis[END_REF][START_REF] Blasco | Hepatic venous pressure gradient identifies patients at risk of severe hepatitis C recurrence after liver transplantation[END_REF] Also the prognostic value of HVPG has been evaluated in patients with HBV-related cirrhosis treated with lamivudine. [START_REF] Manolakopoulos | Antiviral therapy reduces portal pressure in patients with cirrhosis due to HBeAg-negative chronic hepatitis B and significant portal hypertension[END_REF] One of the first papers which explored the relationship between liver histopathology and HVPG was that of van Leeuwen et al., [START_REF] Van Leeuwen | Wedged hepatic venous pressure recording and venography for the assessment of pre-cirrhotic and cirrhotic liver disease[END_REF] who investigated HVPG and evaluated its relationship to histological features of the stage of chronic liver disease. They evaluated 49 patients, divided into four main groups (near normal liver, chronic active hepatitis, chronic hepatitis in transition to cirrhosis and cirrhosis). A HVPG of more than 5 mmHg was always associated with significant liver disease on liver biopsy. Blasco et al. [START_REF] Blasco | Hepatic venous pressure gradient identifies patients at risk of severe hepatitis C recurrence after liver transplantation[END_REF] explored the diagnostic value of HVPG to identify patients at risk of severe HCV recurrence after liver transplantation. A positive association was seen between the median HVPG and the liver disease stage. Although the presence of significant "fibrosis" (METAVIR F2-F3) 1 year after transplantation predicted clinical decompensation (AUC: 0.80), an HVPG ≥6 mmHg more accurately (AUC: 0.96) identified patients at risk of clinical disease progression. These findings were confirmed by Samonakis et al. [START_REF] Samonakis | Hepatic venous pressure gradient to assess fibrosis and its progression after liver transplantation for HCV cirrhosis[END_REF] in 90 consecutive liver transplanted patients for HCV-related cirrhosis who found that HVPG correlated with Ishak stage (r =0.73, p<0.001).

In another paper [START_REF] Carrion | Serum fibrosis markers identify patients with mild and progressive hepatitis C recurrence after liver transplantation[END_REF] the role of three serum markers (hyaluronic acid, amino-terminal propeptide of type-III procollagen, tissue inhibitor of matrix metalloproteinase type-1) at 3, 6 and 12 months after liver transplantation was examined in 133 liver transplanted patients with HCV recurrence, to evaluate patients at risk of developing significant "fibrosis" and portal hypertension. The AUROC curve of the three serum markers for METAVIR stage F≥2 at 3, 6, and 12 months after transplantation was 0.67, 0.77, and 0.78 respectively. In the group with HVPG ≥ 6mmHg at the same time points, AUROC curves were 0.75, 0.87, and 0.90 respectively, which were significantly better than values obtained with the three previously published scores of "non-invasive makers of liver fibrosis" (AST/ALT ratio [START_REF] Toniutto | Role of AST to platelet ratio index in the detection of liver fibrosis in patients with recurrent hepatitis C after liver transplantation[END_REF] (0.55, 0.57, 0.80), AST/platelet ratio [START_REF] Toniutto | Role of AST to platelet ratio index in the detection of liver fibrosis in patients with recurrent hepatitis C after liver transplantation[END_REF] (0.44, 0.45, 0.45), Benlloch score 44 (0.54, 0.58, 0.71)) in this population.

Vizzuti et al. [START_REF] Vizzutti | Liver stiffness measurement predicts severe portal hypertension in patients with HCV-related cirrhosis[END_REF] compared liver stiffness measurement (LSM) and HVPG in predicting severe portal hypertension in 61 consecutive patients with HCV-related chronic liver disease. A strong relationship between liver stiffness measurement and HVPG measurements was found in the overall population (r = 0.81, P < 0.0001). However, although the correlation was excellent for HVPG values less than 10 or 12mmHg (r = 0.81, P = 0.0003 and r = 0.91, P < 0.0001, F o r P e e r R e v i e w respectively), it was weaker for higher values of HVPG. The AUROC for the prediction of HVPG >10 and >12 mm Hg were 0.99 and 0.92, respectively and at LSM cutoff values of 13.6 kPa and 17.6 kPa, the sensitivity was 97% and 94%, respectively. Carrion et al. [START_REF] Carrion | Transient elastography for diagnosis of advanced fibrosis and portal hypertension in patients with hepatitis C recurrence after liver transplantation[END_REF] evaluated the diagnostic accuracy of LSM to assess the severity of hepatitis C recurrence after liver transplantation in the same cohort studied by Blasco et al.. [START_REF] Blasco | Hepatic venous pressure gradient identifies patients at risk of severe hepatitis C recurrence after liver transplantation[END_REF] They found a close direct correlation between liver stiffness and HVPG (Pearson coefficient 0.84; p<0.001) and the AUC for diagnosis of portal hypertension (HVPG≥6 mmHg) was 0.93. In order to discriminate between rapid and slow "fibrosers", the same author 47 evaluated liver stiffness after liver transplantation in 84 liver transplanted patients with HCV-recurrence. The accuracy of liver stiffness at identifying patients with portal hypertension (HVPG≥6 mmHg) at 1 year after liver transplantation improved over time. The AUROC curves at 3,6,9 and 12 months after liver transplantation identifying patients with portal hypertension, were 0.72, 0.77, 0.80 an 0.92 in the test group (n=50) and 0.58, 0.79, 0.84 and 0.93 in the validation group (n=34) respectively. These latter studies did not evaluate histology.

The relationship between collagen proportionate area (CPA) and HVPG has been studied by Calvaruso et al. [START_REF] Calvaruso | Computer-assisted image analysis of liver collagen: relationship to Ishak scoring and hepatic venous pressure gradient[END_REF] in a cohort of 115 liver transplanted patients with HCV recurrence. CPA had a stronger relationship with HVPG than did Ishak stage and CPA was independently associated, by logistic regression, with HVPG≥6mmHg (odds ratio, 1.206; 95%CI [1.094-1.331]; p<0.001), and HVPG ≥10mmHg -clinically significant portal hypertension (odds ratio, 1.105; 95%CI [1.026-1.191]; p<0.009). CPA measured in 1 year post-transplant biopsies in 96 patients after liver transplantation for HCV was found to predict later clinical decompensation [START_REF] Manousou | Computer asisted imagenalysis of liver collagen at one year biopsy can predict clinical outcome in HCV post LT patients[END_REF] . CPA measurement was highly predictive with good sensitivity (90%) and specificity (97.8%), and was better at this prediction than either Ishak stage or HVPG. CPA and HVPG together at 1 year post transplantation correctly identified all but one patient who decompensated at later time points. The relationships between CPA, LSM, HVPG and Ishak stage have been assessed in 45 HCV transplanted patients. [START_REF] Isgro | Correlation of Ishak stage, transient elastography, HVPG and collagen proportionate area in HCV transplanted patients[END_REF] Univariately CPA, Ishak stage, and LSM were associated with portal hypertension (HVPG ≥6mmHg), whereas multivariately CPA was the only independent factor (OR 1.377,95% CI: 1.137-1.169; p=0.001). CPA had a better association than Ishak stage or LSM, with HVPG ≥6mmHg and HVPG ≥10 mmHg. LSM had a better correlation with CPA than with HVPG. As CPA is a continuous variable measuring only collagen, it represents a better histological index for the validation of LSM and other non-invasive markers of fibrosis, than stage scores.

Histological substaging of cirrhosis by measuring fibrosis

Cirrhosis is defined histologically as "a diffuse process characterized by fibrosis and the conversion of normal liver architecture into structurally abnormal nodules". [START_REF] Anthony | The morphology of cirrhosis. Recommendations on definition, nomenclature, and classification by a working group sponsored by the World Health Organization[END_REF] The word "cirrhosis" has been considered sufficient to describe both a pathological and clinical condition, and to have some prognostic importance, with an implication historically that cirrhosis represents "end stage disease". Nowadays however, the "one-stage description" of cirrhosis has become inadequate, especially considering that better prediction of outcome is needed to provide individualized therapy. [START_REF] Garcia-Tsao | Now there are many (stages) where before there was one: In search of a pathophysiological classification of cirrhosis[END_REF] In compensated cirrhosis patients with an HVPG <10mmHg have a 10% probability of developing clinical decompensation during 4 years of follow-up, compared to 40% of patients with an HVPG ≥10mmHg . [START_REF] Ripoll | Hepatic venous pressure gradient predicts clinical decompensation in patients with compensated cirrhosis[END_REF] In patients transplanted for HCV related cirrhosis the cumulative probability of developing clinical decompensation is 22% during the year following a diagnosis of recurrence of compensated cirrhosis with a probability of developing decompensation of 40% at 5 years. [START_REF] Kalambokis | Clinical outcome of HCV-related graft cirrhosis and prognostic value of hepatic venous pressure gradient[END_REF] Specific histological features of cirrhosis have infrequently been linked to clinical outcomes. Any additional histological characterisation, which permits a There is evidence that both HVPG and liver disease grade and stage correspond with haemodynamic (portal hypertension) and clinical features of chronic liver disease. If additional and better correlation can be established between portal pressure and specific histologic features, such as fibrosis (which contributes to the increased intrahepatic vascular resistance in chronic liver disease), the possibility exists of an improved characterisation of cirrhosis, and of quantifying cirrhosis "severity". In this way, we could re-introduce into the histopathological evaluation of cirrhosis the previously recognised important facet of portal hypertension, but which fell into abeyance with the dominance of the current morphological definition of cirrhosis. [START_REF] Anthony | The morphology of cirrhosis. Recommendations on definition, nomenclature, and classification by a working group sponsored by the World Health Organization[END_REF] In patients with liver cirrhosis simple histological features that are undervalued at present may have important prognostic implications. Kutami et al. [START_REF] Kutami | The Laennec grading system for assessment of hepatic fibrosis: validation by correlation with wedged hepatic vein presure and clinical features[END_REF] reported, in abstract form, a series of 302 transjugular liver biopsies, finding that "fibrosis" scores derived from nodule size, and thickness of fibrous septa correlated with HVPG, ascites, INR, bilirubin and inversely with albumin. With a similar approach, Nagula et al. [START_REF] Nagula | Histological-hemodynamic correlation in cirrhosis-a histological classification of the severity of cirrhosis[END_REF] evaluated 43 patients with biopsy-proven cirrhosis and HVPG measurements within 6 months of the biopsy, showing that septal thickness (p=0.03), small nodularity (p=0.003), loss of portal tracts (p=0.01), inflammation (p=0.04) and alcoholic aetiology (p=0.01) were correlated with the presence of clinically significant portal hypertension ≥10 mmHg. However, small nodules and thick septa were the only parameters independently predictive of clinically significant portal hypertension. Kumar et al. [START_REF] Kumar | Histological subclassification of cirrhosis based on histological-haemodynamic correlation[END_REF] significantly associated with the presence of clinically significant portal hypertension. In these studies, neither the exact dimension of the nodules nor the septal thickness is stated. Even within a limited sample of liver, nodules of different sizes, and septa of different thicknesses are apparent, and although an "average" nodular size/septal thickness can be assigned subjectively, this remains an imprecise evaluation. Nodule size was defined in these studies according to the width of needle biopsy specimens (not specified in the studies), with a small nodule being comparable to the width of the biopsy (a large nodule being considered larger than biopsy width).

In their discussion Nagula et al. [START_REF] Nagula | Histological-hemodynamic correlation in cirrhosis-a histological classification of the severity of cirrhosis[END_REF] state that small nodules are <1.0mm and large nodules are >2mm; and thin septa are <0.2mm while thick septa are >0.4mm.

Viola et al. [START_REF] Viola | Quantitative histological assessment in cirrhosis: septal thickness predicts clinical decompensation[END_REF] (from the same group as Nagula et al. [START_REF] Nagula | Histological-hemodynamic correlation in cirrhosis-a histological classification of the severity of cirrhosis[END_REF] ) defined retrospectively the cut-off between thin and thick septa at 169.01 µm, evaluating 115 patients with cirrhotic biopsies. The median septal width of all fibrous septa per slide was greater in patients who developed decompensation compared to those who remained compensated (212.56 µm vs. 156.59 µm, p<0.001). Patients with thick septa (≥169.01 µm) had a significantly higher probability of decompensating. They found that fibrosis area, defined as the percentage of the total biopsy area, was not related to clinical decompensation, but the data are in abstract form, so is not clear how fibrosis area was assessed.

In any event, regardless of the complexities of variations in septal thickness/nodule size within any given biopsy determined either morphometrically or subjectively, a moments reflection reveals that CPA records the reciprocal balance between nodule size and septal fibrosis: a high CPA will integrate and accurately reflect the overall fibrosis in biopsies with small-ish nodules/thick-ish septa, and conversely a low CPA will characterise biopsies with large-ish nodules/thin-ish septa. In the study by Calvaruso et al. [START_REF] Calvaruso | Liver collagen proportionate area and hepatic venous pressure gradient to sub-classify cirrhosis: proof of concept in HCV infected liver transplant recipients with allograft cirrhosis[END_REF] and HVPG (r= 0.65; p<0.001) each had a statistically significant relationship with CPA. The data suggest strongly that CPA is a useful histological variable that substages cirrhosis with a continuous scale and with good applicability and also predicts clinical outcomes. [START_REF] Germani | Histological subclassification of cirrhosis[END_REF] Fibrosis measurement has implications for the currently fashionable idea that cirrhosis might be reversible. [START_REF] Desmet | Cirrhosis reversal: a duel between dogma and myth[END_REF][START_REF] Perez-Tamayo | Cirrhosis of the liver: a reversible disease?[END_REF] Fibrosis should correspond more to the "fixed" component of portal hypertension than to any haemodynamic component of cirrhosis that might respond to vasodilator treatment. When the aetiological agent can be eliminated and when effective antifibrotic agents are available, if fibrosis is measured properly, regression of fibrosis might be documented, with or without reduction of portal hypertension. To date this has not been done. Regression of the cirrhotic "stage" will require another separate evaluation with particular regard to sampling error, decrease of portal venous pressure, and improvement of clinical aspects of cirrhosis and prognosis. Studies of "cirrhosis regression" have not yet addressed these issues, and all too often have confused apparent regression of fibrosis with regression of cirrhosis stage.

Conclusion

Since liver disease "stage" was called "fibrosis" by Knodell et al. [START_REF] Knodell | Formulation and application of a numerical scoring system for assessing histological activity in asymptomatic chronic active hepatitis[END_REF] there has been conflation of the two terms. In the context of chronic liver disease the terms "fibrosis" and "score" should not be contiguous. Liver disease stage definitions include fibrosis, but liver disease stage depends to a great extent on architectural changes as well. Histopathological assignment of liver disease stage is a different process from the measurement of liver fibrosis. The two processes are complementary and each adds value to the liver biopsy assessment. As they are different assessments, the correlation between stage and fibrosis measurement is imperfect. The range of fibrosis (collagen proportionate area, CPA) seen within each liver disease stage represents biological reality, rather than constituting a confusing overlap between stages. The amount of

F o r P e e r R e v i e w

  the fibrosis ratios observed with morphometry correlated poorly with the Ishak staging

  of cirrhosis (i.e. subclassification or substaging of cirrhosis) would be helpful clinically.

  evaluated 47 patients with biopsy-proven cirrhosis and HVPG measurements. Nodule size, loss of portal tracts and central veins, portal inflammation, periportal inflammation, bile duct proliferation, lobular inflammation, ballooning, fatty change, cholestasis and septal thickness were scored. On multiple ordinal regression analysis, small nodule size (p=0.009) and thick septa (p=0.011) were

  in patients with Ishak stage 5 and 6 disease, clinically significant portal hypertension (OR 1.081; CI: 1.001 -1.166; p = 0.046)

  intrahepatic vascular resistance and portal hypertension as CPA correlates with hepatic venous pressure gradient. Thus CPA could help to refine the prognostic value of each liver disease stage in individual patients. CPA measured at 1 year post transplantation in patients with HCV infection predicts clinical decompensation in later years. CPA stratifies clinically important prognostic groups within the stage category of cirrhosis, and can be used to define clinically relevant cirrhosis "substages". Proper measurement of liver fibrosis is relevant for the evaluation of antifibrotic treatments and "non-invasive markers of liver fibrosis" which hitherto have been "validated" incorrectly with liver biopsy stage scores. It is necessary to understand that liver fibrosis is an important aspect of the development of cirrhosis but they are not exactly the same thing.

Figure 1 (

 1 Figure 1 (reproduced from Standish RA et al. 10 ).

  used large tissue blocks (surgical resection cases from patients with HCV infection, without editing of structural collagen, and METAVIR

	staging), while Calvaruso et al.
	F o
	r
	P
	e e r
	R e v i e w

Table 1 . Studies of CPA liver fibrosis (image analysis) vs. stage of liver disease.

 1 ≡66.5±4.6 ; b ≡76.4±6.5; c ≡125.2±13.9; d ≡152.6±11 µg collagen/mg protein[START_REF] Jimenez | Measurement of fibrosis in needle liver biopsies: evaluation of a colorimetric method[END_REF]

	Author	Year Stain	Magnification; and fields	Section	Number of	Liver disease:	CPA %
				measured	thickness	samples	aetiology and stage	(mean±SD)
	Jimenez 14	1993	Masson's	64x;	4µm	38	(Alcoholic) steatosis	4.07±1.07 a
			trichrome	5 random areas per biopsy			Chronic hepatitis 40	9.06±1.04 b
							(Alcoholic) fibrosis	17.2±3 c
							Cirrhosis (various aetiologies,	24.7±2.7 d
							mainly alcohol)
	Nohlgard 20	1993	picroSirius red	6.5x;	4µm	46	Normal	0.9±0.1
				"Four or more different fields			I (Roenigk stage)	11.4±2.1
				corresponding to an average of 25 mm 2 "			IIIA IIIB IV	9.3±1.4 24.0±4.9 11.8±2.7
	Kage 19	1997	Azan	40x;	4µm		("Desmet & Scheuer stage")	Estimated from Fig1 19
				whole area (biopsies with ≥5		74	1 HBV	3±0.8
				portal tracts)			2	3±0.9
							3	6.5±2
						93	4 1 HCV 2 3 4	7.3±2.2 2.1±1.4 2.5±1 5.1±1.9 8.1±4.2
	Duchatelle 18	1998	picroSirius red	25x;	4µm	64	HCV (Knodell stage)
				whole area (biopsies "at least			0-1	1.86±1.56
				1cm long")			3-4	4.07±3.19
	Pilette 22	1998	picroSirus red	100x;	3µm	232	ALD, HBV, HCV	Estimated from Fig1 22
				whole area (biopsies with 12±5 portal tracts)			(Modified Metavir stage) 0 1 2 3 4 5	7±4 8±4 11±4 19±12 20±8 36±9
	O'Brien 21	2000	Masson's	40x;	5µm	230	HCV (Ishak stage)
			trichrome	whole area (biopsies 10mm long			0	3±0.3
				and ≤3 portal tracts)			1	5.6±0.6
							2	7.0±0.6
							3	6.0±0.5
							4	11.9±0.7
							5	15.3±1
							6	28.6±1.7
	Bedossa 16	2003	picroSirius red	100x;	5µm	17	HCV (Metavir stage)

a

Published on behalf of the British Division of the International Academy of Pathology Histopathology

Published on behalf of the British Division of the International Academy of Pathology Histopathology