P Collet

G Gary

B Lundberg

Noise-corrected Estimation of Complex Modulus in Accord With Causality and Thermodynamics: Application to an Impact test

Keywords: Mechanical properties of materials, constitutive modeling of materials, impact, wave propagation. 3

Background. Methods for estimation of the complex modulus generally produce data from which discrete results can be obtained for a set of frequencies. As these results are normally afflicted by noise, they are not necessarily consistent with the principle of causality and requirements of thermodynamics.

Method of Approach:

A method is established for noise-corrected estimation of the complex modulus, subject to the constraints of causality, positivity of dissipation rate and reality of relaxation function, given a finite set of angular frequencies and corresponding complex moduli obtained experimentally. Noise reduction is achieved by requiring that two self-adjoint matrices formed from the experimental data should be positive semidefinite.

Results:

The method provides a rheological model that corresponds to a specific configuration of springs and dashpots. The poles of the complex modulus on the positive imaginary frequency axis are determined by a subset of parameters obtained as the common positive zeros of certain rational functions, while the remaining parameters are 2 obtained from a least squares fit. If the set of experimental data is sufficiently large, the level of refinement of the rheological model is in accordance with the material behaviour and the quality of the experimental data. The method was applied to an impact test with a Nylon bar specimen. In this case, data at the 29 lowest resonance frequencies resulted in a rheological model with 14 parameters.

Conclusions:

The method has added improvements to the identification of rheological models as follows: (1) Noise reduction is fully integrated. (2) A rheological model is provided with a number of elements in accordance with the complexity of the material behaviour and the quality of the experimental data. (3) Parameters determining poles of the complex modulus are obtained without use of a least squares fit.

Introduction

A variety of methods for identification of viscoelastic materials have been proposed

and used since long. Such methods for estimation of complex-valued frequencydependent material parameters such as the complex modulus, generally produce data from which values of these parameters can be obtained for a set of discrete frequencies.

Commercially available machines for dynamic mechanical analysis (DMA) [START_REF] Duncan | Dynamic Mechanical Analysis Techniques and Complex Modulus[END_REF] as well as specialized and laboratory-specific set-ups are used. Some make use of forced or free vibrations of a structure such as a bar, while others involve impact, waves and various transient effects.

Garrett [START_REF] Garret | Resonant Acoustic Determination of Elastic Moduli[END_REF] used a method based on acoustic resonance in a study of the temperature dependency of the Young's modulus. A similar method was proposed by Guo and Brown [START_REF] Guo | Determination of the Dynamic Elastic Moduli and Internal Friction Using Thin Rods[END_REF] for viscoelastic materials. With this method, a 1D theoretical model is fitted to experiments on frequency intervals of finite length which provides values of the complex modulus at discrete frequencies. Some methods [START_REF] Madigowski | Improved Resonance Technique for Materials Characterization[END_REF] make use of random excitation at one end of a beam and measurement of acceleration at the other end. With the assumption of 1D viscoelastic behaviour, the complex Young's modulus is derived from the amplitude ratio and phase shift between acceleration and excitation. Pintelon et al. [START_REF] Pintelon | Identification of Young's Modulus from Broadband Modal Analysis Experiments[END_REF] proposed a method which takes dispersion into account by use of Love's model [START_REF] Love | A Treatise on the Mathematical Theory of Elasticity[END_REF]. They used a resonant frequency method in conjunction with forced vibration. Their method is based on measurement of the force applied at one end of a bar and the axial displacement at a given section. Noise reduction is provided by repetition of measurements and identification of a rheological model. Waves in bars generated by impact have been used

to determine the viscoelastic properties of materials by, e.g., Blanc [START_REF] Blanc | Détermination de l'Équation de Comportement des Corps Viscoélastiques Linéaires par une Méthode d'Impulsion[END_REF][START_REF] Blanc | Transient Wave Propagation Methods for Determining the Viscoelastic Properties of Solids[END_REF], Lundberg and 4

Blanc [START_REF] Lundberg | Determination of Mechanical Material Properties from the Two-point Response of an Impacted Linearly Viscoelastic Rod Specimen[END_REF], Lundberg and Ödeen [START_REF] Lundberg | In Situ Determination of the Complex Modulus from Strain Measurements on an Impacted Structure[END_REF], Hillström et al. [START_REF] Hillström | Identification of Complex Modulus from Measured Strains on an Axially Impacted Bar Using Least Squares[END_REF], Othman [START_REF] Othman | Extension du Champ d'Application du Système des Barres de Hopkinson aux Essais à Moyennes Vitesses de Déformation[END_REF], and Mousavi et al. [START_REF] Mousavi | Identification of Complex Moduli and Poisson's Ratio from Measured Strains on an Impacted Bar[END_REF].

It is not always ensured that the real and imaginary parts of the complex modulus are consistent with the principle of causality. One way to achieve such consistency is to express them in terms of a rheological model constituted by linear springs and dashpots [START_REF] Pintelon | Identification of Young's Modulus from Broadband Modal Analysis Experiments[END_REF][START_REF] Zhao | A Three Dimensional Analytical Solution of Longitudinal Wave Propagation in an Infinite Linear Viscoelastic Cylindrical Bar. Application to Experimental Techniques[END_REF]. An alternative way consists in making sure that the complex modulus is consistent with the Kramers-Kronig relations. Another fundamental constraint on the complex modulus comes from the second principle of thermodynamics: the dissipation per cycle must be positive for any harmonic load. Furthermore, the relaxation function, which is the inverse Fourier transform of the complex modulus, must be real. See, e.g., Landau and Lifchitz [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Landau | Statistical Physics[END_REF] about these constraints.

In Section 2 we present a method for noise-corrected estimation of the complex modulus of a viscoelastic material, subject to the constraints of causality, positivity of dissipation and reality of relaxation function, given an experimental set of frequencies and corresponding values of the complex modulus. It provides the structure, the number and type of elements, and the parameters of a rheological model and is based on results from different areas of mathematics and physics. As far as we know, similar results have not been combined previously into a practical method for producing a noise-corrected rheological model of suitable complexity that satisfies the above mentioned constraints.

The estimation method can be used in conjunction with any test method from which a set of frequencies and corresponding values of the complex modulus can be obtained. In Section 3 the use of the method will be illustrated with such data obtained by impacting a Nylon bar specimen and observing its response at its resonance frequencies.



, where t is time (e.g., [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF]). After Fourier transformation, this relation becomes

   ω ω E = ω σ ˆ, ( 2 )
where

    G ω E î  is the complex modulus,    G ˆ is the Fourier transform of   t G
and  is the angular frequency.

The complex modulus must satisfy the three constraints of causality, positivity of the dissipation rate and reality of the relaxation function. The problem to be considered is that of finding a noise-corrected complex modulus   The constraint of causality implies the Kramers-Kronig relations that connect the real and imaginary parts of the complex modulus. If either the real or the imaginary part 6 can be estimated accurately at a sufficient number of frequencies, these relations can be used to estimate the other part. For several experimental methods based on wave propagation, e.g. [START_REF] Mossberg | Non-parametric Identification of Viscoelastic Materials from Wave Propagation Experiments[END_REF][START_REF] Söderstrom | System Identification Techniques for Estimating Material Functions from Wave Propagation Experiments[END_REF], and the one to be presented in this paper, such a procedure does not work as the experimental noise is rather uniformly distributed over the real and imaginary parts of the complex modulus. Furthermore, results for the complex modulus can be obtained only for a relatively small set of frequencies with the experimental method of this paper. Here, therefore, we adopt the approach of estimating simultaneously the real and imaginary parts of the complex modulus. This will also allow us to extend the results to regions without measurements, especially at low frequencies.

The constraints of causality and of positivity of the dissipation rate [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Landau | Statistical Physics[END_REF][START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF][START_REF] Golden | A Proposal Concerning the Physical Rate of Dissipation in Materials with Memory[END_REF] are equivalent to requiring that the function   ω E be analytic in the lower half plane and that its imaginary part be positive on the positive real axis, respectively. The constraint of reality of the relaxation function is equivalent to the requirement    ω

E = ω E 
. Here we will use the stronger constraint of complete monotonicity of the relaxation function as proposed by Hanyga [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF] and Bouleau [START_REF] Bouleau | Visco-élasticité et Processus de Levy[END_REF] and derived by Beris and Edwards [START_REF] Beris | The Thermodynamics of Flowing Systems[END_REF] for fluids. This constraint, which essentially means that the relaxation function and its derivatives must be monotone, is satisfied by most rheological models [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF]. To which extent it holds for a particular real material can be verified from experimental data.

The property of complete monotonicity of the relaxation function implies that this function is the Laplace transform of a positive function (the Bernstein-Widder Theorem, see [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF]). This means that the complex modulus  

i i ds ω + s s h ω + = ω E   . ( 3
)
Such functions  ω E are analytic outside the positive imaginary axis of the complex ω plane and they satisfy the three constraints. Thus, they are analytic in the lower half plane and therefore satisfy the constraint of causality; their imaginary part is positive on the positive real axis and therefore the constraint of positiveness of the dissipation rate holds.

Finally, the constraint of reality of the relaxation function is satisfied as    ω

E = ω E  . It
should be noted that the function   s h can have singularities such as delta functions.

In order to check whether experimental data are of the form of Eq. ( 3), in particular for the purpose of determining the constants 0  and 0  , and the function h, it is convenient to apply the transformation

  z z E = z f i  . ( 4
)
It has the representation

  ds z s s g + = z f   0  , ( 5
) with 8 8      s s h + s du u u h = s g            0 0  , ( 6 )
where  s is a delta function at the origin, and   0

= s g for 0 < s . As the static complex modulus   ds s s h = E    0 0 0  ( 7 )
is positive and finite, the function   z f is analytic except on the positive real axis, and positive on the negative real axis.

The problem of finding a function   z f with the representation [START_REF] Pintelon | Identification of Young's Modulus from Broadband Modal Analysis Experiments[END_REF] given its values for finitely many z was studied in detail by Krein and Nudelman [START_REF] V Krein | An Interpolation Approach in the Class of Stieltjes Functions and Its Connection with Other Problems[END_REF]. Some of their results, when formulated for the complex modulus   ω E

, are as follows. From the values

1 E , 2 E , …, n E of the function   ω E at n angular frequencies 1  , 2  , …, n  one can form two matrices with elements  k j k j k j, ω + ω E E = M  i 1      k j ω + s ω s ds s h + = i i 0 0     , ( 8
)  k j k k j j k j, ω + ω E E = M   / / 2          . i i 0 0    k j k j ω + s ω s ds s s h + ω ω E = (9) 9 9
It can be verified that the matrix   1 M is positive semi-definite. In particular, it is selfadjoint with non-negative eigenvalues (recall that 0  and h are non-negative). The same

properties hold for  2 M since   0. 0 > E If one of the matrices   1 M or  2 M has a zero eigenvalue, the function  ω E
is completely determined and is a rational fraction (see [START_REF] V Krein | An Interpolation Approach in the Class of Stieltjes Functions and Its Connection with Other Problems[END_REF] and below). However, the occurrence of a zero eigenvalue is not a necessary consequence of general principles, and if none of the matrices   1 M and  2 M has a zero eigenvalue, the function  ω E cannot be completely determined from the data. We refer to Krein Nudelman (1998) for more details.

Assume now that a is an eigenvector corresponding to a zero eigenvalue of   1 M .

Then, since  , = 0 a M 1 we also have

  0 1 = , a M a , i.e.,  0. i 2 1 0 2 1 0 = ds ω + s a s h + a n = k k k n = k k                  ( 1 0 )
Since both terms in the left member are non-negative, they must vanish. In particular, the function  s h must be a linear combination

  l p l l s s      1 with coefficients 0  l  of delta functions   l s s   at the positive zeros p s , , s  1   n < p
of the rational function

 n = k k k ω + s a 1 i . ( 1 1
) 10 10

For the complex modulus given by Eq. ( 3) this implies the model

   p = l = l l l ω + s γ iω + = ω E 1 0 0 i   , ( 1 2 )
where

0 0   and 0  , 1 γ , 2 γ , …, p γ are non-negative.
The same considerations hold for the matrix  

 n = k k k ω + s b 1 i . ( 1 3 )
In particular, we only need to consider the common positive zeros of the rational functions [START_REF] Hillström | Identification of Complex Modulus from Measured Strains on an Axially Impacted Bar Using Least Squares[END_REF] and [START_REF] Mousavi | Identification of Complex Moduli and Poisson's Ratio from Measured Strains on an Impacted Bar[END_REF]. Moreover, if there are several linearly independent eigenvectors corresponding to a zero eigenvalue, we get a set of positive zeros for each of them and should keep only the ones which are common.

Equation ( 12) corresponds to the  

                p l l l l l p l l p l l l l E 1 2 0 1 0 1 0 0 i / / i i / i i                . ( 14
    p l l l s 1 0 0    , l l l s    , 0 0    , 2 l l l s    .  l 1, 2,….. p ( 15
)
The viscoelastic material represented by Eqs. ( 12) and ( 14) has asymptotic

elasticity as  0 0   E is finite, but generally it does not possess instant elasticity. If 0  and 0  vanish, however,  p E         ... 1 0
is finite. Then, the material has also instant elasticity and qualifies as a solid [START_REF] Flugge | Viscoelasticity[END_REF]. This follows from Eq. ( 3) and makes the following approach robust compared to situations with data points on the boundary of the analyticity domain.

It is noted that the points

Implementation.

From experiments, we obtain a finite set of angular

frequencies n ω , , ω  1 and corresponding complex moduli n E , E  1
. By use of these experimental data we form the two matrices (1) k j, M and (2) k j, M given by the first equalities of Eqs. ( 8) and ( 9), respectively. These self-adjoint matrices are positive semi-definite if [START_REF] Othman | Extension du Champ d'Application du Système des Barres de Hopkinson aux Essais à Moyennes Vitesses de Déformation[END_REF] and only if all their eigenvalues are non-negative. However, experiments commonly result in some slightly negative eigenvalues. In the experimental part of this paper, for example, the modulus of the smallest negative eigenvalue is typically a few tenths of a percent of the largest positive eigenvalue. A likely explanation for such small violations of the positive semi-definiteness is experimental noise. As advocated by Gu et al. [START_REF] Gu | Identification in  H using Pick's Interpolation[END_REF] for a similar problem, the effects of such noise can be reduced by searching the smallest possible corrections of ,...,  can be taken as zero while the remaining parameters are identified by a least square fit.

In this way, the method provides the structure, the number of elements and the parameters of the rheological model. Normally, the number of elements is found to be relatively low. We refer to [START_REF] V Krein | An Interpolation Approach in the Class of Stieltjes Functions and Its Connection with Other Problems[END_REF] for an alternative method to complete the identification. 13

Impact Test Method

As stated in the Introduction, an experimental set of discrete angular frequencies and corresponding values of the complex modulus may be obtained with a variety of test methods, commercial as well as specialized and laboratory-specific. Here, as an example, they will be obtained by analysing the wave propagation in a Nylon bar specimen loaded through axial impact. The method is a slightly improved version of that used by Othman et al. [START_REF] Othman | Identification de la Relation de la Dispersion dans les Barres[END_REF]. ). The primary pulse should be shorter than a 2 so that there is no overlap in the measured strain of this pulse and the first pulse reflected from the free end. However, it should be much longer than the diameter of the bar so that approximate 1D conditions prevail (wavelengths much longer than the diameter of the bar [START_REF] Kolsky | Stress Waves in Solids[END_REF].

Theory. Discrete values of the complex modulus

         E E E      i
In the frequency domain, the strain in the bar can be expressed as

         x x e B e A x         i i , ˆ   , ( 1 6 ) 14 14
where is the strain at the impacted end. With these amplitudes and b x  inserted, Eq. ( 16) gives (

      E 2 2  ,           i   k . ( 1 7
       0 -i 1 b b e  . ( 1 8

) 15

Dividing the members of Eq. ( 19) by those of Eq. ( 18) eliminates the strain    0 ˆ at the impacted end which normally cannot be measured. This slightly improves the method previously used by Othman et al. [START_REF] Othman | Identification de la Relation de la Dispersion dans les Barres[END_REF]. Substituting  from the second of Eqs. [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF] into the result, one gets the ratio  can be estimated by minimizing the difference between the resonance peaks represented by the left and right members of Eq. ( 22). From m  and m , an experimental phase velocity m c can be obtained by use of the third of Eqs. [START_REF] Bouleau | Visco-élasticité et Processus de Levy[END_REF]. Finally, by use of the dispersion relation [START_REF] Hanyga | Physically Acceptable Viscoelastic Models[END_REF] and the first of Eqs. ( 21), the complex modulus m E at each resonance frequency can be obtained as

        1 / ˆb b  which gives         l kl a ka e b b b        2 2 2 2 2 2 1 2 sinh sin sinh sin ˆ    . ( 2
          m m m m m m m m b b b l l k a a k e m m                  / sinh / sin / sinh / sin ˆ2 2 2 2 / 2 2 1 2     , ( 22
2         m m m = E    , m m α l m = ξ i   ,  m 1, 2, …. ( 23
)
3.2 Experimental Set-up and Procedure. Impact tests were carried out with a Nylon bar specimen of length  l 3 045 mm, diameter 10.2 mm, and density 1 149 kg/m 3 . Two pairs of strain gauges, one axial and one circumferential, were located at a distance of  a 1 731 mm from the free end and  b 1 314 mm from the impacted end. In order to minimise the effects of friction, the bar was suspended horizontally by means of nine regularly spaced strings with length 300 mm. With this arrangement, the period of oscillation of the system was about one second, which is much longer than the test duration. The striker had length 174 mm and the same diameter 10.2 mm and material as 17 the bar, and its impact velocity was 3.7 m/s. The strain was recorded with 500 kHz sampling frequency, and the signal vanished completely before the end of the recording window.

Results were evaluated up to 8 kHz. With a phase velocity expected to be higher than 1 500 m/s at this frequency, the shortest significant wave length was estimated to be greater than 0.19 m. This wave length is much larger than the diameter 0.01 m of the bar, which means that 3D effects can be neglected as assumed in Section 3.1.

Results and Discussion

The estimation method presented provides a model for the complex modulus corresponding to that shown in Fig. 1, with 1  p pairs of springs and dashpots. This model, represented by Eq. ( 12) or ( 14), satisfies the constraints of causality, positivity of the dissipation rate and reality of the relaxation function. The quantity p , which determines the number   12) which determine the poles 1 is , 2 is , …, p s i of the complex modulus on the positive imaginary frequency axis.

Consider an ideal material which behaves exactly as the rheological model of Fig. 1 with   1 2  q elements, where q represents the complexity of the material behaviour, and let the number n of experimental values of the complex modulus be greater than q .

18 18

Then, in the absence of noise, the method can be shown to provide a model with q p  . If noise is added, the number of common positive zeros, and therefore the value of p , normally decreases. For a given material, therefore, lower quality of the experimental data normally results in a simpler but less accurate rheological model. Thus, for n sufficiently large, the level of refinement of the model is determined by the material behaviour and the quality of the experimental data.

The recorded strain is shown in Fig. 3. Figure 3(a) shows a long-time record, and Fig. 3(b) shows the primary compressive pulse followed by one tensile and one compressive pulse which have undergone one and two free-end reflections, respectively.

As there is no significant overlap of these pulses, it was possible to compute the spectrum 2 1 ˆb  in the right member of Eq. (30) from the measured primary pulse. The spectrum  b ˆ, computed from the long-time strain record, is shown in Fig. 4. The angular frequencies at resonance m  and the corresponding complex moduli m E were determined as described. Figure 5 shows that there was good agreement between the resonance peaks determined from the left and right members of Eq. (30).

For the material of the Nylon bar specimen tested, a rheological model with  p 6

was obtained for  n 29. For this model, the 14 parameters of Eq. ( 12 Figures 6 and7 show three kinds of results for the frequency dependency of the phase velocity, the damping coefficient, and the real and imaginary parts of the complex modulus, viz., discrete experimental results, discrete noise-corrected results, and continuous results for the rheological model given by Eqs. ( 12) and ( 14) with the above sets of parameters. From the discrete experimental results for the phase velocity and the damping coefficient (open circles) shown in Fig. 6, the discrete experimental results for the complex modulus (open circles) in Fig. 7 were obtained as described in Section 3.

The corresponding discrete noise-corrected results (filled circles) and the continuous results for the rheological model obtained as described in Section 2.2 are shown in the same figure.

While the experimental results are scattered due to noise, there is close agreement between the noise-corrected results and the smooth curves representing the rheological model. This indicates that the noise-correction was effective and that the rheological 14parameter model obtained with  p 6 provides an adequate representation of the complex modulus in the frequency range of the test. Within this range, the real part of the complex modulus slowly increases from 3.09 to 3.24 GPa (5 %), while the imaginary part has a minimum 0.064 GPa at about 0.5 kHz and a maximum 0.090 GPa between 3 and 4 kHz.

20

The rheological model obtained by prescribing 0 0   closely agreed with the noise-corrected results for the real part of the complex modulus but markedly disagreed with those for the imaginary part, especially at higher frequencies. Thus, although there is no instant elasticity with 0 0   , a finite positive value of this parameter was found to be important in the frequency range of the test.

At low frequencies, much below the range of the test, the rheological model cannot be expected to give a quantitatively accurate representation of the complex modulus. However, it is still interesting to note that at such frequencies the model predicts a rapid increase with frequency of the real part and a narrow maximum of the imaginary part of the complex modulus. This is explained by the closeness of the pole 1 is to the origin which at frequencies  of the order of 1 s makes the first term of the sum in Eq. ( 12) much larger than the other terms of the sum. Qualitatively, the observed low-frequency behaviour is consistent with the results shown in Fig. 8 of servo-hydraulic tests which were carried out subsequently.

Conclusion

The estimation method of this paper has added some improvements to the identification of rheological models of the type shown in Fig. 1. First, the procedure for noise reduction is fully integrated in the method. Secondly, the method provides a rheological model with a number of elements that is in accordance with the complexity of the material behaviour and the quality of the experimental data. Thirdly, the parameters which determine the poles of the complex modulus on the positive imaginary frequency 21 21 axis are obtained as the common positive zeros of certain rational functions. Only the remaining parameters, on which the complex modulus has linear dependence, need to be identified by a least square fit.

Nomenclature

  the positive real axis such that 7

2 M 2 M

 22 and should lead to the same model. If b is an eigenvector corresponding to a zero eigenvalue of   , we deduce as above that the function  s h must also be a linear combination with non-negative coefficients of delta functions at the positive zeros of the rational function

Fig. 1

 1 Fig. 1 consisting of p full Maxwell elements (each consisting of a spring and a dashpot

  inside the analyticity domain of the function  . ω E

1 ω n ω and 1 E 1 M and  corr 2 M

 1112 , …, n E which restore the non-negativeness of all eigenvalues and establishing noise-corrected sets ,..., at least one of the noise-corrected matrices  corr has a zero eigenvector, one forms the associated rational fraction[START_REF] Hillström | Identification of Complex Modulus from Measured Strains on an Axially Impacted Bar Using Least Squares[END_REF] or (13) and finds its positive zeros. This is repeated for the set of independent eigenvectors, if any, corresponding to zero eigenvalues of both matrices. Then one keeps the common positive zeros rational fractions and obtains the model given by Eq. (12). In this model the the frequency range of interest does not extend far above that of the experimental datap non-negative parameters 1 γ , 2 γ , …, p γ are identified by a least square fit of the model to the noise-corrected data. If the material is required to possess instant elasticity, however, 0

  will be obtained from the resonances of an impacted uniform bar specimen with density  and length l as shown in Fig. 2. At one end, 0  x , the bar is impacted axially by a striker that separates from the bar after the generation of a compressive primary pulse in the bar. The other end of the bar, l x  , is free. The strain     a from the free end and b from the impacted end ( l b a  



  travelling in the directions of increasing and decreasing x , respectively,   k is the wave number and     is the damping coefficient.In order to determine an expression for the recorded strain     associated with the primary pulse alone, the amplitudes A and B are first determined from Eq. (16) and the boundary conditions  

  complete train of pulses is determined similarly. First, the amplitudes A and B are determined from Eq. (16) and the boundary conditions   

  use of the third of Eqs. (21) we then obtain the relation   and angular frequency within the resonance peak. Inserting these expressions for    and   k into Eq. (20) we get

) 16 16 

 16 m 1, 2, … within the m:th resonance peak. In this relation, the spectra   m 1, 2, …, the resonance frequency m  and damping coefficient m

  the rheological model, is obtained as the number of common positive zeros more rational functions established from the noise-corrected data. These zeros constitute the p parameters of Eq. (

Fig. 1 .Fig. 2 .

 12 Fig. 1. Rheological model with   1 2  p parameters.

Fig. 3 . 1 followed

 31 Fig. 3. Recorded strain in the Nylon bar specimen. (a) Long-time record of strain pulses  t b  . (b) Primary strain pulse   t b 1

Fig. 4 .

 4 Fig. 4. Spectrum    b ˆof the recorded strain in the Nylon bar specimen.

Fig. 5 .

 5 Fig. 5. Details of the spectrum   2 ˆ b . The thin and thick curves are based on the left and

Fig. 6 .

 6 Fig. 6. (a) Phase velocity c and (b) damping coefficient  versus frequency. Open circles: discrete experimental results. Continuous curves: results based on the rheological

Fig. 7 .

 7 Fig. 7. (a) Real part E and (b) imaginary part E   of the complex modulus E versus frequency. Open circles: discrete experimental results. Filled circles: discrete noise-

Fig. 8 .

 8 Fig. 8. Real part E and imaginary part E   of the complex modulus E at low frequencies determined from servo-hydraulic tests.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Rheological model with   1 2  p parameters.

Fig. 3 . 1 followedFig. 4 .Fig. 5 .

 3145 Fig. 3. Recorded strain in the Nylon bar specimen. (a) Long-time record of strain pulses  t b  . (b) Primary strain pulse   t b 1

Fig. 6 .

 6 Fig. 6. (a) Phase velocity c and (b) damping coefficient  versus frequency. Open circles: discrete initial results. Continuous curves: results based on the rheological model

Fig. 7 .Fig. 8 .

 78 Fig. 7. (a) Real part E and (b) imaginary part E   of the complex modulus E versus frequency. Open circles: discrete initial results. Filled circles: discrete noise-corrected

23 Superscript corr corrected with respect to noise


INTRODUCTION

So called "brittle materials" first show a brittle behaviour in tension -sometimes in simple compression -and usually a plastic-like behaviour under confined compression. This plasticlike behaviour is more often the response of a damaged material that does not recover its integrity after unloading. It then appears that the notion of behaviour, when applied to brittle materials, is strongly dependant on the loading. It is also dependant on the size of the elementary element in which it will be introduced for the (FEM) modeling of a real structure. Basic physical phenomena may involve a very small scale while a much large scale is required for modeling.

The same situation occurs in the case of testing when, in the opposite way of thinking, one has to go from global measurements to stress-strain relations. Accounting for structural effects in testing is then an evidence as any specimen tested is nothing else than a structure. It will appear in an evident manner for many tests used in brittle material testing (like flexion tests for instance), especially for concrete often requiring big specimens (in "10 cm" range, minimum).

For this reason, we will especially focus on tests for which this aspect does not clearly appear as simple compression and tension tests that are safely processed in a standard way in the case of metals.

In quasi-static testing, going from global measurements -force, displacement, gauge measurement -to the stress-strain relations requires the homogeneity of mechanical fields within the tested area, basically the strain field. Such an assumption cannot exactly be verified in dynamic testing, especially with brittle materials generally described in the range of small strains. This leads to specific approaches that are investigated in the present paper.

This idea can be simply quantified, following [START_REF] Forquin | cConference 24-1[END_REF]. Considering for example the case of HS-Concrete (High performance) in compression, at an average strain rate of 100/s (rather small in dynamics) it would take 10 µs to reach the failure strain of 0.1%. In order to assume equilibrium, waves should run at least 5 round-trips within the specimen during this time, corresponding to a distance of 4 cm (if the speed of wave is 4000 m/s) leading to a maximum specimen size of 0.4 cm which could not be, in any case, representative of the material.

The larger the representative size, the smaller the failure strain, the more difficult dynamic testing.

Structural effects in dynamic testing of brittle materials G. Gary

LMS, Ecole Polytechnique, 91128 PALAISEAU, France

ABSTRACT: Dynamic testing of brittle materials obviously involves the specificities of dynamics and of the special kind of behaviour that describes brittle material. The interaction of both aspects is much more important than for materials like metals which exhibit a plastic behaviour. This interaction is described in the present paper, with a special focus on the Split Hopkinson bar technique commonly used in these fields.

Table 1. Mechanical properties of common brittle materials from Forquin (2013),

Meaning of the word "dynamic"

As distinct from the term "static", "dynamic" implies the influence of time. A test is said to be "quasi-static" -while a purely static test cannot exist -when the effects of time can be neglected. For any real test, the effects of time are typically expressed in two ways: -by inertia forces resulting from the non null acceleration to which elements of structures are submitted.

-by the behaviour of each elementary volume of the material depending on evolution in time of the elementary mechanical values (stress and strain) and possibly of their time derivatives. This dependence is described by the generic name of viscosity.

This distinction is strictly linked to the notion of elementary volume underlying the definition of the behaviour. Actually, the fact that viscosity effects can be the manifestation of inertial microscopic phenomena cannot be excluded.

The behaviour that experimentalists are looking for, to be used in modeling, is supposed to refer to any elementary volume of the studied material free of internal forces.

Specificity of dynamic testing arrangements

The first difficulties encountered in dynamic testing are linked to transient effects inside the machine and the associated sensors: the balancing time of the machine and its sensor array (elastic waves moving back and forth) could be not negligible relative to the length of the test. It has also to be taken care that the acquisition frequency is far higher than the frequency of the transient signals to avoid a possible degradation of the results. Such difficulties mainly concern the faster side of machines providing a range of speeds starting from quasi-static to dynamic loadings.

The response of the machine will be briefly investigated in the special case of SHPB (Split Hopkinson Pressure Bars), as matter of illustration, as it is common knowledge that Hopkinson bars have been indeed especially designed to deal with waves and provide reliable measurements at specimen boundaries.

From global testing to material behaviour.

Recall that the homogeneity of mechanical fields is required in order to derive in a simple way the stress-strain relations from global measurements. This homogeneity depends on the specimen dimensions in regard of the representative size of the material tested. Transient effects in the specimen due to the finite speed of waves lead to non homogeneous stress and strain fields in an increasing manner with the specimen size (as quantified above). The homogeneity also depends on boundary conditions, as for instance friction at specimen ends in 1-D compres- When dealing with brittle materials, especially with concrete, the representative size must be large in comparison with the size of testing devices. This size factor also gives an increased importance to structural forces induced by inertia effects that appear most often in addition to loading forces.

AVAILABLE TESTS AND MEASUREMENTS

It would not be possible to give an extensive list of dynamic tests used for the experimental study of brittle materials. Our paper will then be restricted to the more common ones, with a special attention to those which are more familiar to the author.

Looking for the dynamic material behaviour under compression, SHPB is commonly used (strain-rates ranging from 50 to 500 for concrete). Under such a loading, brittle material are very sensitive to lateral pressure (as shown for instance in fFigure. 1) so that three (complementary) loadings are found: simple compression, compression under controlled pressure, compression of a confined specimen preventing lateral expansion.

Figure 1. Tri-axial quasi-static compression of a ceramic, from [START_REF] Heard | Mechanical behaviour of polycrystalline BeO, Al2O3 and AlN at high pressure[END_REF].

The direct impact test could also be used but it is not well adapted as its processing requires the assumption of equilibrium. At higher strain rates, plate-plate impact tests have been used, but it is shown that they do not provide a direct access to the behaviour and they are limited to very high strain rates (> 10 5 s -1 )

For tension testing, the two more direct approaches are the (modified) SHB for direct tension and spall tests. These last ones, as they start with a compression phase, cannot afford to avoid a transient analysis.

Other tests leading to fracture in tension involve a clear structural response without homogeneity of mechanical fields: Brazilian test, flexion of beams or plates.

Basics of the machine

SHPB suffers from its historical original use introduced by [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]. He proposed his formulas before computers had become generally available for data processing. He used identical input and output bars (same length, diameter and material) and put strain gauges at the middle of each bar. Neglecting the dispersion in the bars and assuming quasi-static equilibrium, Kolsky derived:

) ( ) ( ) ( t t t t r i ε ε ε ≅ + (1) with i ε , r
ε and t ε the incident, reflected and transmitted strains as recorded at the gauges.

The average strain s ε of the specimen is given by

- = = 0 0 ) ( 2 ) ( ) ( τ τ ε τ τ ε ε d l c d t r t s b s s (2)
with b c the speed of waves in bars, s l the specimen length.

The average stress s σ is obtained from the output force (or from the average of input and output forces which provides the same value within the hypothesis of quasi-static equilibrium).

) As soon as the hypothesis of quasi-static equilibrium is not verified, as it is the case for most compression tests on brittle materials, this analysis is not valid.

( ) ( t E A A t t b s b s ε σ = (3)
One has then to go back to basic measurements provided by SHPB. For sake of simplicity, following most authors, we consider identical bars. It does not restrict the generality of the presentation. Considering the values of the strain in bars at specimen ends, forces and displacements at both specimen ends are given by formulas Formulas (4,) and ( 5)

) ( r i b i c v ε ε - - = ) ( t b o c v ε - = (4) ) ( r i b b i E A F ε ε + = t b b o E A F ε = (5) where i v , o v , i F , o
F are input and output speeds and input and output forces at specimen faces, respectively. i ε , r ε , t ε are incident, reflected and transmitted waves, respectively, computed at specimen faces.

When the specimen diameter is less than half of that of the bars, the displacement deduced from speeds (4) is overestimated and must be corrected, especially in the case small strains -see [START_REF] Safa | Displacement correction for punching at a dynamically loaded bar end, IE-1835[END_REF].

The measurement finishes here. In other words, equations Equations ( 2) and (3) are not direct measurements. They are (only) derived from ( 4) and ( 5) on the basis of the hypothesis of quasistatic loading.

Wave shifting. A precise method for SHPB.

The 1-D analysis of the waves implicitly takes account of the Saint-Venant principle: a certain distance is needed between the end of the bar and the strain gauge to insure the homogeneity of the strain across the bar (typically 5 diameters). The three waves i ε , r ε , t ε involved in for-

Strain gauge A Specimen Strain

Striker Incident bar mulas ( 4) and ( 5) being that at specimen faces, one has to take care of the precise shifting in time from gauges to bars ends. One needs then to use a wave theory to deduce the strain (as it would be if this point was not an end) at the end a bar. This shifting involves two aspects: -One is to account for wave dispersion (this is a 3-D effect that is usually modeled in 1-D. The variations of mechanical parameters along the radius of bars are indeed very small at low frequencies involved in standard tests as shown by [START_REF] Davies | A critical study of Hopkinson pressure bar[END_REF] and [START_REF] Merle | On the errors associated with the use of large diameter SHPB, correction for radially non-uniform distribution of stress and particle velocity in SHPB testing[END_REF].

-The other is to correct possible errors in the distance from the gauge to the bar end, or for an imprecise wave speed, or more generally to correct for an imperfect contact between the specimen end and the bar. Note that, with an input speed of 5 m/s, an 0.2 mm thick imperfection induces a 40 µs delay between the first touch and the perfect contact with a 5 m/s input speed.

The input force being proportional to the sum of the incident and reflected waves, it is clear that a relative imperfect shifting in time would induce an error, especially at the beginning of the loading. For an improved shifting, one can use a method, introduced by Zhao & Gary (1996). It is based on the transient simulation of an initial elastic behaviour of the specimen. The incident wave at the input specimen face been known after the dispersion correction process, reflected and transmitted waves can be computed -depending on specimen dimensions, bar dimensions and mechanical properties, specimen Young's modulus. The only unknown is the last one. Using a try and error method, one rapidly finds the Young's modulus that gives shapes of both simulated transmitted and reflected waves similar to those known at the input and output specimen faces. This operation does not work correctly if the dispersion is not taken into account, even with elastic bars, because the elastic response of the specimen concerns the first instants of the loading where the rising time of the waves is strongly affected by the dispersion. An illustration of the method is presented in Ffigures 3-4. Figure 3 shows that the "best" Young modulus fitting the output wave is not realistic at all (0.2 GPa) when dispersion correction is not applied. Furthermore, transmitted and reflected waves cannot be simultaneously fitted. On the contrary, the realistic value of 3.5 Gpa induces a good fitting when dispersion correction is applied, for both reflected and transmitted waves. Furthermore, the separation between the elastic simulated wave and the measured one gives the instant when the specimen starts to have a non elastic response -failure time for a purely elastic material. In the present case, it is around 30 µs. Looking at Ffigure 4, one sees that fracture initiates a long time before equilibrium is reached. Standard SHPB formulas cannot be used to derive the behaviour. Consequently, input and output forces, together with input and output speeds (not shown here) should be the basis of the subsequent analysis towards the behaviour.

Furthermore, the average strain rate varies very rapidly with the strain (as seen in fFigure. 5) so that associating a behaviour to a known strain-rate would not have a clear meaning.

An example for a transient analysis for brittle materials

On the basis of force and velocities measurements at specimen faces, an approach of the specimen behaviour based on an inverse method is theoretically possible, as shown by [START_REF] Rota | An inverse approach of identification of dynamic constitutive equations[END_REF], as these four values are superabundant measurements.

If an appropriate form of the material behaviour with some parameters to be determined is known, using a part of data (two velocities, for example) as input data, another part of data (the two forces) associated with the given parameters can be calculated. The best set of parameter which gives the calculated forces well in agreement with the measured ones can theoretically be found.

An example of a 1-D analysis based on such a method is shown in Ffigure 6, from Gary & Zhao, (1996). It shows that both input and output forces can be recovered. The model used for this simulation helps for the understanding of axial transient effects but, being one-dimensional, it could not account for the increase in strength that could be due to lateral inertia effects. Accounting for this effect would need to make all the parameters of the model strain-rate dependant.

3.1.4

Influence of strain-rate on the apparent strength of concrete.

3.1.4

Many dynamic tests in simple compression show a sharp increase in strength with strain rate, as shown in the famous figure presented by [START_REF] Bischoff | Compressive behaviour of concrete at high strain rates[END_REF], here Ffigure 7.

This important effect have has been proved to be purely structural, as explained underneath. It is due to the change of behaviour induced by an artificial lateral pressure, itself the results of lateral inertia of the specimen preventing its expansion.

This effect is also observed for metals when the apparent plastic response is increased by the tri-axial state of stress induced by inertia. In only leads to a small correction for standard metals, especially because the specimen tested are usually small. Formulas established in this case -for instance by Malinowsky & Klepaczko (1986) show that this effect increases with the square of the specimen radius, the axial strain-rate, the time derivative of the axial strain-rate, and the mass density of the material. The greater importance of the spherical behaviour on the response of brittle materials makes this effect more dramatic in our case. When concrete is concerned, as seen in Ffigure 7, the representative size of the material requires big specimens (at least a few centimeters in length and diameter). The figure shows that the sudden increase can appear in a range of 1 decade, between 10 and 100s -1 .

The case of ceramics. This structural effect being strongly sensitive to the specimen size, testing smaller ceramic specimens in compression would delay this effect towards higher strain rates. Note that these materials being very hard, they induce a special testing difficulty, as they can show an elastic limit higher than that of the bars. When the behaviour is almost perfectly elastic-brittle, there is a huge influence of local imperfections at specimen faces that can induce local stresses much higher than the average measured one, giving an underestimated resistance of the material. For both previous difficulties, a solution is to use dog-bone specimens which need a special processing. Provided that the larger part of the specimen remains elastic, this special processing is possible. . Figure 7. Influence of strain-rate on the apparent increase of the strength of concrete with strain rate.

Compression with SHPB under controlled lateral pressure.

A method proposed by [START_REF] Gary | Behaviour of a quasi-brittle material at high strain rate. Experiment and modelling[END_REF] is briefly recalled here. Following [START_REF] Christensen | Split Hopkinson bar test on rock under confining pressure[END_REF] and [START_REF] Malvern | Dynamic testing of laterally confined concrete[END_REF], the specimen is introduced into a cylindrical quasi-static pressure cell. The bars are acting as pistons and are introduced in the cell through seal rings. A scheme of the complete set-up is shown in fFigure 8. The lateral pressure can be applied with oil (up to 50 MPa), or with air (up to 10 MPa).

Figure 8. Set-up for confined pressure tests.

Sealing problems

Using a significant lateral pressure of oil (20 MPa) a test without specimen (bar against bar) has been performed to investigate the influence of seals on waves propagating in the bars. Input and output forces have been calculated (Ffigure. 9). They are not very different, and it proves that the influence of seals can be disregarded.

Lateral pressure

It seems reasonable to assume that a static confined pressure applied with air will not be significantly affected by the increase in the diameter of the specimen induced by the deformation (in this device, the chamber is 120 mm long and has a diameter of 75 mm so that with a 40 mm long and 40 mm diameter specimen, the volume of the fluid in the chamber is almost 10 times the volume of the specimen).

The situation is not so clear with oil and it is not sure, because of transient effects in the fluid, that a measurement of the oil pressure during the test at a point in the chamber would give an exact measure of the pressure applied to the specimen. To evaluate this question, tests with oil and with air have been performed, using the same initial confinement pressure and other initial conditions. Results under oil pressure look very much like the ones presented in Malvern et& al,. (1991) using a very similar device where water lateral pressure was used. When lateral pressure is applied with air, the stress strain relation shows a much lower apparent strain hardening, as presented in Ffigure 10. The structural effect clearly appears here. Axial compression induces an expansion of the specimen due to its non null Poisson's ratio. The weakest response, which is also the more brittle, is obtained without confinement -case of § 2.1. With air pressure (20 MPa), a significant increase of the maximum stress appears, but the same kind of behaviour is observed. This evolution can be explained with a model proposed by [START_REF] Gary | Behaviour of a quasi-brittle material at high strain rate. Experiment and modelling[END_REF], briefly recalled in Ffigure 11, inspired from the real breaking process of concrete under compression, introducing lateral inertia and lateral pressure.

With oil pressure, inertia of oil presumably prevents the lateral expansion of the specimen, (so that the lateral pressure increases in the vicinity of the specimen) leading to a loading closer to that obtained with oedometric tests (described in § 2.3) Note that, since 1998, more sophisticated models have been developed based on a physical approach - (Desnoual et al. (1997) -accounting in a quantified way related to a [START_REF] Weibull | A Statistical Theory of the Strength of Materials[END_REF] analysis for the nucleation of cracks and their propagation. Extended to dynamic loading by [START_REF] Forquin | Influence of free water on the quasi-static and dynamic strength of concrete in confined compression tests[END_REF] where the finite speed of crack propagation is introduced -this aspect clearly missing in the model of Ffigure 11 -, they account for most structural effects observed in uniaxial compression in particular and in many other dynamic loadings.

Going back to fFigure 11, it is easy to understand that lateral inertia will also prevent, or at least delay, the specimen expansion under pure compression. It explains the apparent increase in strength with average strain rate observed for uniaxial compression tests.

Simple dynamic compression without lateral expansion.

It is guessed from Ffigure 10 that axial compression without a significant lateral expansion (oedometric loading) does not show a brittle behaviour as a global strain-hardening is observed, preventing localization. Very high stresses can be reached under such a loading, presumably breaking the micro-structure when it is associated with high strains. The involved behaviour is then not anymore that of the initial material before loading but it is appropriate to describe its evolution under this kind of state of stress. Such situations are found in military applications or in studies connected with the safety of buildings (power plants) regarding an accidental internal loading or external loading (plane crash). Such tests have been developed in the quasi static regime. Because of high pressure involved, and the huge elastic energy stored in the machine, they have to be done in special buildings, for safety reasons, and are expensive.

At some points of view, the corresponding dynamic test is easier as the energy involved is dissipated in a very short time. Such a test has been developed in our laboratory by [START_REF] Forquin | A testing technique for concrete under confinement at high rates of strain[END_REF] . The specimen is confined in an instrumented metallic ring and loaded by means of a SHPB especially designed for this purpose, with steel bars 80 mm in diameter. The cylindrical specimen embedded in a steel confinement ring is compressed using 2 cylindrical plugs (Ffig.ure 12). The concrete specimen is 30 mm in diameter and 40 mm long. The steel plugs have the same diameter and a thickness of 10 mm. The steel ring has an outer diameter of 65 mm and is 45 mm long. A special interface product ensures that the expansion of the ring is due to internal pressure, allowing for the measurement of the confinement pressure. The typical axial force-displacement response shows a positive strain hardening (as seen in Ffigure. 13). It is then expected that an acceptable homogeneous state of stress and strain is obtained in the specimen. Deriving the average stress from the output force should be then a valid approximation.

Simultaneous gauge measurements made on the elastic ring allow for an evaluation of the radial stress and, consequently, of the lateral pressure applied to the specimen. Furthermore, the radial expansion leading to inertia effects being prevented by the ring, one may think that structural effects can be neglected in such a test. This is not exactly the case as one has to take care of two secondary structural effects. One is the contribution to the axial force of the friction between the ring and the specimen. An other is the difference between forces measured at bar ends and required forces at specimen faces. These points have been carefully studied in [START_REF] Safa | Mise au point d'un essai de compaction dynamique[END_REF] where explicit formulas can be found to derive friction and lateral pressure from gauges measurements.

Figure 14 show the difference between forces at bar ends and forces at specimen ends as deduced from a transient analysis of the response of the plugs. The main result obtained with this test - [START_REF] Forquin | Influence of free water on the quasi-static and dynamic strength of concrete in confined compression tests[END_REF] -is that the material response is very sensitive to the presence of free water, concerning both deviatoric and hydrostatic behaviours. A constant bulk modulus is observed for dried concretes whereas the corresponding hydrostatic behaviour of saturated specimens is non-linear. Moreover, dried concretes show a strong increase of the strength with the loading speed whereas water-saturated specimens exhibit an almost-perfect saturation of the strength. It appears that, supposedly by reducing the level of effective stress applied to the skeleton, the pore-pressure inside the concrete strongly influences the dynamic behaviour of confined concrete.

Very High strain-rates and 1-D (compression) strain. Slab-plates tests.

In the standard case (-Zukas (1982) -, loading is caused by the impact of two identical plates. The impact speed 0 V is known. On the fixed slab, a rear face rate measurement (usually made using laser interferometers) is conducted. The shock induces a plane shock wave propagating at a velocity D. Discontinuities of material rate u, pressure P, the volumic mass or mass volume V and inner energy E are associated with this wave. Assuming the initial conditions are zero, it can be inferred from the Rankine-Hugoniot conservation equations that:

D u D V V / ) ( 0 - = , 0 /V Du P = , ) ( 2 / 0 0 V V P E E - + = (16)
At the time the shock is known, measuring the rear face speed allows to locate the moment when the wave arrives, to measure D and calculate u. Then the test enables to establish a relation between P and V (and also between D and u) giving one point of the so-called "shock polar curve". In order to deduce a uniaxial stress-strain curve one has to make some hypothesis on the behaviour model of the material. In that sense, the structural effect is, in this test, evident. The usual assumption used for metals, neglecting elasticity, that the behaviour is purely deviatoric (without any volume variation) is not valid for brittle materials. The test should then be processed by an inverse method.

TENSION

Tension with SHB.

Following the same basic ideas than in compression, dynamic tension tests for concrete have been developed with Hopkinson bars - [START_REF] Reinhardt | Concrete under impact laoading, Tensile Strength and bond[END_REF]. Referring to Ttable 1, the limits induced by the very small facture strain do not allow for homogeneity of the mechanical parameters at strain-rates greater than 10s -1 , which is hardly in the dynamic range. Furthermore, the specimen holding is difficult, generally requiring the use of glue in the best case, which makes the global measurements unprecise. At lower strain rates, it is then safe to use complementary measurements, with strain gauges for instance.

For all these reasons, the most commonly used method to investigate the dynamic behaviour in tension is the spall test.

Spall test.

Spall tests have been previously introduced for metal and plate-plate impact tests to measure the tension strength (under uniaxial strain) at very high strain rates. In order to avoid gripping problems, the spall test has been introduced (under uniaxial stress) to measure the tension strength of brittle materials, and concrete in particular - [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF].

This smart method is based on the fact that brittle materials have a higher strength in compression than in tension and that they remain in the elastic range in compression. A sketch of the test is given in Ffigure 14. Figure 14. A sketch of the spall test, from [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF].

Using a long thin specimen, a known compression wave is induced at one end. When it reflects at the free other end, a tension wave is produced such that the stress at the free end remains null. If the initial pulse is short enough, a state of tension is induced in the specimen at a certain distance of the free end. If the stress is greater than the material strength in tension, the specimen brakes.

It is very clear, at this stage, that there is no possible direct measurement of the fracture stress. The analysis of the test is indeed based on the previous knowledge of the behaviour. The standard method assumes that the material remains elastic in compression and in tension when the stress is under the fracture stress. Using a 1-D analysis of the waves (where dispersion can be introduced) and knowing the position of the (first) fracture on can go back to the failure strength. Some authors use a measurement of the rear face of the specimen, and the formula (17) established by [START_REF] Novikov | The study of fracture of steel, aluminium and copper under explosive loading[END_REF] Both analyses are based on a pure 1-D elastic response of the material until fracture. In that sense, this test evidently involves structural aspects. The method cannot account for damage occurring before fracture.

A new method based on image correlation and using the principle of virtual work has been recently developed - [START_REF] Pierron | Ultra-High-Speed Full-Field Deformation Measurements on Concrete Spalling Specimens and Stiffness Identification with the Virtual Fields Method[END_REF]. It only works in the dynamic range and needs a fast speed camera but can provide a local measurement (on the surface of the specimen) of the stress and associated strain. Other tension tests that clearly involve structural effects will only be briefly mentioned as the structural response of the specimen is the basis of the processing.

Like in the case of spall tests, their analysis needs assumptions on the material behaviour. The simplest one is the 1-D elastic-brittle behaviour. More complex behaviours can be investigated provided that experimental data are sufficient and precise enough to support inverse methodssee also [START_REF] Erzar | Experiments and mesoscopic modelling of dynamic testing of concrete[END_REF].

The Brazilian test.

We consider a cylinder compressed perpendicularly along two diametric generators. A quasistatic plane strain elastic calculation shows that a constant tension maximum stress is induced in the central plane defined by both previous generators. Achieving quasi-static conditions and punctual contact loading is difficult so that, when compression is applied with SHPB, it is better to carry out a numerical calculation, even with an elastic brittle material - [START_REF] Tedesco | Experimental and numerical analysis of high strainrate splitting tensile tests[END_REF].

The flexion test.

A dynamic flexion test may be conducted with a SHPB with three bars. The output bars support a beam loaded in his middle by the incident bar - [START_REF] Delvare | A non equilibrium approach to processing Hopkinson Bars bending test data:application to quasi-brittle materials[END_REF]. An explicit analysis of the elastic dynamic response of the beam allows for the calculation of the reflected wave from the knowledge of the incident one. It is then assumed that the instant when the real reflected wave separates form the simulated one corresponds to the fracture of the beam. The principle of the method is similar to that described in 3.1.2.

Shock tube test on plates.

The principle is to load the specimen using a shock-tube (as seen in Ffig.ure16). Such a device uses a tube as a wave guide where the loading can be generated by a well-controlled airshock wave -see for example [START_REF] Toutlemonde | Shock-tube tests of concrete slabs[END_REF]. A well known pulse pressure of a given amplitude and duration is applied at one specimen face, allowing for precise initial data for an inverse calculation. The response of the structure is observed with extra sensors (gauges, velocity and displacement measurements). Dynamic testing theoretically cannot provide direct stress-strain relations as transient effects prevent for a homogenous state of stress and strains in the specimen. In order to investigate the strain-rate effects on the material behaviour, that can be significant at low strain-rates, some techniques (like SHPB) use the assumption of a valid equilibrium. The quality of this assumption can be estimated by comparing the duration of the test with the time to equilibrium quantified by the transfer time of the waves through the specimen. As the wave speed is a constant of the material, this time increases with the specimen size, making more difficult the testing of large specimens as required for instance for concrete.

We have seen that this question is especially critical for most brittle materials the behaviour of which in tension is only meaningful at small strains. Consequently, a careful processing, accounting for all measurements -for instance both input and output forces in the case of SHPBmust not be avoided. A pleasant point is that, mainly because of small strains, the temperature increase of the specimen -dynamic tests being adiabatic -can be neglected.

In the special case of simple compression, lateral inertia effects can induce a confinement stress that has a strong influence on the axial response of the specimen and should not be forgotten. The nice thing with this confinement, even more when it is under control with a cylindrical cell, is that the behaviour shows a positive strain hardening making easier the processing of the test.

Assuming the theoretical elastic-brittle behaviour in tension, pure structural tests have been developed (flexion, spall, Brazilian) that directly provide a tension strength. One must keep in mind that the corresponding analysis is not valid as soon as the behaviour is non-linear or shows damage, and that transversal 3-D effects are not taken into account.

These methods are still of interest for a first approach of the material behaviour but they should be validated by inverse methods, in other words by direct calculations -with an adequate model, which is not the less difficult problem -of the measured response of structures submitted to a known loading.

For brittle materials, more than for metals, redundant measurements should be the first basis of a reliable testing.

  Compression with SHPB.

Figure 2 .

 2 Figure 2. Typical SHPB set-up.

  with b A and s A the areas of the bars and the specimen, respectively, b E Young's modulus of the bars.

Figure 3 .

 3 Figure 3. Elastic simulation of the output wave, Figure 4. Input and output forces. with and without dispersion correction.

Figure 5 .

 5 Figure 5. Measured average strain rate.

Figure 6 .

 6 Figure 6. Simulated and measured forces for a test on concrete.

Figure 9 .

 9 Figure 9. Input and Output forces for a confined test without specimen.

Figure 10 .

 10 Figure 10. Comparison between different lateral loading conditions.

Figure 11 .

 11 Figure 11. Analogical rheological model for an elementary volume of concrete.

Figure 12 .

 12 Figure 12. Cylindrical specimen embedded in a steel ring.

Figure 13 .

 13 Figure13. The difference between forces at bar ends and at specimen ends.

Figure 15 .

 15 Figure15. Rear face speed measurement(Novikov, formula 17)

Figure 16 .

 16 Figure 16. Sketch diagram of the shock-tube test from[START_REF] Toutlemonde | Shock-tube tests of concrete slabs[END_REF]


Gérard Gary

Dirk Mohr

Modified Kolsky Formulas for an Increased Measurement Duration of SHPB Systems

Keywords: Split Hopkinson pressure bar, Kolsky formulas, high strain rate testing

The so-called incident, reflected and transmitted strain histories are typically recorded during standard Split Hopkinson Pressure Bar (SHPB) experiments.

Subsequently, the stress-strain curve for the specimen material is determined based on these recordings. Unless wave deconvolution techniques are employed, the reliable measurement of the reflected wave requires an input bar which is at least twice as long as the striker bar (of equal impedance). The present brief technical note elucidates the advantages of a simple alternative configuration which has only been seldom used in the past. Based on the assumption of quasi-static equilibrium at the specimen level, we present a modification of Kolsky's formulas such that the stress-strain curve for the specimen material can be obtained from the measurement of the incident and transmitted strain histories only. As a result, the measurement of the reflected wave may be omitted and a much shorter input bar can be chosen. Conversely, a much longer striker bar may be used for a given input bar length, thereby increasing the valid duration of standard SHPB experiments by up to 100% through the use of the modified Kolky formulas. An example experiment is shown where the duration of valid measurements has been increased by more than 70%.

Motivation

Split Hopkinson Pressure Bar (SHPB) systems are used to investigate the mechanical behavior of materials at high strain rates. The widespread use of SHPB systems in experimental dynamics is mainly due to the simplicity of the experimental procedure.

The experimental technique is based on the early work of [START_REF] Hopkinson | A method of measuring the pressure in the deformation of high explosives by the impact of bullets[END_REF], who recorded a pressure-pulse profile using a slender bar. This approach has been widely adopted since the in-depth analysis of [START_REF] Davies | A critical study of Hopkinson pressure bar[END_REF]. The practical configuration consisting of a short specimen sandwiched between two slender bars is due to [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]. High impedance bars made of steel are typically employed to perform dynamic experiments on metals. After being initially developed for compression tests, the technique was soon extended to tensile loading by Harding et al. (1960 and to torsion loading by [START_REF] Duffy | On the use of a torsional split Hopkinson bar to study rate effects in 1100-0 aluminium[END_REF]. A comprehensive review of developments in classical SHPB testing has been provided by [START_REF] Gray | Classic Split Hopkinson pressure bar testing; ASM Handbook Vol 8, Mechanical testing and evaluation[END_REF].

The stress-strain relationship for the specimen material in a SHPB experiment is typically determined based on the so-called incident, reflected and transmitted waves. It is common practice to measure the incident and reflected waves using a strain gage at the center of the input bar. This imposes a constraint on the minimum length of the input bar to avoid the superposition of the incident and reflected waves at the position of the strain gage. For example, when both bars have the same diameter and are made from the same material, the input bar needs to be at least twice as long as the striker bar to guarantee the validity of the input bar strain history measurements. This limitation can be overcome through the use of multi-point measurements along with a wave deconvolution technique (e.g. [START_REF] Bussac | An optimization method for separating and rebuilding one-dimensional dispersive waves form multi-point measurements. Application to elastic or viscoelastic bars[END_REF]). A valuable alternative approach omitting the use of wave deconvolution has been used by [START_REF] Song | Mechanical characterization at intermediate strain rates for rate effects on an epoxy syntactic foam[END_REF]; they positioned the input bar strain gage near the impact end of the input bar to double the duration of the valid loading pulse duration. It is the purpose of our technical note to elucidate the advantages of this simple alternative configuration. It is presented as a modification of Kolsky's formulas for the use of much shorter input bars without sacrificing measurement accuracy or changing the length and magnitude of the loading pulse. 3

Modified Kolsky formulas

Figure 1 shows the set-up of a SHPB system composed of an elastic striker bar, input bar, specimen and output bar. We differentiate among the corresponding bar lengths i l , cross-sections i A , Young's moduli i E and wave propagation velocities i c through the

subscript i ,

  

at the input bar-specimen boundary ( in x x  ), we may compute the input velocity and force,

) ( re inc in in c v      (1)
) ( re inc in in in A E F     . ( 2
)
Analogously, we may compute the output velocity and force based on the strain history ),

tra out out c v    (3) tra out out out A E F   . ( 4
)
These three strain histories must be clearly identified based on the experimental measurement. The striker impedance must be smaller or equal to that of the input bar (striker rebound) and its length must be less than 50% of that of the input bar.

In a standard SHPB experiment, the assumption of quasi-static specimen equilibrium

) ( ) ( t F t F out in  (5)
needs to hold true. Assuming a homogeneous distribution of the axial stress and strain fields within the specimen, the engineering stress and strain rate histories within the specimen are approximated by the so-called three-wave formulas

s tra out out A A E t    ) ( (6) s tra out re inc in l c c t        ) ( ) (  , (7)
with s A and s l denoting the specimen cross-sectional area and length, respectively. From a signal processing point of view, the assumption of quasi-static equilibrium (5) may be exploited to obtain a relationship among the strain histories at the specimen boundaries.

For a standard system with three bars of identical diameters and material properties, we have

) ( ) ( ) ( t t t tra re inc      , (8a)
while for the more general case of bars of different impedance, the above equation reads

) ( ) ( ) ( t t t tra re inc       with in in out out A E A E   .
(8b) [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF] made use of Eq. (8a) to express the incident wave as a function of the reflected and transmitted strain histories, leading to the "standard Kolsky formulas" which express the stress and strain in the specimen as a function of the reflected and transmitted wave only. As an alternative to removing the incident wave from the final formulas, we make use of Eq. ( 8) to come up with modified Kolsky formulas for the stress and strain history in the specimen that do not require knowledge of the reflected wave. For standard case of identical bars, we have

s tra A A E t    ) ( ( 9a
) s tra inc l c t ) ( ) (       , (10a)
while for the general case, the modified Kolsky formulas read

A A E t tra out out    ) ( (9b) l c c c t tra out in inc in     ) ( 2 ) (     (11b)
This modification is of great advantage from an experimental point of view. In the case of the three-wave and standard Kolsky's formulas, the reflected wave needs to be measured which imposes a much stronger constraint on the validity of the strain history measurement as compared to the new approach, where the measurement of the incident wave is sufficient (in addition to the transmitted wave). Reliable measurements of the incident wave may be obtained from a strain gage positioned near the striker-input bar interface for a duration of approximately

in in c l t / 2  
. Furthermore, striker rebound is no longer required and its impedance may even be greater than that of the input bar. The duration of valid stress-strain curve measurements is therefore greatly increased (by a factor of two in theory) for a given input bar length when using the modified Kolsky formulas. It is emphasized that the above formulas are only valid if the assumption of quasi-static equilibrium holds true. For example, the assumption of quasi-static equilibrium is often not satisfied at the beginning of SHPB experiments or when testing brittle or low impedance materials.

Design of compact SHPB systems

Based on the modified Kolsky formulas, a compact SHPB system can be designed as follows:

1. Chose the duration T of the experiment. It is usually defined through the desired maximum strain max As illustrated in Fig. 1, we reemphasize that the strain gages are no longer positioned at the center of the bars. It is worth comparing the above recommendations with a conventional SPHB system that follows Kolsky's original design. In the case of bars of the same diameters and mechanical properties, the total length of a conventional system is about

st out in st l l l l 5   
. Assuming st in l D  , the length of the proposed compact SHPB system is only

st out in st l l l l 3   
. In other words, for the same experimental duration, the compact system is about 40% shorter than a conventional SHPB system.

Demonstration

We make use of mm 40 diameter aluminum input and output bars to demonstrate the validity of the proposed data processing method. Both bars are mm 2500 long. A first strain gage is positioned on the input bar at a distance of mm 350 from the input barstriker interface, while a second strain gage is positioned near the center of the input bar.

Only one strain gage is glued on the output bar at a distance of mm 300 from the specimen-output bar interface. Compression experiments are performed on a cylindrical lead specimen (diameter 25.5 mm, length 6.5 mm) at a strain rate of a about s / 300 . Two different striker bars will be employed to demonstrate the validity of the modified Kolsky formulas for experiments of short duration (where the reflected wave can be measured directly) and long duration (where a direct measurement of the reflected wave is not possible).

Conventional experiment of short duration

A mm 800
long aluminum striker bar is used to apply the dynamic loading at a striker speed of s m / 9 . 3 for a duration of about s  400 (including the rise and fall time of the incident wave). Figure 2a shows the incident and reflected strain histories as recorded at the center of the input bar along with the transmitted wave recording for the output bar.

Since the input bar is more than twice as long as the striker bar (made from the same material), the incident and reflected waves do not superpose at the center of the input bar.

Note that the maximum duration for valid measurements of the reflected wave is about s  420 for the mm 2500 long aluminum input bar. After reconstructing the waves and evaluating the stress histories at the specimen boundaries (taking material and geometric dispersion into account), we calculate the input and output force based on the Eqs. ( 2) and ( 4). Their comparison in Fig. 2b demonstrates that the assumption of quasi-static equilibrium is approximately satisfied. Subsequently, we compute the stress and strain histories within the specimen using 1. the transported strain histories measured by the first strain gage on the input bar along with the modified Kolsky formulas ( 9) and ( 10), 2. the transported strain histories measured by the second (central) strain gage on the input bar along with the three-wave formulas ( 6) and ( 7).

The corresponding stress-strain curves are shown in Fig. 3. Both curves are in good agreement which is seen as a partial validation of the proposed processing method.

Experiment of long duration

A mm 2500 long steel striker bar (diameter 40 mm) is used for the second experiment. The striker is launched at a speed of 2.35 m/s in order to achieve a similar rate of loading (at the specimen level), with an almost equal amplitude of the incident pulse, as in the previous experiment. However, unlike in the previous experiment, the duration of the loading pulse now exceeds the maximum duration for a valid measurement of the reflected wave, i.e. the incident and reflected wave overlap at the center of the input bar. Standard SHPB processing methods are therefore no longer applicable. Using the strain gage near the striker-input bar interface, the incident wave can be directly measured over a duration of up to about 850µs. The stress-strain curve as determined with the modified Kolsky formulas (9a) and (10a) is shown in Fig. 4.



  and the average strain rate  , the input bar strain gage is positioned at a distance of in D 5 from the striker-input bar 6 interface, with in D denoting the input bar diameter (see for instance[9] for a recommendation of the required minimum distance).

FiguresFigure 1 .

 1 Figures

  Figure 1. Compact SHPB system composed of three bars of (almost) equal length and two strain gages positioned near the striker-input bar interface and the specimen-output bar interface.

Figure 3 :

 3 Figure 3: Experiment of short duration: Stress-strain curve as determined using the standard method (based on measurement of incident, reflected and transmitted waves) and the proposed modified Kolsky formulas (based on measurements of the incident and transmitted waves only).

Figure 4 :

 4 Figure 4: Experiment of long duration: Stress-strain curve as determined using a wave deconvolution method (based on two strain history measurements on the input bar and the transmitted wave) and the proposed modified Kolsky formulas (based on measurements of the incident and transmitted waves only).

8

We make use of the wave deconvolution method proposed by [START_REF] Bussac | An optimization method for separating and rebuilding one-dimensional dispersive waves form multi-point measurements. Application to elastic or viscoelastic bars[END_REF] to validate the measured stress-strain curve. This method operates in the frequency domain; it is used to reconstruct all waves in the input bar based on the recordings of the first and second input bar strain gage. After calculating the strain histories at the specimen boundaries, the three-wave formulas are used to compute the stress-strain curve. The corresponding result is shown as a red curve in Fig. 4. It follows closely the previous result and is therefore seen as another partial validation of the proposed incident wave based processing method.

Conclusions

The duration of a split Hopkinson pressure bar experiment is limited by the duration of the loading pulse plateau which is a function of the striker bar length. The length of the input bar needs to be at least as long as the striker bar (if the same material is used) to avoid any shortening of the duration of the loading pulse plateau. In conventional SHPB systems, the input bar is at least twice as long as the striker bar to allow for valid measurements of the stress-strain curve for the specimen material. [START_REF] Song | Mechanical characterization at intermediate strain rates for rate effects on an epoxy syntactic foam[END_REF] positioned the input bar strain gage near the impact end to increase the duration of valid SHPB measurements. To emphasize the advantages of this configuration, and inspired by Kolsky's work (1949), we propose a simple signal processing formula which expresses the stress and strain history in the specimen as a function of the incident and transmitted waves only. It is thus sufficient to measure the incident strain wave in the input bar, while the measurement of the reflected wave may be omitted. The optimal position of the input bar strain gage moves from the center to the striker-input bar interface. At this position, the incident strain wave may be measured for the full duration of the loading even if the input bar is not longer than the striker bar. This observation may be explored either to build compact SHPB systems or to increase the duration of valid measurements for a system of a given length. Two examples are presented to demonstrate the validity of the proposed modified Kolsky formulas for SHPB experiments. It is emphasized that the proposed configuration is only valid for conventional SHPB experiments where the specimen is in quasi-static equilibrium.