Pierrick Malecot

Christian Lexcellent

Emmanuel Foltete

Manuel Collet

Shape memory alloys cyclic behavior : experimental study and modeling

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Shape Memory Alloys Cyclic Behavior: Experimental Study and Modeling 1 Introduction

The phase transformation from a mother phase called austenite to a product phase called martensite in the shape memory alloys ͑SMA͒ makes them potential candidates for vibration damping systems. SMA are already studied for applications in structural vibration control ͓1͔ and as vibration dampers for example in seismic applications ͓2,3͔. From a mechanical point of view we can distinguish, to explain this potential, two different phenomena linked to the phase transformation: the existence of an hysteresis and a stiffness variation. In order to understand the impact of these two phenomena on the dynamic of a system, a first study was presented in ͓4͔. In this previous work, an equivalent nonlinear complex Young's modulus was used to quantify the balanced harmonic response of the material. It was calculated using a simplified RL model ͓5͔ without taking into account the modification of the behavior with the cycling and the strain rate. A new phenomenological model is developed in the present paper in order to take into account the SMA behavior alteration due to the cycling. Basically, an experimental study is performed in order to quantify the SMA behavior modification generated by the cycling and the strain rate. This experimental study allows to analyze the thermal dissipation and the evolution of the hysteresis width. A review of the literature about experimental works on SMA cyclic behavior ͓6-9͔ led to the conclusion that no previous experimental study had been driven on Cu-Al-Be SMA, so we mainly used this material to enlarge our skills field.

The present paper is divided into 4 parts:

1. A complete experimental study is presented. First, a Ni-Ti alloy was briefly studied by performing a quasistatic cyclic tensile test and a tensile cyclic test at 5 Hz with increasing maximum strain. These tests were mainly performed on JRC-ISPC in ISPRA ͑Italy͒. Then two Cu-Al-Be were studied, the first one was austenitic at room temperature and the second one was martensitic at room temperature. For these two materials, large strain rate amplitudes were applied on cyclic tensile tests. Moreover an infrared camera was used to measure the temperature variation on the sample surface during each test. 2. A phenomenological cyclic thermomechanical model is developed. This modeling is based on the general thermodynamic framework built by Raniecki and Lexcellent ͓5͔ where three internal variables have been introduced: the residual stress, the residual strain, and the volume fraction of residual martensite at free stress state accumulated during the cycling. In order to complete this part, a modeling of the thermal dissipation during cyclic tests is developed by performing a study based on the heat equation. 3. In this part the concept of equivalent complex Young's modulus is introduced in order to describe the damping and the stiffness of the SMA samples. Therefore this modulus is linked to the phenomenological model through the strain energy densities over one period and one quarter of a period. 4. The numerical predictions given by the cyclic thermomechanical model are presented and compared to the experimental results. The impact of the phenomenological cyclic modeling on the equivalent complex Young's modulus real and imaginary parts is discussed. Finally, temperature variation predictions are presented.

1 Corresponding author.

Shape memory alloys (SMA) are good candidates for being integrated in composite laminates where they can be used as passive dampers, strain sensors, stiffness or shape drivers. In order to improve the SMA modeling and develop the use of these alloys in structural vibration control, better understandings of cyclic behavior and thermal dissipation are needed. The present study investigates experimentally the cyclic behavior of SMA and more particularly, the influence of strain rates on three different materials. The thermal dissipation aspect is also studied using an infrared camera. A phenomenological model based on the RL model [START_REF] Raniecki | R L Models of Pseudoelasticity and Their Specifications for Shape Memory Solids[END_REF], "RL Models of Pseudoelasticity and Their Specification for Shape Memory Solids." Eur. J. A/Solids, 13, pp. 21-50) is then presented with the intention of modeling the behavior's alterations due to the cycling. By introducing the thermodynamic first principle, a study of the heat equation is developed in order to predict the temperature evolution during a cyclic tensile test. Furthermore, in order to model the damping effect created by the hysteresis phenomenon and the stiffness variation due to the phase transformation, an equivalent nonlinear complex Young's modulus is introduced. This notion usually used for viscoelastic materials is adapted here to SMA. Moreover, the impact of cycling on the equivalent modulus is presented. As a conclusion, a numerical results panel obtained with the phenomenological cyclic SMA model, the heat equation, and the equivalent complex Young modulus is presented.

Keywords: shape memory alloys, cyclic behavior, thermal dissipation, thermomechanical modeling, nonlinear modeling The first for modeling the cyclic behavior of SMA is to build a database with experimental tests on different materials. Three different materials were studied: a Ni-Ti and two Cu-Al-Be. The Ni-Ti and the first Cu-Al-Be were austenitics at room temperature while the second Cu-Al-Be was martensitic at room temperature. These three materials allowed to explore a large field of SMA behaviors in order to quantify the effect of cycling and the strain rate impact.

Ni-Ti Study.

For this material the tests were performed partly in our laboratory on an electrical powered tensile machine Instron 6250. The other part of the tests was done on the JRC-ISPC laboratory tensile machines. Both of these tests were performed on 2 mm diameter Ni-Ti wires.

2.1.1 Quasistatic Cyclic Tests. This test was performed at a strain rate of 10 -4 s -1 . On the strain-stress curve in Fig. 1 we can observe all the "classical" elements of a cyclic test. The first element to notice is the decrease of the outset of the direct phase transformation with the number of cycles. For the outset of the reverse phase transformation, no significant variation occurs during the cycling ͑Fig. 2͒. The second important fact to point out is a residual strain remaining for a free stress state. At last after a given number of cycles the sample tends to reach a stabilized behavior.

Basically for all the observed parameters the most important variation occurred between the first and the second cycle and less significantly between the second and third cycle, then the parameters variations are slower.

Strain Rate Influence.

In order to study the strain rate impact and the maximum strain influence, the test presented in Fig. 3 was performed. The sample was loaded under a cyclic, tensile deformation with an increasing maximum strain from 3.5% to 8% at a frequency of 5 Hz ͑the equivalent strain rates are given in Table 1͒. The shape of the cycles is very different from the one observed on the quasistatic tests. The phase transformation plateaus for direct and reverse transformations have disappeared. Moreover the area of the cycles is smaller and the stress reached for a strain of 7% is about 900 MPa versus 700 MPa for quasistatic tests.

From a thermal point of view a huge increase of the temperature ͑Fig. 4͒ from 25°C to 60°C can be noticed. Furthermore the slope of the temperature curve increases with the increase of the maximum stress. For this test the temperature variation was measured by a thermal probe. The tests performed on this material were done on an hydraulic powered tensile machine referenced Instron 8501. The maximum strain was imposed and controlled by an extensometer. In order to measure the thermal dissipation a CEDIP infrared system was used. This system consists of an infrared camera referenced Jade MWIR and the ALTAIR software ͓10͔. Cylindrical Cu-Al-Be samples designed for tensilecompression tests were used. The samples were loaded under a cyclic, tensile, sinusoidal deformation between the strains of 0% and 7%. It is important to underline that each test was performed on a virgin sample.

2.2.1 Quasistatic Cyclic Tests. This test was performed at a strain rate of 10 -4 s -1 . On the Cu-Al-Be strain-stress curves ͑Fig. 5͒ almost the same behaviors than for Ni-Ti can be observed: decrease of the outset of the direct phase transformation, decrease of the cycle area with the number of cycles, and no significant variation of the outset of the reverse phase transformation. Nevertheless an important difference can be underlined: the residual strain at a free stress state is more important, 3% for the Cu-Al-Be versus 1% for the Ni-Ti. At last it is important to notice the "memory point" aspect, also called "return memory point" by Ortin ͓11͔͑i.e., at the end of the loading for each cycle the sample finds back the same stress-strain state; 440 MPa for 7%͒.

Like for the Ni-Ti the most important variation for all the observed parameters occurred between the first and the second cycle and less significantly between the second and third cycle.

Strain Rate Influence on Mechanical

Behavior. In order to study the strain rate influence on the cyclic behavior of this material, cyclic, tensile, sinusoidal tests were performed at four different strain rates ͑including quasistatic tests͒ from 10 -4 s -1 to 2 ϫ 10 -1 s -1 , the results are given in Fig. 6. In this figure the shape of the hysteresis for each strain rate are compared after 100 cycles. For Piedboeuf et al. ͓9͔ the mechanical behavior is stabilized after 100 cycles. After the study of the superposition of the four stabilized cycles for this Cu-Al-Be it is clear that the strain rate has no significant influence when it comes to compare of shape, area and apparent slope of the stabilized hysteresis cycle.

Strain Rate Influence on Thermal Dissipation.

For the first test at 10 -4 s -1 Fig. 7͑a͒ shows an isothermal profile during the cycling, and the temperature variation is around 0.2°C which is not significant compared to the room temperature variation. For the second test at 10 -2 s -1 the thermal profile is different ͑Fig. 7͑b͒͒, the first cycle reaches the maximum temperature of 27.5°C.

Then, the maximum temperature decreases cycle after cycle to reach 26.5°C in the stabilized state. During each cycle of this test the temperature variation is about 4 °C. The last two tests at 10 -1 s -1 and 2 ϫ 10 -1 s -1 exhibit the same thermal profile ͑Figs. 8͑a͒ and 8͑b͒͒ the temperature constantly increases during the cycling even after more than 200 cycles. Compared to the room temperature, the increase is about 4 °C after 50 cycles. The differences between the tests come from the different equilibrium between the heat diffusion produced by the phase transformation in the material and the heat dissipation capacity of the boundaries. This experimental study demonstrates that for an increasing strain rate the used Cu-Al-Be has a different behavior than the Ni-Ti. For this Cu-Al-Be there is no significant change of the stabilized cycle when the strain rate is multiplied by 2000, whereas the thermal dissipation is greatly influenced.

Cu-Al-Be

Martensitic at Room Temperature. Finally, in order to complete the experimental database, a Cu-Al-Be martensitic at room temperature was tested ͑Cu-Al-Be ͑2͒͒. The testing conditions for this material are the same than previously. In a practical way, the samples were prestrained at 1.65% and loaded under a cyclic, tensile, sinusoidal deformation between the strains of 1.30% and 2%. It is important to underline that each test was performed on a virgin sample.

Quasistatic Cyclic Tests.

The material behavior is very different from the two others. Indeed this material is martensitic at room temperature and there is no phase transformation during the cycling but martensite reorientation. As shown in Fig. 9, the stress-strain curve given by this quasistatic test at 10 -4 s -1 is very different from the other quasistatic curves ͑Figs. 1 and 5͒. A small decrease of the cycle area and a significant variation of the sample stiffness during the cycling can be observed. It is relevant to notice once again that for all the observed parameters the most important variation occurred between the first and the second cycle and less significantly between the second and third cycle. Finally, after a given number of cycles the sample tends to reach a stabilized behavior.

Strain Rate Influence on Mechanical

Behavior. For this material the previous method was used in order to study the strain rate impact on the cyclic behavior: cyclic, tensile, sinusoidal tests at four different strain rates ͑including quasi static tests͒ from 10 -4 s -1 to 10 -1 s -1 were performed. The results are given in Fig. 10 where the shape of the stabilized cycle obtained after 100 cycles are compared for each strain rate. The superposition of the four stabilized cycles shows no significant influence when it comes to compare shape, area, and apparent slope.

Strain Rate Influence on Thermal Dissipation.

For this material each strain rate gives a different thermal curve. For the quasistatic-test ͑Fig. 11͑a͒͒ a temperature variation between 22.25°C and 22.5°C can be observed for each cycle ͑the room temperature is around 22.3°C͒. For the second test ͑Fig. 11͑b͒͒,at a strain rate 10 times higher, the first cycle reaches the maximum temperature of the test ͑24.75°C͒. Then the maximum temperature decreases during around 10 cycles, afterwards it is stabilized between 24.15°C and 24.55°C ͑the room temperature is around 24.5°C͒. For the third test ͑Fig. 12͑a͒͒ at 10 -2 s -1 the reverse phenomenon can be noticed. Attention can be paid to an increase of the temperature during 20 cycles from the room temperature ͑25.25°C͒ to a maximum temperature of 26°C. After these 20 cycles the temperature variation is stabilized between 25.6°C and 26°C. The last test ͑Fig. 12͑b͒͒ shows a constant increase of the temperature from 26°C to 30°C, and there is no temperature stabilization after more than 200 cycles.

As a conclusion to this experimental set it is important to highlight that for this Cu-Al-Be the heat produced by the sample does not come from phase transformation but from martensite platelets rubbing and less significantly from martensite platelets reorientation. Besides no significant differences are observed for this ma- terial between stabilized cycles when the strain rate is multiplied by 1000, but an important impact of the strain rate on the heat dissipation can be underlined.

Conclusions.

Large amplitudes of strain rates on three different materials have been studied. The three materials show different behaviors. The more sensitive material to the strain rate increase is the Ni-Ti with a huge heat dissipation and a significant decrease of the hysteresis cycle area leading to a decrease of the dissipated mechanical energy. For the two Cu-Al-Be similar conclusions can be done: there is no significant influence of the strain rate on the mechanical behavior but a relevant influence on the thermal dissipation. The martensitic Cu-Al-Be at room temperature proves to be more stable during the cycling, that is why this material seems to have a great potential for structural damping.

Modeling of Cyclic Behavior

A few good models exist in the literature ͓12-15͔ to model the cyclic behavior of SMA. An improved version of the model proposed by Lexcellent and Bourbon ͓12͔ is first developed here. Then as the experimental results presented in this paper show the importance of the thermal diffusion, a complete heat equation study based on the first principle of the thermodynamic ͑the conservation law͒ is presented.

Cyclic Thermomechanical

Modeling. This modeling of the SMA cyclic behavior is based on the general thermodynamic framework used by Raniecki and Lexcellent ͓5͔ in the R L model. Furthermore, the present modeling uses parts of the unified SMA model by Leclerq and Lexcellent ͓16͔. The main hypothesis of this unified model is to distinguish two kinds of martensites: the oriented martensite z and the self-accommodating martensite z T with z = z + z T ͑1͒

Residual Martensite.

A new hypothesis is added to the R L model: the residual strain at a free stress state after a given number of loading and unloading is given by the volume fraction of residual martensite. In order to link this two parameters the same kind of relation than the usual behavior law tr = ␥ • z can be used, where ␥ is the maximum strain associated to a complete phase transformation during a tensile test,

ir = ␥ . h ͑2͒
where ir is the residual strain at =0, h is the fraction of residual martensite. The difference between h and z is similar to the difference between the notions of total and permanent strains in the classical theory of plasticity. The introduction of this new variable involves a few hypotheses about its variation. We assume that the existence and growing of the residual fraction of martensite are dependent on the volume fraction of oriented martensite,

z ˙ Ͼ 0 Þ h ˙ജ 0 z ˙ ഛ 0 Þ h ˙=0 ͑3͒
We also stipulate a second hypothesis concerning the existence of a stabilized cycle; this hypothesis is confirmed by the experiments presented in the previous part of this paper and by the work of Myazaki ͓17͔. This hypothesis imposes a saturation value h ϱ for the residual martensite.

Modeling.

The model is developed by writing the specific free energy of the system at a nonequilibrium state. The system consists of three phases: ͑1͒ the mother phase austenite, ͑2͒ the self-accommodating martensite, ͑3͒ the oriented martensite. The specific free energy of each phase is defined by

͑1͒ = u 0 *͑1͒ -T . s 0 *͑1͒ + E 2. ͑ ͑1͒ -T ͑1͒ ͒ 2 + C v ͩT-T 0 -T •lnͩ T T 0 ͪͪ+ h *͑1͒ ͑2͒ = u 0 *͑2͒ -T . s 0 *͑2͒ + E 2. ͑ ͑2͒ -␥ . z -T ͑2͒ ͒ 2 + C v ͩT-T 0 -T •lnͩ T T 0 ͪͪ+ h *͑2͒ ͑3͒ = u 0 *͑3͒ -T . s 0 *͑3͒ + E 2. ͑ ͑3͒ -␥ . z -T ͑3͒ ͒ 2 + C v ͩT-T 0 -T •lnͩ T T 0 ͪͪ+ h *͑3͒

͑4͒

The term h *͑␣͒ is the contribution of the cycling to the free energy, and is defined by the equation

h *͑␣͒ ͑h,T͒ = u h *͑␣͒ -T . s h *͑␣͒ avec h *͑␣͒ ͑0,T͒ =0 ͑5͒
The specific free energy of the system is written as follows:

= ͑1-z͒ ͑1͒ + z T ͑2͒ + z ͑3͒ + ⌬ ͑6͒ with ⌬ = z T ͑1-z͒ 12 + z ͑1-z͒ 13 + z T z 23 ͑7͒
where ␣␤ is the energy of interaction between phases ␣ and ␤.It is important to underline that the two kinds of martensite cannot be distinguished from a physical point of view so

12 = 13 = it 23 = m it u 0 2 = u 0 3 = u 0 m s 0 2 = s 0 3 = s 0 m ͑8͒
Thus, at a constrained equilibrium state, the specific free energy of the system can be written:

͑,T,z T ,z ,h͒ = E 2. ͑ -␥ . z -T ͒ 2 + C v ͩ͑T-T 0 ͒ -T lnͩ T T 0 ͪͪ+ * ͑T,z͒ + tr ͑T,z,h͒͑ 9͒ with ͩ-. ץ ץ ͪ ˙-. ͩs+ ץ ץT ͪT ˙-ץ ץz z ˙- ץ ץz T z T ˙-. ץ ץh h ˙ജ 0 ͑12͒
So, the entropy and the stress are written

s =- ץ ץT = ␣ 0 + C v lnͩ T T 0 ͪ+z͑1-z͒s 0 + s 0 *͑1͒ -z⌬s * + s h *͑1͒ ͑h͒ -z⌬s h ͑h͒͑ 13a͒ = . ץ ץ = E͑ -␥z -␣ 0 ͑T -T 0 ͒͒ ͑13b͒
The three thermodynamical forces associated to z , z T , and h are

⌸ f =- ץ ץz = ⌸ 0 f + ␥ . -͑1-2z͒ it -z T it m + ⌬⌸ 0 f ͑h,T͒ ⌸ T f =- ץ ץz T = ⌸ 0 f + ␥ . -͑1-2z͒ it -z it m ⌸ h =- ץ ץh =- ץ tr ץh = z ⌬͑ץ⌸ 0 f ͑h . T͒͒ ץh - ץ h *͑1͒ ץh ͑14͒ Moreover, by introducing Ά u h *͑2͒ ͑h͒ = 1 2 u ញ 2 h 2 -u 2 h s h *͑2͒ ͑h͒ = 1 2 s ញ 2 h 2 -s 2 h · ͑15͒ ͭ ញ ͑T͒ = u ញ 2 -T . s ញ 2 ¯͑T͒ = u ¯2 -Ts ¯2 ͮ ͑16͒ we obtain ⌬⌸ 0 f =- 1 2 ញ h 2 + ¯h ͑17͒
The existence of a stabilized cycle implies

when h → h ϱ ⌬ץ⌸ 0 f ץh =0Þ h ϱ = ញ ͑18͒
The complete description of the evolution of the thermodynamical behavior requires the knowledge of the kinetics equations for the two variables z and h. They should be formulated in a way such that the Clausius Duhem inequality is satisfied

dD = ⌸ f dz + ⌸ T f dz T + ⌸ h dh ജ 0 ͑19͒
where dD is the infinitesimal energy dissipation.

Transformations Kinetics.

By analogy with classical plasticity, four yield functions are defined: ⌿ AM ␣ for the direct transformation and ⌿ MA ␣ for the reverse one, with ␣ = for the oriented martensite and ␣ = T for the self-accommodated martensite. These functions are defined by

AM = ⌸ f -k AM MA =-⌸ f + k MA AM T = ⌸ T f -k AM T MA =-⌸ T f + k MA T ͑20͒
Reporting to the model developed by Leclerq ͓16͔, and applied to a cyclic loading, the functions k are defined by

k AM =2 it z -z m -h 1-h - ⌬s * a AM log 1-z + z m 1-h -⌬s * ͫ ͑T -T * ͒ + b AM b m * ͑exp͑-b m * ͑T -M s 0 ͒͒ -exp͑-b m * ͑T * -M s 0 ͒͒͒ ͬ ͑21a͒ k MA =2 it 1-z + z m 1-h - ⌬s * a MA log z -z m -h 1-h -⌬s * ͫ ͑T -T * ͒ + b MA b m ͑exp͑-b m ͑T -A s 0 ͒͒ -exp͑-b m ͑T * -M s 0 ͒͒͒ ͬ ͑21b͒ and k AM T =2 it ͑z T -z T m ͒ - ⌬s * a AM T log͑1-z T + z T m ͒͑ 22a͒ k MA T =2 it ͑z T -z T m -1͒ - ⌬s * a MA T log͑z T -z T m ͒͑ 22b͒ with a ␤ ␣ , b ␤ , b m * , b m ͑␣ = ,
T and ␤ = AM, MA͒ material parameters, and T * is the initial temperature.

The h parameter kinetic must satisfy three properties: 1. h ϱ must exist; 2. dhജ 0 when dz Ͼ 0; 3. dh= 0 when dz ഛ 0.

The simplest kinetic to comply with these properties is

dh = m 0 ͑h ϱ -h͒ p . ͗dz ͘ where ͭ ͗x͘ = xi f x ജ 0 ͗x͘ =0 if x Ͻ 0 ͮ ͑23͒
The identification of all the necessary parameters is fully explained in ͓12,16͔. Equation ͑20͒ permits us to define the transformation phases yield function and direction, so with Eqs.

͑21͒-͑23͒ we are now able to completely model the cyclic behavior of a shape memory alloy.

Thermal Dissipation

Modeling. This part presents a complete study of the heat equation using the model developed in Sec. 3.1. In the following study, the heat equation is used to calculate the temperature variation during the different tests. Indeed, the proposed model aims at linking the temperature to the different mechanical parameters.

Nature of the Problem.

The energy conservation law ͑first thermodynamic principle͒ allows us to write . u ˙= . ˙+ r -div͑q͒͑ 24͒ with u specific internal energy defined by u = + T, s; r the internal heat production volume density; q is the heat current vector, where is defined by Eq. ͑9͒ and s by Eq. ͑13a͒.

In order to obtain the usual expression of the heat equation, the equation u = + Ts has to be introduced in Eq. ͑24͒. Considering

u ˙= ˙+ sT ˙+ Ts ˙͑25͒ with ˙= ץ ץ ˙+ ץ ץT T ˙+ ץ ץz z ˙ + ץ ץz T z ˙T + ץ ץh h ˙͑26͒
and with

s ˙=-ͩ ץ 2 Tץץ ˙+ ץ 2 ץ 2 T T ˙+ ץ 2 ץz ץT z ˙ + ץ 2 ץz T ץT z ˙T + ץ 2 ץhץT h ˙ͪ ͑27͒
we have to calculate

ץ ␣ץ and ץ 2 Tץ␣ץ with ␣ = ͑,T,z ,z T ,h͒ 3.2.2 First Derivative Calculation ץ ץ = ͑28͒ ץ ץT =-s =-C v lnͩ T T 0 ͪ-s 0 *͑1͒ + z . ⌬s * -z͑1-z͒ . s 0 -z . z T . s 0 m -s h *͑1͒ + z . ⌬s h * ͑29͒ ץ ץz = z T . ir m - E . ␥ ͑ -␥ . z -␣ 0 ͑T -T 0 ͒͒ -⌬⌸ 0 f ͑h,T͒ -⌸ 0 f ͑T͒ + ͑1-2.z͒ it ͑30͒ ץ ץz T = z . ir m - E . ␥ ͑ -␥ . z -␣ 0 ͑T -T 0 ͒͒ -⌬⌸ 0 f ͑h,T͒ -⌸ 0 f ͑T͒ + ͑1-2.z͒ it ͑31͒ ץ ץh = ͑u 1 ញ -Ts 1 ញ ͒ . h -͑u 1 -T . s 1 ͒ -.z . ͑ ញ . h + ¯͒͑ 32͒
ץ / ץh is calculated using Eqs. ͑15͒-͑17͒.

Second Derivatives Calculation

ץ 2 ץ 2 T = -C v T ͑33͒ ץ 2 Tץץ =-␣ 0 E ͑34͒ ץ 2 zץץ =-s 0 m z T + E␥␣ 0 + ⌬s h * ͑h͒ + ⌬s * -͑1-2z͒s 0 ͑35͒ ץ 2 zץץ T =-s 0 m z + E␥␣ 0 + ⌬s h * ͑h͒ + ⌬s * -͑1-2z͒s 0 ͑36͒ ץ 2 hץץ =-s 1 ញ h + s 1 -zͩ ץ ញ ץT h + ץ ץ T ͪ ͑37͒
The evolutions of ញ and ¯with T are explained in ͓12,18͔. .If we consider that the heat conduction follows an isotropic, linear Fourier law, the heat current vector becomes q =-k . grad͑T͒͑ 38͒ Thus Eq. ͑24͒ becomes u ˙= ˙+ k⌬T ͑39͒

Heat Equation

where ⌬ is the Laplacian operator. From Eqs. ͑25͒ and ͑39͒ we can write

u ˙= ˙+ k⌬T = ͑ ˙+ sT ˙+ Ts ˙͒͑ 40͒
with T ˙the particular derivative of T as

T ˙= ץT ץt + v . grad͑T͒͑ 41͒
where y is the particle speed ͑negligible for metallic materials load at low rates͒. Substituting Eqs. ͑41͒ and ͑28͒-͑37͒ in Eq. ͑40͒ we obtain the following result, which links the temperature variation to the mechanical parameters:

k⌬T -C v ץT ץt + h ˙Tzͩ ץ ញ ץT h + ץ ץ T ͪ = ͩz T u 0 m - E . ␥ ͑ -␥ . z + ␣ 0 T 0 ͒ -⌬u h * -⌬u * + ͑1-2.z͒u 0 ͪz ˙+ ͩz u 0 m - E . ␥ ͑ -␥z + ␣ 0 T 0 ͒ -⌬u h * -⌬u * + ͑1-2z͒u 0 ͪz T ˙-͑z͑ ញ h + ¯͒͒h ˙. ͑42͒
Numerical results obtained with this equation are presented in Sec. 5.

Nonlinear Behavior Modeling

4.1 Equivalent Complex Young's Modulus Notion. The equivalent complex Young's modulus is a powerful tool usually used to describe the elastic behavior and the damping properties of a viscoelastic material under dynamic loading ͓19,20͔. This equivalent modulus can be generalized to quantify, on the first harmonic, the response of a nonlinear hysteretic material.

First of all let us recall that for an isotropic viscoelastic material, the relaxation modulus E͑t͒ is defined by

͑t͒ = ͵ -ϱ t E͑t -͒ d͑͒ d d ͑43͒
where ͑t͒ and ͑t͒ are the stress and the strain defined for a uniaxial relaxation test. Furthermore, E͑t͒ is the sum of an asymptotic and a time dependant term

E͑t͒ = E ϱ + E * ͑t͒͑ 44͒ where E * ͑t͒ → 0 when t → ϱ.
Thus, Eq. ͑43͒ becomes

͑t͒ = E ϱ͵ -ϱ t d͑͒ d d + ͵ -ϱ t E * ͑t -͒ d͑͒ d d ͑45͒
and can be written

͑t͒ = E ϱ ͑t͒ + ͵ -ϱ t E * ͑t -͒ d͑͒ d d ͑46͒
By using a Fourier transform on the last equation we obtain

͑͒ = E ϱ ͑͒ + iE * ͑͒͑͒͑ 47͒ The complex term E * ͑͒ is defined by E * ͑͒ = E r * ͑͒ + iE i * ͑͒͑ 48͒
Assuming Eq. ͑48͒, Eq. ͑47͒ becomes

͑͒ = ͑E ϱ -E i * ͑͒ + iE r * ͑͒͒͑͒͑ 49͒
which can be written

͑͒ = E ¯͑͒͑͒͑ 50͒
where E ¯͑͒ is the equivalent complex Young Modulus. It can also be written E ¯͑͒ = E͑͒͑1+i E͑͒ ͒͑ 51͒ For the SMA, the hypothesis of an hysteretic damping can be stipulated. This assumption entails E͑͒ = E 0 and E͑͒ = 0 , thus

E ¯͑͒ = E 0 ͑1+i 0 ͒͑ 52͒

Equivalent Complex Youngs Modulus: Application to

SMA. If we consider a given vibrating state, the equivalent stiffness and damping of the material can be modeled by the following expression:

E ¯= E * ͑1+i͒͑ 53͒
In order to reach the harmonical balance on the first harmonic, an energetic equivalence is used. This harmonic balance is a generalization to the Fourier space of the averaging harmonic method developed by Krylov and Bogoliubov ͓21͔ and Bogoliubov and Mitropolsky ͓22͔.

Defining W 1 and W 1/4 the strain energy densities, respectively, accumulated during one period of vibration and one quarter of a period,

W 1 = ͵ 0 T :d W 1/4 = ͵ 0 T/4 :d ͑54͒
the harmonic balance leads to

W 1 = E * m 2 W 1/4 = E * 2 ͩ1+ 2 ͪ m 2

͑55͒

with m the maximum strain during the loading. It appears that the equivalent complex Young's modulus only depends on the chosen model and should be useful for a finite element simulation. Such approach was used in ͓23͔ in order to characterize the pseudoelastic damping behavior of SMA wires. A complete example based on a simplified R L model is also presented in ͓1͔ for the study of a SMA cantilever beam. The simplified R L is illustrated in Fig. 13. The original contribution of our study consists of studying the influence of cycling on the stiffness variation and damping by computing W 1 and W 1/4 with the cyclic model presented in Sec. 3. The numerical results of this study are given in the next section.

Numerical Results

In this section are presented the results of the numerical simulation on the Ni-Ti used in Sec. 2. The parameters used for these simulations are given in the Appendix.

The first result presented in Fig. 14 is the strain-stress curve obtained with the cyclic model presented in Sec. 3 of this paper. For the 20 cycles presented, the simulation seems to be good from a qualitative point of view. Indeed we can notice a decrease of the outset of the direct phase transformation, an increase of the residual strain with the number of cycle. Besides the experimental tests the most important variation occurs between the first and the second cycle and less significantly between the second and third cycle. At last the appearance of a stabilized cycle can be observed.

For the first and the 20th cycles Figs. 15͑a͒ and 15͑b͒ present the comparison between the experimental and the simulated cycles. For most parameters the results are convincing. For the first cycle the simulation is in very good agreement with the experimental curve. For the 20th cycle the agreement is good for the outset of the direct phase transformation and for the residual strain but the result is less obvious for the outset of the reverse phase transformation. This problem is intrinsic to the proposed model and has to be studied further.

In order to confirm the feeling of these first observations, Figs. 16͑a͒ and 16͑b͒ are proposed. For each cycle between the first and the 20th, the outset of the direct phase transformation is well predicted as confirmed by Fig. 16͑a͒. In Fig. 16͑b͒ we can note the good agreement between the computed and the experimental fraction of residual martensite. The residual fraction of martensite is calculated from the residual strain using Eq. ͑2͒.

As a conclusion to the cyclic model, it is important to comment the influence of the cyclic behavior of the Ni-Ti on the nonlinear modeling. In Fig. 17 we can observe on the real part of the equivalent complex Young's modulus that the stiffness decreases with the number of cycles. On the imaginary part the damping seems to increase. This simple observation shows the importance in taking into account the change of the behavior with the cycling to realize a good prediction of the nonlinear behavior thanks to the equivalent complex Young's modulus. At last, predictions of the temperature variations during cyclic tests using the heat equation modeling presented in Sec. 4 are done for the austenitic Cu-Al-Be. These simulations are realized on 50 cycles. The results for the four strain rates from 10 -4 to 2 ϫ 10 -1 s -1 are presented in Figs. 18 and19. We can observe in these figures that the thermal behavior is well predicted excepted for the strain rate of 2 ϫ 10 -1 s -1 where a small drift can be noticed. It is important to highlight that the results observed in Sec. 2 for this material allows us to use a weak coupling between mechanical and thermal behaviors.

Conclusions

The cyclic behavior of the three alloys has been experimentally studied. For the Ni-Ti, the strain rate has an important influence on the mechanical behavior, but for the two Cu-Al-Be there are no significant differences on the stabilized cycles with the strain rate. For these two copper alloys, the most important fact to observe is the huge heat dissipation during the cycling. In the second part a model of the cyclic behavior of the SMA based on the notion of residual fraction of martensite was presented. It was built on an R L model base and a complete study of the heat equation was added in order to be able to predict the temperature variation during the cycling.

In the third part of the paper a simple method was presented in order to model the nonlinear behavior of SMA during dynamical loading. This equivalent complex Young's modulus is a powerful tool to be used with the finite element method for simulations.

At last the results of a few numerical simulations were presented. They show clearly that it is highly important to care about the change of the behavior with the cycling to predict the stiffness and damping variations. Moreover we succeeded in predicting the temperature variations for an austenitic SMA.

In the future, the next step will consist of using the work made on the heat equation in order to compute the temperature during the different tests for martensitic shape memory alloys. Furthermore we want to improve our knowledge of damping on SMA by performing some experiments on Cu-Al-Be wires using a nonlinear identification method based on the wavelet transform.

This study permits us to improve the nonlinear modeling of SMA in order to develop and optimize applications for the control and the damping in civil engineering. MA ϭ outset stress of the reverse phase transforma- tion M → A ␣ ϭ 1 for the austenite ␣ ϭ 2 for the oriented martensite, 3 for the self- accommodated martensite

Mechanical variables

ϭtotal strain ir ϭ residual strain at a free stress state T ϭ total thermal strain ͑␣͒ ϭ ␣ phase strain T ͑␣͒ ϭ ␣ phase thermal strain ϭ strain Thermodynamical variables and functions T ϭ system temperature T 0 ϭ reference temperature C v ϭ specific heat u 0 ϭ system configuration internal energy s 0 ϭ system configuration entropy it ϭ system configuration free specific energy ͑␣͒ ϭ ␣ phase free specific energy ϭ biphased system free specific energy u 0 *͑␣͒ ϭ ␣ phase internal energy at reference state s o *͑␣͒ ϭ ␣ phase entropy at reference state ⌸ 0 f ϭ temperature induce phase transformation at free stress state driving force ⌸ ␣ f ϭ phase transformation driving force ͑associated with z ␣ , ␣ = T, ͒ ⌸ h ϭ phase transformation driving force ͑associated with h͒ ͑␣͒ ϭ ␣ phase transformation yield function k ͑␣͒ ϭ ␣ phase transformation kinetic s ϭ system specific entropy u ϭ system free energy q ϭ system specific heat transfer ⌬⌸ 0 f ϭ thermodynamic force associated with h ញ , ¯ϭ T functions defining ⌬⌸ 0 f m 0 , p ϭ constants associated with h kinetics h ϱ ϭ h saturation value

Appendix: Simulations Main Parameters

Ni-Ti

ϭ 6500 kg m -3 E ϭ 60,000 MPa ␣ 0 ϭ 11e -6 K -1 ␥ ϭ 6% M s 0 ϭ 213 K A s 0 ϭ 248 K u 0 ϭ 1997 J Kg -1 s 0 ϭ 3.13 J kg -1 K -1 ⌬u ¯ϭ 15,898 J kg -1 ⌬s ¯ϭ 68 J kg -1 K -1 ញ ϭ 70,000 J kg -1 h ϱ ϭ 0.2 m 0 ϭ 2.1 p ϭ 1

Cu-Al-Be(1)

ϭ 7500 kg m -3 E ϭ 75,000 MPa ␣ 0 ϭ 17e -6 K -1 ␥ ϭ 6% M s 0 ϭ 213 K A s 0 ϭ 253 K C v ϭ 794 J kg -1 K -1 @25°C u 0 ϭ 472 J kg -1 s 0 ϭ 0Jk g -1 K -1 ⌬u ¯ϭ 5600 J kg -1 ⌬s ¯ϭ 24 J kg -1 K -1 ញ ϭ 100,000 J kg -1 h ϱ ϭ 0.3 m 0 ϭ 10 p ϭ 1.1 u 1 ϭ 9.7e 4 Jkg -1 u 1 ញ ϭ 1.8e 6 Jkg -1

FigFig. 2

 2 Fig. 1 Ni-Ti cyclic test at 25°C,

Fig. 4

 4 Fig. 4 Ni-Ti dynamic test "5H z …: temperature variation

Fig

  Fig. 5 Cu-Al-Be"1… cyclic test at 22°C, ˙=10 -4 s -1 Fig. 6 Cu-Al-Be"1… stabilized cycles comparison after 100 cycles

Fig. 8

 8 Fig. 8 Cu-Al-Be"1… temperature variation: "a… test at 10 -1 s -1 ; "b… test at 2 Ã 10 -1 s -1

Fig. 12

 12 Fig. 12 Cu-Al-Be"2… temperature variation: "a… test at 10 -2 s -1 ; "b… test at 10 -1 s -1 Fig. 11 Cu-Al-Be"2… temperature variation: "a… test at 10 -4 s -1 ; "b… test at 10 -3 s -1

Fig. 13

 13 Fig. 13 Simplified nonlinear model of the adiabatic transition

Fig. 14

 14 Fig. 14 Ni-Ti quasistatic cyclic test: numerical simulation

Acknowledgment

  Fig. 18 Cu-Al-Be"1… temperature variation simulation: "a… test at 10 -4 s -1 ; "b… test at 10 -2 s -1 Fig. 19 Cu-Al-Be"1… temperature variation simulation: "a… test at 10 -1 s -1 ; "b… test at 2 Ã 10 -1 s -1

Table 1 Equivalent strain rate for a frequency of 5 Hz

	Maximum strain ͑%͒	3.5	5	6.5	8
	Equivalent strain rate 10 -1 s -1	1.2	2.7	4.2	5.7


David Naccache
email: david.naccache@ens.fr

Ofer Yifrach-Stav

A Note on Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

. Note that it is now known [1] that E|Sx| ≍ √ x log log x .

Introduction

Introduction

At section 7 of [START_REF] Heap | Moments of random multiplicative functions and truncated characteristic polynomials[END_REF], Heap and Lindqvist provide a conjectured bound:

E(|S x |) ≤ (1 + o(1)) • 0.9036020036 . . .

√ x

To avoid unnecessarily reproducing the context we refer the reader to [START_REF] Heap | Moments of random multiplicative functions and truncated characteristic polynomials[END_REF] and include here only novel ideas improving the above bound. This note can hence be read as a continuation of [START_REF] Heap | Moments of random multiplicative functions and truncated characteristic polynomials[END_REF].

We denote by p i the i-th prime. Note that it is now known [START_REF] Harper | Moments of random multiplicative functions, i: Low moments, better than square root cancellation, and multiplicative chaos[END_REF] that

E|S x | ≍ √ x
log log x which reduces the interest of this note to exploring the limits of the Cauchy-Schwarz strategy.

The improvement strategy

For understanding the improvement strategy we remark the following:

Let P (X ℓ(1) , . . . , X ℓ(t) ) be a multivariate polynomial. Consider the operator Replace taking P and substituting all occurrences of X 2 ℓ(i) by 1 and all occurrences of X ℓ(i) by 1/ℓ(i).

Form the rational fraction F in the variables v ℓ(1) , . . . , v ℓ(t) :

F (v ℓ(1) , . . . , v ℓ(t) ) = Replace t i=1 (1 -v ℓ(i) X ℓ(i) ) 2 t i=1 (1 -v 2 ℓ(i) )
The minimum of F over [0, 1] t is reached for:

vℓ(i) = ℓ(i) -ℓ(i) 2 -1
To see why, develop first

∆ = Replace t i=1 (1 -v ℓ(i) X ℓ(i) ) 2 = Replace t i=1 (1 -2v ℓ(i) X ℓ(i) + v 2 ℓ(i) X 2 ℓ(i) = Replace t i=1 (1 -2v ℓ(i) X ℓ(i) + v 2 ℓ(i) ) = t i=1 (1 -2 v ℓ(i) ℓ(i) + v 2 ℓ(i) )
Which gives:

F (v ℓ(1) , . . . , v ℓ(t) ) = t i=1 1 -2 v ℓ(i) ℓ(i) + v 2 ℓ(i) 1 -v 2 ℓ(i)
Hence:

∂F ∂v ℓ(u) = 4ℓ(u)v ℓ(u) -2(1 + v 2 ℓ(u) ) ℓ(u)(v 2 ℓ(u) -1) 2 i̸ =u 1 -2 v ℓ(i) ℓ(i) + v 2 ℓ(i) 1 -v 2 ℓ(i)
And:

∂F ∂v ℓ(u) = 0 ⇒ vℓ(u) = ℓ(u) ± ℓ(u) 2 -1
Substituting back the second root ℓ(u) -ℓ(u) 2 -1 we get:

F (v ℓ(1) , . . . , vℓ(t) ) = t i=1 ℓ(i) 2 -1 ℓ(i) 2 < 1
Hence, when ℓ(i) = p i we have that:

lim t→∞ F (v ℓ(1) , . . . , vℓ(t) ) = 1 ζ(2) = √ 6 π
The above improves the upper bound on the Steinhaus expectation for k = 1/2 given in [START_REF] Heap | Moments of random multiplicative functions and truncated characteristic polynomials[END_REF].

To see why, let 0 ≤ v ℓ(1) , . . . , v ℓ(t) ≤ 1 and let S x = n≤x X n as in [START_REF] Heap | Moments of random multiplicative functions and truncated characteristic polynomials[END_REF]. By the Cauchy-Schwarz inequality we have:

E[|S x |] 2 ≤ E[| t i=1 (1 -v ℓ(i) X ℓ(i) )S x | 2 ] • E(| t i=1 (1 -v ℓ(i) X ℓ(i) )| -2 ) = E[| t i=1 (1 -v ℓ(i) X ℓ(i) )S x | 2 ] t i=1 (1 -v 2 ℓ(i) ) = F (v ℓ(1) , . . . , v ℓ(t) )
whose minimum was given before.

  We thank Winston Heap for his remarks concerning a preliminary version of this note.

	t	bound	exact expression	comment
	2	0.816497	2/3		bound of [2]
	3 4	0.800000 0.791795	4/5 √ 16 3/35
	5	0.788516	96 2/5/77
	6	0.786180	384 3/35/143
	7	0.784818	4608 6/35/2431
	8 9	0.783731 0.782989	55296 3/7/46189 663552/(96577 √ 77)
	10 0.782524	1327104 30/11/2800733
	. . . ∞ 0.779697 . . . Table 1. Convergence to . . . √ 6/π	√	6/π
	3 Acknowledgment


Krisha Mehta

Talha Ahmad Khan

Marouen Ben Guebila

A state-of-the-art database enables a macro perspective on tumor microenvironment

The dramatic growth in the use of single-cell assays in the study of various cancers has produced a wealth of new data, but methods for integrating and interpreting the resulting complex data have limited the use of these data and interpretation of their potential clinical significance. IMMUcan scDB, developed by Camps, Noël, and colleagues in "Meta-Analysis of Human Cancer Single-Cell RNA-Seq Datasets Using the IMMUcan Database" (PMID: 36459564), is a database that includes 144 single-cell datasets covering 56 cancers in a single online resource. IMMUcan scDB aims to enable biological discovery by allowing combined analyses of these heterogeneous datasets. The authors illustrate the value of IMMUcan scDB for meta-analyses of single-cell data and their associations with cancer phenotypes in two examples by identifying markers of response to immunotherapy and unique cell compositions in the microenvironment of specific cancer types.

There is a growing body of evidence that the interaction between tumor cells and the local tumor microenvironment (TME) plays a significant role ins cancer progression and response to therapy. The TME is an ecosystem consisting of diverse cell types whose interactions drive chemical changes within both tumor and host cells that include modulating immune and stromal cells, blood vessels, and extracellular matrices in ways that to enhance tumor growth. [START_REF] Anderson | The tumor microenvironment[END_REF] The heterogeneity of the TME has been found to hinder the efficacy of immunotherapy treatment for many patients, possibly by confounding the recognition of tumor cells. [START_REF] Pardoll | The blockade of immune checkpoints in cancer immunotherapy[END_REF] These and other lines of evidence suggest that investigating TME cell type composition and characterizing gene expression dynamics could both improve our understanding of how tumors develop and suggest ways to improve immunotherapy response. The advent of single-cell RNA sequencing 3 (scRNA-Seq) has given us unprecedented information about the composition of tumors and their microenvironments and the way in which those cells contribute tumor progression and metastasis [START_REF] Wang | Single-cell genomic and transcriptomic landscapes of primary and metastatic colorectal cancer tumors[END_REF] and how cell-cell interactions alter tumor properties, [START_REF] Luecken | Current best practices in single-cell RNA-seq analysis: a tutorial[END_REF] and allowed detection of rare cell types. More than 1300 studies have released scRNA-Seq data with samples from a variety of species, tissues, cell counts, and data formats. There are about 190 cancer-related scRNA-Seq data sets available, but analyzing these together presents a number of challenges including an absence of standardization for data processing [START_REF] Suvà | Single-cell RNA sequencing in cancer: lessons learned and emerging challenges[END_REF] and the difficulty of clinical interpretation given the sparsity of annotation, not to mention the inherent challenges of single-cell data analysis.

Databases such as Recount3, [START_REF] Wilks | recount3: summaries and queries for large-scale RNA-seq expression and splicing[END_REF] ReMap2022, 8 and ARCHS4 9 (Table1) have demonstrated the value of genomic data standardization, providing researchers with uniformly processed and curated data with efficient search and user-friendly interfaces that facilitate multi-study analyses. Similar standardized databases for scRNA-Seq data have begun to appear. CancerSEA 10 was one of the first databases with tumor sample datasets consisting of 41,900 cells from 25 cancer types focused on identifying functional states (stemness, invasion, metastasis, proliferation, EMT, angiogenesis, apoptosis, cell cycle, differentiation, DNA damage, DNA repair, hypoxia, inflammation and quiescence) of the collected cells. CancerSCEM 11 provides data on 638,341 cells across 20 types of cancer and provides tools for analyzing cell types within samples and the expression of genes within and between samples. TISCH 12 has assembled data from both human and mouse studies representing over two million cells with an emphasis on exploring the tumor microenvironment. But all of these databases primarily enable analyses that are somewhat limited in scope.

IMMUcan scDB [START_REF] Camps | Meta-analysis of human cancer single cell RNA-seq datasets using the IMMUcan database[END_REF] is the newest entry in the world of scRNA-Seq databases and represents a significant step forward. Camps, Noël, and colleagues have collected and standardized scRNA-Seq data from 144 single-cell studies representing 56 types of cancer. The IMMUcan initiative, of which IMMUcan scDB is a part, is a public-private partnership to profile cancer and its microenvironment in more than 3000 patients. As such, it has been engineered to allow complex, multi-study analyses and to be able to scale to store thousands of single-cell profiling datasets, in a large-scale, annotated, and searchable database that can be accessed by nonspecialists.

IMMUcan scDB builds on its parent project's well-designed database that includes manually derived annotation that has been extended using accurate automatic data processing algorithms. IMMUcan scDB is populated using a four-step pipeline that collects, processes, clusters, and annotates scRNA-Seq data. First, data are collected from peer-reviewed and nonpeer-reviewed datasets in other public databases using keywords as filters. Next, to ensure that the database consists of uniform, high-quality metadata that supports effective search, the authors captured metadata and organized it into four broad categories: study-wise information, sample-specific attributes, information about the applied single-cell technology workflow, and annotated, standardized information about the single-cell data (when available). Datasets were downloaded in raw read counts format to better facilitate downstream analyses and analyzed using scProcessoR, a bespoke R pipeline that performs normalization, variable gene selection, PCA transformations and scaling, clustering, and dimensionality reduction. Finally, cell types were annotated using the CHETAH [START_REF] De Kanter | CHETAH: a selective, hierarchical cell type identification method for single-cell RNA sequencing[END_REF] algorithm to annotate cells automatically and further analyzed using CopyKAT [START_REF] Gao | Delineating copy number and clonal substructure in human tumors from single-cell transcriptomes[END_REF] to identify malignant cells.

IMMUcan scDB is unique in providing information about patient treatment (available for 56% of the datasets) and co can be used to study therapeutic response (Figure 1). In their paper, they show the utility of their system by conducting a cell-centric analysis of response to anti PD-1 treatment in which they found an enrichment for B cells (and their CD20marker gene). They also reported gene-centric analysis of CXCL9 and found it to be expressed primarily in melanoma and basal cell carcinoma, an interesting result in that it had previously been reported as expressed in T follicular helper cells and exhausted CD8 + T cells and implicated in response to immunotherapy. They also found a correlation across datasets between expression of CXCL13-another immunotherapy marker gene that has been associated with the levels of PD1 expression, particularly in exhausted CD8 + T cells. Finally, they assessed cell type frequency in 25 datasets and found regulatory T cells in the tissue microenvironment but not in normal tissue; further analysis identified a tumor-specific pattern for Treg activation and that might be linked to cells responding to PD-1 blockade therapy. These exemplar analyses clearly demonstrate the value of the resource as it allows complex analyses through an intuitive web portal that also offers visualizations such as scatterplots and Venn diagrams.

IMMUcan scDB is valuable addition to the set of annotated scRNA-Seq databases and its transparent analytical pipeline and intuitive a query an analysis tools that support crossexperiment. The extensive collection of cellular data and phenotypic information allows complex analyses exploring the interface between gene expression, tumor composition, and response to therapy. These assets put it ahead of its competitors and its robust and extensible software framework position it to continue to be a leading resource in the field for a long time to come.

Table 1 -Transcriptomics databases and resources

Figure 1 -

 1 Figure 1 -IMMUcan scDB enables large-scale analyses of therapeutic response phenotypes and single-cell dissection of the tumor microenvironment based on its cell composition, marker genes, and gene coexpression.

Database Summary URL Tumor microenvironment single-cell databases IMMUcan scDB Tumor microenvironment single-cell data https://immucanscdb.vital-it.ch/ TISCH Tumor microenvironment single-cell data http://tisch.comp-genomics.org/ CancerSCEM Tumor microenvironment single-cell data https://ngdc.cncb.ac.cn/cancerscem/ CancerSEA Cancer single-cell data http://biocc.hrbmu.edu.cn/CancerSEA/ Uniformly processed genomic databases Recount 3 RNA-Seq data ARCHS4 RNA-Seq and ChIP-Seq data https://maayanlab.cloud/archs4/ REMAP2022 ChIP-Seq data https://remap2022.univ-amu.fr/ Data availability Database resources were summarized in Table 1.

Competing interests

The authors declare that they have no competing interests.


Francisco J López-Fraguas

Stephan Merz
email: stephan.merz@loria.fr

Juan Rodríguez-Hortalá
email: juanrh@fdi.ucm.es

A Formalization of the Semantics of Functional-Logic Programming in Isabelle ⋆

Modern functional-logic programming languages like Toy or Curry feature non-strict non-deterministic functions that behave under call-time choice semantics. A standard formulation for this semantics is the CRWL logic, that specifies a proof calculus for computing the set of possible results for each expression. In this paper we present a formalization of that calculus in the Isabelle/HOL proof assistant. We have proved some basic properties of CRWL: closedness under c-substitutions, polarity and compositionality. We also discuss some insights that have been gained, such as the fact that left linearity of program rules is not needed for any of these results to hold.

Introduction

Fully formalizing the (meta)theory of a programming language can be beneficial for developing its foundations. There is an increasing number of researchers (see e.g. [START_REF] Aydemir | Mechanized metatheory for the masses: The PoplMark challenge[END_REF]) sharing the conviction that the combination formalization+mechanized theorem proving must (and will) play a prominent role in programming languages research and technology. In particular, formalizations help to clarify overlooked aspects, to discover pitfalls, and even to provide new insights; moreover, formalized metatheories lead to mechanized reasoning about programs, giving reliable support to tools like certifying compilers or certified program transformations.

In this paper we formalize the semantics of functional logic programming (FLP), a well established paradigm (see [START_REF] Hanus | Multi-paradigm declarative languages[END_REF]) integrating features of logic and functional languages. In modern FLP languages such as Curry [START_REF]Curry: An integrated functional logic language (version 0.8.2)[END_REF] or Toy [START_REF] López-Fraguas | A multiparadigm declarative system[END_REF] programs are constructor based rewrite systems that may be non-terminating and non-confluent. Semantically this leads to the presence of non-strict and nondeterministic functions. The semantics adopted for non-determinism is call-time choice [START_REF] Hussmann | Non-Determinism in Algebraic Specifications and Algebraic Programs[END_REF][START_REF] González-Moreno | An approach to declarative programming based on a rewriting logic[END_REF], informally meaning that in any reduction, all descendants of a given subexpression must share the same value. The semantic framework CRWL 3 was proposed in [START_REF] González-Moreno | A rewriting logic for declarative programming[END_REF][START_REF] González-Moreno | An approach to declarative programming based on a rewriting logic[END_REF] to accomodate this view of non-determinism, and is nowadays considered the standard semantics of FLP. For the purpose of this paper, the most relevant aspect of CRWL is a proof calculus devised to prove reduction statements of the form P ⊢ e t , meaning that t is a possible (partial) value to which e can be reduced using the program P.

We have chosen Isabelle/HOL as concrete logical framework for our formalization. Using such a broadly used system is not only easier, but also more flexible and stable than developing language specific tools like has been done, e.g., for logic programming [START_REF] Stärk | The theoretical foundations of lptp (a logic program theorem prover)[END_REF] or functional programming [START_REF] De Mol | Theorem proving for functional programmers[END_REF].

The remainder of the paper is organized as follows: Sect. 2 contains some preliminaries about the CRWL framework , Sect. 3 presents the Isabelle theories developed to formalize CRWL, and Sect. 4 gives the mechanized proofs of some important properties of CRWL. Finally, Sect. 5 summarizes some conclusions and points to future work.

An extended version of this paper can be found at http://gpd.sip.ucm.es/ juanrh/pubs/isabell-crwl-report.pdf. The Isabelle code underlying the results presented here is available at https://gpd.sip.ucm.es/trac/gpd/wiki/ GpdSystems/IsabelleCrwl.

Preliminaries

Constructor-based term rewrite systems

We consider a first-order signature Σ = CS ∪ FS , where CS and FS are two disjoint sets of constructor and defined function symbols respectively, each with associated arity. We write CS n (FS n resp.) for the set of constructor (function) symbols of arity n. The set Exp of expressions is inductively defined as Exp ∋ e ::= X | h(e 1 , . . . , e n ), where X ∈ V, h ∈ CS n ∪ FS n and e 1 , . . . , e n ∈ Exp. The set CTerm of constructed terms (or c-terms) is defined like Exp, but with h restricted to CS n (so CTerm ⊆ Exp). The intended meaning is that Exp stands for evaluable expressions, i.e., expressions that can contain function symbols, while CTerm stands for data terms representing values. We will write e, e ′ , . . . for expressions and t , s, . . . for c-terms. The set of variables occurring in an expression e will be denoted as var (e). We will frequently use one-hole contexts, defined as The set Subst of substitutions consists of finite mappings θ : V -→ Exp (i.e., mappings such that θ(X ) = X only for finitely many X ∈ V), which extend

Cntxt ∋ C ::= [ ] | h(e 1 , . . . , C, . . . , e n ) for h ∈ CS n ∪ FS n .
(θ) = {X ∈ V | X θ = X }.
In most cases we will use c-substitutions θ ∈ CSubst, for which X θ ∈ CTerm for all X ∈ dom(θ).

A CRWL-program (or simply a program) is a set of rewrite rules of the form f (t ) → e where f ∈ FS n , e ∈ Exp and t is a linear n-tuple of c-terms, where linearity means that each variable occurs only once in t . Notice that we allow e to contain extra variables, i.e., variables not occurring in t. CRWLprograms often allow also conditions in the program rules. However, CRWLprograms with conditions can be transformed into equivalent programs without conditions, therefore we consider only unconditional rules.

The CRWL framework

In order to accomodate non-strictness at the semantic level, we enlarge Σ with a new constant constructor symbol ⊥. The sets Exp ⊥ , CTerm ⊥ , Subst ⊥ , CSubst ⊥ of partial expressions, etc., are defined naturally. Notice that ⊥ does not appear in programs. Partial expressions are ordered by the approximation ordering ⊑ defined as the least partial ordering satisfying

θ, σ ∈ Subst ⊥ we say θ ⊑ σ if X θ ⊑ X σ for all X ∈ V.
The semantics of a program P is determined in CRWL by means of a proof calculus (see Fig. 1) for deriving reduction statements P ⊢ e t , with e ∈ Exp ⊥ and t ∈ CTerm ⊥ , meaning informally that t is (or approximates) a possible value of e, obtained by iterated reduction of e using P under call-time choice. Rule B (bottom) allows us to avoid the evaluation of any expression, in order to get a non-strict semantics. Rules RR (restricted reflexivity) and DC (decomposition) allow us to reduce any variable to itself, and to decompose the evaluation of an expression whose root symbol is a constructor. Rule OR (outer reduction) expresses that to evaluate a function call we must first evaluate its arguments to get an instance of a program rule, perform parameter passing (by means of a CSubst ⊥ θ) and then reduce the instantiated right-hand side. The use of partial c-substitutions in OR is essential to express call-time choice, as only single partial values are used for parameter passing. Notice also that by the effect of θ in OR extra variables in the right-hand side of a rule can be replaced by any c-term, but not by any expression. The CRWL-denotation of an expression e ∈ Exp ⊥ is defined as

[[e]] P = {t ∈ CTerm ⊥ | P ⊢ CRWL e t }.
3 Formalizing CRWL in Isabelle Signatures are represented by a datatype that provides two constructors cs and fs to distinguish between constructor and function symbols. The type varId is used to represent variable identifiers, which will be employed to define substitutions. Then the datatype exp is naturally defined following the inductive scheme of Exp ⊥ , therefore with this representation every expression is partial by default. Substitutions (type subst) are represented as partial functions from variable identifiers to expressions, using Isabelle's option type. Hence the domain of a substitution ϑ will be the set of elements from varId for which ϑ returns some value different from None. Note that this representation does not ensure that domains of substitutions are finite. Our proofs do not rely on this finiteness assumption. Finally we represent a program rule as a pair of expressions, where the first element is considered the left-hand side of the rule and the second the right-hand side, and a program simply as a set of program rules. The set of valid CRWL programs is characterized by a predicate crwlProgram :: "program ⇒ bool" that checks whether the restrictions of left-linearity and constructor discipline are satisfied.

We define a function apSubst :: "subst ⇒ exp ⇒ exp" for applying a substitution to an expression. The composition of substitutions is defined through a function substComp :: "subst ⇒ subst ⇒ subst". The following lemma ensures the correctness of this definition. lemma subsCompAp : "(apSubst ϑ (apSubst σ e)) = (apSubst (substComp ϑ σ) e)"

Just as ML, the Isabelle type system does not support subtyping, which could have been useful to represent the sets of c-terms and c-substitutions. Instead, we define predicates cterm and csubst characterizing these subtypes. We prove the expected lemmas, such as that the composition of two c-substitutions is a c-substitution, or that the application of a c-substitution to a c-term yields a c-term.

Approximation order and contexts

Two key notions of CRWL have not yet been formalized: the approximation order ⊑, which will be used in the formulation of the polarity of CRWL, and the notion of one-hole context, which will be used in the compositionality.

The following inductively defined predicate ordap (with concrete infix syntax ⊑) models the approximation order. Rule B asserts that perp ⊑ e holds for every e; rule V is needed for ⊑ to be reflexive; finally rule Ap ensures closedness under Σ-operations, and thus compatibility with context [START_REF] Baader | Term Rewriting and All That[END_REF], because ⊑ is reflexive and transitive, as we will see.

The following results state that our formulation of ⊑ defines a partial order.

lemma ordapRefl : "e ⊑ e" lemma ordapTrans : assumes "e1 ⊑ e2" and "e2 ⊑ e3" shows "e1 ⊑ e3" lemma ordapAntisym :

assumes "e1 ⊑ e2" and "e2 ⊑ e1"

shows "e1 = e2" definition ordap_less ("_ < _" [51,51] 50) where "e < e' ≡ e ⊑ e' ∧ e = e'" interpretation exp : order [ordap ordap_less] Contexts are represented as the datatype cntxt, defined as follows:

datatype cntxt = Hole | Cperp | CVar varId
| CAp signat "cntxt list" Note that cntxt cannot follow the inductive structure of Cntxt with precision, because the type system of Isabelle is not expressive enough to allow us to specify that only one of the arguments of CAp will be a context and the others will be expressions. Then our contexts are defined as expressions with possibly some holes inside. Therefore the datatype cntxt represents contexts with any number of holes, even zero holes, and the function apCon :: "exp ⇒ cntxt ⇒ exp" is defined so it puts the argument expression in every hole of the argument context. In order to characterize contexts with just one hole, we define a function numHoles :: "cntxt ⇒ nat" that returns the numbers of holes in a context.

Using it we can define define predicates oneHole and noHole and prove the following lemmas. definition den :: "program ⇒ exp ⇒ exp set" where "den P e = {t. P ⊢ e → t}"

Reasoning about CRWL in Isabelle

The first interesting property that we are proving about CRWL expresses that evaluation is closed under c-substitutions: reductions are preserved when terms are instantiated by c-substitutions.

theorem crwlClosedCSubst : assumes "prog ⊢ e → t" and "csubst ϑ"

shows "prog ⊢ apSubst ϑ e → apSubst ϑ t"

The proof of this lemma proceeds by induction on the CRWL-proof of the hypothesis, therefore we will have one case for each CRWL rule. The first three cases are proved automatically. However, to prove the case for rule OR Isabelle needs some help from us. We need to prove prog ⊢ (Ap (fs f) (map (apSubst ϑ) es)) → (apSubst ϑ t)

and then let the simplifier apply the definition of apSubst. In the proof for that subgoal we used lemma CSubsComp to ensure that the c-substitution µ used for parameter passing composed with the c-substitution ϑ in the hypothesis yields another c-substitution, and lemma subsCompAp to guarantee the correct behaviour of the composition for those c-substitutions. Note that for this result to hold no additional hypotheses about the program or the expressions involved are needed. In particular, this implies that the result holds even for programs that do not follow the constructor discipline or that have non left-linear rules. The Isabelle proof clearly shows that the important ingredients are the use of c-substitutions for parameter passing and the reflexivity of CRWL wrt. c-terms, expressed by lemma ctermRefl, which allows us to reduce to itself any expression Xϑ coming from a premise X → X.

The second property that we address is the polarity of CRWL. This property is formulated by means of the approximation order and roughly says that if we can compute a value for an expression then we can compute a smaller value for a bigger expression. Here we should understand the approximation order as an information order, in the sense that the bigger the expression, the more information it gives, and so more values can be computed from it.

theorem crwlPolarity : assumes "prog ⊢ e → t" and "e ⊑ e'" and "t' ⊑ t"

shows "prog ⊢ e' → t'" using assms proof (induct arbitrary: e' t') The idea of the proof is to construct a CRWL-proof for the conclusion from the CRWL-proof of the hypothesis, hence it is natural to proceed by induction on the structure of this proof (method induct). The qualifier arbitrary is used to generalize the assertion for any expressions e' and t'. The proof also relies on the following additional lemmas about the approximation order, which were proved automatically by Isabelle.

lemma ordapPerp: assumes "e ⊑ perp" shows "e = perp" lemma ordapVar: assumes "Var v ⊑ e" shows "e = Var v" lemma ordapVar_converse:

assumes "e ⊑ Var v" shows "e = perp ∨ e = Var v" lemma ordapAp:

assumes "Ap h es ⊑ e'" shows "∃ es'. e' = Ap h es' ∧ size es = size es' ∧ (ALL i < size es. es!i ⊑ es'!i)" lemma ordapAp_converse:

assumes "e' ⊑ Ap h es"

shows "e' = perp ∨ (∃ es'. e' = Ap h es' ∧ size es = size es' ∧ (ALL i < size es. es'!i ⊑ es!i))"

The inductive proof for theorem crwlPolarity again considers each CRWL rule in turn. In the case for B we have t = perp, hence we just have to apply ordapPerp to get t' = perp, and then use the CRWL rule B. Regarding RR, as then t = Var v, by ordapVar_converse we get that either t' = perp or t' = Var v. The first case is trivial, and in the latter we just have to apply ordapVar getting e' = Var v, which is enough for Isabelle to finish the proof automatically. The case of DC is more complicated. Again we obtain two cases for t' = perp and t' a constructor application, by using lemma ordapAp_converse.

While the first case is trivial, the second one requires some involved reasoning over the list of arguments, using the information we get from applying lemma ordapAp. Finally, the proof for OR is similar to the second case of the proof for DC, with a similar manipulation of the list of arguments, and the use of lemma ordapAp to obtain the induction hypothesis for the arguments.

Once again we find that this proof does not require any hypothesis on the linearity or the constructor discipline of the program: this is indeed quite obvious because this property only talks about what happens when we replace some subexpression by perp.

Finally we will tackle the compositionality of CRWL, that says that if we take a context with just one hole and an expression, then the set of values for the expression put it that context will be the union of the set of values for the result of putting each value for the expression in that context. theorem compCRWL :

assumes "oneHole xC"

shows "den P (apCon e xC) = ( t∈den P e. den P (apCon t xC))"

We have proved the two set inclusions separately as auxiliary lemmas compCRWL1 and compCRWL2. The proofs of these lemmas are quite laborious but essentially proceed by induction on the CRWL-proof in their hypothesis, using it to build a CRWL-proof for the statement in the conclusion. In these proofs, Lemma noHoleApDontCare from Subsect. 3.2 is fundamental. Again, while theorem compCRWL requires the context to have just one hole, it does not assume the linearity or constructor discipline of the program. This came as a surprise to us, and initially made us doubt about the accuracy of our formalization of CRWL. But it turns out that although CRWL is designed to work with CRWL-programs, that fulfil these restrictions, it can also be applied to general programs. For those programs some properties, such as the theorems crwlClosedCSubst, crwlPolarity, and compCRWL still hold, but other fundamental properties do not, in particular the strong adequacy results w.r.t. its operational counterparts of [START_REF] González-Moreno | An approach to declarative programming based on a rewriting logic[END_REF][START_REF] López-Fraguas | A simple rewrite notion for call-time choice semantics[END_REF][START_REF] Albert | Operational semantics for declarative multi-paradigm languages[END_REF]. The point is that for those programs CRWL does not deliver the "intended semantics" anymore. And this is not strange, because that semantics was intended with CRWL-programs in mind. For example, consider the non linear program P = {f (X , X ) → a}. There is a CRWL-proof for the statement P ⊢ f (a, b) a but this value cannot be computed in any of the operational notions of [START_REF] González-Moreno | An approach to declarative programming based on a rewriting logic[END_REF][START_REF] López-Fraguas | A simple rewrite notion for call-time choice semantics[END_REF][START_REF] Albert | Operational semantics for declarative multi-paradigm languages[END_REF] nor in any implementation of FLP, in which the independence of the matching process of the arguments -derived from leftlinearity of program rules -is assumed. It is also not very natural that f (a, b) could yield the value a for the arguments a and b being different values, which implies that the semantics defined by CRWL for non left-linear programs is pretty odd. But that is not a big problem, because we only care about the properties of CRWL for the kind of programs it has been designed to work with. And if it enjoys some interesting properties for a bigger class of programs that is fine, because that nice properties will be inherited by the class of CRWL-programs.

On the other hand, for programs not following the constructor discipline, we will not even be able to have a matching for an argument of a rule which is not a constructor, because in the rule OR we have to reduce every argument of a function call to a value, which will be a c-term by Lemma ctermVals (see the extended version of this paper), and so could never be an instance of expression containing function symbols. Thus, the rule OR could not be used for program rules not following the constructor discipline.

Conclusions

This paper presented a formalization of the essentials of CRWL [START_REF] González-Moreno | A rewriting logic for declarative programming[END_REF][START_REF] González-Moreno | An approach to declarative programming based on a rewriting logic[END_REF], a wellknown semantic framework for functional logic programming, in the interactive proof assistant Isabelle/HOL. We chose that particular logical framework for its stability and its extensive libraries. The Isar proof language allowed us to structure the proofs so that they become quite elegant and readable, as can be observed by looking at the Isabelle code.

Our formalization is generic with respect to syntax, and includes important auxiliary notions like substitutions or contexts. This is in contrast to previous work [START_REF] Cleva | A logic programming approach to the verification of functional-logic programs[END_REF][START_REF] Cleva | Verification of CRWL programs with rewriting logic[END_REF] that focused on formalizing the semantics of each concrete program. In contrast, our paper focuses on developing the metatheory of the formalism, allowing us to obtain results that are more general and also more powerful: we formally prove essential properties of the paradigm like polarity or compositionality of the CRWL-semantics. We plan to extend our theories so that we will be able to reason about properties of concrete programs by deriving theorems that express verification conditions in the line of those stated in [START_REF] Cleva | A logic programming approach to the verification of functional-logic programs[END_REF][START_REF] Cleva | Verification of CRWL programs with rewriting logic[END_REF].

While developing the formalization we realized an interesting fact not pointed out before: properties like polarity or compositionality do not depend on the constructor discipline and left-linearity imposed to programs. However, such requirements will certainly play an essential role when extending our work to formally relate the CRWL-semantics with operational semantics like the one developed in [START_REF] López-Fraguas | A simple rewrite notion for call-time choice semantics[END_REF], one of our intended subjects of future work. We think that could be interesting in several ways. First of all it would be a further step in the direction of challenge 3 of [START_REF] Aydemir | Mechanized metatheory for the masses: The PoplMark challenge[END_REF], "Testing and Animating wrt the Semantics", because we would end up getting an interpreter of CRWL during the process. We should then also formalize the evaluation strategy for the operational semantics, obtaining an Isabelle proof of its optimality. Finally there are precedents [START_REF] López-Fraguas | Rewriting and call-time choice: the HO case[END_REF][START_REF] López-Fraguas | A simple rewrite notion for call-time choice semantics[END_REF] of how the combination of a denotational and operational perspective is useful for general semantic reasoning in FLP.

  The application of a context C to an expression e, written C[e], is defined inductively by [ ][e] = e and h(e 1 , . . . , C, . . . , e n )[e] = h(e 1 , . . . , C[e], . . . , e n ).

⊥

  ⊑ e and e ⊑ e ′ ⇒ C[e] ⊑ C[e ′ ] for all e, e ′ ∈ Exp ⊥ , C ∈ Cntxt This partial ordering can be extended to substitutions: given

3. 1

 1 Basic definitionsWe describe our formalization of CRWL in Isabelle. The first step is to define elementary types for the syntactic elements. datatype signat = fs string | cs string datatype varId = vi string datatype exp = perp | Var varId | Ap signat "exp list" types subst = "varId ⇒ exp option" rule = "exp * exp" program = "rule set"

  inductive ordap :: "exp ⇒ exp ⇒ bool" ("_ ⊑ _" [51,51] 50) where B: "perp ⊑ e" | V: "Var x ⊑ Var x" | Ap: "[[ size es = size es' ; ALL i < size es. es!i ⊑ es'!i ]] =⇒ Ap h es ⊑ Ap h es'"

  lemma noHoleApDontCare : assumes "noHole xC" shows "apCon e xC = apCon e' xC" lemma oneHole : assumes "oneHole (CAp h xCs)" shows "∃ xC yCs zCs. xCs = (yCs @ xC # zCs) ∧ oneHole xC ∧ (∀ c ∈ set (yCs @ zCs). noHole c)" 3.3 The CRWL logic in Isabelle/HOL The CRWL logic has been formalized through the inductive predicate clto with infix notation "_ ⊢ _ → _". The rules defining clto faithfully follow the inductive structure of the definition of CRWL as it is presented in Fig. 1. inductive clto :: "program ⇒ exp ⇒ exp ⇒ bool" ("_ ⊢ _ → _" [100,50,50] 38) where B[intro]: "prog ⊢ exp → perp" | RR[intro]: "prog ⊢ Var v → Var v" | DC[intro]: "[[size es = size ts; ∀ i < size es. prog ⊢ es!i → ts!i ]] =⇒ prog ⊢ Ap (cs c) es → Ap (cs c) ts" | OR[intro]: "[[(Ap (fs f) ps, r) ∈ prog ; csubst ϑ ; size es = size ps ; ∀ i < size es. prog ⊢ es!i → apSubst ϑ (ps!i); prog ⊢ apSubst ϑ r → t ]] =⇒ prog ⊢ Ap (fs f) es → t" Using clto we can easily define the CRWL denotations in Isabelle as follows.

  Exp -→ Exp. We write eθ for the application of θ to e, and θθ ′ for the composition of substitutions, defined by X (θθ ′ ) = (X θ)θ ′ . The domain of θ is defined as dom

	(RR) X	X	X ∈ V		(B) e ⊥
	(DC)	e1 c(e1, . . . , en ) t1 . . . en c(t1, . . . , tn ) tn		c ∈ CS n
	(OR)	e1	p1θ . . . en f (e1, . . . , en ) pn θ r θ t	t	f (p1, . . . , pn ) → r ∈ P θ ∈ CSubst ⊥
				Fig. 1. Rules of CRWL
	naturally to θ :

⋆ This work has been partially supported by the Spanish projects TIN2005-09207-C03-03 (MERIT-FORMS-UCM), S-0505/TIC/0407 (PROMESAS-CAM) and TIN2008-06622-C03-01/TIN (FAST-STAMP). 3 CRWL stands for "Constructor-based ReWriting Logic".