Ivan A Volkov

Leonid A Igumnov
email: igumnov@mech.unn.ru

Francesco Dell'isola
email: fdellisola@gmail.com

Svetlana Yu Litvinchuk
email: litvinchuk@mech.unn.ru

Victor A Eremeyev
email: eremeyev.victor@gmail.com

Keywords: Modeling, Thermal cyclic strength, Complex deformation, Damaged medium mechanics, Fatigue life, Strength, Failure

A continual model of a damaged medium used for analyzing fatigue life of polycrystalline structural alloys under thermal-mechanical loading

Introduction

The general trend in the development of structures and machines of modern mechanical engineering is characterized by the increasing number of their working parameters, the decreasing metal consumption due to optimal design and use of novel high-strength materials, the increasing relative share of nonstationary loading regimes. The requirements to reliability and accident-free life of both the entire structures and their separate elements are becoming tougher. As a result, one of the main tasks of the development and exploitation of structures and machines of modern technologies at present is the task of reliable evaluation of their service life. This task is especially vital for objects with service lives of several tens of years (atomic power plants, aviation gas-turbine engines, spaceship engines, etc.). as a rule, the exploitation conditions of such structures and machines are characterized by multiparametric nonstationary thermal-mechanical loading, effects of external fields leading to the degradation of the initial strength properties of structural materials and, finally, to exhausting the life of the structural units of the object [START_REF] Halford | Low cycle thermal fatigue[END_REF][START_REF]Issledovanie malotsiklovoy prochnosti pri vysokih temperaturah[END_REF][START_REF] Collins | Damage of Materials in Structures. Analysis, Prediction, Prevention[END_REF][START_REF] Eorum | Otsenka sovremennoy metodologii proektirovaniya vysokotemperaturnyh elementov konstruktsiy na osnove eksperimentov po ih razrusheniyu (in Russ[END_REF][START_REF] Bernard-Connolly | Low-cycle fatigue behaviour and cumulative dormage effect of SA-516-70 steel at room and high temperature[END_REF][START_REF] Gusenkov | Prochnost pri malotsiklovom i dlitelnom tsiklicheskom nagruzhesnii i nagreve (in Russ.)[END_REF][START_REF]Sentoglou: Vliyanie ogranicheniy na termomehanicheskuyu ustalost (in Russ[END_REF].

Effects of deformation under thermal-mechanical loading are determined by the dependence of the physical-mechanical properties of structural materials on temperature, by the presence of limitations of thermal strains and the effect of the ratio of mechanical and thermal strain rates. As frequencies and phases of variation cycles of temperature and mechanical strain tensor components as a rule do not coincide, nonisothermal cyclic loading is generally irregular, nonproportional, multiaxial and is accompanied by the rotation of the main areas of stress and strain tensors.

Cyclic nonisothermal deformation of materials, in contrast to isothermal one, is a nonbalanced cyclic process. The degree of imbalance is a function of the range of the temperature cycle and phase shift in the variation of temperature and deformation. Even in the elastic region of the tension-compression cycle, due to the dependence of the elasticity modulus on temperature (a nonlinear decreasing function of the growing temperature), the deformation process depends on the phase of the variation of the temperature and mechanical strain and is hard at the end of the cold temperature cycle and more elastic at its high-temperature end [START_REF] Halford | Low cycle thermal fatigue[END_REF]. In the case of nonisothermal elastoplastic deformation, the deformation process will have a more complex character as the elastoplastic behavior of the material depends on temperature.

As a rule, a volumetric thermal strain results in a multiaxial deformation process. This leads to new sources of imbalance: the stressed-strained state in any point of the cycle differs from that in other points. Phasing by varying the temperature and mechanical strain is one of the main factors also connected with thermal fatigue.

The multiaxial nature of the stressed-strained state also causes imbalances in thermal-mechanical loading. A combined effect of mechanical and thermal loads results, as a rule, in substantial rotation of the main areas of stress and strain tensors (nonproportional loading), which, in the presence of plastic deformation leads to noncoaxiality of stress and total and plastic strains tensors. To model such processes, reliable models of cyclic thermal plasticity are required.

Nonisothermal deformation of structural materials, as compared with isothermal one, is characterized by a number of specific features [START_REF] Halford | Low cycle thermal fatigue[END_REF]:

-the nonlinear stress-strain relation even in the elastic region, as elasticity moduli depend on temperature; -the imbalanced nature of the diagrams for symmetric cycles of variation of mechanical strain and strong dependence of the form of the stress-strain hysteresis loop on the combination of the variation phases of temperature and mechanical strain.

Thermal-mechanical fatigue cycling may initiate damage mechanisms that do not develop under the effect of isothermal fatigue. The main parameters determining the rates of thermal-mechanical fatigue processes are: temperature gradients, maximal temperature of the cycle, geometric characteristics and conditions of joining of structural elements, physical-mechanical characteristics of materials (especially the linear expansion coefficient), inhomogeneity (anisotropy) of the structural composition of such materials.

Thus, despite the apparent similarity of the phenomena of isothermal and nonisothermal types of fatigue, the results on isothermal low-cycle fatigue have to be used with due care when evaluating nonisothermal fatigue life.

Damage and failure of materials is, for the most part, caused by nucleation of microdefects, their growth and merging to form macroscopic cracks. The tasks of describing the mechanical behavior of such microdefects are as important as the tasks of describing the growth of macrodefects (cracks); so, in the recent years, a new scientific direction, mechanics of damaged media (MDM), is being successfully developed to solve these tasks [START_REF] Bodner | An incremental criterion for time-dependent failure of materials[END_REF][START_REF] Lemaitre | A continuous damage mechanics model for ductile fracture[END_REF][START_REF] Murakami | Mechanical description of creep damage and its experimental verification[END_REF][START_REF] Chaboche | Continuous damage mechanics a tool to describe phenomena before crack initiation[END_REF][START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF].

In [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF], in the framework of MDM, a mathematical model was developed that describes processes of cyclic thermal-plastic deformation and fatigue damage accumulation in structural materials (metals and their alloys) under multiaxial nonproportional paths of combined thermal-mechanical loading. To assess qualitatively and quantitatively the reliability of the model, the present paper investigates the effect of the laws of change of mechanical strength and temperature (the type of deformation trajectory) on fatigue life of heat-resistant alloys (Nimonic 80A, Haynes 188) under nonproportional regimes of thermal-mechanical loading.

The methods presented in this paper can also be suitably applied to a large class of metamaterials (see [START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Di Cosmo | Acoustic metamaterials based on local resonances: homogenization, optimization and applications[END_REF] for a general discussion about metamaterials). In particular, continuum models for describing the emerging of damage and fracture have been investigated in [START_REF] Cuomo | Continuum damage model for strain gradient materials with applications to 1D examples[END_REF][START_REF] Cuomo | Continuum model of microstructure induced softening for strain gradient materials[END_REF][START_REF] Spagnuolo | Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments[END_REF][START_REF] Battista | Numerical investigation of a particle system compared with first and second gradient continua: deformation and fracture phenomena[END_REF].

The constitutive relations of mechanics of damaged media

The damaged medium model developed in [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] consists of three interrelated parts:

-relations determining thermal-plastic behavior of materials, accounting for its dependence on the failure process; -evolutionary equations describing damage accumulation kinetics; -a strength criterion of the damaged material.

(a) Constituting relations of plasticity

The governing relations of plasticity are based on the following main assumptions [START_REF] Mitenkov | Prikladnaya teoriya plastichnosti[END_REF]:

-the tensor components of strains e ij and strain rates ėij include elastic e e ij , ėe ij and plastic strains-e p ij , ė p ij , i. e., reversible and irreversible components; -the initial yield surface for different temperatures is described by a von Mises-type surface. The evolution of the yield surface is described by the variation of its radius C p and the displacement of its center ρ ij ; -the body changes its volume elastically; -initially isotropic media are considered. Only the anisotropy caused by processes of plasticity are accounted for; -processes characterized by small deformations are considered.

In the elastic region, the correlation between spherical and deviatoric components of stress and strain tensors and of their rates is established with the Hooke law:

σ = 3K[e -α(T -T 0 )], σ ij = 2Ge e ij , σ = 3K( ė -αT -α Ṫ ) + K K σ, σ ij = 2G ė e ij + Ġ G σ ij , ( 1
)
where T is temperature, T 0 is an initial (referential) temperature, K(T ) is the bulk modulus, G(T ) is the shear modulus, α(T ) is the coefficient of linear thermal expansion of the material.

To model the effects of monotone and cyclic deformation, a yield surface is introduced:

F s = S ij S ij -C 2 p = 0, S ij = σ ij -ρ ij . ( 2
)
To describe complex cyclic deformation regimes in the stress space, a cyclic 'memory' surface is introduced. The equation of the 'memory' surface is:

F ρ = ρ ij ρ ij -ρ 2 max = 0, (3)
where ρ max is maximal value of modulus ρ ij during the loading history.

It is assumed that the evolutionary equation for the yield surface radius has the following structure:

Ċ p = [q χ H (F ρ ) + a(Q s -C p )Γ (F ρ )] χ + q 3 Ṫ (4) C p = C 0 p + t 0 Ċ p dt, χ = 2 3 ė p ij ė p ij 1 /2 , χ m = t 0 χ H F ρ dt, χ = t 0 χdt. ( 5
)
q χ = q 2 Aψ 1 + (1 -A)q 1 Aψ 1 + (1 -A) , Q s = Q 2 Aψ 2 + (1 -A)Q 1 Aψ 2 + (1 -A) , 0 ≤ ψ i ≤ 1, i = 1, 2. A = 1 -cos 2 θ, cos θ = n e ij n s ij , n e ij = ė ij ( ė ij ė ij ) 1 /2 , n s ij = S ij (S ij S ij ) 1 /2 , H (F ρ ) = 1, F ρ = 0 ∧ ρ ij ρij > 0 0, F ρ < 0 ∨ ρ ij ρij ≤ 0 , Γ (F ρ ) = 1 -H (F ρ ), (6)
where q 1 , q 2 , q 3 are experimentally determined constitutive isotropic hardening moduli, Q 1 and Q 2 are cyclic isotropic hardening moduli, a is a constant determining the rate of the stationing process of the hysteresis loop of cyclic deformation of the material, Q s is stationary value of the yield surface radius for the assigned ρ max and T , C 0 p is initial value of the yield surface radius [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Mitenkov | Prikladnaya teoriya plastichnosti[END_REF]. The evolution of internal variable ρ ij is assumed to have the form [START_REF] Volkov | Modelirovanie ustalostnoy dolgovechnosti polikristallicheskih konstruktsionnyh splavov pri blochnom nesimmetrichnom malotsilkovom nagruzhenii (in Russ.)[END_REF]:

ρij = g1 ė p ij -g 2 ρ ij χ + ρ * ij + g 5 ρ ij Ṫ , ρ ij = t 0 ρij dt, (7)
g1 = g 1 + k 1 1 -e -k 2 χ m cos β , cos β = ρij ρ ij ρij ρij 1 /2 ρ ij ρ ij 1 /2 , (8) ρ
* ij = g 3 ė p ij H F ρ -g 4 ρ ij χ, (9)
where g 1 , g 2 , g 3 , g 4 , g 5 , k 1 and k 2 are experimentally determined material parameters.

For nonsymmetric hard and soft types of cyclic loading, Eq. ( 7) describes the processes of placing and ratcheting of the cyclic plastic hysteresis loop with the help of term ρ * ij . If g 3 = g 4 = k 1 = 0 in [START_REF]Sentoglou: Vliyanie ogranicheniy na termomehanicheskuyu ustalost (in Russ[END_REF], one has a special case of Eq. ( 7) that is the Armstrong-Frederik-Kadashevich equation [START_REF] Mitenkov | Prikladnaya teoriya plastichnosti[END_REF]:

ρij = g 1 ė p ij -g 2 ρ ij χ. (10)
To describe the evolution of the 'memory' surface, it is necessary to formulate an equation for ρ max :

ρmax = (ρ ij ρij )H (F ρ ) (ρ mn ρ mn ) 1 /2 -g 2 ρ max χ -g 5 ρ max Ṫ . ( 11
)
The plastic strain rate components obey the rule of orthogonality of the plastic strain rate vector to the yield surface in the loading point:

ė p ij = λS ij , ( 12
)
where λ is proportionality coefficient determined from the condition that a new yield surface passes through the end of the stress deviator vector at the end of the loading stage. At the stage of defects scattered over the volume, the effect of the damage degree on the physical-mechanical properties of the material is observed. This effect can be taken into account by introducing effective stresses:

σ i j = F 1 (ω) σ i j = G G σ i j = σ i j (1 -ω) 1 -(6K +12G) (9K +8G) ω , ( 13
) σ = F 2 (ω) σ = K K σ = σ 4G (1 -ω) / (4G + 3K ω) , ( 14
)
where G, K are effective moduli of elasticity defined by McKenzie formulas [START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF].

Effective variable ρij is determined in a similar way:

ρij = F 1 (ω) ρ ij . ( 15
)

(b) Evolutionary equation of fatigue damage accumulation

In the framework of the mechanics of damaged media, the final failure stage corresponds to the nucleation of a macroscopic crack (∼ 0.1 cm), that is, leading to material discontinuity, which is rather large at the level of heterogeneity of the material (grain, structure blocks, etc.). The simplest visual physical measure of material damage is the relative volumetric fraction of defects in the reference elementary volume of the material ω [START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF]:

ω = V /V f , 0 ≤ ω ≤ 1,
where V is the current volumetric fraction of defects, V f is the critical volumetric fraction, corresponding to nucleation of a macrocrack in this volume of the material (complete loss of the bearing capacity of the material in this elemental volume).

To formulate evolutionary equations for ω, we use an approach based on establishing the correlation of velocities ω with certain mechanical parameters, depending on the material deformation process, the critical value of which determines the moment of complete failure of the elementary volume of the material. The most effective approach to solving this problem is the energy approach [START_REF] Collins | Damage of Materials in Structures. Analysis, Prediction, Prevention[END_REF][START_REF] Bodner | An incremental criterion for time-dependent failure of materials[END_REF][START_REF] Lemaitre | A continuous damage mechanics model for ductile fracture[END_REF][START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF].

It is not a trivial task to extract from the total dissipation energy in the volume the part of it that is directly spent on the damage formation and accumulation.

It is established (see [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF] and the references therein) that, at low-cycle fatigue, the energy, spent on nucleation of defect during transient elastoplastic deformation, correlates well with the work of W the microstress tensor (center coordinates of the yield surface F) ρ ij on plastic deformations e p ij :

Ẇ = ρ ij ė p ij , W = ρ ij de p ij . ( 16
)
The multiaxial nature of stress state significantly affects the durability of the material, and this effect manifests in two ways: the effect of multiaxial nature under proportional loading (when all components of the strain tensor vary proportionally to one parameter) and the rotation effect of the main stress tensor areas (when stress components vary out of phase).

Experimental and theoretical studies of the influence of multiaxial nature in other types of stress states (biaxial tension-compression, triaxial tension) showed a significant effect of volumetric stressed state of the material on its durability, characterized by the intensity of the stress tensor σ u and its ball (hydrostatic) component σ [START_REF] Bodner | An incremental criterion for time-dependent failure of materials[END_REF][START_REF] Lemaitre | A continuous damage mechanics model for ductile fracture[END_REF][START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF].

Parameter β = σ/σ u or functions of β are used as parameters of the volumetric stressed state. Summarizing the data available in the literature, it can be confirmed that the rate of damage accumulation ω depends on volumetric stressed state, characterized by some function f 1 (β): ω increases with β > 0 and β → +∞ (threedimensional tension) and decreases with β < 0 and β → -∞ (in this case partial healing of the accumulated damage may occur). With β = 0 (pure shift, σ = 0), the normalized function f 1 (β)should be equal to 1.

The effect of nonproportional loading, under which the components of the stress-strain tensors change out of phase (the main tensor areas rotate), and the stress and plastic strain tensors are not coaxial, is as follows [START_REF] Lemaitre | A continuous damage mechanics model for ductile fracture[END_REF][START_REF] Chaboche | Continuous damage mechanics a tool to describe phenomena before crack initiation[END_REF][START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF]:

-the shape of the deformation path is a parameter that significantly affects the fatigue life; -structural materials exhibit complex cyclical behavior under multiaxial nonproportional loading conditions -additional cyclic hardening or softening; -under nonproportional cyclic loading, the criterion of equivalent deformations or strain intensity e u are not equivalence criteria and may lead to nonconservative evaluations.

Summarizing the above results, the general structure of an evolutionary equation of the damage accumulation in elementary volume of the material can have the following form [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF]:

ω = f 1 (β) f 2 (ω) f 3 (W ) f 4 Ẇ . ( 17
)
where functions f i , i = 1...4 account for: the volumetric stressed state ( f 1 (β)), level of accumulated damage ( f 2 (ω)), accumulated relative damage energy, spent on nucleation of defects ( f 3 (W )) and rate of change of damage energy ( f 4 Ẇ ).

In ( 16):

f 1 (β) = exp(β), f 2 (ω) = ⎧ ⎨ ⎩ 0, W ≤ W a , ω 1 /3 (1 -ω) 2 /3 ∧ W > W a ∧ ω ≤ 1 /3, 3 √ 16 9 ω -1 /3 (1 -ω) -2 /3 ∧ W > W a ∧ ω > 1 /3, f 3 (W ) = W -W a W f -W a , f 4 Ẇ = Ẇ / W f -W a . ( 18
)
where β is parameter of volumetric stressed state, W a is value of the damage energy at the end of the stage of nucleation of scattered defects for LCF, and W f is value of the energy corresponding to the nucleation of a macroscopic crack. The duration of the microdefect nucleation phase will be related with the value of parameter W a .

When the size of microdefects becomes comparable with the average distance between them, the merging process (breakage of the remaining continuous spaces between the defects) begins. The present paper was not aimed at constructing a detailed model of defect merging; instead, this process was accounted for by formulating the kinetic equation in such a way (due to using term f 2 (ω)) that when the damage degree reaches the value of ω = 1 /3 relation ω = f 1 (ω) accounts for the avalanche-like increase in the damage degree value.

c) The strength criterion of the damaged material

The condition when damage degree ω reaches its critical value

ω = ω f ≤ 1 ( 19
)
is taken as the criterion of termination of the phase of growth of scattered microdefects.

Numerical results

The results on thermal cyclic deformation of the heat-resistant nickel alloy Nimonic 80A are presented in [START_REF] Liang | Thermomechanical fatique behavior of a nickel base superalloy[END_REF]. Isothermal (Fig. 1a) and nonisothermal (Fig. 1b,c) tests were conducted with hard loading, the amplitude of strain e 11 in the tests with constant temperature T being 1%. For the tests with a varying temperature, the results are given for the amplitudes from 0.45 to 1%. The temperature in nonisothermal tests varied both in-phase (Fig. 1b) and in counter-phase (Fig. 1c) relative to the variation of the strains.

The main physical-mechanical characteristics of the heat-resistant nickel alloy Nimonic 80A and the material parameters of the damaged medium model [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF] are listed in Tables 1, 2 and 3.

When conducting the experiments, the selected samples should ensure a uniform distribution of stress, strain and temperatures within the operating part of the sample. If this condition is met, then, for numerical modeling of experimental processes, there is no need in formulation of the initial-boundary problem with the statement of boundary conditions and the use of computer complexes of finite element analysis [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF]. The calculation of the processes of deformation and failure of laboratory samples is carried out by numerical integration 1)- [START_REF] Di Cosmo | Acoustic metamaterials based on local resonances: homogenization, optimization and applications[END_REF]. In numerical simulation of experimental data, experimental deformation trajectories (laws of changes in the components of the strain tensor e ij (t) and temperature T (t)) are specified, while stress trajectories σ ij (t) and damage values ω(t)are obtained by integrating the defining relations of MDM.

Figure 2a compares the results of modeling and the experimental data for a stabilized cyclic deformation loop for the temperature of 700 • C (the loading type is shown in Fig. 1a), for the strain amplitude of 1%. Here and in what follows the markers show the experimental results, and the solid lines depict the numerical results obtained using the MDM model introduced by the authors of [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF].

Figure 2b presents the comparison of the stabilized loops of cyclic nonisothermal deformation for in-phase loading (see Fig. 1b): the strain amplitude is 0.6%, the maximal temperature in the cycle was 823 • C (achieved in tension), the minimal temperature in the cycle was 571 • C (achieved in compression). Figure 2c compares the results of modeling the nonisothermal cyclic loading for the counter-phase loading (see Fig. 1c) (the strain amplitude and the maximal and minimal temperature values are the same as in the previous case). The 67-th and 68-th loading half-cycles are shown as stabilized cyclic deformation loops in all the three cases.

It can be seen that the model of cyclic thermal-plastic deformation presented in the paper adequately describes the experimental results and can be used in future for analyzing the kinetics of the fatigue damage accumulation process under cyclic in-phase and counter-phase thermal-mechanical effects.

To assess the applicability of the introduced evolutionary equation of fatigue damage accumulation [START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] and to analyze the effect of the deformation trajectory on low-cycle fatigue, fatigue life of specimens made of the heat-resistant alloy Nimonic 80A was analyzed for different laws of change of axial deformation e 11 and temperature T (see Fig. 1). The numerical results were compared with the available test data [START_REF] Liang | Thermomechanical fatique behavior of a nickel base superalloy[END_REF]. The material parameters of the evolutionary equation of damage accumulation were determined using point P on the fatigue curve for isothermal cyclic deformation (Fig. 3). The material parameters of the evolutionary equation of damage accumulation ( 16) of the heat-resistant alloy Nimonic 80A are summarized in Table 1.

The numerical results are compared with the available test data in Figs. 3,4 and Table 4. Figure 3a, b depicts the isothermal fatigue curves, and Fig. 4a, b the nonisothermal fatigue curves for the in-phase and counter-phase loading regimes; the isothermal tests for the different temperatures T = 571 • C and 823 • C (Fig. 3a,b, respectively) and the nonisothermal in-phase (Fig. 4a) and counter-phase (Fig. 4b). The solid lines represent the corresponding calculated fatigue life curves. The experimental results are shown by square markers.

The numerical and experimental results show that:

-for known parameters of cyclic thermal plasticity equations, material parameters of evolutionary equations of damage accumulation [START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF][START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Di Cosmo | Acoustic metamaterials based on local resonances: homogenization, optimization and applications[END_REF] are determined using a single experimental point P on the fatigue curve Table 2 Modulus of monotone isotropic hardening q 1 (MPa) as a function of the length of the plastic deformation path on monotone parts χ m for the heat-resistant alloy Nimonic 80A (q 2 = 0) (see Fig. 3), which are used to reconstruct numerically low-cycle fatigue curves for various deformation trajectories with a high accuracy; -the damaged medium model developed in [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] describes accurately enough for engineering purposes the processes of isothermal and nonisothermal deformation and fatigue damage accumulation under combined thermal-mechanical loading and makes it possible to evaluate low-cycle fatigue life of heat-resistant alloys for random deformation trajectories; -for the isothermal deformation regimes with a strain amplitude of e 11 = 0.45%, the temperature substantially affects fatigue life of the alloy: fatigue life decreases as the temperature increases (thus, for T = 571 • C number of cycles to failure N f = 367, whereas for T = 823 • C N f = 140). For an increasing strain this trend does not work (see Table 4); -for the nonisothermal deformation regimes in this temperature range (571 • C-823 • C) the heat-resistant alloy Nimonic 80A is insensitive to the phase pattern of the change in temperature and mechanical deformation (see Table 4).

T = 571 • C χ m 0 0.
The next example presents the results of comparing the numerical and experimental data on thermal cyclic deformation of thin-walled tubular specimens of the heat-resistant cobalt alloy Haynes 188 in the temperature interval of 316-760 • C. Isothermal and nonisothermal experiments were conducted with hard loading: the amplitude of axial strain e 11 was 0.4%, and that of shear strain e 12 amounted to 0.9% [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF].

In the experiments described in [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF], the following parameters varied:

-phase shift angle θ between axial strain e 11 and shear strain e 12 (for isothermal deformation with θ = 0proportional loading; for θ = 90 • , the axial and shear strains vary in counter-phase); -phase shift angle ψ between the amplitudes of axial strain e 11 and temperature T (for nonisothermal loading with ψ = 0 the axial strain and temperature vary in-phase, for ψ = 180 • in counter-phase).

In the course of complex isothermal and nonisothermal deformation described in [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF] eight types of tests were implemented:

-experiment T316MIP: mechanical deformations vary in-phase (θ = 0), temperature T = 316 Assigning different laws of change of axial deformation e 11 , shear deformation e 12 and temperature T , different deformation trajectories can be realized (see Table 5).

The main physical-mechanical characteristics of the heat-resistant cobalt alloy Haynes 188 [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF] and the material parameters of the damaged medium model [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] are listed in Tables 6, 7 and 8.

Figure 5 compares the results of numerical modeling and experimental data on the cyclic hysteresis loops for N = 50-th cycle of nonisothermal loading (experiments MIPTIP, MOPTOP). Here and in what follows, Table 5 Laws of change of axial strain e 11 , shear strain e 12 and temperature T (deformation trajectories) Table 6 The main physical-mechanical characteristics and the material parameters of the damaged medium model for the heat-resistant cobalt alloy Haynes 188

Characteristics

Temperature T ( the solid lines indicate the results of numerical modeling using the defining relations of MDM [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF], and the dotted lines show the corresponding experimental data (the comparison was done for all the test data presented in [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF]). The analysis of the numerical results shows that during isothermal cyclic loading (experiments T316MIP, T316MOP, T760MIP, T760MOP) stabilization of the cyclic thermal-plastic hysteresis loops is observed.

During nonisothermal thermal cyclic loading (experiments MIPTIP, MIPTOP, MOPTIP, MOPTOP) the form of axial σ 11 ∼ e 11 and shear σ 12 ∼ e 12 thermal-plastic hysteresis loops depends on phase shift θ Table 7 Monotone isotropic hardening modulus q 1 (MPa) as a function of the length of plastic deformation path over monotone parts χ m for the heat-resistant alloy Haynes 188 (q 2 = 0) between the amplitudes of axial strain e 11 and shear strain e 12 . Thus, in experiment MFTF during the 50-th cycle relaxation of stresses (both normal and tangential) near the maximal temperature of the cycle (see Fig. 5).

T = 316 • C χ m 0 0.
The plastic hysteresis loops in experiment MOPTOP qualitatively differ from the in-phase mechanical tests MIPTIP (Fig. 5).

Thus, the position and degree of relaxation of both axial and shear thermal-plastic hysteresis loops are greatly determined by phase shift angle ψ between axial strain e 11 and temperature T .

Figure 6 presents computational and experimental diagrams of the amplitudes of axial σ 11 and shear σ 12 stresses as a function of number of loading cycles. The solid line shows the numerical modeling results; the markers depict the corresponding experimental data. It can be seen that the degree of cyclic hardening for nonisothermal thermal cyclic loading is considerably higher than that observed in the isothermal tests with The analysis of the obtained results makes it possible to conclude that the model of cyclic thermal-plastic deformation introduced in the paper adequately describes the experimental results and thus can be used in future for analyzing the kinetics of the fatigue damage accumulation process under cyclic thermal-mechanical effects.

To study the effect of deformation trajectory on low-cycle fatigue of heat-resistant alloys, fatigue damage accumulation processes were numerically analyzed for various assigned laws of change of axial strain e 11 , shear strain e 12 and temperature T (see Table 5). The computational results were compared against the available test data [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF]. The isothermal fatigue curves for temperatures T = 571 and 760 • C (solid lines) were obtained using spline-approximation (quadratic spline) of the fatigue curves for temperatures pri temperaturah T = 425,650 and 870 • C [27]. To determine material parameters of the evolutionary equation of damage accumulation, point P on the fatigue curves (Fig. 7) was used. The material parameters of the evolutionary equation of damage accumulation [START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] for the heat-resistant alloy Haynes 188 are listed in Table 6.

The computational results are presented and compared with the available test data in Table 9. The comparison of the numerical results and experimental data show that:

-the damaged medium model developed in [START_REF] Volkov | Uravneniya sostoyaniya vyazkouprugoplasticheskih sred s povrezhdeniyami[END_REF][START_REF] Volkov | Vvedenie v kontinualnuyu mehaniku povrezhdennoy sredy[END_REF][START_REF] Volkov | Modelirovanie slozhnogo plasticheskogo deformirovaniya i razrusheniya metallov pri mnogoosnom neproportsionalnom nagruzhenii (in Russ.)[END_REF][START_REF] Volkov | Chislennoe modelirovanie nakopleniya povrezhdeniy pri slozhnom plasticheskom deformirovanii[END_REF][START_REF] Volkov | Modelirovanie osnovnyh zakonomernostey protsessa deformirovaniya i nakopleniya povrezhdeniy v konstruktsionnyh materialah na baze kontseptsii mehaniki povrezhdennoi sredy (in Russ.)[END_REF] accurately enough for engineering purposes describes the processes of cyclic isothermal and nonisothermal deformation and fatigue damage accumulation under combined thermal-mechanical loading and makes it possible to evaluate low-cycle fatigue life of heatresistant alloys for arbitrary deformation trajectories (when comparing experimental and test data it should be kept in mind that the information in [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF] is presented without accounting for the scatter of experimental data within each individual test); -during isothermal cyclic thermal-plastic deformation T = 316 • C, nonproportional mechanical deformation (test T316MOP) leads to an almost threefold decrease in fatigue life as compared with the proportional one (test T316MIP). With the increasing temperature T = 760 • C this dependence does not work; -nonisothermal cyclic thermal-plastic deformation leads to an increased fatigue life as compared with the isothermal one; -the lowest fatigue life is observed when the maximal value of the temperature cycle is achieved in a halfcycle of tension for mechanical strains e 11 and e 12 , whereas the minimal value of the temperature cycle is achieved in a half-cycle of compression (test MIPTIP), which agrees with the experimental data of [START_REF] Kalluri | An axial-torsional termomechanical fatique testing technique[END_REF].

The highest fatigue life (test MOPTOP) is observed when mechanical strains and temperature vary in counter-phase (θ = 90 • and ψ = 180 • ).

Conclusions

In the framework of mechanics of damaged media, a mathematical model has been developed, which describes processes of thermal-plastic deformation and fatigue damage accumulation in polycrystalline structural alloys under arbitrary complex regimes of low-cycle loading The reliability of the modes has been assessed by comparing the computational results with the test data on thermal cyclic fatigue life of heat-resistant alloys (Nimonic 80A, Haynes 188) under isothermal and nonisothermal regimes of combined thermal-mechanical loading, which corroborated the adequacy of modeling and determining of the material parameters of the defining relations of MDM. Some interesting analyses of damage and fracture can be carried with very performing results also in case of microstructured architectured materials. These materials exhibit different architectures and geometries. A large class of them can be considered as constituted by fibers, which may be modeled as Euler-Bernoulli beams [START_REF] Andreaus | A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams[END_REF][START_REF] Spagnuolo | A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling[END_REF][START_REF] Eugster | Constraints in structural and rigid body mechanics: a frictional contact problem[END_REF][START_REF] Eugster | Determination of the transverse shear stress in an Euler-Bernoulli beam using non-admissible virtual displacements[END_REF][START_REF] Eugster | An intrinsic geometric formulation of the equilibrium equations in continuum mechanics[END_REF]. In particular, one can develop continuum models for the description of fiber reinforced materials, as [START_REF] Franciosi | Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates[END_REF], or array of fibers planarly aligned, also known as pantographic structures [START_REF] Boutin | Linear pantographic sheets: asymptotic micro-macro models identification[END_REF][START_REF] Misra | Pantographic metamaterials show atypical Poynting effect reversal[END_REF][START_REF] Turco | Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations[END_REF][START_REF] Giorgio | Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis[END_REF][START_REF] Placidi | A review on 2D models for the description of pantographic fabrics[END_REF][START_REF] Barchiesi | A review on models for the 3D statics and 2D dynamics of pantographic fabrics[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF][START_REF] Placidi | An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations[END_REF][START_REF] Turco | Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model[END_REF][START_REF] Golaszewski | Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions[END_REF][START_REF] Lekszycki | Application of energy measures in detection of local deviations in mechanical properties of structural elements[END_REF]. For 2D lattices with uniform triangular microarchitecture (being possibly the simplest possible 2D configuration and providing an isotropic material in the classical sense), the material properties related to bending has been determined in [START_REF] Khakalo | Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics[END_REF]. For a more complicated 2D triangular lattice, higher-order material parameters associated with the third gradient elasticity, have been determined by tension, shear and vibration tests in [START_REF] Khakalo | Form II of Mindlin's second strain gradient theory of elasticity with a simplification: for materials and structures from nano-to macro-scales[END_REF]. Before studying damage and fracture in generalized material, it will be useful to perform some dynamic analyses as the one presented in [START_REF] Abbas | Wave propagation in a generalized thermoelastic plate using eigenvalue approach[END_REF][START_REF] Abd-Alla | Plane waves and eigenfrequency study in a transversely isotropic magneto-thermoelastic medium under the effect of a constant angular velocity[END_REF]. In particular, first experimental results about dynamic behavior in pantographic metamaterials has been discussed in [START_REF] Laudato | First experimental observation of the dynamical behavior of a pantographic metamaterial[END_REF].

Analysis of damage and fracture can be also performed in the field of generalized materials, suitably described by higher gradient models. Some properties of higher gradient models can be involved in such analyses. In particular, it is possible to refer to [START_REF] Mohammed | Discrete and continuous aspects of some metamaterial elastic structures with band gaps[END_REF][START_REF] Placidi | Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second gradient elasticity[END_REF][START_REF] Placidi | A second gradient formulation for a 2D fabric sheet with inextensible fibres[END_REF] for detailed descriptions of these models and their principal features when the considered material is composed by inextensible fibers. A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model is reported in [START_REF] Placidi | A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model[END_REF].

Strain gradient modeling has proven to be useful also in regularizing mesh-dependent problems in continuum damage and fracture mechanics, where localization of deformations is frequently observed. Implementation of such regularization in second-gradient materials would be straightforward, also in view of parameters identification carried out in [START_REF] Placidi | Energy approach to brittle fracture in strain-gradient modelling[END_REF][START_REF] Placidi | Two-dimensional strain gradient damage modeling: a variational approach[END_REF][START_REF] Placidi | A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results[END_REF][START_REF] Placidi | Simulation results for damage with evolving microstructure and growing strain gradient moduli[END_REF]. In [START_REF] Barchiesi | Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation[END_REF] experimental results relative to displacement-controlled large deformation shear loading tests of pantographic structures are reported. A generalized plate model is validated by means of these experimental results. In particular, five differently sized samples are analyzed up to first rupture. Results show that the deformation behavior is strongly nonlinear, and the structures are capable of undergoing large elastic deformations without reaching complete failure.

The thermo-mechanical approaches are, in general, very useful. Apart the applications we have studied in this paper, these methods can be suitably applied also in case of beams. An independent and novel approach to understand the large deformation of beams consists in regarding a beam as the boundary curve of a twodimensional manifold in a three-dimensional space. In doing so, not only is the curve endowed with its own energy similar to that in the context of lower-dimensional energetics [START_REF] Javili | Thermomechanics of solids with lower-dimensional energetics: On the importance of surface, interface, and curve structures at the nanoscale. A unifying review[END_REF] but also in a geometrically nonlinear framework [START_REF] Javili | A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology[END_REF] and in accordance with higher gradient elasticity accounting for boundary energetics elaborated in [START_REF] Javili | Geometrically nonlinear higher-gradient elasticity with energetic boundaries[END_REF]. The advantage of this approach, particularly from a computational viewpoint, is that the bulk material acts to regularize the behavior of the beam especially important to analyze the instabilities associated with thin beams similar to the instabilities of thin films on an elastic foundation [START_REF] Javili | Computational aspects of growth-induced instabilities through eigenvalue analysis[END_REF]. Obviously, in the limiting case of the vanishing bulk one would recover exactly the beam theory.

In the field of polycrystalline structural alloys, a further application of the methods developed in this paper could be could be implemented in the case of the Potts model, whose 1D case is exactly solvable [START_REF] De Masi | Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles[END_REF][START_REF] De Masi | Coexistence of ordered and disordered phases in Potts models in the continuum[END_REF].

The above-discussed methods, especially for generalized materials, can be validated by numerical simulations. One of the currently most adopted techniques in finite elements coding is based on isogeometric analyses [START_REF] Niiranen | Isogeometric analysis for sixth-order boundary value problems of gradientelastic Kirchhoff plates[END_REF][START_REF] Khakalo | Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software[END_REF][START_REF] Greco | An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods[END_REF][START_REF] Cuomo | An implicit strong G1-conforming formulation for the analysis of the Kirchhoff plate model[END_REF][START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF][START_REF] Cazzani | Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes[END_REF].

Fig. 1 Fig. 2

 12 Fig. 1 Laws of change of axial strain e 11 and temperature T

Fig. 3

 3 Fig. 3 Isothermal fatigue curves of the heat-resistant alloy Nimonic 80A

Fig. 4

 4 Fig. 4 Nonisothermal fatigue curves for the in-phase (a) and counter-phase (b) cyclic loading of the heat-resistant alloy Nimonic 80A

  • C remains constant (isothermal tests); -experiment T316MOP: mechanical strains vary in counter-phase (θ = 90 • ), temperature T = 316 • C remains constant (isothermal tests); -experiment T760MIP: mechanical deformations vary in-phase (θ = 0), temperature T = 760 • C remains constant (isothermal tests); -experiment T760MOP: mechanical strains vary in counter-phase (θ = 90 • C), temperature T = 760 • C remains constant (isothermal tests); -experiment MIPTIP: mechanical strains and temperature vary in-phase (θ = 0 and ψ = 0)-nonisothermal tests; -experiment MIPTOP: mechanical strains vary in-phase, temperature varies in counter-phase (θ = 0 and ψ = 180 • )-nonisothermal tests; -experiment MOPTIP: mechanical strains vary in counter-phase, temperature varies in-phase (θ = 90 • and ψ = 0)-nonisothermal tests; -experiment MOPTOP: mechanical strains and temperature vary in counter-phase (θ = 90 • and ψ = 180 • )-nonisothermal tests.

Fig. 5

 5 Fig. 5 Results of comparing the experimental and numerical data for cyclic plastic hysteresis loops (N = 50) for nonisothermal loading

Fig. 6

 6 Fig. 6 Computed and experimental diagrams of the amplitudes of axial σ 11 and shear σ 12 strains

  Figure 7 demonstrates the experimental fatigue curves of the heat-resistant alloy Haynes 188 [27] under uniaxial-tension-compression for temperatures T = 425,650 and 870 • C, respectively (dotted lines).

Fig. 7

 7 Fig. 7 Fatigue curves of heat-resistant alloy Haynes 188 under isothermal loading

Table 1

 1 The main physical-mechanical characteristics and the material parameters of the damaged medium model of the heatresistant alloy Nimonic 80A

	Characteristics	Temperature T ( • C)
		571	700	823
	K (MPa)	150,500	139,666	106,000
	G (MPa) α (1/ • ) C • p (MPa)	70,450 0.0000148 384	64,461 0.0000155 363	48,333 0.0000164 340
	g 1 (MPa)	70,850	56,300	15,500
	g 2	308	290	271
	a	65	58	45
	W f (MJ/m 3 )	41.5	39	35.5
	W a (MJ/m 3 )	0	0	0

Table 3

 3 Cyclic hardening modulus Q 1 (ρ max ) (MPa) as a function of the value of maximal displacement of the yield surface center ρ max for the heat-resistant alloy Nimonic 80A (Q 2 = 0)

			0005	0.001	0.002	0.003	0.004	0.005	0.0075	0.01
	q 1	-1981	-5362	-8255	-11,002	-8321	-7006	-8243	-14,420	-20,600
	T = 700 • C
	χ m	0	0.0005	0.001	0.002	0.003	0.004	0.005	0.0075	0.01
	q 1	2666	-3667	-4202	-3847	-3049	-8346	-8112	3072	14,260
	T = 823 • C
	χ m	0	0.0005	0.001	0.002	0.003	0.004	0.005	0.0075	0.01
	q 1	-4974	-4405	-3692	-2124	-3716	-3463	-2890	-1722	-553
	T = 571 • C
	ρ max		266.6		232.7		160.9		88.5
	Q 1		384			395		420		450
	T = 700 • C
	ρ max		160.5		140.9		106.6		63.2
	Q 1		303			309		281		278
	T = 823 • C
	ρ max		68.7			65.8		57.86		45.9
	Q 1		230			219		125		120

Table 4

 4 Number of cycles to failure for different types of thermal cyclic loading

	Loading regime		Strain amplitude, e 11 , %	Number of cycles to failure, N f
				Experimental	Numerical
	Isothermal	T = 571 • C	0.45	319	367
			0.6	66	66
			0.8	24	20
		T = 823 • C	1 0.45	1 3 128	1 0 140
			0.6	65	64
			0.8	51	42
			1	3 0	2 5
	Nonisothermal in-phase	0.45	230	255
			0.6	73	65
			0.8	32	31
			1	2 0	1 7
	Nonisothermal counter-phase	0.45	325	255
			0.6	60	64
			0.8	35	30

Table 8

 8 Cyclic hardening modulus Q 1 (ρ max ) (MPa) as a function of the maximal displacement value of the yield surface center ρ max for the heat-resistant alloy Haynes 188 (Q 2 = 0)

			0002	0.0004	0.0005	0.001	0.002	0.003	0.005	0.01
	q 1	-2401	-2716	-3032	-3190	-3979	-5296	-6089	-6186	-2133
	T = 760 • C
	χ m	0	0.0002	0.0004	0.0005	0.001	0.002	0.003	0.005	0.01
	q 1	-2124	-2129	-2135	-2138	-2152	-2470	-3367	-5207	-1471
	T = 316 • C
	ρ max			24.35			46.81			49.16
	Q 1			520			520			500
	T = 760 • C
	ρ max			43.88			46.51			60.75
	Q 1			350			440			330

Table 9

 9 Number of cycles to loading for different regimes of thermal cyclic loading

	Loading regime	Number of loading cycles, N
		Experiment	Numerical
	Isothermal
	T316MIP	9448	10,512
	T316MOP	3410	2860
	T760MIP	910	1205
	T760MOP	1089	785
	Isothermal
	MIPTIP	282	143
	MIPTOP	569	830
	MOPTIP	270	341
	MOPTOP	1004	1307

Acknowledgements This work was supported by a grant from the Government of the Russian Federation (Contract No. 14.Y26.31.0031).

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Man Liang

Daniel Delahaye
email: daniel@recherche.enac.fr

Drone Fleet Deployment Strategy for Large Scale Agriculture and Forestry Surveying

Keywords: Drone fleet management, Agriculture and forestry surveying, cooperative path planning

Agriculture drones offer clear advantages over other monitoring methods including satellite imaging, manned scouting, and manned aircraft. However, for large scale areas, such as large forestry and agriculture mapping problems, the single drone is hard to accomplish its mission of mapping in a relatively short time period of 30 to 45 minutes. In addition, in large forestry mapping, camera, communication, and payload settings may further reduce the maximum endurance of drones in the air. With a single drone, the total required mission time to cover all the area is prolonged, not only producing a high cost for a drone service provider but also having more uncertainty. While with multiple drones, or a fleet of drones, it is possible to identify a globally optimized solution to reduce the total required mission time. In this paper, we mainly discuss the strategy of drone fleet deployment for large scale area surveying. Three key parts are analyzed, including a fleet of drones, cooperative coverage path planning, communication and data processing. The associated state-of-the-art solutions are listed and reviewed. In addition, in this paper, the key operational constraints for large scale agriculture and forestry surveying are analyzed. It should be pointed out that, from a comprehensive point of view, a drone fleet deployment for large scale surveying could attract more attention from the commercial drone industry.

I. INTRODUCTION

Unmanned Aerial Vehicles (UAV) or drones have gained popularity in many industries due to their vast potential applications, such as survey, search and rescue, agriculture, and forestry. In a 2016 report, Goldman Sachs estimated that drone technology will reach a total market size of $100 billion between 2016 and 2020 [START_REF] Goldman | Drones[END_REF]. Commercial drones currently only represent 6% of the market in term of units, but their price tags of approximation of $100,000 are projected to representing 60% of the industrial revenue. According to PWC, the most promising industry seems to be infrastructure, with a global potential valued at $45 billion, followed by agriculture, at $32 billion and transportation at $13 billion [START_REF]Global report on the commercial applications of drone technology[END_REF].

Agriculture drones offer more compelling advantages over other monitoring methods. Satellite imaging, manned scouting and manned aircraft, for example: 1) Cheaper imaging: for fields less than 50 hectares in size, drones are considerably less expensive than satellites or manned aircraft surveillance. 2) Greater precision: drone cameras take centimeter-level images. 3) Earlier detection of problems: because drones survey more frequently, abnormalities are detected earlier [START_REF] Andrew | Best drones for agriculture 2019: The ultimate buyers guide[END_REF].

In previous research, significant effort has been applied on studying the single drone mission flight within line-ofsight. However, for large scale area, such as large forestry and agriculture mapping problems, single drones are hard to accomplish its mission of mapping in a relatively short amount of time, 30 to 45 minutes and within line-of-sight range. With multiple drone, it is capable of accomplishing tasks which single drone cannot. In addition, large forestry mapping and assessment, camera, communication, and payload settings determine the maximum available flying time. With single drone, the possible mapping time is restrained, while multiple drones, it is possible to identify the global optimized solution. Thus, in this paper, we aims to discuss drone fleet deployment strategy on large scale land surveying, which is beneficial in reducing the cost efficiency of commercial drone service providers.

II. STATE-OF-THE-ART A. Coverage planning with single drone

In operational reality, many of the latest agriculture drones come with flight planning software that lets the users to draw a box around the field they want to survey on Google Earth map. Otherwise, an automated flight plan to be created for the user. Some even automatically plan the camera shots for them. Then, the users would upload the flight path onto their drone and get ready to fly. If the users are using a home-built platform or a ground control station, then they may need to manually code or set for each points to turn [START_REF] Andrew | Best drones for agriculture 2019: The ultimate buyers guide[END_REF].

B. Coverage planning with multiple drones

In the sight of academic research field, researchers spot their attention closely on the swarm behavior of drones. They aim to use networked swarm model for drone operations. For example, Swarm Robotics for Agricultural Applications (SAGA). Multi-rotor drones are deployed to collectively monitor a sugar beet field and cooperatively map the presence of volunteer potatoes. The major threat is that they spread diseases (e.g., late blight) and facilitate harmful soil nematodes [START_REF]Saga: Swarm robotics for agricultural applications[END_REF]. Swarm robotics, in particular, is considered extremely relevant for precision farming and large-scale agricultural applications. However, swarm robotics research is yet confined,only a few application in the field is currently available [START_REF]Saga: Swarm robotics for agricultural applications[END_REF]. SAGA demonstrated for the first time the application of swarm robotics principles to the agricultural domain. Specifically, SAGA targets a decentralized monitoring/mapping scenario, and implements the use case for the detection and mapping of weeds in field by a group of small drones.

In terms of industrial usage, most of the popular drone fleet management models are designed simply. It models with a number of independent missions. There is no communication between drones, but there maybe a conflict alert service for remote pilots. However, one start-up company SkyX is special, they are in the field of Precision Agriculture Spraying Swarm (PASS). It enables the modular swarm of autonomous rotor drones for spraying. SkyX is developing a proprietary communications and computing hardware that will be installed on each drones to allow for real-time planning and swarming. Unfortunately, there is no application of fixedwing swarm drones found available for precision agriculture.

C. Contributions

Note that line-of-sight drone flying rule makes swarm drones for large scale agriculture application concept operationally less achievable to be implemented in reality. However, it should be emphasized that restriction for drones has its publicity support by the Beyond Visual Line Of Sight (BVLOS) missions in nowadays. In case of resulting in injury to person due to unmanned aircraft crash that is below a certain severity threshold, it may permit to do a BVLOS mission with small drones [5]. Facing the near future with BVLOS drones in the national airspace and the great revenue of drone services in agriculture, new drone fleet management system is needed for accident avoidance, such as life-threatening collision. Efficiently planning paths with multiple BVLOS drones, requires collaboratively communicating among drones, etc. In this paper, we will address the problem of BVLOS drone fleets deployment for large scale agriculture and forestry surveying from a strategic point of view. Three topics are: fleets of drone, coverage path planning, the communication and data processing.

III. STRATEGY OF DRONE FLEET DEPLOYMENT FOR LARGE SCALE AREA

A. Fleet of drones

Fixed-wing drones are the best choice to cover a large scale of area. Fixed-wing agriculture drones can cover up to 10 times of the acreage then a typical quadcopter can cover in a single flight [START_REF] Andrew | Best drones for agriculture 2019: The ultimate buyers guide[END_REF]. The most popular ready-to-fly agriculture drone systems in the world with fixed-wing and battery powered are listed in Table I. Most of the agriculture drones are powered by electric batteries, when in need for heavy duty, or heavy payload, drones with gas powered are available. Gas powered agriculture drones could fly much longer than the battery powered drones, also more stable in windy weather for aerial photography purpose. However, operating cost of gas powered drones is more expensive and more pollute to the air. Hybrid drones merge the benefits of fixed-wing drones with the ability to hover, Vertical Take-Off and Landing (VTOL) same as rotor drones and cruise ability as fixed-wing drones. These three kinds of fixed-wing agriculture drones are shown in Fig. 3. In this paper, battery powered fixed-wing agriculture drones are selected as the study case for drone fleets deployment.

B. Coverage path planning 1) Coverage decomposition:

The shape of the area could be convex polygon, irregular polygon, or irregular line etc. For a large area, a single drone with generally maximum of 60 minutes flying endurance can only cover some part of the large survey area. The solution could be numbers of separated missions with individual drone at different time, or using multiple drones to cover the entire area at the same time, see ATM is a collaborative way to ensure that safety and efficient of aircraft operation in the sky. The entire airspace is divided into several sectors by considerating the traffic complexity and controller's workload. The objective of sectorization is to find the best solution with minimum number of sectors and relatively balanced workload for controller. In addition, the airspace sectorization optimization include both static and dynamic airspace management [START_REF] Delahaye | Genetic algorithms for automatic regroupement of air traffic control sectors[END_REF], [START_REF] Treimuth | A branch-and-price algorithm for dynamic sector configuration[END_REF]. As shown, the related algorithms in ATM airspace management could give some experience in developing a suitable model for large scale survey with multiple drones. Here, for large area of agriculture or forestry surveying with drones, the coverage decomposition is to divide the area of interest into several smaller cells, as shown in Fig. 3. Some operational constraints in coverage decomposition are discussed as following:

• The available number of drones in a fleet. Different drone service providers have different number of drones available for a specific mission in large area agriculture or forestry surveying. • The required mission time constraint. Due to the uncertainty of weather in local or tropical area,day-time drone operational rules, and priority levels of mission, the possible mission time for drone fleet varies. Especially, most of the agriculture drones have low resistance to water, available mission time could be a tremendous constraint during the optimization process. • Take-off and land site. Fixed-wing drones need to take off at a specific site, the space of this site has to be wide enough, especially in forestry mapping where the density of the trees is high, it is challenging to locate suitable taking-off sites. Constraint in the optimization is: at least one available site is available for one drone for each small sector after coverage decomposition. • Endurance. Battery volume is fixed, and it is impossible to recharge battery while airborne in today's technology. Moreover, the endurance of battery is sensitive to the weather condition. In the optimization problem, the shortest endurance and the longest endurance should be transferred into a maximum coverage area in consideration of the ground speed of drones. Meanwhile, the data solution with camera and sensor system consume high level of energy, contributing to the endurance. Last but not least, it is important that emergency procedure is available for drone return to safe place with its maximum endurance. • Radio frequency. To manage a fleet of drones, decentralized and centralized architecture are feasible. In a centralized approach, radio communication between drone and ground station about commands and control links must be emphasized. The remote pilot, who is on the ground, can control the aircraft using commands and control links, receiving real-time information about the drone on-board system. The radio communication signal range and the availability of radio frequencies are constraints for drone fleet deployment. Once signal is lost, disconnection between drones and ground station occurs. In this case, it is dangerous. Thus, in the optimization problem, a constraint was considered, that is: the signal should 100% cover all the sectors after coverage decomposition. • The controlling capacity of remote pilot. Most of the remote pilot only control one drone. In order to control a fleet of drones, decision making supporting tools should be developed to help remote pilot to handle more drones at a time. For example, conflict detection and warning, conflict resolution manoeuvring advice, multiple flight trajectory management system etc. If remote pilot could handle two drones at a time, the mission time could be reduced to half of the mission time completed by a drone, which is beneficial to drone service providers.

2) Shape of mission area and cooperative path planning: For conventional agriculture surveying and mapping problem, geometric approach is popularly. As shown in Fig. 4, in convex polygon and concave polygon case, a conventional back-and-forth path planning is designed to cover all of the surveying area, and with different cooperative path planning with multiple drones, the strategy is different.In the case of polygon with no-fly zone, firstly we decompose the entire irregular polygon shape into several small polygon area by the blue cutting line, then a conventional rectangular backand-forth pattern to be applied to do the survey on each smaller cell. The path planning consists of several nodes and legs, either straight legs or curved legs. For curved legs, it requires drones to have a better manoeuvrability.

There is another strategy to do the cooperative path planning on the irregular area with no-fly zone. A grid based approach is introduced by [START_REF] Valente | Near-optimal coverage trajectories for image mosaicing using a mini quad-rotor over irregular-shaped fields[END_REF]. Firstly, the area of interest is divided into several regular grids. Each cell represents Fig. 4. Shape of mission area and cooperative path planning a way-point of the path and its size, depending on the picture dimensions of the on-board camera. Then a heuristic algorithm is applied to invedtigate the links between different nodes. As shown in Fig. 5, the solution with single drone and the solution with multiple drones are totally different. With multiple drones, it is possible to cover all the area of interest in a short time, and these drones could share the same takeoff and landing area. This kind of cooperative strategy with a fleet of drones requires networked based deployment. This problem is also similar to Open Vehicle Routing Problem (OVRP). OVRP is a version of the well known Capacitated Vehicle Routing Problem (CVRP), in which each route ends with the last serviced user. More precisely, a vehicle does not return to the depot after servicing the last user as is the case of classical CVRP. There are another cooperative path planning with multiple drones, such as spiral, line formation, and decentralized technique based strategy [START_REF] Cabreira | Survey on coverage path planning with unmanned aerial vehicles[END_REF]. They significantly contributing to the reduction of mission time in single drones.

3) Weather effect: Besides seamless routing constraints on the coverage path planning problem, rising expectations to the new outdoor applications are aspect to be considered, such as the weather forecast and energy consumption.

The capability and application of new drones require high demand of power which triggers the need of developing strategies to promote efficient energy usage. Exploiting windenergy is one of the most important ways to extend flight duration for drones. To support functions of all sensor system on board to the best usage, wind is of foremost awareness. The importance to affect migrating birds' optimal flying speed and altitude, and their optimal migratory routes. With the same concept, we could develop an optimal coverage path planning algorithm in consideration of multiple objectives: energy saving, good quality of sensor data collection and bigger coverage area etc. With the benefit of wind, the ground speed of fixed wing drone could satisfy the requirement of surveying speed without consuming excess energy on board. Thus, searching for a feasible and optimal path, planning solution with the precise wind field prediction could definitely benefit a lower cost on large scale aerial survey. [START_REF] Holvoet | Planning for active mapping of crop fields using uavs[END_REF] proposes a novel method based on meta-heuristics produce an optimized flight time trajectory to survey a non-convex poly-gon area in wind with a fixed-wing drone and curved path. While offering better results than the previous approach, this algorithm is proved to greatly reduce execution time. Thus, it can be used for online re-planning tasks where the plan goals are deduced from a vision-based simultaneous localization and mapping (SLAM), which exhibits areas where additional data must be gathered. This work is developed and validated within a realistic simulation framework.

Rain, another weather condition that affect large area surveying missions. Especially in tropical area, due to the higher uncertainty of precipitation prediction, some on-thejob mission has to be stopped, and it may cause damage to drones. Thus, 1-3 hours' weather forecast data should be provided to remote pilot for a better path planning.

C. Communication and data processing

Pre-flight path planning is important. Due to the complexity of weather and other uncertainty factors, an online planning decision making is required to dynamically adapt to the environmental conditions which drones operate. Meanwhile, agriculture and forestry surveying need to use cameras to extract high quality photos. All this operations require power. However, the amount of energy that battery carries on drone is constrained.

Firstly, drones are aircraft flown in absence of physically present humans on board and can either be operated remotely or with pre-programmed script to execute planned missions on given data. The challenge is to find a smart and intelligent approach to efficiently execute the mission with smart online communication and data processing. To process image data, one solution is to use drone equipped Odroid-X motherboard 1, which consumes a high amount of energy for processing images than other boards, such Rapsberry PI. Another alternative is the possibility of integrating them with Cloud processing. Depending on the kind of processing required by the drone, Cloud offloading is possible, that is transfer the processing of some images from the drone to the Cloud, so that it can be processed properly, without overloading the drone. This entails the drone sending the images to Cloud by 3G network service and being given the result of the processing afterwards. The outcome from image processing by Cloud can be sent back to the drone or be forwarded directly to a base station. This solution needs an embedded intelligence in the drone so that it can make decisions while taking images and deciding whether it should process images locally or convey it to a Cloud. In addition, compared with expert image data processing and local self-processing methods, cloud-based automated processing normally is at low cost, fast turnaround time, and high reliability, but the accuracy may be relatively low. Secondly, as each drone has limited coverage and computational capabilities, for large scale surveying mission with a fleet of drones, it is necessary to consider how to keep effective communication between drones and a relay station. A relay station is a broadcasting system that uses communications signals to direct drones. Relay stations can be a satellite, command vehicle, or ground station. In [START_REF] Wang | Taking drones to the next level: Cooperative distributed unmannedaerial-vehicular networks for small and mini drones[END_REF], Flying Ad-hoc NETwork (FANET) concept is proposed to bolster the performance and range of small drones, see Fig. 6. FANET is based on the Vehicular Ad-Hoc Network (VANET) wireless communication system created to help self-driving cars position themselves in relation to other self-driving cars in the area. In FANET, multiple drones are part of a cooperative system, which contains a sensor unit, a communication unit, and an information-processing unit. The range and efficiency of a FANET is enhanced by the coordination and collaboration of its drones, which allows the network to cover more ground. When the communication session between a single drone and its relay station is interrupted, the drone becomes isolated. With FANETs, the communication between drones and relay station will remain more resilient due to the support of wireless communication gateways through select drones, which disseminate information throughout the network. The gateway selection is a critical component of FANET. To optimize gateway selection, [START_REF] Wang | Taking drones to the next level: Cooperative distributed unmannedaerial-vehicular networks for small and mini drones[END_REF] has also developed a dynamic gateway-selection algorithm for ensuring that the gateway function in the network is sustained. If a gateway drone goes down, another gateway drone in the cluster will take over relaying the information. The simulations demonstrated that the proposed dynamic gateway clustering was capable of significantly increasing the average gateway-retention probability.

IV. CONCLUSION

With multiple drones, it is possible to largely reduce the mission time, which gives great benefits for drone service providers. In this paper, the strategy of drone fleet deployment for large scale agriculture and forestry surveying problem was mainly analyzed. Firstly, a short review about stateof-the-art coverage path planning is done. Then, several key aspects in drone fleet deployment was explained in details, including fleet of drones, coverage decomposition, cooperative path planning for regular and irregular area, weather effect, communication and data processing. In future works, we are going to build up a mathematical optimization model to solve a real surveying problem and get the numerical results.

Fig. 1 .

 1 Fig. 1. Type of agriculture drones

Fig. 3 .

 3 Fig. 3. Coverage decomposition wit cells

Fig. 5 .

 5 Fig. 5. Grid based approach to cover the irregular area (Note: figures from [9])

Fig. 6 .

 6 Fig. 6. FANET concept

TABLE I

 I

			AGRICULTURE DRONES SPECIFICATIONS
	Type of drone	Weight [lb] Max. Time [min] Max. Cov. [acres/flight]	Cruise speed[m/s]	Wind resis. [m/s]
	eBee SQ	2.4	55	500	11-31	12
	Birdseyeview Firefly6 Pro	9.9	59	600	15-18	null
	AgDrone	4.75	55	600	13-23	9
	AgEagle RX60	7	60	400	13.8-29.9	19
	Delair UX11	3.1	59	600	15	12.5
	Fig. 2. No matter which approach, the problem of decomposi-
	tion method is important. It is found that this decomposition
	problem is immensely similar to the airspace sectorization
	optimization problem in Air Traffic Management (ATM).

ACKNOWLEDGEMENT

The authors would like to thank Dr. Geraldo Pereira from the University of Brasilia on discussions about the cloud based data processing, and Dr. Patrick Ribeiro from Open-Forests for discussions about the drone services in forestry mapping and surveying.

AUTHORS