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XXIIth Winter School on Continuous Media Mechanics

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Procedia Structural Integrity 32 (2021) 1–2

the BigBlueButton platform during the breakout sessions. The School's topics covered the following scientific areas:  physics and mechanics of meso- and nano-structured systems,  convection, hydrodynamic stability and turbulence,  hydrodynamics of multiphase media,

* Corresponding author. Tel.: +7-342-237-8461; fax: +7-342-237-8487. E-mail address: mvp@icmm.ru

Valerii P. Matveenko et al. / Procedia Structural Integrity 32 (2021) 1–2 Author name / Structural Integrity Procedia 00 (2019) 000–000

 hydrodynamics of non-Newtonian liquids and liquids with special characteristics,  computational continuum mechanics,  coupled problems of solid mechanics,  interdisciplinary studies,  mining mechanics.

The materials of the reports presented at the School are published in the issues of the Journal of Physics: Conference Series, Computational Continuum Mechanics and in this issue containing the works on the topics related to the mechanics of a deformable solid body. The Organizing Committee of the XXII Winter School on Continuous Media Mechanics is grateful to the European Society for Structural Integrity (ESIS) for the opportunity to publish the works presented at the School.

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© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” Abstract The paper proposes a mathematical model for estimating changes in the strain fields caused by the installation of a substrate with an optical fiber sensor on the surface of the controlled structure. It also analyzes the information on changes in the strain tensor components in the zone of sensor location at di ff erent dimensions of the structure. It has been shown that these changes depend on the ratio of mechanical characteristics of the substrate and the material of the structure with the fiber optic strain sensor placed on its surface. A comparative analysis of strain changes is made under di ff erent loading conditions of the structure with the substrate-bonded optical fiber sensor placed on its surface. © 2021 The Authors. Published by Elsevier B.V. his is an open access article under the CC BY- C-ND license (http: // cr ativecommons.org / licenses / by-nc-nd / 4.0 / ) P e ie unde responsibility of the scientific committee of the XXIIth Winter Sch ol on Continuous Media Mechanics. Keywords: fiber optic sensors; substrate; strain redistribution; finite element method Fiber-optic strain sensor (FOSS) based on a fiber Bragg grating is a relatively new strain measurement tool com pared to other types of sensors. In the article by Lee et al. (2003), fiber Bragg gratings (FBGs) were applied to perform real-time measurements of dynamic strains inside a small-scale airplane wing model subjected to wind tunnel tests. During these tests the flutter instability was detected with the aid of FBG sensors embedded in the wing skin and the e ffi ciency of this sensor system for monitoring the stress-strain state of an aircraft wing under flight conditions was es timated. The article by Ghoshal et al. (2015) surveys experimental studies of di ff erent types of sensors, including fiber Bragg grating sensors embedded in composite components of army rotorcrafts. The paper also discusses the results of dynamic loading experiments on a composite flexbeam of army rotorcraft with an embedded optical fiber, which were conducted with the aim to detect the beginning of delamination in the flexbeam. A review article by Wymore et al. (2015) analyzes the current state and the main problems of monitoring the mechanical condition of wind turbines. It has been noted that fiber optic sensors can be used to measure the deformation of such structures. The article by Abstract The paper proposes a mathematical model for estimating changes in the strain fields caused by the installation of a substrate with an optical fiber sensor on the surface of the controlled structure. It also analyzes the information on changes in the strain tensor components in the zone of sensor location at di ff erent dimensions of the structure. It has been shown that these changes depend on the ratio of mechanical characteristics of the substrate and the material of the structure with the fiber optic strain sensor placed on its surface. A comparative analysis of strain changes is made under di ff erent loading conditions of the structure with the substrate-bonded optical fiber sensor placed on its surface. © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics. Keywords: fiber optic sensors; substrate; strain redistribution; finite element method 1. Introduction Fiber-optic strain sensor (FOSS) based on a fiber Bragg grating is a relatively new strain measurement tool com pared to other types of sensors. In the article by Lee et al. (2003), fiber Bragg gratings (FBGs) were applied to perform real-time measurements of dynamic strains inside a small-scale airplane wing model subjected to wind tunnel tests. During these tests the flutter instability was detected with the aid of FBG sensors embedded in the wing skin and the e ffi ciency of this sensor system for monitoring the stress-strain state of an aircraft wing under flight conditions was es timated. The article by Ghoshal et al. (2015) surveys experimental studies of di ff erent types of sensors, including fiber Bragg grating sensors embedded in composite components of army rotorcrafts. The paper also discusses the results of dynamic loading experiments on a composite flexbeam of army rotorcraft with an embedded optical fiber, which were conducted with the aim to detect the beginning of delamination in the flexbeam. A review article by Wymore et al. (2015) analyzes the current state and the main problems of monitoring the mechanical condition of wind turbines. It has been noted that fiber optic sensors can be used to measure the deformation of such structures. The article by XXIIth Winter School on Continuous Media Mechanics Analysis of strain changes caused by placing the substrate-bonded optical fiber sensor on the structure surface Andrey Yu. Fedorov a, ∗ , Elizaveta R. Vindokurova b a Institute of Continuous Media Mechanics UB RAS, 1 Academician Korolev Street, Perm 614018, Russian Federation b Perm National Research Polytechnic University, 29 Komsomolsky prospekt, Perm, 614990, Russian Federation XXIIth Winter School on Continuous Media Mechanics Analysis of strain changes caused by placing the substrate-bonded optical fiber sensor on the structure surface Andrey Yu. Fedorov a, ∗ , Elizaveta R. Vindokurova b a Institute of Continuous Media Mechanics UB RAS, 1 Academician Korolev Street, Perm 614018, Russian Federation b Perm National Research Polytechnic University, 29 Komsomolsky prospekt, Perm, 614990, Russian Federation 1. Introduction

∗ Corresponding author. Tel.: + 7-342-273-8330 ; fax: + 7-342-237-84-87. E-mail address: fedorov@icmm.ru ∗ Corresponding author. Tel.: + 7-342-273-8330 ; fax: + 7-342-237-84-87. E-mail address: fedorov@icmm.ru

2452-3216 © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” 10.1016/j.prostr.2021.09.028 2210-7843 © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics. 2210-7843 © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics.

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Hong et al. (2016) provides a review of current developments and applications of FBG sensors for health monitoring of key geotechnical structures, including slopes, piles, and soil nail systems. Possible technical di ffi culties with the use of FBG sensors for geotechnical monitoring are discussed. In the article by Anoshkin et al. (2016), the example of a rectangular plate with ”butterfly” shaped cuts made of polymer composite material was concidered to demonstrate that with the use of fiber-optic strain sensors embedded in the material, it is possible to measure the gradient strain field. Matveenko et al. (2020) presented a technique for detecting the onset and development of local material damage based on the strain values measured by a limited number of fiber-optic sensors and the results of numerical simulation of the stress-strain state. Interest in these sensors and their fast-growing applications to di ff erent engineering facilities is due to a number of advantages they o ff er, including small size, resistance to environmental factors and corrosion, insensibility to electromagnetic interference, short response time, in-line location of several sensors and some others. At the same time, a justified commercial application of fiber-optic strain sensors requires solving a number of new problems. One of these problems is related to the evaluation of strain redistribution caused by the incorporation of sensors into the monitored structure. Fiber-optic strain sensors can be embedded in the material of the structure or located on its surface. In the second option, the most common sensor design is that of an optical fiber mounted on a substrate of metallic or polymeric materials, which, in turn, must be rigidly attached to the surface of the controlled structure. In this case, it is natural to expect that the installation of this substrate on the structure surface will lead to a local redistribution of strains near the installation site. In the last decades, there have been dozens of papers dealing with the estimation of strain transfer from the material of the measured structure to the core of the optical fiber. As a rule, these are the analytical studies, in which three- (or more) layer models based on certain assumptions are used to estimate the characteristic features of strain transfer between the host material and the embedded optical fiber. Ansari et al. (1998) developed a simplified analytical three layer concentric model to estimate the actual strain level based on the interpretation of measurements made with the fiber optic sensors (with protective coating) embedded in the material. The mathematical expressions developed in the framework of this model were used to evaluate the level of strain losses within the protective coating of the optical fiber, and the amount of strain transferred to the optical fiber core depending on the mechanical properties of the fiber core and the protective coating, and the gauge length of the optical fiber. A more advanced n -layer concentric model (Li et al. (2009)) makes it possible to estimate the amount of strain transferred to the optical fiber core for a fiber optic sensor placed in a adhesive filled steel tube embedded in a homogeneous material, as well as for a fiber optic sensor embedded in a multilayer composite. The five-layer (Wu et al. (2014); Shen et al. (2018)), six-layer (Wang et al. (2012)), seven-layer (Falcetelli et al. (2020)) analytical models were proposed to analyze the strain transfer from the structure material to the core of the substrate-bonded FBG sensor located on the surface of the structure. These models also provide analytical formulas for estimating the relationship between the strain in the core of the FBG sensor and the strain in the measured structure. More detailed review of these (and other similar) analytical models can be found in the introduction to the article by Falcetelli et al. (2020). The authors of Shen et al. (2018) in order to simplify the analysis of performance a substrate-bonded FBG sensor, split whole process of strain transfer into two parts: 1) strain transfer from the substrate to the fiber core, and 2) strain transfer from the measured structure to the substrate. Thus, the total amount of strain transfer (as %) for a substrate-bonded FBG sensor is evaluated by taking into account the strain transfer ”losses” during the two processes. In present paper, in order to determine the strain changes introduced by the installation of the substrate with fiber optic sensor, we consider only the second process. We propose a mathematical model to estimate changes in the strain fields and present information on changes in the strain tensor components in the zone of substrate location depending on the ratio of the mechanical characteristics of the substrate and the material of the controlled structure for di ff erent dimensions of the structure and under di ff erent loading conditions.

2. Some information about fiber optic sensors based on the fiber Bragg grating

A Bragg grating is a distributed reflector within the core of an optical fiber. When light is allowed to pass from a source with a broadband spectrum through an optical fiber, part of the light wave is reflected from the Bragg grating (Fig. 1). The resonant wavelength of the reflected spectrum λ ∗ depends on the refractive index of the optical fiber n and the length of the period of the the Bragg grating L λ ∗ = 2 nL .

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Quarz cladding

Bragg grating

Broadband spectrum

Transmitted spectrum

fiber core

Reflected spectrum

Fig. 1. Principle of operation of the Bragg grating.

A change in the Bragg grating length leads to a change in the wavelength of the reflected signal ∆ λ . Under isother mal conditions, the relationship between the change of reflected wavelength and the fiber strain in the zone of the Bragg grating is defined by the following relations (Luyckx et al. (2010))

1 2 n

2 ( p

∆ λ 1 λ ∗ = ε 3 − ∆ λ 2 λ ∗ = ε 3 −

11 ε 1 + p 12 ( ε 2 + ε 3 )) ,

(1)

1 2 n

2 ( p

11 ε 2 + p 12 ( ε 1 + ε 3 )) ,

where ε 3 is the strain along the fiber; ε 1 , ε 2 are the principal strains in the plane perpendicular to the optical fiber; ∆ λ 1 = λ 1 − λ ∗ , ∆ λ 2 = λ 2 − λ ∗ are the di ff erences in the resonance wavelengths of the reflected spectrum at the current ( λ 1 , λ 2 ) and initial ( λ ∗ ) instants of time, and p 11 , p 12 are the Pockels coe ffi cients. In the uniaxial stress state of the optical fiber, which does not interact with the environment, the strain in the optical fiber is ε 1 = ε 2 = − νε 3 , where ν is Poisson’s ratio for the optical fiber. In this case, ∆ λ 1 = ∆ λ 2 = ∆ λ and ∆ λ λ ∗ =   1 − n 2 1 2 ( p 12 − ν ( p 11 + p 12 ))   ε 3 , (2) or ∆ λ/λ ∗ = k · ε 3 . For the optical fibers used k = 0 . 78. Thus, when the fiber is in a complex stress state there are two resonant peaks in the reflected spectrum (1). Then, the problem of unambiguous determination of the strain in the fiber core in terms of a change in the wavelength of the reflected spectrum is unsolvable. In the case when an optical fiber is in the uniaxial stress state it is possible to uniquely determine the strain along the fiber through the change in the wavelength of the reflected spectrum (2). Therefore, the development of fiber-optic sensors is associated, as a rule, with a search for the variant of implementation of the uniaxial stress state in the Bragg grating zone. One of the variants of such implementation is the installation of the fiber-optic sensors on a substrate. Numerical analysis of strain changes caused by the installation of the substrate-bonded fiber optic sensors on the surface of a structure was performed for a square plate with a rectangular substrate of a fiber optic sensor located on its surface (Fig. 2). The plate is subjected to distributed tensile forces P x and P y applied to its side edges. The dimensions of the plate are determined by the length of its sides a and thickness t . The dimensions of the substrate of the fiber optic sensor are determined by the length l x , width l y and thickness t 1 . In the study, the dimensions of the substrate were fixed. The dimension a was also fixed, but in such a way as to capture the entire zone of influence of the substrate placed on the surface of the plate. From the computational model (Fig. 2), it is evident that with the substrate dimensions fixed, the ratios of the mechanical characteristics of the structure and substrate materials, as well as the plate thickness t are the parameters, which will determine the strain changes near the substrate. In this case, 3. Mathematical model and its finite element implementation

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P y

l y

t 1

l x

P x

Fig. 2. Computational scheme of a plate with a substrate.

provided that the behavior of the materials is linear, the mechanical parameters are Poisson’s coe ffi cients ( ν , ν 1 ) and the ratio of elastic moduli ( E / E 1 ) of the structure and substrate materials. The assumptions used in this study are as follows: 1. All materials are linearly elastic and isotropic. 2. All materials are perfectly connected and there is no relative slipping. These assumptions are part of the basic assumptions used in the studies by Wu et al. (2014); Shen et al. (2018); Wang et al. (2012). The following ratios of geometric parameter were used in the calculations: a / l x = 10, l y / l x = 0 . 15, t 1 / l x = 0 . 0075. In terms of geometry, we can change the substrate dimensions whenever it is necessary , but with fixed substrate dimensions, the strains near the substrate are determined by the plate-to-substrate thickness ratio t / t 1 . The stress-strain state of the plate with a substrate is calculated by the finite element procedure implemented with the use of the ANSYS package. The examined plate is a composite structure, which is made of elastic isotropic materi als. The modeling domain was discretized into prismatic 20-node elements SOLID186 with a quadratic approximation of the unknowns. A variant of the finite-element mesh near the substrate is shown in Fig. 3. The essence of the numerical study is to compare the strain values obtained at the center of the outer surface of the substrate (point with coordinates [0 , 0 , 0]) and at the center of the plate without a substrate (point with coordinates [0 , 0 , − t 1 ]) under the same conditions of tensile loading. Two loading conditions were considered: extension along the x -axis ( P x = 1, P y = 0) and extension along the y -axis ( P x = 0, P y = 1). Before the numerical study, the convergence of the solution was estimated depending on the number of elements of the finite-element mesh. Since the computational scheme contains singular points and lines, at which di ff erent materials come into contact, the convergence of the solution was estimated by considering the extreme ratios of elastic moduli of the plate and substrate materials ( E / E 1 = 0 . 001 and E / E 1 = 1000). Fig. 4 shows the dependences of stresses σ x / P x obtained at the center of the substrate surface on the number of elements of the finite-element mesh at t / t 1 = 10.

Fig. 3. Variant of the finite-element mesh near the substrate (a quarter of the mesh is shown).

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0.00163970

70.5

0.00163965

 x  P x

0.00163960

69.5

0.00163955

0.00163950

68.5

0 50000 100000 150000 200000

Number of elements

Fig. 4. Dependences of stresses σ x on the substrate surface on the number of elements of the finite-element mesh at t / t 1 = 10 for di ff erent ratios of the plate and substrate elastic moduli: a) E / E 1 = 0 . 001; b) E / E 1 = 1000.

Fig. 5. Patterns of strain distribution ε x on the plate surface at t / t 1 = 10, E / E 1 = 0 . 1, ν = ν 1 = 0 . 3: a) without substrate; b) with substrate.

In our calculations, the characteristic size of elements near the substrate that corresponds to that of the mesh with 54000 elements for t / t 1 = 10. Figure 5 shows the strain distributions ε x on the plate surface with and without a substrate at t / t 1 = 10, E / E 1 = 0 . 1, ν = ν 1 = 0 . 3 and under load P x = 1, P y = 0. These results show that the embedding of a substrate has a rather significant e ff ect on the redistribution of strains.

4. Results

Changes in the strain fields caused by the installation of the substrate will be defined by the following quantity:

ε x 0 − ε x 1 ε x 0

100% ,

(3)

δ x =

where ε x 0 is the strain ε x at the center of the plate (without the substrate), ε x 1 is the strain ε x at the center of the outer surface of the substrate.

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 

 x , %

 

 x , %   0

5 10 25 50 75 100



E/E 1

0.1

0.2

 

 

 

 1

0.3



0.4

 

 

 

E/E 1

0.49

 

0.01

0.49

0.1 0.2 0.3 0.4  1

Fig. 6. Dependence of ε x on the ratio of elastic moduli E / E 1 and Poisson’s ratio ν of the plate at ν 1 = 0 . 3, t / t 1 = 1 and at the load P x = 1, P y = 0: a) as 3D wireframe; b) as contour plot.

At the first stage of the study we investigated the influence of mechanical characteristics of the plate material on the redistribution of the strain field at fixed mechanical characteristics of the substrate ( E 1 = const , ν 1 = 0 . 3). It was found that in this case, the mechanical characteristics E and ν of the plate material has the greatest e ff ect on ε x at the smallest plate thickness. In this study, we did not consider the plate thicknesses smaller than the substrate thickness ( t / t 1 ≥ 1). Figure 6 shows the dependence of ε x on the ratio of elastic moduli E / E 1 and the Poisson’s ratio of the plate ν at ν 1 = 0 . 3, t / t 1 = 1, and under the load P x = 1, P y = 0. The analysis of these results leads to the following conclusions. The orientation of the sensor with respect to the load has practically no e ff ect on the dependence of the redistribution of strain caused by the attachment of the FBG sensor substrate to the surface of the controlled structure. Poisson’s ratio of the material of the controlled structure also does not a ff ect the dependence of the strain redistribution when the substrate-bonded FBG sensor is pasted to its surface. Therefore, it makes sense to evaluate the changes in the value of ε x as a function of the ratio of the elastic moduli E / E 1 and the ratio of the thicknesses t / t 1 . Figure 7 shows the dependences ε x on the ratio of the moduli E / E 1 and the thicknesses t / t 1 at ν = ν 1 = 0 . 3 for two di ff erent load conditions: a) P x = 1, P y = 0; b) P x = 0, P y = 1. We also evaluated the influence of the substrate width l y at a fixed length l x . Figure 8 shows ε x as a function of the ratios of moduli E / E 1 and thicknesses t / t 1 at ν = ν 1 = 0 . 3 when the aspect ratio of the substrate is l y / l x = 0 . 3 for the same load conditions: P x = 1, P y = 0 and P x = 0, P y = 1. A comparison of the dependencies in Fig. 7 and Fig. 8 demonstrates that the width of the substrate a ff ects the strain redistribution relationship. The range of small values of ε x in Fig. 7 is larger than in Fig. 8. A model has been developed to analyze numerically the strain changes caused by the installation of substrates with fiber optic sensors on the surface of structures. The performed simulation has revealed an insignificant influence of the Poisson’s coe ffi cient of the material of the measured structure on the strain redistribution. The obtained results allow us to estimate changes in the structure strains due to the presence of fiber-optic sensors on its surface in terms of the relationship between the mechanical characteristics and geometrical dimensions of the structure and the contact surface of the sensor. The results obtained make it possible to determine the main characteristics of the structure, at which the installation of the substrate with the selected sensor does not introduce significant errors in the measured 5. Conclusion

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 

 x , %

 



E/E 1

5 10 25 50 75

10 25 50 75

 

 

100

 

3 5 10 12 2 4 6 7 8 9 11



3 5 10 12 2 4 6 7 8 9 11



t/t 1

Fig. 7. Dependences of ε x on the ratio of elastic moduli E / E 1 and the ratio of thicknesses t / t 1 at ν = ν 1 = 0 . 3, l y / l x = 0 . 15 for two load conditions: a) P x = 1, P y = 0; b) P x = 0, P y = 1.

 

 x , %

 



5 10 25 50 75

E/E 1

10 25 50 75

 

 

100

 

3 5 10 12 2 4 6 7 8 9 11



3 5 10 12 2 4 6 7 8 9 11



t/t 1

Fig. 8. Dependences of ε x on the ratio of elastic moduli E / E 1 and the ratio of thicknesses t / t 1 at ν = ν 1 = 0 . 3, l y / l x = 0 . 3 for two load conditions: a) P x = 1, P y = 0; b) P x = 0, P y = 1.

strains. These results can be used for estimating the possibility of application of the selected substrate-bonded FBG sensor, as well as other sensors on substrates.

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© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” Abstract Stress intensity factors, T-stresses and higher order coefficients in the Williams series expansion are the fundamental concepts of continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear regime. It is well-known that critical values of stress intensity factors stipulate the of materials to growth a crack. Nowadays the modern multi-purpose computational tools such as Simulia Abaqus allow us to calculate T-stresses in cracked specimens and structures along with stress intensity factors. The aim of this study is to understand if one can obtain this fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics method. The ability to describe fracture processes at atomic scale by means of stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, chemical compositions and concentrations and others. In this study the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are determined using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large scale Atomic/Molecular Massively Parallel Simulator. The over-deterministic method is used to determine SIFs, T-stresses and coefficients of higher-order terms from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions coincide qualitatively. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” Keywords: Stress intensity factors; T-stress; atomistic modelling; molecular dynamics method; continuum fracture mechanics, overdeterministic method. Abstract Stress intensity factors, T-stresses and higher order coefficients in the Williams series expansion are the fundamental concepts of continuum fracture mechanic for characterizing the stress fields around the crack tip in a h mogeneous material in the linear regime. It is well-known that critical values of stress int n ity facto s stipulate the of materials to rowth a cr ck. Nowadays the modern multi-purpose computational tools such a Simulia Abaqus allow us o calcul T- tresses in cra ked specimens and structures a ong with stress in ensity factors. T e aim of this study is to understand if on can obtain this fracture ara ters of conventional fracture mechanics from atomistic si ula ions based on molecular dynamics method. The ability to d scribe fracture processes at atomic scale by eans of stress intensity f ctors, T-stresses an higher-ord r coeffici nts will provide th opportunity to take int ac ount many ffects such as material microstruc ure , chemical compositi ns and concentrations and thers. In this study the values of str ss intensity factors, T-stress and coeffici nts of the Williams series xp nsion are det mined using atomistic simulations ba ed o molecula dynamic method with a classical molecular dynamics code Large scale Atomic/Molecular Ma sively Parallel Simulat r. The over-deter inistic met od is used to d termine SIFs, T-stresses and coefficients of higher-order terms from molec lar dynamics modelling of a plate w th a central c ack. The accuracy of the proposed approach is tested fo this ather simple cracked configuration. There is the theoretical nalytical solution with all coefficients of the h gh r-order terms in the Williams series expansion. The exist ng theoretical solution allows s to comp re angular di tributions of the st ess tensor components for a large pla e with the central cra k. It is shown that results obtained from molecular dy amics simulations and the theore ical ana ytical solutions coincide qualit tively. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review u der re ponsibility of t scientific committe of the XXIIth Winter Sch ol n Continuous Media Mechanics” Keywords: Stress intensity factors; T-stress; atomistic modelling; molecular dynamics method; continuum fracture mechanics, overdeterministic method. XXIIth Winter School on Continuous Media Mechanics Atomistic Determination of Fracture Mechanics Parameters Stepanova L.V. a *, Belova O.N. a , Bronnikov S.A. a a Department of Mathematical Modelling in Mechanics, Samara National Research University, Moskovskoye shosse, 34, 443086, Samara, Russia XXIIth Winter School on Continuous Media Mechanics Atomistic Determination of Fracture Mechanics Parameters Stepanova L.V. a *, Belova O.N. a , Bronnikov S.A. a a Department of Mathematical Modelling in Mechanics, Samara National Research University, Moskovskoye shosse, 34, 443086, Samara, Russia

* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: stepanova.lv@ssau.ru * Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: stepanova.lv@ssau.ru

2452-3216© 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” 2452-3216© 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review u der responsibility of t e scientific committee of the XXIIth Winter School on Continuous Media Mechanics”

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1. Introduction Computational modeling of fracture has significant engineering relevance as a predictive capability to quantify material failure under load, an essential consideration during design (Wilson et al. (2019), Singh (2019), Stepanova and Bronnikov (2020)). For the last few decades, advances in modeling have continually demonstrated this predictive capability. As it is noted in (Wilson et al. (2019))while linear elastic fracture mechanics provides a continuum representation of fracture, there are a number of significant phenomena related to atomic scale that can’t be described by continuum theory. Nowadays there are some attempts to determine stress intensity factors from molecular dynamics modelling. Thus, in (Wilson et al. (2019)) a novel numerical method for determination of SIFs from atomistic simulations is presented. The authors Wilson et al. (2019)) of Using atomiccoordinates to describe the displacement field about a crack tip, the authors Wilson et al. (2019)) projected observed displacements onto theset of continuous displacement fields defined by the Williams expansion, with the expansion coefficients determiningthe stress intensity factors. The authors found agreement with experimental data of fracture toughness in silica glass. A computational scheme to analyze the mixed-mode fracture of grain boundaries in polycrystalline solids was developed in (Mai et al. (2018)). The authors extract the atomic -level J-based integral to estimate stress intensity factors for Mode I and mode II loadings. The individual SIFs were obtained from the atomic-level J integral and J based mutual integral, respectively. The singular K-field near an interfacial crack in anisotropic bi-materials as an auxiliary field was incorporated with discrete atomic information obtained from MD simulations of crack propagation along grain boundaries in polycrystalline solids. As the authors notice this technique is advantageous for studying the inter-granular fracture of brittle polycrystalline solids at an atomic scale, as crack propagation along grain boundaries can generally be considered as a mixed-mode fracture.

Nomenclature ij σ

stress tensor components around the crack tip

, r θ

polar coordinates of the system with its origin at the crack tip coefficients of the terms of the Williams series expansion

k a

, I II K K , ( ) k m ij f θ , ( ) k m i g θ ( ) ( )

mode-I stress intensity factors

angular functions included in stress distribution related to the geometric configuration, load and mode angular functions included in displacement distribution related to the geometric configuration, load and mode

m G

index associated to the fracture mode

shear modulus mixity parameter

e M

In (Dehaghani et al. (2020)) fracture toughness and crack propagation behavior of nanoscale beryllium oxide graphene-like structures is analyzed and the critical value of stress intensity factor according to the linear fracture mechanics under Mode I is calculated. Thus, the values stress intensity factors are determined by the conventional macroscopic linear fracture mechanics and the possibilities of atomic scale simulation are not used. In (Tsai et al (2010)) the fracture behavior of a graphene sheet with a center crackwas characterized using atomistic simulation and linear elasticfracture mechanics (LEFM). In the atomistic simulation, the graphene was regarded as an atomistic structure, containing discretecarbon atoms; nevertheless, in the LEFM, it was modeled as an isotropic homogeneous media. Results from atomistic simulationindicated that because of the discrete attribute, there is no stresssingularity near the crack tip. Therefore, the concept of stressintensity factor, which is generally employed in the continuummechanics, may not be suitable for modeling the crack behaviorin the atomistic graphene sheet. In order to validate the strain energy release rate concept, the energy variation before and after thecrack extension were evaluated in both continuum and atomisticmodel. For the discrete atomistic model, two methods, i.e., the global energy method and the crack closure method, were employedto compute the energy variation as well as the strain

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energy re-lease rate. On the other hand, the finite element analysis was per-formed in the continuum graphene sheet for the estimation of thestrain energy release rate. It was denoted that the strain energy re-lease rate calculated based on the global energy method and crackclosure method are almost the same. Furthermore, it was demonstrated that the global energy method is a simple manner in theatomistic simulation for the calculation of the strain energy releaserate. A comparison of atomistic simulation with finite element results illustrated that the strain energy release rates obtained fromcontinuum model coincided with that in the discrete model associated with the same loading condition. As a result, the conceptof strain energy release rate is regarded as a physical quantity thatcan establish connections between the atomistic simulation andcontinuum modeling for modeling the fracture of covalentlybonded graphene sheet. The overarching objective of (Roy and Roy (2019)) was to investigate the validity of application of continuum based linear elastic fracture mechanics methodology at the nanoscale, and to extend the concept of the continuum J integral to the atomistic domain, by addressing the following key issues: (a) computation of continuous variables, such as displacement and their derivatives, from discrete atomistic quantities, (b) including nonlocality in J that is inherent in atomistic computations due to long range inter-atomic forces, and (c) incorporating entropic effects due to thermal motion in a atomistic system which is not present in a conventional continuum description. an atomistic J-integral is implemented as a nano-scale fracture metric to investigate flaw-tolerance at the nanoscale. Predictions obtained using the atomistic J are compared with linear elastic fracture mechanics predictions for the case of a single (zig-zag) graphene sheet with a center-crack under tensile loading at room temperature, and show significant deviation from LEFM for crack lengths below a certain threshold. However, the authors conclude that thecritical J integral value obtained from the methodology discussed in this paper were found to be in good agreement with the valueavailable in literature. In (Cheng and Sun (2011)) it is noted that stress intensity factor is one of the most significant fracture parameters in linear elastic fracture mechanics (LEFM). Due to its simplicity, many researchers directly employed this concept to explain their results from molecular simulation. However, stress intensity factor defines the amplitude of the singular stress, which is the product of continuum elasticity. Under atomistic systems without the stress singularity, the concept of stress intensity factor must be examined. In addition, the difficulty of studying the stress intensity factor in atomistic systems may be traced back to the ambiguous definition of the local atomistic stress. In this study, the definition of the local virial stress is adopted. Subsequently, through the consideration of K-dominance, the approximated stress intensity factor based on the atomistic stress can be projected within a reasonable region. Moreover, the influence of cutting interatomic bonds to create traction free crack surfaces and the critical stress intensity factor is also discussed. The paper (Gallo (2020)) reviews recent molecular statistics numerical experiments of cracked samples and discusses the crack-tip region stress in ideal brittle materials. Continuum-based linear elastic fracture mechanics breaks down at extremely small scales where the discrete nature of materials has to be considered. However, as it is noted in (Gallo (2020)) recent results have shown that the concept of stress intensity factor is still valid. In (Gallo (2020)) by means of molecular statistics simulations on single-edge cracked samples of ideal brittle silicon it is shown that the stress intensity factor derived from the virial stress may be useful to describe the fracture at extremely small dimensions and to quantify the breakdown of continuum-based linear elastic fracture mechanics. Thus, one can notice that conclusions are very different and versatile. This is due to 1) differences in basic concepts of continuum fracture mechanics and discrete atomistic approach and 2) lack of right understanding and approaches for correct comparison of results of simulation obtained by two different schemes. Therefore, the goal of this study is to obtain the fracture mechanics parameters (stress intensity factor, T-stress and coefficients of higher order terms in the Williams series expansion of the crack-tip stress and displacement fields) from atomistic simulations and to compare continuum fracture mechanics parameters and the same parameters obtained from molecular dynamics modelling. 1.1. Details of molecular dynamics simulations All atomistic simulations are based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) using an embedded atom model EAM potential a copper crystal (Cu_u3.eam). FCC copper is considered.The plate with a central crack under tensile loading and mixed mode loading has been modeled. For the full MD model simulations start with the static

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