PSI - Issue 23

Volume 2 3 • 201 9

ISSN 2452-3216

ELSEVIER

9th International Conference Materials Structure & Micromechanics of Fracture (MSMF9)

Guest Editors: J aroslav P okluda P avel Š andera

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9th International Conference on Materials Structure and Micromechanics of Fracture Editorial Jaroslav Pokluda* Brno University of Technology, Brno, Czech Republic This issue of the Procedia Structural Integrity contains papers from the 9 th International Conference on Materials Structure and Micromechanics of Fracture (MSMF9) that was held in Brno, Czech Republic, June 26-28, 2019. The series of MSMF conferences was established in Brno, June 1995. The basic idea was to create a periodical international forum for multiscale approaches in fatigue and fracture of materials in the middle Europe. Therefore, respective sections focused on atomistic models, models based on crystal defects, numerical and statistical continuum models, advanced experimental methods and relationships between microstructure and mechanical properties appeared during the MSMF2 conference in 1998. The power of atomistic, mesoscopic and multiscale approaches in fracture and fatigue was then demonstrated by participants at next MSMF meetings organized in Brno in 2001 - 2016. Many world-leading experts in the field of fracture and fatigue attended the MSMF conferences as plenary speakers. The conference MSMF9 has successfully carried on the tradition of previous conferences with 188 scientists from 27 countries all over the world who presented 180 contributions on fundamental relations between structural and mechanical characteristics of materials. There were five invited plenary talks delivered by Prof. Reinhard Pippan (Leoben, AT), Prof. Malcolm Neal James (Plymouth, UK), Prof. Takeshi Ogami (Tenesee, USA), Prof. Elias Eifantis ( Thessaloniki, GR) and Prof. Ludvík Kunz (Brno, CZ). T he 13 th workshop of the ESIS Technical Committee on Micromechanisms (TC2) was organized by Prof. A. Jivkov (Manchester, UK) as a sub-symposium of the conference. After a peer-review procedure, as many as 103 papers based on atomistic, mesoscopic, macroscopic and multiscale approaches were included in the PSI volume devoted to MSMF9. I would like to thank all the members

* Corresponding author. Tel.: +420-541-142-827; E-mail address: pokluda@fme.vutbr.cz

2452-3216 © 2019 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 IC MSMF organizers.

2452-3216 © 2019 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 ICMSMF organizers 10.1016/j.prostr.2020.01.053

Jaroslav Pokluda / Procedia Structural Integrity 23 (2019) 1–2 Jaroslav Pokluda / Structural Integrity Procedia 00 (2019) 000 – 000

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of the International Advisory Board for their support that was essential to the success of the conference. My thanks also go to all session chairpersons who successfully guided the program as well as to reviewers who thoroughly helped many authors to improve the quality of their manuscripts. Special thanks belong to the local organizing team chaired by Prof. Pavel Šandera for its perfect work. I would also like to thank all participants for their active presence and contribution to a friendly atmosphere during this successful event. Last but not least, it is my pleasure to thank all the leading personalities of Elsevier and ESIS relevant for production of this volume.

Prof. Jaroslav Pokluda, chair, on behalf of the organizers of the 9 th International Conference on Materials Structure and Micromechanics of Fracture

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Procedia Structural Integrity 23 (2019) 396–401

© 2019 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 ICMSMF organizers © 201 9 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 IC MSMF organizers. Moving vectors, motors and screws are in roduced and the mathematical background is explained: du l numbers are us d for their des ription. Further, the pap r deals with the dual space and curves in it. Some exa ples (in particular h lic s) are given. N wly, s called Spivak's dual curve is udied from the point of vi w of its natural p rameteriza ion; it is presented that curvature and torsion at zer ar not able to disti guish this curve from the plane analogy again – as in the real case. It is also mentioned the applicability of the theory in mechanics. © 201 9 The Authors. Published by Elsevier B.V. This is an ope acces article under CC BY-NC-ND lic nse (http://creativecommon org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. 9th International Conference on Materials Structure and Micromechanics of Fracture A characterization of sliding vectors by dual numbers, some dual curves and the screw calculus Miroslav Kureš a* a B rno University of Technology, Technická 2, 61669 Brno, Czechia 9th International Conference on Materials Structure and Micromechanics of Fracture A characterization of sliding vectors by dual numbers, some dual curves and the screw calculus Miroslav Kureš a* a B rno University of Technology, Technická 2, 61669 Brno, Czechia Abstract Abstract Moving vectors, motors and screws are introduced and the mathematical background is explained: dual numbers are used for their description. Further, the paper deals with the dual space and curves in it. Some examples (in particular helices) are given. Newly, so called Spivak's dual curve is studied from the point of view of its natural parameterization; it is presented that curvature and torsion at zero are not able to distinguish this curve from the plane analogy again – as in the real case. It is also mentioned the applicability of the theory in mechanics. 1. Sliding vectors Let us start with a very apposite introduction to sliding vectors, as written in the book Kinetics of Human Motion of Vladimir M. Zatsiorsky. See Zatsiorsky (2002). Force is a measure of the action of one body on another. Force is a vector quantity. A force can be treated as either a fixed vector or as a sliding vector. When a force is treated as a fixed vector, it is defined by its (a) magnitude, (b) direction, and (c) point of application. When a force is considered a sliding vector, the line of force action rather than the point of application defines the force. Forces are considered sliding vectors when (a) the body of interest is rigid and (b) the resultant external effects, rather than the internal forces and the deformations, are investigated. In this paper, we deal with dual numbers that very well represent gliding vectors, dual space, and curves in it. 1. Sliding vectors Let us start with a ver apposite introduction to sliding vectors, as written in the book Kinetics of Human Motion of Vladimir M. Zatsiorsky. See Zatsiorsky (2002). Force is a measure of the action of one body on another. Force is a vector quantity. A force can be tre ted as either a fixed vector or as a sliding vector. When a force is treate as a fixe vector, it is defi ed by its (a) magnitude, (b) direction, and (c) point of application. When a force is considered a sliding vector, the line of force action rather than the point of application defines the force. Forces are considered slidi g vectors when (a) the body of int rest is rigid and (b) the resultant external effects, rather t an the internal forces an the deformations, are investigated. In this paper, we deal with dual numbers that very well represent gliding vectors, dual space, and curves in it. Keywords: sliding vector; dual number; dual space; curves in dual space; motor; screw; curvature; torsion Keywords: sliding vector; dual number; dual space; curves in dual space; motor; screw; curvature; torsion

* Corresponding author. Tel.: +420-541-142-714. E-mail address: kures@fme.vutbr.cz * Corresponding author. T l.: +420-541-142-714. E-mail address: kures@fme.vutbr.cz

2452-3216 © 2019 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 IC MSMF organizers. 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an ope acces article under CC BY-NC-ND lic nse (http://creativecommon org/licenses/by-nc-nd/4.0/)

Peer-review under responsibility of the scientific committee of the IC MSMF organizers.

2452-3216 © 2019 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 ICMSMF organizers 10.1016/j.prostr.2020.01.119

Miroslav Kureš / Procedia Structural Integrity 23 (2019) 396–401 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1.1. Lines

We consider the affine space with the set of points A = R 3 and the vector space V = R 3 possessing the Euclidean inner product. A line L is usually given by two distinct points X = [ x 1 , x 2 , x 3 ], Y = [ y 1 , y 2 , y 3 ] or by a point X = [ x 1 , x 2 , x 3 ] and a non-zero direction vector u = ( u 1 , u 2 , u 3 ), so in total by 6 numbers. However, one can determine a line quite comfortably with only 4 numbers and an information on the choosen axis. So, let a quintuplet ( a , b , c , d ,   is given, where a , b , c and d  are four real numbers and   { x, y, z }. The chosen axis represents an axis in which the direction vector u has component 1 (there exists as u is non-zero) and the corresponding component of the point X lying on L is 0 (there exists such X because the point coordinate in the direction of the chosen axis is linearly growing). E.g. (5, 6, 7, 8 , z ) determines the line with u = (5, 6, 1) and X = [7, 8, 0]. There is also possibility to represent a line in R 3 by 4 or less numbers. First, let us start with lines in R 2 . Let us consider two parallel reference lines L 0 given by x = 0 and L 1 given by x = 1. Then, any line L not parallel to L 0 and L 1 will intersect the two reference lines and the second coordinates of the points of intersection will determine L uniquely. The only exceptions are lines parallel to parallel to L 0 and L 1 . In that case, the first coordinate determines the line uniquely. Thus, lines in R 2 are represented by 2 or 1 number. Now, we can extend this to lines in R 3 . Consider two reference planes P 0 and P 1 planes given by x = 0 and x = 1, respectively. Then any line L not parallel to P 0 and P 1 will intersect each of the two reference planes and each of the two points of intersection are given by coordinates of which the first coordinate is already fixed, and the remaining two coordinates of each point uniquely specify the line L with 4 numbers. The only exceptions are lines parallel to P 0 and P 1 . In that case, the first coordinate specifies a plane that contains the line L and, in that plane, by the case for lines in R 2 , that is uniquely determined by 2 numbers, with exceptions noted already. In summary, lines in R 3 are represented by 4, 3 or 2 numbers by this way. (The author thanks a Stack Exchange user nicknamed Somos that suggested this idea.) E.g. (5, 6, 7, 8) determines the line going through X = [0, 5, 6] and Y = [1, 7, 8], (5, 6, 7) the line going through X = [5, 0, 6] and Y = [5, 1, 7] and (5, 6) the line given by x = 5 and y = 6. A vector u  V is also called the free vector . A bound vector is a pair ( X , u ), where X  A and u  V . We will write u ( X ) . A non-zero sliding vector is a pair ( L , u ) consisting of a line L in A together with a non-zero vector u  V that leaves L invariant. We will write u (( L )) . The zero can be also considered as a sliding vector. We have natural projections pr 1 sending a bound vector onto a sliding vector, pr 2 sending a sliding vector onto a free vector and pr 3 which is the composition of the previous two projections and sends a bound vector onto a free vector. In R 3 a sliding vector can be determined by five numbers, e.g., by the coordinates of the point intersection M of one of the coordinate planes and the line containing the vector (two numbers), by the magnitude of the vector (one number) and by two independent angles α and β between the vector and two of coordinate axes (two numbers), see Borisenko and Tarapov (1968). Let u ( X ) be a bound vector. The moment 0 of u ( X ) is defined as the cross product 0 = ( ) × 3 ( ( ) ) where ( ) is a (free) radius vector of the point X . The following assertion holds. ( ) × 3 ( ( ) ) = ( ) × 3 ( ( ) ) if and only if Y = X + k u . Proof .  : Let Y = [ x 1 + ku 1 , x 2 + ku 2, x 3 + ku 3 ]. Then we observe that ( ) × 3 ( ( ) ) equals ( x 2 u 3 – x 3 u 2 , x 3 u 1 – x 1 u 3 , x 1 u 2 – x 2 u 1 ).  : Let ( x 2 u 3 – x 3 u 2 , x 3 u 1 – x 1 u 3 , x 1 u 2 – x 2 u 1 ) = ( y 2 u 3 – y 3 u 2 , y 3 u 1 – y 1 u 3 , y 1 u 2 – y 2 u 1 ) and let us denote ( v 1 , v 2 , v 3 ) = ( y 1 – x 1 , y 2 – x 2 , y 3 – x 3 ). Then components of the equality read as − = − = − = 1.2. Sliding vectors

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and hence ( v 1 , v 2 , v 3 ) = ( ku 1 , ku 2 , ku 3 ) for an arbitrary k  R . Q. E. D. It reads as the bound vectors u ( X ) and u ( Y ) possess the same moment u

0 if and only if X and Y lie on a line with the

direction vector u .

1.3. Motors and screws By a motor we understand a couple ( u , u 0 ) . “Motor” is a combination of words “moment” and “vector” , Dimentberg (1968). It is a representation of a vector system, expressed by the principal vector and principal moment of the system. However, our radius vectors have been so far based on the origin O of the coordinate system, but another point may be such a reference point. Every motor can be brought to such a reference point that its moment and vector parts become colinear, which turns a motor into an equivalent screw . A parallel line through this point is the screw axis , see Brodsky and Shoham (1999). The screw calculus , Dimentberg (1968), is based on the basis of the apparatus of modern vector algebra using dual numbers. Traditional mechanics of continua endows particles of a material body with translational degrees of freedom, the Cosserat brothers' approach endows them with both translational and rotational degrees of freedom. In elementary approach a body B of dimension 1 (rods, beams) or 2 (plates, shells) or 3 in R 3 is considered. For each particle of such a body we consider its initial position (a radius vector) and its initial settings of 3 orthonormal directors. Such approach is based on differential geometry theory applied to mechanics and there is no doubt that Cosserat continuum theory is suitable e.g. for describing the kinematics of granular media. The mathematical description can be based on motors, as stated in the monograph of Vardoulakis (2018) in which the basic theorems used to formulate the Cosserat continuum, together with the appropriate kinematic fields conjugate to the motor vectors; kinematic motors are compound vectors including linear velocity and spin (angular velocity), fully describing a rigid body motion in the new reduced geometric representation. 2.1. Dual vectors and dual curves The dual numbers are defined as numbers of a form a = a + A ; a, A  R , which extend the real numbers by adjoining new (“infinitesimal” ) element with the property 2 = 0. The set of dual numbers is denoted by D and it forms a two dimensional commutative unital associative R-algebra. The arithmetic of dual numbers has several specifics, so we refer e.g. to the paper Kure š (to appear) for details. Further, a dual function f : D  D of a dual variable x = x + X can be represented in the form ( + ε) = φ( , ) + Φ( , ) Dual functions which are also differentiable in a neighborhood of a point are called synektic . This is satisfied if and only if ∂ ∂ φ = ∂ ∂ Φ and = 0. It is an analogy with the analytic functions over C which comply with the Cauchy-Riemann conditions. Then we have 1.4. Application: Cosserat media 2. Dual curves

Miroslav Kureš / Procedia Structural Integrity 23 (2019) 396–401 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 d d = ∂ ∂ φ + ∂ ∂ Φ ε.

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Dual vectors will now be understood as elements of ( D ) 3 = D  D  D . Then ( D ) 3 is a free D -module, dim D ( D ) 3 = 3, and an R -vector space, dim R ( D ) 3 = 6. We will call ( D ) 3 the dual space and its elements will be denoted by = (a 1 + A 1 ε, a 2 + A 2 ε, a 3 + A 3 ε). Dual numbers introduced by W. Clifford were deeply investigated by E. Study who used dual numbers and dual vectors in his research on line geometry and kinematics. He devoted special attention to the representation of oriented lines by dual unit vectors and defined the famous mapping: the set of oriented lines in a Euclidean three-dimension space R 3 is one-to-one correspondence with the points of a dual space D 3 of triples of dual numbers. Of course, from the time these classic results came into being, research has expanded from straight lines to more curves. So, let us give a definition. Synektic curves in the dual space have a form = ( + ε) ↦ (t + Tε) = ( 1 ( , ) + 1 ( , )ε, 2 ( , ) + 2 ( , ) , 3 ( , ) + 3 ( , ) ), where dual functions (of a dual variable) 1 ( , ) + 1 ( , ) , 2 ( , ) + 2 ( , ) , 3 ( , ) + 3 ( , ) are synektic. We present two examples of synektic curves: 1 ( , ) + 1 ( , )ε = r + ( − ) 2 ( , ) + 2 ( , ) = + ( + ) 3 ( , ) + 3 ( , ) = + ( + ) is a circular helix in the dual space while 1 ( , ) + 1 ( , ) = e cos + (( e + e ( + )) cos − e sin ) ε 2 ( , ) + 2 ( , ) = e + (( e + e ( + )) sin + e cos ) 1 ( , ) + 1 ( , ) = e + ( e + ( + )) is a conic helix in the dual space . From the viewpoint of practical applications, the helices in the micro-scale are important and interesting; the fabrication of microhelices from different materials is categorized and their novel properties and applications are summarized and reviewed in Huang and Mei (2015). Smaller helices, i.e. the helices in nano-scale, on the contrary, are even difficult to be fabricated and investigated, although the new sciences in such small scale may suggest even significant potentials in the future. The author’s paper Kureš (to appear) highlights the importance of study of helices in dual space and the curvature and torsion of these two types of helices as synektic curves over dual numbers are derived. 2.2. Frenet – Serret formulas Synectic curves may be parametrized, as in the real case, by the arc length. Details of this reparameterization are described in Nav rátil (2017) . We denote the dual natural parameter by s = + . Then the unit tangent vector of a curve ( s ) is the vector = d d and the unit normal vector perpendicular to g with the same orientation as d d is denoted by n . Furthermore, the unit vector b with the same orientation as × is called the binormal vector . Then for the dual curvature κ and for the dual torsion τ , the following relations are satisfied: d d = , d d = κ , d d = −κ + τ , d d = − . These are Frenet – Serret formulas for dual curves.

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2.3. Spivak’s dual curve

Michael Spivak has described in his famous book Spivak (1970) a real curve lying in the xy plane for t > 0 and in the xz plane for t < 0, effectively switching planes at t = 0, while remaining smooth. We will call this curve Spivak's (real) curve for the purposes of this paper.

Fig. 1. An illustration of the real Spivak’s curve. For the negative t we use the dashed curve and for the positive t we use the full curve. So we start at the top left, switch in the middle and finish at the bottom right. Components are connected in t = 0 smoothly.

And now, we will consider this curve in the dual version.

2 = 2 + 2 ε, − 1 2 = − 1 2 + 2 3 e − 1 2 = − 1 2 + − 1 2 2 3 , ( ) = ( , , )

As

and

we generalize Spivak's curve by

for t = 0, 1 2 + − 1 2 2 3 , ) for t > 0 and ( ) = ( + , , − 1 2 + − 1 2 2 3 ) for t < 0. ( ) = ( + , −

This curve will be called Spivak's dual curve . (We remark that we obtain Spivak's real curve for T = 0.) The original question of Michael Spivak how one can use the curvature and the torsion to distinguish between this curve and the curve ̃( ) = ( , , ) for t = 0, ̃( ) = ( + ε, − 1 2 + − 1 2 2 3 ε, ) for ≠ 0 is now naturally reformulated for dual space. This question requires the expression of a natural parameter for dual case for a calculation of the curvature and torsion. As a new result, let us conclude with the final expression of this parameter:

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6 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 + = e − 2 2 (4 2 ( − 3 ) + e 2 2 9 + 8 ) √ 1 + 4 − 2 2 6 8 This expression of the natural parameter is equal for both parts of the curve. But we can deduce from that, quite easily, that this is the property highlighted by Spivak remains in force in the dual case, too.

2.0

1.5

1.0

0.5

0.5

1.0

1.5

2.0

Fig. 2. The natural parameter for T = 0 from the previous expression on the inteval (0,2] (monotony and thus invertibility of the function are obvious).

References

Borisenko A. I. and Tarapov, I. E., 1968. Vector and Tensor Analysis with Applications. Courier Corporation. Brodsky, V. and Shoham, M., 1999. Dual Numbers Representation of Rigid Body Dynamics. Mechanism and Machine Theory 34 (5), 693 – 718. Dimentberg, F. M., 1968. The Screw Calculus and Its Applications in Mechanics (No. FTD-HT-23-1632-67). Foreign Technology Div Wright Patterson AFB Ohio. Translation from Russian. Huang, G. and Mei, Y., 2015. Helices in Micro-World: Materials, Properties, and Applications. Journal of Materiomics 1 (4), 296 – 306. Kureš, M., Helices over Dual Numbers, to appear (2020). In: Kuczma, M. (Ed.), Foundations of Shape-Memory Materials and Structures. Springer. Navr átil, D., 2017. Curves in D 3 1 . Bachelor Thesis (in Czech). Brno University of Technology. Spivak, M. D., 1970. A Comprehensive Introduction to Differential Geometry. Publish or Perish. Vardoulakis, I., 2018. Cosserat Continuum Mechanics: With Applications to Granular Media. Vol. 87. Springer. Zatsiorsky, V. M., 2002. Kinetics of Human Motion. Human Kinetics.

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© 2019 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 ICMSMF organizers © 201 9 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 IC MSMF organizers. Keywords: Molecular dynamics approach; EAM; crack propagation direction angle; classical fracture mechanics; fracture criteria; maximum tangential stress criterion; maximum tangential strain; strain energy density criterion. Abstract The ver rching objective of the paper is to analysis the mixed-mode crack p opagation direction angles by molecular dynamic meth d and to investigate the validity of continuum- ased linear elastic fr cture m chanics crack growth criteria. For this purpose, an embedded tom potential (EAM) available in LAMMPS (Large-scale Atomic/Molecular Ma sively Parallel Simulator) molecular dynamics (MD) s ftware is utiliz d to accurately pinpoint mixed-m de crack growth. The study is focused on he application of the differe t approaches for t e determination of th initial crack prop gation angle. Copper and aluminum plates with the central crack under complex mechanica tresses (Mode I and Mode II loading) ar studi d by extensive MD s mulations. Williams ’ exp nsion for the crack tip fi lds containing the h gher-order t r s is used. The crack propag tion direction angles for combinations of Mode I and Mode II loadings are obtained by 1) the multi-par meter frac ure mechanics app oach based on three fr cture mechanics crite ia: maximu tangential stress (MTS), aximum tangential s in and strain energy density (SED); 2) atomistic modeling for the mixed-mode loading of the plane medi m with the central crack. The temp rature effects during frac ure processes in MD simulations are considered and the temperature field distributions for mixed mode crack propagation are obtained. © 201 9 The Authors. Published by Elsevier B.V. This is an ope acces article under CC BY-NC-ND lic nse (http://creativecommon org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. Keywords: Mole ula dyna ics appro ch; EAM; crack propagation irec ion angle; classical fracture mechanics; fracture criteria; maximum tangential stress criterion; maximum tangential strain; strain energy density criterion. 9th International Conference on Materials Structure and Micromechanics of Fracture A computational study of the mixed-mode crack behavior by molecular dynamics method and the multi-parameter crack field description of classical fracture mechanics Stepanova Larisa a *, Bronnikov Sergey a a Samara National Research Univeristy, Moskovskoe shosse, 34, Samara 443086, Russia Abstract The overarching objective of the paper is to analysis the mixed-mode crack propagation direction angles by molecular dynamics method and to investigate the validity of continuum-based linear elastic fracture mechanics crack growth criteria. For this purpose, an embedded atom potential (EAM) available in LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics (MD) software is utilized to accurately pinpoint mixed-mode crack growth. The study is focused on the application of the different approaches for the determination of the initial crack propagation angle. Copper and aluminum plates with the central crack under complex mechanical stresses (Mode I and Mode II loading) are studied by extensive MD simulations. Williams ’ expansion for the crack tip fields containing the higher-order terms is used. The crack propagation direction angles for combinations of Mode I and Mode II loadings are obtained by 1) the multi-parameter fracture mechanics approach based on three fracture mechanics criteria: maximum tangential stress (MTS), maximum tangential strain and strain energy density (SED); 2) atomistic modeling for the mixed-mode loading of the plane medium with the central crack. The temperature effects during fracture processes in MD simulations are considered and the temperature field distributions for mixed mode crack propagation are obtained. 9th International Conference on Materials Structure and Micromechanics of Fracture A computational study of the mixed-mode crack behavior by molecular dynamics method and the multi-parameter crack field description of classical fracture mechanics Stepanova Larisa a *, Bronnikov Sergey a a Samara National Research Univeristy, Moskovskoe shosse, 34, Samara 443086, Russia

* Corresponding author. Tel.: +7-927-752-2102. E-mail address: stepanovalv@samsu.ru * Correspon ing au hor. Tel.: +7-927-752-2102. E-mail address: stepanovalv@samsu.ru

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1. Introduction

Modelling particle interactions at the nanoscale requires approaches that can account for the complex, nonlinear constitutive behavior at this length scale (Roy and Roy (2019)). Molecular dynamics method has been proven effective in capturing the typical phenomena related to fracture at the nanoscale (Roy and Roy (2019), Chowdhury et al. (2019), Liu et al. (2019)). MD approach is a powerful tool in characterizing the inception and evolution of plastic deformation and associated fracture mechanism at the atomic scale. Applied to the problem of crack propagation and growth, the method has been used extensively to analyze the fundamental mechanisms of material’s fracture in the past (Tang et al. (2010), Sung et al. (2015), Tang et al. (2014), Wu et al. (2015), Chandra et al. (2016)) and at present (Zhou et al. (2019), Han et al. (2019), Roy and Roy (2019), Chowdhury et al. (2019), Singh et al. (2019), Belova and Stepanova (2018)) . The study is aimed at the determination of the crack propagation direction angle in a wide range of mixed-mode loading using 1) molecular dynamics simulation of the crack propagation behavior under mixed mode loading; 2) the multi-parameter strain energy density criterion; 3) the multi-parameter maximum tangential strain criterion; 4) the multi-parameter maximum tangential stress criterion in a full range of the mixity parameter which is defined as       22 12 2/ arctan , 0 / , 0 . e M r r         The mixity parameter equals 0 for pure mode II; 1 for pure mode I, and 0 1 e M   for different mixities of modes I and II. There are several methods of simulating crack behavior, including conventional fracture mechanics analysis, molecular mechanics, molecular dynamics, finite element method and methods based on density functional theory (DFT). Focus of this study is comparison of molecular dynamics and continuum fracture mechanic approach. Atomistic simulations of the central crack growth process in an infinite plane medium under mixed-mode loading using LAMMPS, a classical molecular dynamics code, are performed. In the framework of MD approach two types of materials are considered. The first one is hcp-copper and the second one is fcc - aluminum. The inter-atomic potential used in this investigation is Embedded Atom Method potential. The plane specimens with initial central crack were subjected to Mixed-Mode loadings. The simulation cell contains from 300000 to 800000 atoms. The crack propagation direction angles under different values of the mixity parameter in a wide range of values from pure tensile loading to pure shear loading in a wide range of temperatures (from 0.1 К to 800 К ) are obtained and analyzed. Periodic boundary conditions were implemented in all three directions of the cell. To neglect the effect of neighbouring cells we choose the size of the central crack to be relatively small (1:10 ratio) to the size of the simulation cell. Furthermore, we added small non-interacting boundaries to the edge of the plane. Before mixed loading is applied, plate is optimized to the minimal energy with conjugated gradient method. When minimum energy state is achieved, we apply mixed strain. During all 50000 steps of simulation, we collect data of the state of all atoms in the cell. The results of simulations for Mode I loading in the copper plate with the central crack are shown in Figs. 1-8, color coding is obtained by OVITO tool. Brighter colors correspond to higher stress. Figs. 1-3 show the distribution of the stress component 11  in the copper plate with Mode I crack at different times. Figs. 3-6 show the distribution of the stress component 22  in the copper plate with Mode I crack at different times. Figs. 6-8 show the temperature distribution in the copper plate with Mode I crack at different times. The temperature distributions and temperature values obtained via the molecular dynamics method are in a good agreement with the experimental data (Bui (2006)). It can be seen from Figs. 6-8 that the temperature rise in the vicinity of the crack tip is very high for crack propagation. The hot region near the crack tip is observed. This hot zone is due to inelastic dissipation distributed over a small, but non vanishing area around the moving crack tip. From MD simulations one can get crack propagation directions angles for different values of the mixity parameter. Calculations of MD method have been performed for seven different values of the mixity parameter e M : 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8. Calculated values of crack direction propagation angles were -51.6 ° , -44 .5° , -40.7 ° , -36 .5° , -30.3 °, -24.5 ° and -20 .2° respectively. 2. Approaches of crack growth simulation: molecular dynamics

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Fig. 1. The stress components 11  (GPa) after initialization (left), at 10 ps (center) and at 11 ps (right)

Fig. 2. The stress components 11  (GPa) at 12 ps (left), at 13 ps (center) and at 14 ps (right)

Fig. 3. The stress components 11  (GPa) at 15 ps (left) and at 16 ps (center), the stress components 22  (GPa) after initialization (right)

Fig. 4. The stress components 22  (GPa) at 10 ps (left), at 11 ps (center) and at 12 ps (right)

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Fig. 5. The stress components 22  (GPa) at 13 ps (left), at 14 ps (center) and at 15 ps (right)

Fig. 6. The stress components 22  (GPa) at 16 ps (left), the temperature distribution after initialization (K) (center) and the difference between the temperature (K) in the cracked plate at 10 ps (right) and at the initial state

Fig. 7. The difference between the temperature (K) in the plate at 11 ps (left), 12 ps (center), 13 ps (right) and at the initial state respectively

Fig. 8. The difference between the temperature (K) in the plate at 14 ps (left), 15 ps (center), 16 ps (right) and at the initial state respectively

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3. Conventional fracture mechanics approach

The Williams asymptotic description for the stress field in the vicinity of the crack tip has the form

2      1 m k  m

( ) r,   ij

m k a r f

( ), 

/2 1 ( ) , k m ij 

k



where index m associated to the fracture mode; m

k a amplitude coefficients related to the geometric configuration,

load and mode; ( ) , ( ) k m ij f  angular functions depending on stress component and mode. Analytical expressions for angular eigenfunctions are available (Hello (2012), Hello (2018)). The multi-parameter fracture mechanics concept consists in the idea that the crack-tip stress field is described by means of the Williams expansion. Hello (2012) considered the central crack in an infinite plane medium, where analytical determination of coefficients in crack-tip expansion for a finite crack in an infinite plane medium is given. Since all the coefficients in the Williams asymptotic series expansion for the geometry considered are known it is possible to estimate the crack propagation direction angle by means of the multi-parameter fracture mechanics concept. In this paper two fracture criteria were chosen for the estimation of the initial crack growth direction: maximum tangential stress (MTS) criterion and strain energy density (SED) criterion. The results obtained by MTS and SED criteria are given in (Stepanova and Bronnikov (2018), Stepanova et al. (2017)). The maximum tangential strain criterion postulates that crack propagation initiates from the crack tip in direction of the maximum tangential strain 2 2 / 0, / 0.             The results given by the maximum tangential strain criterion for plane strain conditions are shown in Table 1 and Table 2 where the first column shows the crack propagation direction angle given by the one-term asymptotic expansion ( 1 N  ). The columns 2-9 in Table 1 and Table 2 show the results from multi-parameter fracture criterions where 100 terms in the Williams asymptotic series expansion have been kept. Table 1. Crack propagation direction angles obtained from MTS criterion at various radial distances from the crack tip, 0.3   . 1 N  0.1 c r  0.2 c r  0.3 c r  0.4 c r  0.5 c r  0.6 c r  0.7 c r  0.8 c r  e M -59.99 -52.30 -48.54 -46.38 -45.05 -44.18 -43.60 -43.21 -42.93 0.1 -57.70 -48.70 -45.14 -43.22 -42.07 -41.35 -40.89 -40.57 -40.36 0.2 -55.21 -44.89 -41.58 -39.91 -38.95 -38.37 -38.01 -37.78 -37.63 0.3 -52.35 -40.72 -37.73 -36.32 -35.56 -35.12 -34.86 -34.70 -34.62 0.4 -48.91 -36.05 -33.44 -32.31 -31.74 -31.44 -31.28 -31.19 -31.16 0.5 -44.51 -30.69 -28.53 -27.68 -27.30 -27.13 -27.06 -27.04 -27.06 0.6 -38.50 -24.44 -22.80 -22.23 -22.02 -21.96 -21.96 -22.01 -22.05 0.7 -29.82 -17.17 -16.08 -15.76 -15.68 -15.69 -15.74 -16.98 -17.02 0.8 -17.00 -8.91 -8.37 -8.24 -8.22 -8.26 -8.30 -9.56 -9.62 0.9 Table 2. Crack propagation direction angles obtained from MTS criterion at various radial distances from the crack tip, 0.5   . 1 N  0.1 c r  0.2 c r  0.3 c r  0.4 c r  0.5 c r  0.6 c r  0.7 c r  0.8 c r  e M -54.52 -48.12 -45.39 -43.92 -43.06 -42.54 -42.23 -42.03 -41.92 0.1 -52.70 -44.90 -42.31 -41.01 -40.30 -39.90 -39.67 -39.54 -39.48 0.2 -50.70 -41.47 -39.06 -37.96 -37.40 -37.10 -36.95 -37.78 -37.63 0.3 -48.39 -37.71 -35.53 -34.63 -34.21 -34.03 -33.96 -34.70 -31.16 0.4 -45.56 -33.47 -31.56 -30.87 -30.61 -30.53 -30.54 -31.20 -34.62 0.5 -41.86 -28.58 -26.99 -26.52 -26.40 -26.41 -26.49 -27.04 -27.06 0.6 -36.69 -22.83 -21.63 -21.36 -21.35 -21.44 -21.56 -21.68 -21.80 0.7 -28.92 -16.08 -15.29 -15.18 -15.24 -15.36 -15.49 -15.68 -15.73 0.8 -16.89 -8.36 -7.97 -7.95 -8.01 -8.10 -8.19 -8.27 -8.34 0.9

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4. Conclusions

The paper is aimed at the different approaches for the determination of the initial crack propagation direction angle. The crack propagation angle is obtained by 1) the multi-parameter fracture mechanics approach based on three fracture mechanics criteria: MTS, SED and maximum circumferential strain criteria; 2) atomistic modeling for the mixed-mode loading of the plane medium with the central crack. It is shown that the initial crack propagation angles given by the both approaches are very close especially for the case when the higher order terms in the Williams series expansion are kept. One can conclude either that considerable efforts still remain while modeling the mechanical behavior of crystalline materials with empirical potential in the framework of MD simulations. The data obtained from these atomistic studies can be employed in hierarchical multiscale modelling of advanced materials.

Acknowledgements

Financial support from the Russian Foundation of Basic Research (project No. 19-01-00631) is gratefully acknowledged.

References

Belova, O., Stepanova, L., 2018. Estimation of crack propagation direction angle under mixed mode loading in linear elastic isotropic materials by generalized fracture mechanics criteria and atomistic modeling (molecular dynamics method). Journal of Physics: Conference Series 1096, 012060. Bui, H.D., 2006. Fracture mechanics. Inverse problems and solutions. Springer, Dordrecht, 384 p. Chandra, S., Kumar, N.N., Samal, M.K., Chavan, V.M., Patel, R.J., 2016. Molecular dynamics simulations of crack growth behavior in Al in the presence of vacancies. Computational Materials Science 117, 518. Chowdhury, S.C., Wise, E.A., Ganesh, R., Gillespie, J.W., 2019. Effects of surface crack on the mechanical properties of Silica: A molecular dynamics simulation study. Engineering Fracture Mechanics 207, 99. Han, X., Liu, P., Sun, D., Wang, Q., 2019. Molecular dynamics simulations of the tensile responses and fracture mechanisms of Ti2AlN/TiAl composite. Theoretical and Applied Fracture Mechanics 101, 217. Hello, G., Tahar, M.B., Roelandt, J.-M., 2012. Analytical determination of coefficients in crack-tip expansions for a finite crack in an infinite plane medium. International Journal of Solids and Structures 49, 556. Hello, G., 2018. Derivation of complete crack-tip stress expansions from Westergaard-Sanford solutions. International Journal of Solids and Structures 144-145, 265. Liu, Q.Y., Zhou, J., Long, Y.H., 2019. A semi-empirical fracture model for silicon cleavage fracture and its molecular dynamics study. Theoretical and Applied Fracture Mechanics 100, 86. Roy, S., Roy, A., 2019. A computational investigation of length-scale effects in the fracture behavior of a graphene sheet using the atomistic J integral. Engineering Fracture Mechanics 207, 165. Singh, D., Sharma, P., Parashar, A., 2019. Atomistic simulations to study crack tip behavior in single crystal of bcc niobium and hcp zirconium. Current Applied Physics 19, 37. Stepanova, L., Bronnikov, S., 2018. Mathematical modeling of the crack growth in linear elastic isotropic materials by conventional fracture mechanics approaches and by molecular dynamics method: crack propagation direction angle under mixed mode loading. IOP Conference Series: Journal of Physics 973, 012046. Stepanova, L.V., Bronnikov, S.A., Belova, O.N., 2017. Estimation of crack propagation direction angle under mixed-mode loading (mode I and mode II): generalized fracture mechanics criteria and atomistic modeling (molecular dynamics method). PNRPU Mechanics Bulletin 4, 189. Sung, P.H., Chen, T.C., 2015. Studies of crack growth and propagation of single-crystal nickel by molecular dynamics. Computational Materials Science 102, 151. Tang, F.L., Cai, H.M., Rui, Z.Y., 2014. Molecular dynamics simulations of void growth in  -TiAl single crystal. Computational Materials Science 84, 232. Tang, T., Kim, S., Horstemeyer, M.F., 2010. Fatigue crack growth in magnesium single crystals under cyclic loading: Molecular dynamics simulation. Computational Materials Science 48, 426. Wu, W.P., Li, Y.L., Sun, X.Y., 2015. Molecular dynamics simulation-base cohesive zone representation of fatigue crack growth in a single crystal nickel. Computational Materials Science. 109, 66. Zhou, X., Yu X., Jacobson, D., Thompson, G.B., 2019. A molecular dynamics study on the stress generation during thin film growth. Applied Surface Science 469, 537.

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© 2019 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 ICMSMF organizers © 201 9 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 IC MSMF organizers. Pip line sy tem failures confirm th t spec al attention must be paid to the main components f nuclear p wer plants n particular. One of the significa t degradation factors in terms of integrity and residual life of these components is ero ion-corro ion in piping syst ms and cr ep in hi h pressure pipelines of thermal pow r plants. This articl deals with analysis of a set of steel samples with iffere t deg es of degrad tion using acou tic emis ion ethod based on detection of elastic-st ss waves in a m t rial. Time domain and frequency domain characteristics of oustic emission signals ge erated by diff rent creep mechanisms are analyzed. The main task s to find a relationship between crack reation and propagation and acoustic emission response. Part of t e solution is also the design and implementation of a diagnostic m thod for operation m itoring of the eteriora ion of the high- ressure piping systems a igh temperature. The benef t shou d b a significant reduction in h i k of damage to important components and reduction of the probability of damaging pipe wall integrity potentially sensitive to erosion-corrosion. © 201 9 The Authors. Published by Elsevier B.V. This is an ope acces article under CC BY-NC-ND lic nse (http://creativecommon org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. 9th International Conference on Materials Structure and Micromechanics of Fracture Acoustic Emission Response to Erosion-Corrosion and Creep Damage in Pipeline Systems Libor Nohal a , Pavel Mazal a , Frantisek Vlasic a *, Marie Svobodova b a Brno University of Technology, Faculty of Mechanical Engineering, Technicka 2, Brno, 616 69, Czech Republic b UJP Praha, a.s., Nad Kaminkou 1345, Prague, 156 10, Czech Republic 9th International Conference on Materials Structure and Micromechanics of Fracture Acoustic Emission Response to Erosion-Corrosion and Creep Damage in Pipeline Systems Libor Nohal a , Pavel Mazal a , Frantisek Vlasic a *, Marie Svobodova b a Brno University of Technology, Faculty of Mechanical Engineering, Technicka 2, Brno, 616 69, Czech Republic b UJP Praha, a.s., Nad Kaminkou 1345, Prague, 156 10, Czech Republic Abstract Abstract Pipeline system failures confirm that special attention must be paid to the main components of nuclear power plants in particular. One of the significant degradation factors in terms of integrity and residual life of these components is erosion-corrosion in piping systems and creep in high pressure pipelines of thermal power plants. This article deals with analysis of a set of steel samples with different degrees of degradation using acoustic emission method based on detection of elastic-stress waves in a material. Time domain and frequency domain characteristics of acoustic emission signals generated by different creep mechanisms are analyzed. The main task is to find a relationship between crack creation and propagation and acoustic emission response. Part of the solution is also the design and implementation of a diagnostic method for operation monitoring of the deterioration of the high-pressure piping systems at high temperature. The benefit should be a significant reduction in the risk of damage to important components and reduction of the probability of damaging pipe wall integrity potentially sensitive to erosion-corrosion.

Keywords: acoustic emission method; creep; crack; steam piping material Keywords: acoustic emission method; creep; crack; steam piping material

1. Introduction 1. Introduction

Secondary circuit failures in nuclear power plants confirm that there is a need to pay due attention to the main components of this part of the nuclear power plant. One of the major degradation factors in terms of integrity and Secondary circuit failures in nuclear power plants confirm that there is need to pay due attention to the main components of this part of the nuclear power plant. One of the major degradation factors in terms of integrity and

* Corresponding author. Tel.: +420-541-143-240 . E-mail address: vlasic@fme.vutbr.cz * Corresponding author. Tel.: +420-541-143-240 . E-mail address: vlasic@fme.vutbr.cz

2452-3216 © 2019 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 IC MSMF organizers. 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an ope acces article under CC BY-NC-ND lic nse (http://creativecommon org/licenses/by-nc-nd/4.0/)

Peer-review under responsibility of the scientific committee of the IC MSMF organizers.

2452-3216 © 2019 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 ICMSMF organizers 10.1016/j.prostr.2020.01.091