CC BY
93
Dedicated to Professor Siegfried Schmauder on the occasion of his 65th birthday
УДК 539.5
Моделирование ударного поведения и фрагментации каркаса вспененного карбида кремния
E. Postek1, T. Sadowski2, J. Bienias2
1 Институт фундаментальных технологических исследований ПАН, Варшава, PL-02-106, Польша 2 Люблинский технический университет, Люблин, PL-20-618, Польша
Вспененный карбид кремния является перспективным материалом для использования в условиях высоких температур, поскольку обладает высокой термостойкостью, прочностью, химической стойкостью и низкой теплопроводностью. Свойства вспененного карбида кремния изучались в основном при квазистатическом сжатии. В данной статье впервые предпринята попытка численного исследования вспененного карбида кремния в условиях ударного нагружения. В рамках перидинамического подхода описаны зарождение и развитие повреждений, а также фрагментация материала. Результаты исследований показали, что при высоких скоростях ударного воздействия повреждения возникают значительно раньше, чем при низкоскоростном воздействии, причем режимы разрушения имеют качественные отличия.
Ключевые слова: карбид кремния, каркас, ударное поведение, моделирование, перидинамический подход
DOI 10.24412/1683-805X-2021-5-89-98
Simulation of impact and fragmentation of SiC skeleton
E. Postek1, T. Sadowski2, and J. Bienias2
1 Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, PL-02-106, Poland 2 Lublin University of Technology, Lublin, PL-20-618, Poland
Silicon carbide foam is a promising material for high-temperature use because it has excellent thermal shock resistance and strength, low thermal conductivity, and excellent chemical resistance. Up till now SiC foam was analyzed under quasi-static compression and 4- and 3-point bending. This paper attempts for the first time to numerically analyze the SiC foam properties under impact conditions. Using the peridynamical approach it was possible to describe damage initiation, its dynamical growth, and the final fragmentation of the foam. The major conclusions resulting from the analysis are that for high impact velocities damage initiates much earlier compared to low-velocity impact, and the modes of failure are qualitatively different.
Keywords: silicon carbide, skeleton, impact, fragmentation, simulation, peridynamical approach
1. Introduction
The matrix materials in classical and advanced composites play a crucial role in designing their optimal thermomechanical properties. The most popular matrix materials are polymers [1-3], ceramics [415], cement or metal [10, 11, 16-23], which joint different parts of the second phases, particles, fibres,
© Postek E., Sadowski T., Bienias J., 2021
whiskers, nanoreinforcements, etc. Advanced composites are the most materials used for various branches of innovative industry like space, aircraft, automobile, military, and civil or mechanical engineering.
Mixing different phases, i.e. matrixes with various kinds of reinforcements one can get novel multiphase materials having in general disordered internal struc-
tures. However, controlling of manufacturing process, it is possible to create the required internal structure of composites necessary to get specific ther-momechanical properties, which are optimal for particular engineering applications. Various artificial structures with different phases contents and reinforcement geometries can be fabricated [1-30]. One can classify different internal structures of advanced composites:
- homogeneous composites consisting of multi-phases uniformly distributed in the volume including porosity [2-5], nanoparticles [1, 6, 7], particles, short fibers, whiskers [1, 8] etc.,
- irregular composites, i.e. continuous matrix with randomly distributed of the other phases [9-12] or reinforcements [1]. The important cases are foam materials [31-44] and interpenetrating composites created of multiphases, e.g. ceramic or metal foam filled with metal [44],
- layered or lamellar composites having regular or irregular arrangements [4, 30, 45],
- functionally graded composites with continuous variation of physical and mechanical properties [2738] or stepwise variations [30, 45].
In this paper, we focus on foam materials with their further application for the manufacturing of interpenetrated composites. The important type of these materials is silicon carbide foam (SCF). The SCFs can be treated as a promising material for high-temperature use because the silicon carbide ceramic has excellent thermal shock resistance and strength. On the other hand, it has low thermal conductivity and thermal expansion coefficient, and very good chemical resistance [31-36]. The SCFs are fabricated with different methods [35], which are able to create different levels of porosity volume content during the technological process. Moreover, the important is the type of porosity, i.e. open or closed which greatly impacts the macroscopic thermomechanical features. The applications of the SCFs with open porosity is generally various filters in diesel engines of cars, in metallurgy, water purification, fuel cells, bioimplants, or petrochemistry. To get higher mechanical strength and oxidative, and thermal insulation closed porosity is required.
Mechanical response of the SCFs is strongly related to the microstructure described by several parameters like pores interconnection, pore orientation, pore size distribution, and different impurities—second phase. All these characteristics are strongly related to applied manufacturing technology, rate of heating during foaming and ultimate ceramization tempera-
ture, and composition (multiphase) of the starting raw material. The higher level of porosity leads to better thermal insulation but lower mechanical strength.
The mechanical characterization of the SCFs is usually limited to quasi-static compression or 3-point bending, or 4-point bending. Therefore, this paper is the first attempt to elaborate numerical analysis of the dynamic behaviour of the SCFs subjected to external impact. In addition, the gradual degradation process of the SCFs was modelled and assessed.
In the paper, a presentation of the impact process of the SiC foam under low- and high-velocity is presented. The analysis demonstrates the qualitatively different behaviour of the foam that depends on the range of initial velocities.
This paper is the first attempt for numerical analysis of the SCF under impact conditions. Using the pe-ridynamical approach, it was possible to describe damage initiation, its dynamical growth, and the final fragmentation of the SCF. The major conclusion resulting from the analysis is that for high impact velocities, damage initiates much earlier in comparison to low-velocity impact.
In the calculations, the peridynamics (PD) method is used. The method is nonlocal and originates from the mechanics of crystals [46, 47]. The nonlocal field theories are presented in [48]. The first paper on the method concerning the PD approach in the theory of elasticity appeared 21 years ago [49]. A bond-based model in application to brittle materials is presented in [50]. The state-based model was elaborated in [51]. Further applications to the dynamics of brittle materials are given in [52-59]. Finally, recent reviews of peridynamics are given in [58-60].
2. Problem solution method
Let us assume that the ceramic foam is manufactured from purely elastic material SiC. The material model is made dependent on peridynamic states [61] (Fig. 1). The left side concerns the initial configuration and to assess deformation of the body we select two points Q and x.
The deformation of the material depends on the change of bond length % (distance) between positions Q and x in this nonlocal model:
% = Q-x. (1)
In the initial configuration, X is a function acting on the bond X(%), for example an initial stretch. However, for simplification, it is omitted. The deformation state is dependent on the new position of
Cracks initiation and growth is related to fracture energy evaluation [4, 48]:
Fig. 1. Peridynamics approach in definition of body deformation (color online)
the coordinate x in the deformed body y(x) and coordinate Q in the deformed configuration y(Q):
Y(x, $) = y (x + $) - y (x), (2)
Y(x, $) = y (Q) - y (x). (3) Then the displacements can be expressed:
U(x, $) = u(x + $) - u(£), (4)
U(x, $) = u(Q) - u(x). (5)
The scalar extension of the bond e(Y) is equal to
e(Y) = |Y| - |X|. (6)
The scalar extension state can be split up to spherical e1 and deviatoric ed parts [51, 62]:
e = ei + ed. (7)
Similarly, the force state t(Y) is the sum of its spherical and deviatoric parts:
3 k 6 d
t(Y) =-©x + a©e .
m
(8)
In the above equation, k is the bulk modulus, 0 is the dilatation, m is the weighted volume, © is the influence function, x = is the basic scalar state and a = (15^)/m is the coefficient that depends on the shear modulus ^ [51].
The considered bonds in peridynamics fail when their elongation overcomes the critical value ecr; the total damage treated as irreversible process is defined as a sum of broken bonds.
A special case of the state-based model is the bond-based model [47, 50, 56]. The force in a bond reads:
f = ceq(x, t, 0, (9)
where c= 18k/(nh4) depends on the bulk modulus k and the horizon h. The force f is maximal if the bond elongation is lower than ecr, f = 0 when e > ecr. The function q is equal to
flfor e < ecr,
H0f (10)
10for e = e,.
9kh
(11)
The fracture energy Gci related to failure mode I reads:
G =
(1 -V2) K ! 2E
(12)
where E is the Young's modulus, v is the Poisson's ratio and KI is the fracture toughness.
The damage variable at a point in peridynamics is defined as
fnq(x, t, £,)dQ d(x, t) = 1 V —. (13)
in dn
When the d = 0, the material is virgin without cracks. If d is in <0, 1>, the material is partly cracked. For d = 1, the material fails. The integration in the Eq. (13) is done over the domain Q, a part of the considered body surrounding the considered point with a sphere of the radius h, namely the horizon.
3. Material, geometry and numerical model
A SiC foam is presented in Fig. 2. A piece of foam stands for an impactor. The piece of foam of the dimensions 34.7 mm height, 8.9 mm thick and 18.6 mm width hits a steel base plate with an initial velocity V (Fig. 3, a).
The geometry of the sample is obtained from mi-croCT slices. The scanning device is Sky Scan 1174 (Bruker) [63]. The geometry is reconstructed with MIMICS program [64]. An initial tetrahedral mesh is obtained with the same software. Further, the initial tetrahedral structured mesh is converted into an unstructured one and refined. The latter steps are done with GMSH program [65]. To convert the initial mesh into the tetrahedral unstructured, a "skin" of triangular mesh is taken with GiD program [66]. Before
Fig. 2. Sample of a SiC foam used in the analysis
ecr
entering the GMSH program, the triangular ("skin") mesh is smoothed with MSC Patran program [67]. The tetrahedral mesh is generated in GMSH basing on the smoothed triangular mesh. The volumes of the tetrahedral elements are associated with the calculation points placed the tetrahedron centres. The volume calculation is done in the peridynamics program. Since all the field variables are calculated at the centres of the tetrahedrons, the calculation points and the associated volumes are presented as spheres of the equivalent volume to the tetrahedrons.
Figures 3 and 4 presents a fragment of the SCF fixed at its bottom at the steel plate. The general contact conditions are applied between the foam and the base with a friction coefficient of 0.3. The contact is searched between all surfaces of the contacting bodies. The contact search includes self-contact that is important since the structure is porous and contains several openings of complex shape. The penalty
number is 1012.
x
Fig. 4. The discretized system, top view (color online)
The SCF sample is discretized with 261496 volumes, while the discretization of the steel base counts 100000 volumes. In the calculations, the horizon value for the foam is assumed of 30 x 10-4 m, and for the base is 6.5 x 10-4 m. The horizon radius should be two to three times the distance between calculation points [53, 68]. The assumed values are taken basing on the evaluated maximum distance between the points. The higher values of the horizon do not deteriorate the results [53].
During the analysis, the damage variable is followed at three points placed along with the height of the SCF sample. The approximate location of the points is shown in regions A, B and C (Fig. 3, b). Then, the exact position of the points K, L and M within the magnified regions is shown in Fig. 5.
The material properties of SiC are the following: Young's modulus is 430 GPa, Poisson's ratio is 0.37, mass density is 3200 kg/m3, and fracture toughness is 4.1 MPa m1/2 [69]. It gives the critical value ecr = 9.5716 x 10-5. The base is made of steel of Young's modulus 210 GPa and Poisson's ratio 0.3.
The system undergoes the impact with the velocities of the range 15 up to 800 m/s. The dynamic solver of the program Peridigm [68, 70] is used in the calculations. The system was validated on several examples; therefore, it can be treated as reliable [54, 61, 70]. The time of the process is 19.8 x 10-6 s. The explicit time integration method is used, with a fixed time step of 0.02 x 10-6 s. The stable time step in the solution is 0.0331 x 10-6 s. It is evaluated at the beginning of the process. The stable time is significantly higher than the time step applied in the solu-
Fig. 5. Positions of the points in the magnified regions (cf. Fig. 3, b): A (a), B (b), C (c) (color online)
tion. The calculations are performed on the Cray XC60 Linux cluster. It has been employed 480 cores. The time of a production run is 7400 s.
4. Numerical analysis and results
The foam hits the plate with a range of velocities. During the impact, the sample undergoes damage and fragmentation. When observing the total damage of the sample in time, the structure damage is qualitatively different for the low and high impact velocities (Fig. 6). Namely, for the lower velocities of range up to 100 m/s, the total damage starts to grow significantly faster at 15 x 10-6 (Fig. 6, a), while for the high velocities, the damage grows practically linearly
100
80
8 60 Qh
8) 40
20-
+*+> > ♦ ♦♦ / ♦ / V* / * *»s y
* * m *v /
... • /
----------- [a
8 12 Time, 10"6 s
16
20
Fig. 6. Percent of damage of the sample for low velocities V = 15 (1), 30 (2), 45 (3), 100 m/s (4) (a) and for high velocities V = 100 (5), 300 (6), 500 (7), 800 m/s (8) (b)
with slight deviations (Fig. 6, b). Thus, the groups of the curves are qualitatively different. The effect is confirmed in Fig. 7, where the damage versus velocity curves are plotted at three chosen time instants. For the velocities higher than 100 m/s, damage grows almost linearly. Therefore, the damage of the foam sample is described in terms of low and high velocities.
The dependence of the damage variable on time at the selected points given in Fig. 5 is shown in Fig. 8. Damage curves calculated in points K and L for low velocities are close to each other. It is similar to curves calculated for high velocities. However, the progress of these curves for low and high velocities is slightly different. When concerning the low velocities, in point K, the curves reach their maximum and rest on their plateau in 6 x 10-6 s that happens for 15 and 45 m/s. In the high velocities, 300 and 800 m/s the plateau is reached in 3 x 10-6 s. The damage growth is abrupt in both cases, and before reaching the plateau, a transition zone appears starting at 0.5 x 10-6 s approximately. When observing point L placed in about one-third of the height of the sample after rapid growth of damage, the plateau is reached at about 2 x 10-6 s. In the case of the lowest velocity, the damage slightly grows further. For the high range of
Velocity,
Fig. 7. Percent of damage at the selected time instants t = 6 x 10-6 (1), 12 x 10-6 (2), 18 x 10-6 s (3)
Fig. 8. Damage at the observed points: K (1, 2), L (3, 4), M (5, 6)—low velocities V = 15 (1, 3, 5), 45 m/s (2, 4, 6) (a) and high velocities V = 300 (1, 3, 5), 800 m/s (2, 4, 6) (b)
the velocities, the plateau is higher than for the low range. Finally, close to the end of the process, the curves grow rapidly, indicating total damage at the point (d = 1). While noting point M placed close to the top of the sample, the damage in the low-velocity range develops smoother than in the high velocities
case. The difference is in a jump of damage up to 0.45 level when the damage starts to develop. In the case of high velocities, the damage starts earlier than in the low-velocity range.
The damage development in the foam for the low-velocity range is shown in Figs. 9 and 10. The subse-
Fig. 9. Impact velocity 15 m/s, damage at time 2.4 x 10 6 (a), 11.4 x 10 6 (b), 19.8 x 10 6 s (c) (color online)
Fig. 10. Impact velocity 15 m/s, points of damage d< 0.8 at time 2.4 x 10 6 (a), 11.4 x 10 6 (b), 19.8 x 10 6 s (c) (color online)
Fig. 11. Impact velocity 800 m/s, damage distribution at time 2.4 x 10 (a), 11.4 x 10 6 (b), 19.8 x 10 6s (c) (color online)
Fig. 12. Impact velocity 800 m/s, points of damage d< 0.8 at time 2.4 x 10 6 (a), 11.4 x 10 6 (b), 19.8 x 10-6 s (c) (color online)
quent snapshots of the process indicate the damage initiates at the bottom of the structure; then, the process progresses towards the top of the foam. In Fig. 10, the points in the structure where the damage parameter is higher than 0.8 are shown. The comparison of Figs. 9, b and 10, b shows that the damage is distributed practically in the entire sample but is not critical. At the end of the process, most of the sample is strongly damaged (Figs. 9, a and 10, a). However, the displacements in the structure are not large. There is no visible significant deformation of the foam (deformation scale is 1.0).
An example of the high-velocity impact is shown in Figs. 11 and 12. The behaviour of the foam is qualitatively different from the low-velocity impact. The sample undergoes large visible deformations and fragmentation. There are several regions of fragmenting, for example, F1 and F2 (Fig. 11, b). Another effect appears due to the contact phenomenon in the course of the process. When comparing Fig. 11, b
and Fig. 11, c, it has been found that the self-contact in the pores exists. A representative region is indicated as C1 (Fig. 11, b). The damage progresses in the structure more rapidly than in the low-velocity impact. A comparison of the damage progress can be made while observing Figs. 10 and 12. In the highvelocity impact, the lower part of the structure is already strongly damaged in half of the process (Fig. 12, b), which is in contrast to the low-velocity impact (Fig. 10, b).
5. Summary
A qualitative comparison of the high and low-velocity impact of SiC foam has been presented in the paper. The analysis is performed using the peridyna-mics method. It has been found that the behaviour of the foam is different when the impact velocity is low in comparison to the high-velocity response. The distinction of the low- and high-velocity process is based
on evaluating total damage accumulation in the structure. The basic features of the process are as follows:
In the high-velocities impact range, the total damage of the sample grows almost linearly with slight deviations from the linearity.
When the sample is subjected to the high-velocity impact, the structure undergoes self-contact in the pores.
The destruction of the SCF into fragments of the structure appears in the course of a high-velocity impact.
While the impact is performed with the low-velocity, the fragmentation and self-contact do not appear.
In the low-velocities impact, the damage of the structure is mainly due to local microcracks in the material.
Further analysis will concern infiltrated composites with the SiC skeleton including the interface properties of the SiC phase and metal phase.
Funding
This work was funded by the National Science Centre (Poland) under projects No UM0-2019/33/B/ ST8. The calculations were done using PL-GRID national computational resources at the Interdisciplinary Centre for Mathematical and Computational Modeling, University of Warsaw, Poland. The license for the MSC Patran program was provided by the Academic Computer Centre in Gdansk, Poland.
References
1. Fracture Failure Analysis of Fiber Reinforced Polymer Matrix Composites / Ed. by S. Mavinkere Rangappa, T.P. Satishkumar, M.M.M. Cuadrado, S. Siengchin, C. Barile. - Heidelberg: Springer, 2021. - doi 10.1007/ 978-981-16-0642-7
2. Sadowski T., Golewski P., Craciun M.E. Internal structure influence on the impact strength and dynamic fracture toughness of hybrid polymer matrix composites with integrated elastomer interlayers // Compos. Struct. - 2021. - V. 258. - P. 113375. - doi 10.1016/j. compstruct.2020.113375
3. Golewski P., Sadowski T. Description of thermal protection against heat transfer of carbon fiber reinforced plastics (CFRP) coated by stiffened ceramic mat (TBC) // Compos. Struct. - 2019. - V. 229. - P. 111489. - doi 10.1016/j. compstruct.2019.111489
4. Huang L.J., Geng L., Peng H-X. Microstructurally inho-mogeneous composites: Is a homogeneous reinforcement distribution optimal? // Progr. Mater. Sci. -2015. - V. 71. - P. 93-168. - doi 10.1016/j.pmatsci. 2015.01.002
5. Rice R.W. Porosity of Ceramics. - New York: Marcel Dekker Inc., 1998. - doi 10.1016/S0921-5093(99)00696-6
6. Min C, Yang X., Xue M, Li Q, Wang W, Mei X. Mi-cromachining porous alumina ceramic for high quality trimming of turbine blade cores via double femtosecond laser scanning // Ceram. Int. - 2021. - V. 47. - P. 461469. - doi 10.1016/j.ceramint.2020.08.153
7. Chen S, Cai W.-H, Wu J.-M, Ma Y.-X., Li C.-H, Shi Y.-C., Yan C.-Z., Wang Y.-J., Zhang H-X. Porous mullite ceramics with a fully closed-cell structure fabricated by direct coagulation casting using fly ash hollow spheres/kaolin suspension // Ceram. Int. - 2020. -V. 46. - P. 17508-17513. - doi 10.1016/j.ceramint. 2020.04.046
8. Koch C.C., Ovid'ko I.A., Seal S., Veprek S. Structural Nanocrystalline Materials: Fundamentals and Applications. - Cambridge: Cambridge Univ. Press, 2007. - doi 10.1017/CB09780511618840
9. Sha J.J, Wang S.H., Dai J.X., Zu Y.F., Li W.Q., ShaR.Y. Improved microstructure and high temperature mechanical properties of C/C-SiC composites by introduction of ZrC nanoparticles // Ceram. Int. - 2020. - V. 46. -P. 8082-8091. - doi 10.1016/j.ceramint.2019.12.034
10. Tjong S.C. Recent progress in the development and properties of novel metal matrix nanocomposites reinforced with carbon nanotubes and graphene nanosheets // Mater. Sci. Eng. R. - 2013. - V. 74. - P. 281-350. - doi 10. 1016/j.mser.2013.08.001
11. Szutkowska M., Podsiadlo M., Sadowski T., Figiel P., Boniecki M., Polczyl T. A novel approach by spark plasma sintering to the improvement of mechanical properties of titanium carbonitride-reinforced alumina ceramic // Molecules. - 2021. - V. 26. - P. 1344. - doi 10.3390/ molecules26051344
12. Azizian-Kalandaragh Y., Namini A.S., Ahmadi Z., Asl M.S. Reinforcing effects of SiC whiskers and carbon nanoparticles in spark plasma sintered ZrB2 matrix composites // Ceram. Int. - 2018. - V. 44. - P. 1993219938. - doi 10.1016/j.ceramint.2018.07.258
13. Sequeira S., Fernandes M.H., Neves N., AlmeidaM.M. Development and characterization of zirconia-alumina composites for orthopedic implants // Ceram. Int. -2017. - V. 43. - P. 693-703. - doi 10.1016/j.ceramint. 2016.09.216
14. Sadowski T. Gradual degradation in two-phase ceramic composites under compression // Comput. Mater. Sci. -2012. - V. 64. - P. 209-211. - doi 10.1016/j. commatsci.2012.01.034
15. Clayton J.D., Knap J. Geometric micromechanical modelling of structure changes, fracture and grain boundary layers in polycrystals // J. Micromech. Molecular Phys. - 2018. - V. 3. - P. 1840001. - doi 10.1142/ S2424913018400015
16. Sadowski T., Golewski G. A failure analysis of concrete composites incorporating fly ash during torsional loading // Compos. Struct. - 2018. - V. 183. - P. 527-535. -doi 10.1016/j.compstruct.2017.05.073
17. Ravichandran K.S. Fracture toughness of two phase WC-Co cermets // Acta Metal. Mater. - 1994. - V. 42. -P. 143-150. - doi 10.1016/0956-7151(94)90057-4
18. Honle S., Schmauder S. Micromechanical simulation of cracks growth in WC/Co using embedded unit cells // Comput. Mater. Sci. - 1998. - V. 13. - P. 56-60. - doi 10.1016/S0927-0256(98)00045-7
19. Li W, Wang H, Wang L, Hou C, Song X., Liu X., Han X. In situ of fracture behavior of ultrafine WC-Co cemented carbide // Mater. Res. Lett. - 2017. - V. 5. -P. 55-60. - doi 10.1080/21663831.2016.1208300
20. Sadowski T., Hardy S., Postek E. A new model for the time-dependent of polycrystalline ceramic materials with metallic inter-granular layers under tension // Mater. Sci. Eng. A. - 2006. - V. 424. - P. 230-238. -doi 10.1016/j.msea.2006.03.004
21. Postek E., Sadowski T. Thermomechanical effects during impact testing of WC/Co composite material // Compos. Struct. - 2020. - V. 241. - P. 112054. - doi 10.1016/j.compstruct.2020.112054
22. Postek E., Sadowski T. Impact model of WC/Co composite // Compos. Struct. - 2019. - V. 213. - P. 231242. - doi 10.1016/j.compstruct.2019.01.084
23. Postek E., Sadowski T. Temperature effects during impact testing of a two-phase metal-ceramic composite material // Materials. - 2019. - V. 12. - P. 1629. - doi 10.3390/ma12101629
24. Postek E., Sadowski T. Impact model of the Al2O3/ZrO2 composite by peridynamics // Compos. Struct. - 2021. -P. 114071. - doi 10.1016/j.compstruct.2021.114071
25. Sadowski T., Pankowski B. Numerical modelling of two-phase ceramic composite response under uniaxial loading // Compos. Struct. - 2016. - V. 143. - P. 388394. - doi 10.1016/j.compstruct.2016.02.022
26. Postek E., Sadowski T. Distributed microcracking process of WC/Co cermet under dynamic impulse compres-sive loading // Compos. Struct. -2018. - V. 194. -P. 494-508. - doi 10.1016/j.compstruct.2018.04.014
27. Saleh B., Jiang J., Fathi R., Al-Hababi T., Xu Q., Wang L., Song D., Ma A. 30 years of functionally graded materials: An overview of manufacturing methods, applications and future challenges // Composites. B. - 2020. - V. 201. - P. 108376. - doi 10.1016/ j.compositesb.2020.108376
28. Suresh S., Mortensen A. Fundamentals of Functionally Graded Materials. - Cambridge: The University Press, 1998.
29. Ivanov V., Sadowski T., Pietras D. Crack propagation in functionally graded strip under thermal shock // Eur. Phys. J. Special Topics. - 2013. - V. 222. - P. 15871595. - doi 10.1140/epj st/e2013-01947-3
30. Sadowski T., Golewski P. Multidisciplinary analysis of the operational temperature increase of turbine blades in combustion engines by application of the ceramic thermal barrier coatings (TBC) // Comput. Mater. Sci. -2011. - V. 50. - P. 1326-1335. - doi 10.1016/j. commatsci.2011.02.027
31. Mishra M.K., Kumar S., Ranjan A., Eswara Prasad N. Processing, properties and microstructure of SiC foam derived from epoxy-modified polycarbosilane // Ceram. Int. - 2018. - V. 44. - P. 1859-1867. - doi 10.1016/j. ceramint.2017.10.123
32. Zhang B., Huang H., Lu X., Xu X., Yao J. Fabrication and properties of SiC porous ceramics using a polyurethane preparation process // Ceram. Int. - 2018. -V. 44. - P. 16589-16593. - doi 10.1016/j.ceramint. 2018.06.083
33. Colombo P., Hellmann J.R., Shelleman D.L. Mechanical properties of silicon oxycarbide ceramic foams // J. Am. Ceram. Soc. - 2001. - V. 84. - P. 2245-2251. - doi 10. 1111/j.1151-2916.2001.tb00996.x
34. FukushimaM., Colombo P. Silicon carbide-based foams from direct blowing of polycarbosilane // J. Eur. Ceram. Soc. - 2012. - V. 32. - P. 503-510. - doi 10.1016/ j.jeurceramsoc.2011.09.009
35. Eom J.H., Kim Y.W., Raju S. Processing and properties of macroporous silicon carbide ceramics: A review // J. Asian Ceram. Soc. - 2013. - V. 1. - P. 220-242. - doi 10.1016/j.jascer.2013.07.003
36. Chen F., Yang Y., Shen Q., Zhang L. Macro/micro structure dependence of mechanical strength of low temperature sintered silicon carbide ceramic foams // Ceram. Int. - 2012. - V. 38. - P. 5223-5229. - doi 10.1016/j. ceramint.2012.03.030
37. Atinkok N., Demir A., Ozsert I., Findik F. Compressive behavior of Al2O3-SiC ceramic composite foams fabricated by decomposition of aluminium sulfate aqueos solution // J. Compos. Mater. - 2007. - V. 41. - P. 13611373. - doi 10.1177/0021998306068081
38. Lacroix M., Nguyen P., Schwiech D., Huu C.P., Savin-Poncet S., Edouard D. Pressure drop measurements and modelling on SiC foams // Chem. Eng. Sci. - 2007. -V. 62. - P. 3259-3267. - doi 10.1016/j.ces.2007.03.027
39. Liu J., Rong Y., Lu Y., Zhao Y., Wang L., Xi X., Yang J., Huang Y. Ultralight and mechanically robust SiC foams with interconnected cellular architecture // Ceram. Int. -2020. - V. 46. - P. 17962-17968. - doi 10.1016/j. ceramint.2020.04.108
40. Sun X., Zeng T., Zhou Y., Zhang K., Xu G., Wang X., Cheng S. 3D printing of porous SiC ceramics added with SiO2 hollow microspheres // Ceram. Int. - 2020. -V. 46. - P. 22797-22804. - doi 10.1016/j.ceramint. 2020.06.047
41. Kovarik T., Hajek J., Pola M., Rieger D., Svoboda M., Benes J., Sutta P., Deshmukh K., Jandova V. Cellular ceramic foam derived from potassium-based geopoly-mer composite: Thermal, mechanical and structural properties // Mater. Design. - 2021. - V. 198. -P. 109355. - doi 10.1016/j.matdes.2020.109355
42. Tang Z., Liang J., Jiang W., Liu J., Jiang F., Feng G., Lao X., Tang H., Wang T. Preparation of high strength foam ceramics from sand shale and steel slag // Ceram. Int. - 2020. - V. 46. - P. 9256-9262. - doi 10.1016/j. ceramint.2019.12.179
43. Xu S.C., Zhang N.L. Yang J.E., Wang B., Kim C.Y. Silicon carbide-based foams derived from foamed Sic-filled phenolic resin by reactive infiltration of silicon // Ce-ram. Int. - 2016. - V. 42. - P. 14760-14764. - doi 10. 1016/j.ceramint.2016.06.104
44. Maj J., Basista M., Wqglewski W., Bochenek K., Stro-jny-Nqdza A., Naplocha K., Panzner T., TatarkovaM.,
Fiori F. Effect of microstructure on mechanical properties and residual stresses in interpenetrating aluminum-alumina composites fabricated by squeeze casting // Mater. Sci. Eng. A. - 2018. - V. 715. - P. 154-162. -doi 10.1016/j.msea.2017.12.091
45. Sadowski T., Nakonieczny K. Thermal shock response of FGM cylindrical plates with various grading patterns // Comput. Mater. Sci. - 2008. - V. 43. - P. 171-178. -doi 10.1016/j.commatsci.2007.07.051
46. Rogula D. Nonlocal Theory of Material Media. - Wien: Springer, 1982. - doi 10.1007/978-3-7091-2890-9
47. Kunin A. Elastic Media with Microstructure, One Dimensional Models. - Berlin: Springer, 1982. - doi 10. 1007/978-3-642-81748-9
48. Eringen A.C. Nonlocal Continuum Field Theories. -New-York: Springer, 2001. - doi 10.1007/b97697
49. Silling S.A. Reformulation of elasticity theory for discontinuities and long-range forces // J. Mech. Phys. Solids. - 2000. - V. 48. - P. 175-209. - doi 10.1016/ S0022-5096(99)00029-0
50. Silling S.A., Askari E. A meshfree method based on the peridynamic model of solid mechanics // Comput. Struct. - 2005. - V. 83. - P. 1526-1535. - doi 10.1016/ j.compstruc.2004.11.026
51. Warren T.L., Silling S.A., Askari A., Weckner O., Ep-tonM.A., Xu J. A non-ordinary state-based peridynamic method to model solid material deformation and fracture // Int. J. Solids Struct. - 2009. - V. 46. - P. 11861195. - doi 10.1016/j.ijsolstr.2008.10.029
52. Littlewood D.J. Simulation of Dynamic Fracture Using Peridynamics, Finite Element Modeling, and Contact // Proceedings of the ASME 2010 International Mechanical Engineering Congress and Exposition, Vancouver, Canada, Nov. 12-18, 2010. - IMECE2010-40621 (9 p.).
53. Ha Y.D., Bobaru F. Studies of dynamic crack propagation and crack branching with peridynamics // Int. J. Fracture. - 2010. - V. 162. - P. 229-244. - doi 10.1007/ s10704-010-9442-4
54. Littlewood D.J., Volgler T. Modeling Dynamic Fracture with Peridynamics, Finite Element Modeling and Contact // 11th US National Congress on Computational Mechanics, July 25-28, 2011. - https://www.osti.gov/ servlets/purl/1106568 (accessed on 12 January 2021)
55. Ye L.Y., Wang C., Chang X., Zhang H.Y. Propeller-ice contact modeling with peridynamics // Ocean Eng. -
2017. - V. 139. - P. 54-64. - doi 10.1016/j.oceaneng. 2017.04.037
56. Foster J.T. Dynamic Crack Initiation Toughness: Experiments and Peridynamic Modeling // Sandia Report SAND2009-7217, 2009.
57. Demmie P.N., Silling S.A. An approach to modeling extreme loading of structures using peridynamics // J. Mech. Mater. Struct. - 2007. - V. 2. - P. 1921-1945. -doi 10.2140/jomms.2007.2.1921
58. Madenci E., Oterkus E. Peridynamic Theory and Its Applications. - New York: Springer, 2014. - doi 10.1007/ 978-1-4614-8465-3
59. Bobaru F., Foster J.T., Geubelle P.H., Silling S.A. Handbook of Peridynamic Modeling. - Boca Raton: CRC Press, 2017. - doi 10.1201/9781315373331
60. Voyadjis G.Z. Handbook of Nonlocal Continuum Mechanics for Materials and Structures. - Springer, 2019.
61. Littlewood D.J. Introduction to peridynamic modeling and applications // Sandia Report SAND2017-13417C. -Sandia National Laboratories. - https://www.osti.gov/ servlets/purl/1511976 (accessed on 19 May 2021)
62. Mitchell J.A., Silling S.A., Littlewood J.A. A position-aware linear solid constitutive model for peridynamics // J. Mech. Mater. Struct. - 2015. - V. 10. - P. 539-557. -doi 10.2140/jomms.2015.10.539
63. https://www.bruker.com (accessed on 19 May 2021)
64. Materialise Mimics. https://www.materialise.com (accessed on 19 May 2021)
65. Geuzaine C., Remacle J-F. A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. - https://gmsh.info/ (accessed on 10 Oct. 2020)
66. GiD The personal pre- and postprocessor. - https:// www.gidhome.com/ (accessed on 19 May 2021)
67. MSC Software. - https://www.mscsoftware.com/pro duct/patran (accessed on 19 May 2021)
68. Parks M.L., Littlewood D.J., Mitchell J.A., Silling S.A. Peridigm Users Guide // Sandia Report 2012-7800. -Sandia National Laboratories, 2012. - https://old-www. sandia.gov/~djlittl/docs/PeridigmV1.0.0.pdf (accessed on 19 May 2021)
69. AZO Materials. - https://www.azom.com/article.aspx? ArticleID=3735 (accessed on 19 May 2021)
70. Open Source Peridigm program repository. - https://gi thub.com/peridigm/peridigm (accessed on 19 May 2021)
Received 24.05.2021, revised 30.07.2021, accepted 30.07.2021
Сведения об авторах
Eligiusz Postek, PhD, DSc, Adiunkt, Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland, epostek@ippt.pan.pl
Tomasz Sadowski, PhD, DSc, Prof., Head of Department, Lublin University of Technology, Poland, t.sadowski@pollub.pl Jaroslaw Bienias, PhD, DSc, Assoc. Prof., Head of Department, Lublin University of Technology, Poland, j.bienias@pollub.pl
