Non-local damage modelling of quasi-brittle composites

Jiří Vala; Vladislav Kozák

Applications of Mathematics (2021)

  • Volume: 66, Issue: 6, page 815-836
  • ISSN: 0862-7940

Abstract

top
Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering.

How to cite

top

Vala, Jiří, and Kozák, Vladislav. "Non-local damage modelling of quasi-brittle composites." Applications of Mathematics 66.6 (2021): 815-836. <http://eudml.org/doc/297977>.

@article{Vala2021,
abstract = {Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering.},
author = {Vala, Jiří, Kozák, Vladislav},
journal = {Applications of Mathematics},
keywords = {quasi-brittle composite; steel fibre concrete; micro- and macro-fracture; non-local viscoelasticity; cohesive interface; partial differential equations of evolution; method of discretization in time; extended finite element method},
language = {eng},
number = {6},
pages = {815-836},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Non-local damage modelling of quasi-brittle composites},
url = {http://eudml.org/doc/297977},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Vala, Jiří
AU - Kozák, Vladislav
TI - Non-local damage modelling of quasi-brittle composites
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 6
SP - 815
EP - 836
AB - Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering.
LA - eng
KW - quasi-brittle composite; steel fibre concrete; micro- and macro-fracture; non-local viscoelasticity; cohesive interface; partial differential equations of evolution; method of discretization in time; extended finite element method
UR - http://eudml.org/doc/297977
ER -

References

top
  1. Altan, S. B., Existence in nonlocal elasticity, Arch. Mech. 41 (1989), 25-36. (1989) Zbl0725.73022MR1065652
  2. Babuška, I., Melenk, J. M., 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N, Int. J. Numer. Methods Eng. 40 (1997), 727-758. (1997) Zbl0949.65117MR1429534DOI10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
  3. Belytschko, T., Black, T., 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S, Int. J. Numer. Methods Eng. 45 (1999), 601-620. (1999) Zbl0943.74061DOI10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
  4. Castro, A. Bermúdez de, 10.1007/3-7643-7383-0, Progress in Mathematical Physics 43. Birkhäuser, Basel (2005). (2005) Zbl1070.74001MR2145925DOI10.1007/3-7643-7383-0
  5. Bhatia, G. S., Arora, G., 10.17485/ijst/2016/v9i45/105079, Indian J. Sci. Technol. 9 (2016), Article ID 45, 18 pages. (2016) DOI10.17485/ijst/2016/v9i45/105079
  6. Bouhala, L., Makradi, A., Belouettar, S., Kiefer-Kamal, H., Fréres, P., 10.1016/j.compositesb.2012.12.013, Composites B, Eng. 55 (2013), 352-361. (2013) DOI10.1016/j.compositesb.2012.12.013
  7. Bunkure, J. K., Lebesgue-Bochner spaces and evolution triples, Int. J. Math. Appl. 7 (2019), 41-52. (2019) 
  8. Cianchi, A., Maz'ya, V., 10.1016/j.aim.2016.02.012, Adv. Math. 293 (2016), 644-696. (2016) Zbl1346.46022MR4074620DOI10.1016/j.aim.2016.02.012
  9. Clark, D. S., 10.1016/0166-218X(87)90064-3, Discrete Appl. Math. 16 (1987), 279-281. (1987) Zbl0612.39004MR0878027DOI10.1016/0166-218X(87)90064-3
  10. Maso, G. Dal, Lazzaroni, G., 10.3934/dcds.2011.31.1219, Discrete Contin. Dyn. Syst. 31 (2011), 1219-1231. (2011) Zbl1335.74051MR2836349DOI10.3934/dcds.2011.31.1219
  11. Piero, G. Del, Owen, D. R., 10.1007/BF00375133, Arch. Ration. Mech. Anal. 124 (1993), 99-155. (1993) Zbl0795.73005MR1237908DOI10.1007/BF00375133
  12. Dlouhý, L., Pouillon, S., Application of the design code for steel-fibre-reinforced concrete into finite element software, Beton 116 (2020), 8-13. (2020) 
  13. Drábek, P., Milota, J., 10.1007/978-3-0348-0387-8, Birkhäuser Advanced Texts. Basler Lehrbücher. Birkhäuser, Basel (2013). (2013) Zbl1264.35001MR3025694DOI10.1007/978-3-0348-0387-8
  14. Eliáš, J., Vořechovský, M., Skoček, J., Bažant, Z. P., 10.1016/j.engfracmech.2015.01.004, Eng. Fract. Mech. 135 (2015), 1-16. (2015) DOI10.1016/j.engfracmech.2015.01.004
  15. Emmrich, E., Puhst, D., 10.1088/0951-7715/28/1/285, Nonlinearity 28 (2015), 285-307. (2015) Zbl1312.35163MR3297136DOI10.1088/0951-7715/28/1/285
  16. Eringen, A. C., Theory of Nonlocal Elasticity and Some Applications, Technical Report 62. Princeton University Press, Princeton (1984). (1984) 
  17. Eringen, A. C., 10.1007/b97697, Springer, New York (2002). (2002) Zbl1023.74003MR1918950DOI10.1007/b97697
  18. Evgrafov, A., Bellido, J. C., 10.1177/1081286518810745, Math. Mech. Solids 24 (2019), 1935-1953. (2019) Zbl1425.74093MR3954360DOI10.1177/1081286518810745
  19. Fries, T.-P., Belytschko, T., 10.1002/nme.1761, Int. J. Numer. Methods Eng. 68 (2006), 1358-1385. (2006) Zbl1129.74045DOI10.1002/nme.1761
  20. Gao, Z., Zhang, L., Yu, W., 10.1016/j.engfracmech.2017.10.019, Eng. Fract. Mech. 189 (2018), 481-500. (2018) DOI10.1016/j.engfracmech.2017.10.019
  21. Giry, C., Dufour, F., Mazars, J., 10.1016/j.ijsolstr.2011.08.012, Int. J. Solids Struct. 48 (2011), 3431-3443. (2011) DOI10.1016/j.ijsolstr.2011.08.012
  22. Grija, S., Shanthini, D., Abinaya, S., A review on fiber reinforced concrete, Int. J. Civil Eng. Technol. 7 (2016), 386-392. (2016) 
  23. Hashiguchi, K., 10.1007/978-3-642-35849-4, Lecture Notes in Applied and Computational Mechanics 69. Springer, Berlin (2014). (2014) Zbl1318.74001MR3235845DOI10.1007/978-3-642-35849-4
  24. Havlásek, P., Grassl, P., Jirásek, M., 10.1016/j.engfracmech.2016.02.029, Eng. Fract. Mech. 157 (2016), 72-85. (2016) DOI10.1016/j.engfracmech.2016.02.029
  25. Hoekstra, A., Design methodologies for steel-fibre-reinforced concrete and a new methodology for a real time quality control, Beton 116 (2020), 44-49. (2020) 
  26. Horgan, C. O., 10.1016/0022-247X(79)90190-2, J. Math. Anal. Appl. 69 (1979), 231-242. (1979) Zbl0412.35073MR0535293DOI10.1016/0022-247X(79)90190-2
  27. Javili, A., Morasata, R., Oterkus, E., Oterkus, S., 10.1177/1081286518803411, Math. Mech. Solids 24 (2019), 3714-3739. (2019) Zbl07273389MR4000179DOI10.1177/1081286518803411
  28. Jirásek, M., 10.1007/978-3-7091-0897-0_1, Numerical Modeling of Concrete Cracking CISM Courses and Lectures 532. Springer, Wien (2011), 1-49. (2011) Zbl1247.74055DOI10.1007/978-3-7091-0897-0_1
  29. Kaliske, M., Dal, H., Fleischhauer, R., Jenkel, C., Netzker, C., 10.1007/s00466-011-0578-5, Comput. Mech. 50 (2012), 303-320. (2012) Zbl1398.74347MR2967876DOI10.1007/s00466-011-0578-5
  30. Kawde, P., Warudkar, A., 10.5281/zenodo.233321, Int. J. Eng. Sci. Res. Technol. 6 (2017), 130-133. (2017) DOI10.5281/zenodo.233321
  31. Khoei, A. R., 10.1002/9781118869673, Wiley Series in Computational Mechanics. John Wiley & Sons, New York (2015). (2015) Zbl1315.74001DOI10.1002/9781118869673
  32. Kozák, V., Chlup, Z., 10.4028/www.scientific.net/KEM.465.231, Key Eng. Materials 465 (2011), 231-234. (2011) DOI10.4028/www.scientific.net/KEM.465.231
  33. Kozák, V., Chlup, Z., Padělek, P., Dlouhý, I., 10.4028/www.scientific.net/SSP.258.186, Solid State Phenomena 258 (2017), 186-189. (2017) DOI10.4028/www.scientific.net/SSP.258.186
  34. Lazar, M., Maugin, G. A., Aifantis, E. C., 10.1016/j.ijsolstr.2005.04.027, Int. J. Solids Struct. 43 (2006), 1404-1421. (2006) Zbl1120.74342MR2200992DOI10.1016/j.ijsolstr.2005.04.027
  35. Lazzaroni, G., 10.1007/s10231-010-0145-2, Ann. Mat. Pura Appl. (4) 190 (2011), 165-194. (2011) Zbl1215.35156MR2747470DOI10.1007/s10231-010-0145-2
  36. Li, X., Chen, J., 10.1016/j.cma.2016.11.029, Comput. Methods Appl. Mech. Eng. 315 (2017), 744-759. (2017) Zbl1439.74334MR3595276DOI10.1016/j.cma.2016.11.029
  37. Li, X., Gao, W., Liu, W., 10.1177/1056789518823876, Int. J. Damage Mech. 28 (2019), 1299-1322. (2019) DOI10.1177/1056789518823876
  38. Macek, R. W., Silling, S. A., 10.1016/j.finel.2007.08.012, Finite Elem. Anal. Des. 43 (2007), 1169-1178. (2007) MR2393568DOI10.1016/j.finel.2007.08.012
  39. Majdisova, Z., Skala, V., 10.1016/j.apm.2017.07.033, Appl. Math. Modelling 51 (2017), 728-743. (2017) Zbl07166287MR3694560DOI10.1016/j.apm.2017.07.033
  40. Mielke, A., Roubíček, T., 10.1007/978-1-4939-2706-7, Applied Mathematical Sciences 193. Springer, New York (2015). (2015) Zbl1339.35006MR3380972DOI10.1007/978-1-4939-2706-7
  41. Moradi, M., Bagherieh, A. R., Esfahani, M. R., 10.1177/1056789519851159, Int. J. Damage Mech. 29 (2020), 388-412. (2020) DOI10.1177/1056789519851159
  42. Morandotti, M., 0.1007/978-981-10-6283-4_11, Mathematical Analysis of Continuum Mechanics and Industrial Applications II Springer, Singapore (2018), 125-136. (2018) DOI0.1007/978-981-10-6283-4_11
  43. Nakamura, N., 10.3389/fbuil.2016.00014, Front. Built Environ. 2 (2016), Article ID 14, 13 pages. (2016) DOI10.3389/fbuil.2016.00014
  44. R. H. J. Peerlings, R. de Borst, W, A. M. Brekelmans, M. Geers, 10.1002/(SICI)1099-1484(1998100)3:4<323::AID-CFM51>3.0.CO;2-Z, Mech. Cohesive-frictional Mater. 3 (1998), 323-342. (1998) DOI10.1002/(SICI)1099-1484(1998100)3:4<323::AID-CFM51>3.0.CO;2-Z
  45. Pijaudier-Cabot, G., Mazars, J., 10.1016/B978-012443341-0/50056-9, Handbook of Materials Behavior Models. Volume II Academic Press, London (2001), 500-512 J. Lemaitre. (2001) DOI10.1016/B978-012443341-0/50056-9
  46. Pike, M. G., Oskay, C., 10.1016/j.finel.2015.07.007, Finite Elem. Anal. Des. 106 (2005), 16-31. (2005) DOI10.1016/j.finel.2015.07.007
  47. Povstenko, Yu. Z., 10.1007/BF02364923, J. Math. Sci. 97 (1999), 3840-3845. (1999) DOI10.1007/BF02364923
  48. Ray, P., 10.1098/rsta.2017.0396, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 377 (2019), Article ID 20170396, 13 pages. (2019) DOI10.1098/rsta.2017.0396
  49. Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, Mathematics and Its Applications 4. D. Reidel, Dordrecht (1982). (1982) Zbl0505.65029MR0689712
  50. Roubíček, T., Nonlinear Partial Differential Equations with Applications, ISNM. International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005). (2005) Zbl1087.35002MR2176645
  51. Roubíček, T., 10.1137/080729992, SIAM J. Math. Anal. 42 (2010), 256-297. (2010) Zbl1213.35279MR2596554DOI10.1137/080729992
  52. Šilhavý, M., 10.2140/memocs.2017.5.191, Math. Mech. Complex Syst. 5 (2017), 191-215. (2017) Zbl1447.49026MR3669123DOI10.2140/memocs.2017.5.191
  53. Su, X. T., Yang, Z. J., Liu, G. H., 10.1016/j.ijsolstr.2010.04.031, Int. J. Solids Struct. 47 (2010), 2336-2345. (2010) Zbl1194.74313DOI10.1016/j.ijsolstr.2010.04.031
  54. Sumi, Y., 10.1007/978-4-431-54935-2, Mathematics for Industry (Tokyo) 2. Springer, Tokyo (2014). (2014) Zbl1395.74001MR3234571DOI10.1007/978-4-431-54935-2
  55. Svenning, E., Larsson, F., Fagerström, M., 10.1016/j.compstruc.2018.08.003, Comput. Struct. 211 (2019), 43-54. (2019) DOI10.1016/j.compstruc.2018.08.003
  56. Swati, R. F., Wen, L. H., Elahi, H., Khan, A. A., Shad, S., 10.1007/s00542-018-4021-0, Microsyst. Technol. 25 (2019), 747-763. (2019) DOI10.1007/s00542-018-4021-0
  57. Vala, J., Structure identification of metal fibre reinforced cementitious composites, Algoritmy: 20th Conference on Scientific Computing STU Bratislava, Bratislava (2016), 244-253. (2016) 
  58. Vala, J., Kozák, V., 10.1016/j.tafmec.2020.102486, Theor. Appl. Fract. Mech. 107 (2020), Article ID 102486, 8 pages. (2020) DOI10.1016/j.tafmec.2020.102486

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.