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=== Fishnet Statistics for Quasibrittle Materials with Nacre-Like Alternating Series and Parallel Links: Design for Failure Probability < 1/1,000,000 ===
=== Fishnet Statistics for Quasibrittle Materials with Nacre-Like Alternating Series and Parallel Links: Design for Failure Probability < 1/1,000,000 ===


==== [https://www.swansea.ac.uk/staff/engineering/a.j.gil/ Zdeněk P. Bažant, Northwestern University, Evanston, Illinois, USA ====
==== Zdeněk P. Bažant, Northwestern University, Evanston, Illinois, USA ====


'''Room B-366, [https://goo.gl/maps/9fgUfCGjn2k Faculty of Civil Engineering, CTU in Prague]'''  
'''Room B-366, [https://goo.gl/maps/9fgUfCGjn2k Faculty of Civil Engineering, CTU in Prague]'''  
Line 7: Line 7:
'''Monday, 16 September 2019, 11:00-12:00'''  
'''Monday, 16 September 2019, 11:00-12:00'''  


Dielectric Elastomers (DE) are a class of Electro-Active Polymers with outstanding actuation properties. Voltage induced area expansions of 1980% on a DE membrane have been recently reported. In this case, the electromechanical instability is harnessed as a means for obtaining these electrically induced massive deformations with potential applications in soft robots. Computational simulation in this context becomes extremely challenging and must be addressed ''ab initio'' by the definition of well-posed constitutive models. <br>
The failure probability of engineering structures such as bridges, airframes and MEMS ought to be <10-6. This is a challenge. For perfectly brittle and ductile materials obeying the Weibull or Gaussian failure probability distribution functions (pdf) with the same coefficient of variation, the distances from the mean strength to 10-6 differ by about 2:1. For quasibrittle or architectured materials such as concrete, composites, tough ceramics, rocks, ice, foams, bone or nacre, this distance can be anywhere in-between. Hence, a new theory is needed. The lecture begins with a review of the recent formulation of Gauss-Weibull statistics derived from analytical nano-macro scale transitions and equality of probability and frequency of interatomic bond ruptures governed by activation energy. Extensions to the lifetime pdf based on subcritical crack growth is pointed out. Then, motivated by the nanoscale imbricated lamellar architecture of nacre, a new probability model with alternating series and parallel links, resembling a diagonally-pulled fishnet, has been developed. After the weakest-link and fiber-bundle models, it is the third model tractable analytically. It allows for a continuous transition between Gaussian and Weibull distributions, and is strongly size-dependent. The original fishnet model for strength of fishnet with brittle links is extended to quasibrittle links and is handled by order statistics. The size effect on the mean fishnet strength is a new kind of Type 1 size effect. It is found to consist of a series of intermediate asymptotes of decreasing slope and can be used for calibrating the fishnet distribution. Finally it is observed that random particulate materials such a concrete may follow the fishnet statistics in the low probability range. Comparisons with experimental histograms and size-effect tests support the theory.          
In this presentation, we postulate a new Convex Multi-Variable (CMV) variational framework for the analysis of these materials exhibiting massive deformations [2, 3, 4]. This extends the concept of polyconvexity [1] to strain energies which depend on non-strain based variables introducing other physical measures such as the electric displacement.     A new definition of the electro-mechanical internal energy is introduced, being expressed as a Convex Multi-Variable (CMV) function of a new extended set of electromechanical arguments. Crucially, this new definition of the internal energy enables the most accepted constitutive inequality, namely ellipticity, to be extended to the entire range of deformations and electric fields and, in addition, to incorporate the electromechanical energy of the vacuum, and hence that for ideal dielectric elastomers, as a degenerate case. Spurious numerical instabilities can then effectively be removed from the model whilst maintaining real physical instabilities. Hyperbolicity, variational principles, and Finite Element functional spaces will be shown prior to demonstrating the potential of the new paradigm through extremely challenging numerical examples involving wrinkling and the onset of instabilities [5, 6].


 
'''Selected references''' <br>
'''References''' <br>
# Luo, Wen, and Bažant, Z.P. (2017). ``Fishnet model for failure probability tail of nacre-like imbricated lamellar materials." Proc. Nat. Acad. of Sciences 114 (49), 12900--12905.  
[1]  J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis 63 (1976), 337–403. <br>
#    Luo, Wen, and Bažant, Z.P. (2017). ``Fishnet statistics for probabilistic strength and scaling of nacreous imbricated lamellar materials." J. Mech. Phys. of Solids 109, 264—287.
[2]  A.J. Gil, R. Ortigosa, A new framework for large strain electromechanics based on convex multi-variable strain energies: Variational formulation and material characterisation, CMAME 302 (2016), 293–328. <br>
#    Luo, Wen, and, Z.P. Bažant  (2018). ``Fishnet model with order statistics for tail probability of failure of nacreous biomimetic materials with softening interlaminar links." JMPS 121, 281—295.
[3]  R. Ortigosa, A.J. Gil, A new framework for large strain electromechanics based on convex multi-variable strain energies: Finite  Element discretisation and computational implementation, CMAME 302 (2016), 329–360. <br>
#    Z.P. Bažant and J.-L. Le (2017) Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime and Size Effect, Cambridge
[4] R. Ortigosa, A.J. Gil, A new framework for large strain electromechanics based on convex multi-variable strain energies:  Conservation laws, hyperbolicity and extension to electro-magneto-mechanics, CMAME 309 (2016), 202–242. <br>
#    Bažant, Z.P. (2004). “Scaling theory for quasibrittle structural failure.” Proc., Nat. Acad. of Sciences 101 (37), 14000—14007.
[5]  R. Poya, A.J. Gil, R. Ortigosa, A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics, CPC 216 (2017), 35–52. <br>
#    Bažant, Z.P., and Pang, S.-D. (2006). “Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors.” Proc. of the National Academy of Sciences 103(25), 9434-9439.
[6]  R. Poya, A.J. Gil, R. Ortigosa, R. Sevilla, J. Bonet, W. Wall, A curvilinear high order finite element framework for electromechanics: from linearised electro-elasticity to massively deformable dielectric elastomers, CMAME 329 (2018), 75–117.
#    Bažant, Z.P., Le, J.-L., and Bazant, M.Z. (2008). “Scaling of strength and lifetime distributions based on atomistic fracture mechanics.” Proc. of the Nat. Academy of Sciences, 106 (28), 11484-11489.
#    Le, J.-L., Bažant, Z.P., and Bazant, M.Z. (2011). “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures.” J. of the Mechanics and Physics of Solids 59, 1291—1321.
#    Le, Jia-Liang, and Bažant, Z.P. (2014). “Finite weakest-link model of lifetime distribution of quasibrittle structures under fatigue loading." Mathematics and Mechanics of Solids 19(1), 56—70.
#  Bažant, Z.P. (2019). ``A precis of fishnet statistics for tail probability of failure of materials with alternating series and parallel links."  Physical Mesomechanics 22 (1) (Special Issue in memory of G.I. Barenblatt), in press.

Revision as of 07:58, 15 July 2019

Fishnet Statistics for Quasibrittle Materials with Nacre-Like Alternating Series and Parallel Links: Design for Failure Probability < 1/1,000,000

Zdeněk P. Bažant, Northwestern University, Evanston, Illinois, USA

Room B-366, Faculty of Civil Engineering, CTU in Prague

Monday, 16 September 2019, 11:00-12:00

The failure probability of engineering structures such as bridges, airframes and MEMS ought to be <10-6. This is a challenge. For perfectly brittle and ductile materials obeying the Weibull or Gaussian failure probability distribution functions (pdf) with the same coefficient of variation, the distances from the mean strength to 10-6 differ by about 2:1. For quasibrittle or architectured materials such as concrete, composites, tough ceramics, rocks, ice, foams, bone or nacre, this distance can be anywhere in-between. Hence, a new theory is needed. The lecture begins with a review of the recent formulation of Gauss-Weibull statistics derived from analytical nano-macro scale transitions and equality of probability and frequency of interatomic bond ruptures governed by activation energy. Extensions to the lifetime pdf based on subcritical crack growth is pointed out. Then, motivated by the nanoscale imbricated lamellar architecture of nacre, a new probability model with alternating series and parallel links, resembling a diagonally-pulled fishnet, has been developed. After the weakest-link and fiber-bundle models, it is the third model tractable analytically. It allows for a continuous transition between Gaussian and Weibull distributions, and is strongly size-dependent. The original fishnet model for strength of fishnet with brittle links is extended to quasibrittle links and is handled by order statistics. The size effect on the mean fishnet strength is a new kind of Type 1 size effect. It is found to consist of a series of intermediate asymptotes of decreasing slope and can be used for calibrating the fishnet distribution. Finally it is observed that random particulate materials such a concrete may follow the fishnet statistics in the low probability range. Comparisons with experimental histograms and size-effect tests support the theory.

Selected references

  1. Luo, Wen, and Bažant, Z.P. (2017). ``Fishnet model for failure probability tail of nacre-like imbricated lamellar materials." Proc. Nat. Acad. of Sciences 114 (49), 12900--12905.
  2. Luo, Wen, and Bažant, Z.P. (2017). ``Fishnet statistics for probabilistic strength and scaling of nacreous imbricated lamellar materials." J. Mech. Phys. of Solids 109, 264—287.
  3. Luo, Wen, and, Z.P. Bažant (2018). ``Fishnet model with order statistics for tail probability of failure of nacreous biomimetic materials with softening interlaminar links." JMPS 121, 281—295.
  4. Z.P. Bažant and J.-L. Le (2017) Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime and Size Effect, Cambridge
  5. Bažant, Z.P. (2004). “Scaling theory for quasibrittle structural failure.” Proc., Nat. Acad. of Sciences 101 (37), 14000—14007.
  6. Bažant, Z.P., and Pang, S.-D. (2006). “Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors.” Proc. of the National Academy of Sciences 103(25), 9434-9439.
  7. Bažant, Z.P., Le, J.-L., and Bazant, M.Z. (2008). “Scaling of strength and lifetime distributions based on atomistic fracture mechanics.” Proc. of the Nat. Academy of Sciences, 106 (28), 11484-11489.
  8. Le, J.-L., Bažant, Z.P., and Bazant, M.Z. (2011). “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures.” J. of the Mechanics and Physics of Solids 59, 1291—1321.
  9. Le, Jia-Liang, and Bažant, Z.P. (2014). “Finite weakest-link model of lifetime distribution of quasibrittle structures under fatigue loading." Mathematics and Mechanics of Solids 19(1), 56—70.
#   Bažant, Z.P. (2019). ``A precis of fishnet statistics for tail probability of failure of materials with alternating series and parallel links."  Physical Mesomechanics 22 (1) (Special Issue in memory of G.I. Barenblatt), in press.
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