Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave
Results of theoretical and experimental study of failure wave phenomena are presented. A description ofthefailure wavephenomenon wasproposed in terms ofa self-similar solutionfor the microshear density. The mechanisms offailure wave generation andpropagation were classified as a delayedfailure with...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2008
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| Цитувати: | Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave / O. Naimark, O. Plekhov, W . Proud, S. Uvarov // Проблемы прочности. — 2008. — № 1. — С. 105-108. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-484492013-08-19T19:26:27Z Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave Naimark, O. Plekhov, O. Proud, W. Uvarov, S. Научно-технический раздел Results of theoretical and experimental study of failure wave phenomena are presented. A description ofthefailure wavephenomenon wasproposed in terms ofa self-similar solutionfor the microshear density. The mechanisms offailure wave generation andpropagation were classified as a delayedfailure with the delay time corresponding to the time ofexcitation ofself-similar blow-up collective modes in a microshear ensemble. Experimental study of the mechanism of the failure wave generation andpropagation was carried out using afused quartz rod and included the Taylor test with high-speed framing. The results obtained confirmed the "delayed” mechanism of the failure wave generation and propagation. Представлены результаты теоретических и экспериментальных исследований явления волны разрушения. Предложено описание явления волны разрушения на основе автомодельного решения для плотности микросдвигов. Механизмы возникновения и распространения волны разрушения классифицировали как замедленное разрушение, при чем время задержки соответствовало времени возбуждения автомодельных взрывных коллективных колебаний во множестве микросдвигов. Экспериментальное исследование механизма возникновения и распространения волны разрушения проводилось с использованием расплавленного кварцевого стержня и включало испытание по методу Тейлора с высокоскоростным фотографированием. Полученные результаты подтверди ли “замедленный” механизм возникновения и распространения волны разрушения. 2008 Article Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave / O. Naimark, O. Plekhov, W . Proud, S. Uvarov // Проблемы прочности. — 2008. — № 1. — С. 105-108. — Бібліогр.: 11 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48449 539. 4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Naimark, O. Plekhov, O. Proud, W. Uvarov, S. Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave Проблемы прочности |
| description |
Results of theoretical and experimental study of failure wave phenomena are presented. A description ofthefailure wavephenomenon wasproposed in terms ofa self-similar solutionfor the microshear density. The mechanisms offailure wave generation andpropagation were classified as a delayedfailure with the delay time corresponding to the time ofexcitation ofself-similar blow-up collective modes in a microshear ensemble. Experimental study of the mechanism of the failure wave generation andpropagation was carried out using afused quartz rod and included the Taylor test with high-speed framing. The results obtained confirmed the "delayed” mechanism of the failure wave generation and propagation. |
| format |
Article |
| author |
Naimark, O. Plekhov, O. Proud, W. Uvarov, S. |
| author_facet |
Naimark, O. Plekhov, O. Proud, W. Uvarov, S. |
| author_sort |
Naimark, O. |
| title |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave |
| title_short |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave |
| title_full |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave |
| title_fullStr |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave |
| title_full_unstemmed |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave |
| title_sort |
collective modes in the microshear ensemble as a mechanism of the failure wave |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2008 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/48449 |
| citation_txt |
Collective Modes in the Microshear Ensemble as a Mechanism of the Failure Wave / O. Naimark, O. Plekhov, W . Proud, S. Uvarov // Проблемы прочности. — 2008. — № 1. — С. 105-108. — Бібліогр.: 11 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-04T08:57:44Z |
| last_indexed |
2025-07-04T08:57:44Z |
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1836706131904299008 |
| fulltext |
UDC 539. 4
C o lle c t iv e M o d e s in th e M ic r o s h e a r E n s e m b le a s a M e c h a n is m o f th e F a ilu r e
W a v e
O . N a im ark , 1 O . P lek h ov ,1 W . P rou d ,2 and S. U v a ro v 1,a
1 Institute o f Continuous Media Mechanics, Russian Academy o f Sciences, Perm, Russia
2 Cambridge University, Department o f Physics, Cavendish Laboratory, Cambridge, UK
a usv@icmm.ru
Results o f theoretical and experimental study o f failure wave phenomena are presented. A
description o f the failure wave phenomenon was proposed in terms o f a self-similar solution fo r the
microshear density. The mechanisms o f failure wave generation and propagation were classified as
a delayed failure with the delay time corresponding to the time o f excitation o f self-similar blow-up
collective modes in a microshear ensemble. Experimental study o f the mechanism o f the failure
wave generation and propagation was carried out using a fused quartz rod and included the Taylor
test with high-speed framing. The results obtained confirmed the "delayed” mechanism o f the
failure wave generation and propagation.
K e y w o rd s : m esodefect evolution, failure w aves.
In trod u ction . The phenom enon o f a failure w ave in brittle materials has been the
subject o f intensive study during the last tw o decades [1 -3 ]. The term “failure w ave” was
introduced by G alin and Cherepanov [4] as the lim it case o f damage evolution, w here the
number o f m icroshears is large enough for the determ ination o f the front w ith a
characteristic group velocity. This front separates the structured material from the failed
area. Rasorenov et al. [1] were the first to observe the phenom enon o f delayed failure
behind an elastic w ave in glass. Such a w ave w as introduced b y Brar and B less in [5],
where the concept o f a fracture w ave w as d iscussed to explain the nature o f the elastic
limit. A failure w ave appeared in shocked brittle materials (glasses, ceram ics) as a
particular failure m ode in w hich they lose strength behind the propagating front. Generally,
the interest to the failure w ave phenom enon is initiated by the still open problem o f
physical interpretation o f traditionally used material characteristics such as the Hugoniot
elastic lim its, dynam ic strength, and relaxation m echanism o f elastic precursor.
Qualitative changes in silicate g lasses behind the failure w ave, e.g., an increase in
the refractive index, allow ed G ibbons and Ahrens (1971) to qualify this effect as the
structural phase transformation. T hese results stim ulated C lifton [6 ] to propose a
phenom enologica l m odel in w hich the failure front w as assum ed to be a propagating
phase boundary. A ccording to this m odel, the m echanism o f failure w ave nucleation and
propagation results from the local densification fo llow ed by shear failure around the
inhom ogeneities triggered by the shock.
U sing h igh-speed photography, Paliw al et al. [7] obtained real-tim e data on the
damage kinetics during dynam ic com pressive failure o f a transparent A lO N . The results
suggest that final failure o f the A lO N under dynam ic loading w as due to the formation o f
a damage zone w ith unstable propagation o f the critical crack.
S ta tistica l M od el. The description o f the failure w ave phenom enon w as proposed
by Naim ark et al. [8 , 9] after analyzing the damage localization dynam ics in terms o f a
self-similar solution for the microshear density. This solution describes qualitative changes
in the m icroshear density kinetics that allow s defining failure w aves as a specific (“slow
dynam ics”) collective m ode in the m icroshear ensem ble that could be excited due to the
pass o f a shock w ave. Structural parameters associated w ith typical m esodefects were
introduced as a m acroscopic tensor o f the defect density , w hich coincides w ith the
© O. N A IM A R K , O. PLEK H O V , W. PR O U D , S. U V A R O V , 2008
ISSN 0556-171X. Проблемы прочности, 2008, № 1 105
mailto:usv@icmm.ru
O. Naimark, O. Plekhov, W. Proud, and S. Uvarov
deform ation induced by defects. Taking into account the large number o f m esoscopic
defects and the influence o f thermal and structural fluctuations involved in the damage
accum ulation process, the form ulation o f a statistical problem concerning the defect
distribution function w as proposed by Naim ark [9] in terms o f the solution to the
Fokker-Plank equation in the phase space o f characteristic m esodefect variables.
The statistical description allow ed us to propose a m odel o f a solid w ith defects
based on the appropriate free energy form. A sim ple phenom enological form o f the part o f
free energy caused by defects (for the uniaxial case ) is g iven by a sixth order expansion,
w hich is similar to the G inzburg-Landau expansion in the phase transition theory [9]:
F = 1 A ( 1 - 5 /5 * ) p 2 - 1 B p 4 - 1 C (l-<5/<5c ) p 6 - D o p + X ( V l p )2 . ( 1)
2 4 6
Here the gradient term describes non-local interaction in the defect ensem ble; A , B , C,
and D are positive phenom enological material parameters, and % is the nonlocality
coefficient. The damage kinetics is determ ined by the evolution inequality
d F / dt = (d F / d p ) p + (d F / d5 )5 < 0, (2)
that leads to kinetic equations for the d efect density p and scaling parameter (5:
p = - r p (dF/ dp - d/ dxi (% dp l dxi (3)
5 = - ^ dF/ d5, (4)
where r p and are kinetic coefficients. A nalysis o f Eqs. (3) and (4) show s that the
scaling parameter 5 determ ines the reaction o f a solid to the defect growth. I f 5 < 5 c , the
evolution o f the defect ensem ble is governed by spatial-temporal structures (S 3) o f a
qualitatively new type characterized by an exp losive (“b low -up”) accum ulation o f defects
as t ^ r c in the spectrum o f spatial scales. The “blow -up” self-sim ilar solution is the
precursor o f the crack nucleation due to a specific kinetics o f damage localization,
p = g ( t ) f ( £ X £ = Xl Lc , g ( t ) = G( 1 - tj r c ) m , (5)
where r c is the so-called “peak tim e” (p at t c ), Lc is the scale o f localization,
and G > 0 and m > 0 are the parameters o f non-linearity, w hich characterise the free
energy release rate for 5 < 5 c . The function determ ines the defect density distribution in
the damage localization area. Equation (3) describes the characteristic stages o f damage
evolution. A s the stress at the shock w ave front approaches the critical value o c , the
properties o f the kinetic equation (3) change qualitatively (for p ^ p c ) and the damage
kinetics is subject to the self-sim ilar solution [Eq. (5)]. The m ethod for the solution o f this
problem w as developed by Kurdjumov [10]. It allow ed the estim ation o f £ f and the
definition o f the failure front propagation kinetics:
X f = £ ^ 0/2S - “/[2(/3- 1)]t (^ -“ + 1)/[2(/3-1)]. (6)
Equation (6 ) determ ines self-sim ilar regim es o f the failure w ave propagation, w hich
depends on the values o f the parameters /3 and o>. For instance, for the values o f the
parameters /3 ~ « + 1, a failure w ave w ill be generated as the subsequent excitation o f a
“blow -up” dam age localization area arising after the shock w ave pass w ith the delay tim e
t c .
106 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N9 1
Collective Modes in the Microshear Ensemble
N um erical sim ulation o f the damage kinetics [11] based on Eq. (6 ) for the conditions
o f the plate im pact test confirm ed the m echanism o f the failure w ave generation predicted
b y the aforem entioned self-sim ilar solution (Fig. 1).
Fig. 1. Simulation of the shock (S) and failure (F) wave propagation for the condition o f the plate
impact test. The photos correspond to different times o f the shock and failure wave propagation.
E xp erim en t. A n experim ental study o f the failure w ave generation and propagation
w as realized for the sym metric Taylor test perform ed on 25 m m -diam eter fused-quartz
rods [11]. Figure 2 show s processing o f photos obtained by a h igh-speed photography for
an experim ent w ith a flyer rod traveling at 534 m /s at impact. The flyer rod w as traveling
from the left to the right. In the first frame (0.3 i s after im pact), tw o vertical dark lines are
observed. The line on the left is the im pact surface. The line to the right is a shock w ave
that can be clearly seen propagating at a higher velocity in front o f other w aves in the
subsequent frames.
T----------------------1---------------------- 1----------------------1---------------------- 1----------------------1----------------------T
0 0.5 1 1.5 2 2.5 3 3.5 4
Time, (.is
Fig. 2. Processing of high-speed photos o f the shock and failure wave propagation. Three dark
zones correspond to the images o f the impact surface (A ), failure wave (■ ), and shock wave (♦ ) .
B ased on the measurem ents from the photographs, the first front was calculated to
slow dow n from the velocity approxim ately equal to the longitudinal w ave speed in fused
quartz (5 .96 km /s) during the initial 2.1 i s after im pact to 5 .2 ± 0 .3 m m /is after 3.9 [is.
Another front is observed in the frames labeled 1.5 and 1.8 i s after impact. B y the 2.1 i s
after impact, it becam e the failure front (marked by a square). The 1D strain state w ill
exist until the release w aves from the outer edges converge along the center o f the
specim en. Therefore, the developm ent o f failure is under the sam e conditions as those
experienced during the plate impact, including the transition to the 1D stress state.
The second front appears at the 1.2 i s (0.6 i s after the first (elastic) front passes this
point). It is interesting to note that the second front appearing at the 1.2 i s does not
advance significantly until the material behind it becom es fully com m inuted (opaque).
During this tim e the front velocity is Vfw ~ 1 5 7 km /s, w hich is c lose to that traditionally
m easured in the plate im pact test. H ow ever, the fo llow ing scenario reveals an increase in
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N 1 107
O. Naimark, O. Plekhov, W. Proud, and S. Uvarov
the failure front velocity up to Vfw ~ 4 km /s. The fact that the failure w ave front velocity
approaches the shock front ve locity supports the theoretical result concerning the failure
w ave nature as “delayed failure” w ith the lim it o f the “delay tim e” corresponding to the
“peak tim e” in the self-sim ilar solution (5). The loss o f transparency is caused by the
defect nucleation and occurs during the “blow -up” tim e after the induction tim e r i (the
tim e o f the formation o f the self-sim ilar profile o f defect distribution). Failure occurs after
the delay r d , w hich is the sum o f the induction tim e r i , and the “peak tim e” r c (the
tim e o f the “blow -up” damage kinetics). The steady-state regim e o f the failure w ave front
propagation can be associated w ith the successive activation o f the “b low -up” dissipative
structures under the condition w here r d ~ r c .
The research w as supported by the RFBR projects (Nos. 07-08-96001 and 05-01
00863).
1. S. V. Rasorenov, G. I. Kanel, V. E. Fortov, and M. M. Abasenov, High Press. Res., 6 ,
225-232 (1991).
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3. G. I. Kanel, A. A. Bogach, S. V. Rasorenov, and Zhen Chen., J. Appl. Phys., 92, 5045-5052
(2002).
4. L. A. Galin and G. P. Cherepanov, Sov. Phys. Doklady, 167, 543-546 (1966).
5. N. K. Brar and S. J. Bless, High Press. Res., 10, 773 (1992).
6. R. J. Clifton, Appl. Mech. Rev., 46, 540-546 (1993).
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8. O. B. Naimark and V. V. Belayev, Phys. Combust. Explos., 25, 115 (1989).
9. O. B. Naimark, V. A. Barannikov, M. M. Davydova, et al., “Crack propagation: Dynamic
stochasticity and scaling,” Tech. Phys. Lett., 26, No. 3, 254-258 (2000).
10. S. P. Kurdjumov, in: Dissipative Structures and Chaos in Non-Linear Space, Utopia,
Singapure (1988), Vol. 1, P. 431.
11. O. B. Naimark, S. V. Uvarov, D. D. Radford, et al., in: Proc. Fifth Int. Symp. on Behavior o f
Dense Media under High Dynamic Pressures, Saint Malo, France (2003), Vol. 2, pp. 65-74.
Received 28. 06. 2007
108 ISSN 0556-171X. npo6neMbi npouHocmu, 2008, № 1
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