Failure Fronts in Brittle Materials and 

Their Morphological Instabilities 

by Michael A. Grinfeld, Scott E. Schoenfeld, and Tim W. Wright 

ARL-TR-3660 September 2005 

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Army Research Laboratory 

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ARL-TR-3660 September 2005 

Failure Fronts in Brittle Materials and 

Their Morphological Instabilities 

Michael A. Grinfeld, Scott E. Schoenfeld, and Tim W. Wright 

Weapons and Materials Research Directorate, ARL 

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Failure Fronts in Brittle Materials and Their Morphological Instabilities 

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Michael A. Grinfeld, Scott E. Schoenfeld, and Tim W. Wright 

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14. ABSTRACT 

This report outlines the results of the effort to suggest a theoretical method of describing failure fronts in glasses and ceramics. 

There are various observations and experiments showing that in addition to standard shock-wave fronts, which propagate with 

transonic velocities, other much slower wave fronts can propagate within glass or ceramic substances undergoing intensive 

damage. These moving fronts propagate into intact substance, leaving intensively damaged substance behind them. They have 

been called failure waves. In this report, we model them as sharp interfaces separating two states—the intact and comminuted 

states. The approach is based on an analogy between failure fronts and fronts of slow combustion. Two main results are 

announced. One of them concerns the speed of a failure wave driven by oblique impact of a brittle target, and the other 

establishes a criterion for morphological instability of failure fronts.

15. SUBJECT TERMS

ceramic armor, brittle fracture, failure waves, morphological instability 

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Michael A. Grinfeld 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. The Main Idea of the Model 1

3. Formal Statement of the Problem 2

4. Morphological Instability of the Failure Front 4

5. References 7

Distribution List 9 
iv

List of Figures 

Figure 1. The energy balance for a damaged spring.......................................................................2 

Figure 2. Oblique impact of a brittle substance. .............................................................................2
1

1. Introduction 

Extensive experiments with glasses and brittle ceramic materials made by different groups in 

Britain, Russia, and the United States in the 1990s (1–8) proved that failure waves exist. This 

conclusion is backed by other observations in geomechanics and engineering (9–12). 

The problem of failure waves demands significant progress of the relevant experiment (theory 

and numerical modeling). These three pillars grow simultaneously and in close interaction with 

each other. The experimenters find an analogy of such waves with solid-solid phase 

transformation fronts (13) or with fronts of slow combustion (14). These two analogies are not 

antagonistic. Modeling of both phenomena includes changes in the thermodynamic potentials of 

the solid states involved. Although damage is definitely not a solid/solid phase transformation, 

both problems have the same roots—it is the minimization of accumulated energy by big 

changes in the microstructure. For us, the analogy with combustion looks somewhat more 

relevant and appealing. 

Our model is based on the analogy with the simplified theory of slow combustion. In the theory 

of slow wave with phase transformations, such an approach was quite successfully applied about 

two decades ago in geophysics and celestial physics (15) and in low-temperature physics (16). 

Certain difficulties of the simplified theory of phase transformation waves have been discussed 

in Grinfeld (17) and Glimm (18). The discussion given there is equally relevant for modeling of 

failure waves. According to the simplified theory, the internal wave “structure” of the wave 

front is ignored, and the failure front is treated as a mathematical surface. (Relevant discussions 

can be found in the classical monographs in Landau and Lifshitz [19] and Courant and Friedrichs 

[20].) 

2. The Main Idea of the Model 

To justify the main idea of the energetic approach, let us consider an elementary spring-like 

system like that shown in figure 1. The right edge of the rod is fixed. The left edge is under the 

action of an axial load P . The load is large enough to cause a considerable damage in the rod. 

The rod is divided by the failure front F into the intact domain, shown in white, and the 

damaged domain, shown in black. For the sake of simplicity, we assume that the intact 

substance has the Young’s modulus Ei , whereas the weaker damaged substance has the Young’s 

modulus Ed < Ei . The total energy of the model system is assumed to be equal to 

∆ ( ) ( ) 2 2 E = P+ E total d d b i i ε /2+e x+ E ε L – x /2 , (1) 
2

0 F 

P 

x L 

Figure 1. The energy balance for a damaged spring. 

where ∆ is the displacement of the left end of the rod, i ε and d ε are the deformations in the two 

domains of the combined rod, and b e is the energy of the broken bonds per unit length. In the 

quasistatic evolution, we obviously get the two following equations: 

∆= εi d x+ε (L–x), (2) 

and 

P=Edd ii ε = E ε . (3) 

Using the last two equations, we can rewrite the total energy of the system as follows: 

( ) 1 12 2 2 2 – – E = e – E – E P / x+ P L/ E total b d i i ⎡ ⎤ ⎣ ⎦ . (4) 

The last formula implies that at a sufficiently large load, P , such that ( ) 1 12 2 – – Edi b –E P > e , the 

growth of the length x of the damaged zone becomes energetically favorable. 

3. Formal Statement of the Problem 

We limit our study with two-dimensional propagation and consider an initially resting uniform 

half plane x ≥ 0 experiencing an impact at x = 0 by the oblique force P (figure 2). 

Figure 2. Oblique impact of a brittle substance. 
3

Within each of the domains, the energy densities int e and dam e per unit mass of the intact and 

damaged states, respectively, are given by the following formulas: 

( ) = (1/2) ijkl

int m,n int i, j k,l ρe u c uu , (5) 

and 

( ) = (1/2) + ijkl

dam m,n dam i, j k,l b ρe u c uu q , (6) 

where m,n u is the displacement gradients, int

ijkl c and ijkl

dam c are the elasticity tensors of the two states, 

and ρ is the original mass density ( ,i a is the symbol of differentiation with respect to the spatial 

coordinates i x ). Limiting ourselves with the approximation of linear elasticity, we ignore all 

effects of mass density change. The positive constant b q takes into account the energy required 

to produce various defects (interfaces, vacancies, shear bands, holes, etc.) distributed within unit 

mass of the bulk of damaged substance. This term is analogous to the constant used in the theory 

of slow combustion, which takes into account the energy release/consumption due to chemical 

reactions. (See Grinfeld and Wright [21] for a detailed discussion of the model.) 

In addition to appropriate initial and (external) boundary conditions, the master system includes 

the bulk equilibrium equation within each of the bulk domains (equation 7), the displacement 

continuity equation across the failure front (equation 8), and the traction continuity across the 

failure front (equation 9). 

0

ji

j

p = x∂

∂ , (7) 

0 + i

j – ⎡ ⎤ u n= ⎣ ⎦ , (8) 

0 + ji

j – ⎡ ⎤ p n = ⎣ ⎦ , (9) 

where ji

m,n i, j p = ρ∂ ∂ e(u )/ u is the stress tensor and i n is the unit normal to the failure front. 

The last equation (equation 10) across the failure front describes the kinetics of failure as 

follows: 

+ j. k

.k j – c = –K ⎡ ⎤ μ n n ⎣ ⎦ , (10) 

where c is the velocity of the failure front, the tensor j.

. k μ is defined as ( ) j. j –1 ji μ. k = – eδ ρ k ik i ,k p δ +u — 

it plays the same role as the scalar Gibbs chemical potential μ of a liquid substance—and K is a 

positive (kinetic) constant or function with dimension [velocity]—1. We assume explicitly that the 

displacements and traction remain continuous across the interface. This is not the only 

reasonable option. Another reasonable option, especially when dealing with pulverized states, 
4

would be the model of a friction-free interface with discontinuous displacements. In this case, 

the last kinetic, constitutive equation (equation 10) should be modified as well (a similar system 

was analyzed in Parshin [16] and Grinfeld [17] in the context of phase transformations). 

The system (equations 7–10) allows piecewise linear solutions of the following forms: 

at i ii

+ ++ u (x,z,t)= d t +a z x ct ≥ , (11) 

and 

 at i ii

– –– u (x,z,t)= d t +a z x ct ≤ , (12) 

where i

± d , i± a , and c are certain constants. The sign ± marks the quantities related to the intact 

(damaged) state. The constant c gives the speed of the failure front across which the 

displacement gradients i

± a suffer the finite discontinuities. The solution just mentioned allows 

considering the quasistatic problem oblique impact (loading) when the applied force has the 

following components: 

xx

± p = π , (13) 

and 

zx

± p = τ . (14) 

In this case, the system (equations 7–10) leads to the following formula of the velocity of the 

failure front: 

2 2 1 1 11

22 22 b

– –+ + – +

ρπ τ c= – + – –q K λ + μ λ +μ μ μ

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

. (15) 

4. Morphological Instability of the Failure Front 

In order to explore morphological stability of the piecewise linear solution, we present the 

elastic displacement ) u (x,z,t i

± and the speed of the interface c(z,t) in the following forms: 

() () i ii ° ± ±± u x,z,t = u (x,t)+u x,z,t % , (16) 

and 

 () () c z,t = c +c z,t ° % , (17) 

where i

± u (x,z,t) % and c(z,t) % are small disturbances. These equations should be substituted in the 

bulk equations and in the boundary conditions (equations 7–10), which should then be linearized with 
5

respect to the small disturbances. We then look for the solutions of the linearized system in the 

following forms ( k is the in-plane wave-number and η is the rate of growth): 

( ) ( ) i i ikz+ηt ° ± ± u x,z,t =W x – c t e % , (18) 

and 

 () ikz+ηt c z,t = Se % , (19) 

where the functions ( ) i W X ± should decay exponentially at X → ∞± . The dispersion equation 

for η is too lengthy to be presented here. It becomes relatively compact and instructive in the 

limit of an incompressible damaged state ( = 1/2 – ν ) and = 1/3 +ν (ν±± ±± = λ λ /2( ) + μ ) when it 

reads as follows: 

( ) ( )

2 2 2 2

2 22 2 2

Τ 4 3 4

8 58 3 Π 58 3

++ – ++ – – +

– –+ + – – –+ +

ρ μ μ –μ μ μ –μ μ+μ η= – Kk μ + μμ+ μ μ μ+ μμ+ μ . (20) 

The wave front is morphologically unstable if the linearized master equation has solutions of the 

previous form and with the rate η corresponding to exponential growth in time. The last formula 

shows that shear stresses T play a stabilizing role (similar to the case of morphological 

instabilities of solid-solid phase interfaces [17]). In addition, at T=0 , the failure front is 

morphologically unstable in the most interesting case when the shear modulus – μ of the damaged 

state is less than the shear modulus +μ of the intact state. Speaking very schematically, the 

instability has a simple, physical meaning. It means that penetration of fingers of damaged 

material into intact material is the fastest way of releasing accumulated elastic energy from the 

system. 

The morphological failure-front instability has the potential to explain appearance of fingering 

cracks on disintegration fronts in Prince Rupert drops reliably documented in experiments of 

Chandrasekar and Chaudhri (4). This interpretation, however, requires further (numerical) 

studies of deeply nonlinear stage of the instability. Another appealing possibility of the 

instability applications is to the failure-front waviness that was detected in the numerical 

modeling of Resnyansky et al. (22). These authors use a formally different, but conceptually 

close, model. Appearance of the waviness in their simulations could be not a numerical artifact 

but a manifestation of the instability discussed here. 

In conclusion, we have presented a simple model for a failure front based on an analogy with 

slow combustion and a formula for the failure wave generated by oblique impact on brittle 

material. We then demonstrated that under rather general assumptions, the flat failure front is 

morphologically unstable. The last conclusion shows an additional analogy between failure 

fronts and slow combustion fronts (but, of course, the mechanisms of destabilization are totally 

different). The morphological instability discussed here may shed some light on the appearance 
6

of corrugations on failure fronts and crack bifurcations as observed in experiments on static 

indentation of brittle materials, the appearance of radial cracks in dynamic experiments with 

penetration of projectiles through brittle materials, the appearance of corrugations on the 

disintegration fronts in Prince Ruppert drops (4), and the appearance of failure front waviness 

detected in numerical simulations (22). 
7

5. References 

1. Kanel G. I.; Molodets, A. M.; Dremin, A. N. Combust. Explos. Shock Waves 1977, 13, 772. 

2. Brar, N. S.; Bless, S. J.; Rozenberg, Z. J. Appl. Phys. 1991, 59, 3396. 

3. Bless, S. J.; Brar, N. S.; Kanel, G.; Rozenberg, Z. J. Amer. Ceramic Soc. 1992, 75, 1002. 

4. Chandrasekar, S.; Chaudhri, M. M. Phil. Mag. B 1994, 70 (6), 1195–1218. 

5. Dandekar, D. P.; Beaulieu, P. A. Metallurgical and Materials Applications of Shock-Wave 

and High-Strain-Rate Phenomenon; Murr, L. E., Staudhammer, K. P., Meyers, M. A., Eds.; 

Elsevier Science: New York, 1995; pp 211–218. 

6. Bourne, N.; Rosenberg, Z.; Field, J. E. J. Appl. Phys. 1995, 78, 3736. 

7. Bourne, N. K.; Forde, L.; Field, J. E. Speed Photography and Photonics–1996; Paisley, 

D. L. Ed.; Proc. SPIE 2869 1997, 626. 

8. Bourne, N.; Millett, J.; Rosenberg, Z. J. Appl. Phys. 1997, 81, 6670. 

9. Grigoryan, S. S. PMM 1967, 31, 667. 

10. Slepyan, L. I. MTT 1977, 1, 181. 

11. Chaudhri, M. M. Phil. Mag. Lett. 1998, 78 (2), 153–158. 

12. Grady, D. E. Mech. Mat. 1998, 29, 181. 

13. Clifton, R. J. Appl Mech. Rev. 1993, 46, 540. 

14. Kanel, G. I.; Bogatch, A. A.; Razorenov, S. V.; Chen, Z. J. Appl. Phys. 2002, 92, 5045. 

15. Grinfeld, M. A. Dokl. AN SSSR 1982, 262, 1339. 

16. Parshin, A. Ya. Low Temperature Physics; Borovik-Romanov, A. S., Ed.; Mir, Moscow, 

1985. 

17. Grinfeld, M. A. Thermodynamic Methods in the Theory of Heterogeneous Systems; 

Longman: Sussex, 1991. 

18. Glimm, J. Lecture Notes in Phys. 1989, 344, 177. 

19. Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: New York, 1987. 

20. Courant, R.; Friedrichs, K. O. Supersonic Flow and Shock Waves; Springer-Verlag: New 

York, 1948. 
8

21. Grinfeld, M. A.; Wright, T. W. Metall. Mater. Trans. A 2004, 35 (9), 2651–2661. 

22. Resnyansky, A. D.; Romensky, E. I.; Bourne, N. K. Journal Appl. Phys. 93 2003, 1537. 
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