On temporal-structural dynamic failure criteria for rocks

1 Introduction

Deformation and failure of rock mass exhibit interesting temporal and spatial features. Temporal features relate with not only the real time history of deformation, but also the temporal scales over which the internal structure evolves and defines macroscopic behavior in time. Spatial features relate not only to the spatial extension of bodies, but also to the length scales over which the internal structure redistributes to form macroscopic patterns.

Over the last hundred years various disciplines have been developed, such as fracture mechanics, damage mechanics, and defect/dislocation theory in connection with the kinetic and statistical mechanics of strength. Various dynamic failure criteria have been developed and a number of models have been advanced focusing on some temporal-structural aspects from the viewpoint of internal structural hierarchy of materials and the finiteness of propagation velocity of cracks in solids.

In section 2 we provide a critical review of various failure criteria, and discuss their connection with temporal and structural aspects of the underlying microstructure. In section 3 we discuss the issues of finiteness of crack propagation velocity, the structural hierarchy or rock mass, and the essence of incubation time. In section 4 we address the question of whether or not the dynamic failure strain is constant and whether an “impulse” or “energy” fracture criterion is more appropriate.

2 On some temporal-structural failure criteria of solids: a critical review

Traditionally, the approaches to the investigation of deformation and failure of solids were phenomenological, according to which when a certain combinations of stresses or strains at one point in the solid reaches a critical value failure occurs. In configuration space, e.g. stress or strain space, these critical values constitute a surface beyond which the material fails, e.g. yield surface, failure surface, etc. The selection of such combination of stresses and strains and/or the determination of the critical threshold value is the basis of strength of materials theories. There are at least five classical failure criteria: maximum tensile stress criterion, maximum tensile strain criterion, maximum shear stress criterion, maximum distortion energy criterion, and the Mohr-Coulomb criterion. These failure criteria have been applied widely in engineering practice.

The accumulated experimental data until the 1940s suggested that the strength of materials depends on loading time and the strain rate. Experiments also showed that even for small stress, failure occurs eventually, if the loading time is long enough. For the interpretation of this phenomenon, two hypotheses were advanced. The first is due to Murgatroid [1], according to which the stress redistribution during plastic flow leads to the formation of stress concentrators, where stresses increase with time eventually leading to fracture. The second hypothesis is proposed by Orowan [2], according to which because of the environmental action and the aging of material, the strength limit decreases with time. When the material strength limit reaches the level of applied loading, then fracture occurs. The interpretation of Murgatroid seems to contradict the second law of thermodynamis [3], while Orowan’s hypothesis is limited to Rebinder’s effect (stress corrosion in surface-active media) [4].

In 1960s Zhurkov [5], based on a large number of experimental data, proposed the following empirical (experimentally determined) formula (Zhurkov’s formula) for the determination of the lifetime of samples of materials

where U₀ is the activation energy; σ is the uniaxial tensile stress; γ is the activation volume; k is the Boltzmann’s constant; T is the absolute temperature; and τ₀ is a temporal parameter of the order of Debye’s vibration period of atoms (about 10^-12 s). Zhurkov’s formula is valid for a very wide range of materials, including metals and alloys, halogenic and semiconducting crystals, glasses and polymers, composites and rocks. It shows that time-dependence is a common property of failure of solids. A similar formula had been advanced a bit earlier by Stroh [6], based on dislocation theory and metal plasticity. He postulated that fracture is controlled by dislocation process such as motion, accumulation and emission from pre-existing or newly nucleated crack tips. In particular, emission of dislocations is controlled by the activation energy U(σ). The probability of dislocation emission per unit time is γ₀ exp(-U(σ)/kT), and the dislocation emission time is given by the relation

A similar formula has been proposed by Bartenev [7] who investigated polymer materials by using linear fracture mechanics and kinetic theory. He proposed the relation

In contrast to Zhurkov’s empirical relation, the above relation includes the following structural properly parameters: χ₀, denoting the overloading coefficient of individual polymer molecular chain induced by the heterogeneity in the polymer structure; β, denoting the stress concentration coefficient near the crack tip; and ν_A , denoting the activation volume.

Kalthoff and Shocky [8] were the first to introduce the concert of incubation time. They proposed the so-called minimum-time failure criterion, the main feature of which is the introduction of the “internal time”; i.e. the parameter t_inc accounting for the incubation process preceding macrofracture. According to this concept, fracture occurs when the current stress intensity factor K_I (t) exceeds the dynamic fracture viscosity K_Id during some minimal time needed for macrocrack development.

About ten years earlier a time-dependent fracture criterion was proposed by Tuler and Butcher [9], in the following integral form

where ρ is the material density; c is the elastic wave propagation speed; σ(t) is the applied stress; σ_th is a corresponding threshold value of the stress; and K_m denotes a critical value. The exponent m is a material property. For m=1 Eq. (4) may be viewed as an impulse criterion, whereas for m=2, it may be viewed as energy or work criterion. As m becomes large, Eq. (4) gives a constant stress failure criterion.

Nikiforovsky and Shemyakin [10] proposed the impulse criterion, according to which failure occurs when the integral of local stress impulse σ(t) over a time internal [0, t₀] exceeds a critical value J_c , i.e.

This criterion is especially applicable to the analysis of failure caused by short-term intensive impulse loading. It can explain qualitatively the strength enhancement of materials with the increase of strain rate. The advantage of the criterion lies in that it can directly account for loading history, and can explain a few principal effects of fast dynamic rupture of solids. Its disadvantage is that it can not reduce to a static failure criterion in the limit case.

When failure is controlled by distributed cracks or collective microcrack activity, a phenomenological damage parameter Ψ is introduced in the theory of damage mechanics (e.g. Kachanov [11]) to describe the damage evolution expressed as follows

where A, n are constants and σ₀ is a threshold stress below which damage does not occur. When Ψ reaches sdefinite critical value (the maximum value of Ψ~1) failure occurs. Integrating Eq. (6), considering that Ψ≤1, we have

For n=1 we obtain Eq. (5).

In the above criteria, spatial dependencies and nonlocal effects due to long-range interactions have not been explicitly accounted for. The first scientists who seem to have recognized the importance of nonlocal effects to fracture were Neuber [12] and Novozhilov [13] who at different times and on different bases, arrived at the following fracture criterion

where σ is the governing tensile stress near the crack tip; σ_c is the stress limit of “intact material”, and d is a characteristic internal material length. The introduction of the spatial structural parameter d in Eq. (8), in fact, overcomes the difficulty that one may have with a local stress criterion due to the stress singularity predicted by classical elasticity at the crack tip. In this connection, reference is made to nonlocal and gradient theory of elasticity (such as those suggested by Eringen and Aifantis) which produce non-singular fields at the crack and allow the formulation of corresponding local fracture criteria. The discussion of such nonlocal failure criteria for stationary and propagating cracks within the framework of nonlocal and gradient theories is postponed for a future publication. For completeness a brief summary with relevant references is provided in the last section along with our conclusion. On returning to the nonlocal criterion of Eq. (8), it is pointed out that in classical fracture mechanics a parameter having dimension of length exists by combining various key physico-mechanical parameters as follows

where E is the Youg’s elastic modulus, Γ is the specific surface energy, and σ_c is a critical stress, K_IC donates the so-called critical stress intensity factor.

According to Morozov and Petrov [14] d may also be interpreted as a linear dimension characterizing the size of fracture cell at given scale level

In this case the criterion given by Eq. (8) coincides with the Griffith-Irwin criterion in simple geometry/loading cases. These authors generalized the criterion given by Eq. (8) to include both spatial and temporal parameters at given scale level. They propose the following fracture criterion [14]:

where τ is a characteristic fracture incubation time, d is a characteristic internal length parameter, σ_C is the static material strength, and σ(t, x) is the stress near the crack tip.

The Morozov-Petrov spatio-temporal structural criterion has the advantage that it can describe the dependence of strength on the loading duration by using only two parameters τ and σ_C . For a given scale level a criterion similar to the impulse criterion may be used [15], i.e.

The advantage of the criterion given by Eq. (12) lies in that it does not need to introduce additional quantities such as dynamical strength and toughness of the material.

The aforementioned failure criteria reflect to some extent the main features of time-dependent dynamic failure at low strain rates (creep failure), as well as high strain rates (dynamic fracture). They recognize the fact that failure of materials under loading needs some incubation period to occur which depends on both structural and mechanical properties of materials. In the following section we will discuss some aspects of dynamic failure of rocks from the viewpoint of the underlying structural hierarchy and the finiteness of propagation of cracks in the rock mass.

3 Some aspects of dynamic failure in rocks

3.1 The finiteness of crack propagation velocity

The propagation speed v of a crack depends on loading conditions, as well as on the environment. Experimental observations of crack propagation in corrosive environments [16–19] indicate that crack may grow for energy supplies lower than the critical limit of fracture. The mechanism for such subcritical crack propagation is often related to tensile failure of microcracks at the micro-scale, a reason for creep at the macro-scale. The dependence of the fracture propagation rate on the stress intensity factor in mode I may be approximated by the tri-modal curve shown in Figure 1.

Figure 1:

The dependence of crack propagation speed on stress intensity factor.

In regime I, the rate of stress corrosion reaction controls the speed of crack growth. Regime II is mainly determined by the rate of transport of reactive species to the crack tip. In regime III the speed of crack growth increases drastically up to catastrophic failure, it is relatively independent of the chemical environment, and is controlled by mechanical rupture. Real materials usually exhibit viscosity and plasticity properties. Theoretical studies of Freund and Lee [20] showed that when these factors are considered, for a fixed value of the viscosity parameter, the applied dynamic stress intensity factor increases with crack speed. Only when the viscosity is zero or very small, the dynamic stress intensity factor decreases with the increase of crack speed.

Experiments and theoretical investigations showed the finiteness of maximum crack growth speed, which is generally smaller than the Rayleigh wave speed C_R [21–32]. Figure 2 shows the dependence of crack propagation speed on the applied stress for glass [25]. For mode II cracks, the maximum crack growth speed is limited. Even for the case of interfacial super-shear crack propagation, the maximum crack growth speed is also limited by the longitudinal wave speed [33]. At high strain rates, when the crack growth speed exceeds a certain value, branching may occur, and in this case the growth of crack is retarded. At high strain rates the failure mode may be mixed, but the resulting effective crack propagation speed is limited. Generally speaking, we may assume that we have two characteristic crack propagation velocities: the crack growth velocity v_th corresponding to K_th , and the limit crack growth velocity v_C .

Figure 2:

The dependence of crack propagation speed on applied stress.

4 Choosing the proper failure criterion: limit strain, stress impulse and energy release

4.1 Is the dynamic limit failure strain constant?

In solids under mechanical shock, the following relation holds

(18)σ=ρDv (18)

where ρ is the material density; v is the particle velocity, and D is the velocity of the shock wave (In the case of a stress wave, a similar relation holds: σ=ρCv, where C is the stress wave velocity). Hence, for the pulse criterion of Eq. (5), we have

(19)∫0t∗σ(t)dt=∫0t∗ρDvdt=ρDu=Jc (19)

where u is the particle displacement. Equation (19) shows that, when the macroscopic displacement of particles reaches a critical value, failure occurs.

The displacement is a macroscopic measure of the deformation of materials. If the characteristic dimension covered by the shock wave is L_shock, then the displacement can be expressed by a corresponding strain quantity (ε), as

(20)u=Lshock⋅ε (20)

Therefore, Eq. (19) becomes

(21)∫0t∗σ(t)dt=ρDLshockε=Jc (21)

i.e. when the strain reaches a critical value ε_cr , failure occurs.

In creep experiments with polymers as early as in 1945, Aleksandrov [37] obtained the following equation for the strain rate

(22)ε˙=ε˙0exp(-U0-γσkT) (22)

where ε˙0 denotes a limit strain rate, and the physical meaning of the other parameters in Eq. (22) are the same as for those in Eq. (1). Multiplying Eq. (1) by Eq. (22) yields

(23)τε˙=τ0ε˙0=εst=const (23)

which shows that the critical strain at failure should be constant. Experimental data under static loading conditions or at low strain rates show that, indeed, the limit strain at failure is approximately constant [3]. But these results are not valid for dynamic loading conditions.

For rocks with complex hierarchy of internal structure under dynamic loading conditions the situation is more complicated. With the increase of strain rate, more and more scale levels are involved into deformation and fracture processes, and the critical strain magnitude increases. For example, if the shock wave covers the j-th scale level element, and if only the structural surfaces at this level are fractured, then the limit strain at the j-th scale level is very close to the value of the aforementioned “geo-mechanical invariant”, i.e.

(24)εjcr≈μΔ (24)

If the next smaller scale level, i.e. the (j+1)-th level, is also activated, and Δ_j+1/Δ_j =α<1, then the limit strain will be

(25)εjcr≈μΔΔj+αμΔΔjΔj=(1+α)μΔ (25)

If the next two smaller scale levels, i.e. the (j+1)-th and (j+2)-th levels, are activated, then the limit strain will be

(26)εjcr≈(1+α+α2)μΔ (26)

and so on.

Experiments in [38] show that the limit failure strain actually increases with the strain rate (see Figure 3). This can also be seen from Table 1, indicating that the failure strain is, indeed, close to the “geo-mechanical invariant” μ_Δ.

Figure 3:

Dependence of failure strain on strain rate of amphibolites under compression [38].

Table 1

The SHPB test results on amphibolites samples (see Figure 3) [38].

Mean strain rate (s-1)	79.62	87.19	95.33	112.56	142.53
log	1.9	1.94	1.98	2.05	2.15
Failure strain (10-3)	8.02	8.33	7.47	11.44	11.53
Dynamic strength (MPa)	199.02	209.13	233.37	262.19	299.04

SHPB, Split Hopkinson pressure bar.

With more and more scale levels being activated, the limit strain approach the following limit value

(27)εcr→μΔ1-α (27)

For example, if α=1/2, then ε_cr ≈3.4 μ_Δ. Therefore, the dynamic increase factor of failure strain DIF_ε =ε_dyn/ε_st (where ε_dyn is the dynamic critical strain, ε_st is the static critical strain) is DIF_ε ≈3.4. Obviously, Eq. (23) is valid for fracture at one specific structure level because of the “geo-mechanical invariant” relation given by Eq. (15). For quasi-brittle materials, the dynamic increase factor of strength DIFσ=σ_dyn/σ_st (where σ_dyn is the dynamic strength, and σ_st is the static strength) varies from 3 to 5. Hence, we see that the dynamic elastic modulus is approximately constant: Edyn=σdynεdyn=σstDIFσεstDIFε≈Est. Experimental data in [38, 39] show that the initial elastic modulus of hard rocks is not sensitive to strain rate, and the growth of elastic modulus beyond the initial loading stage is very weak.

4.2 Impulse or energy failure criterion?

The dependence of dynamical strength on strain rate in quasi-brittle materials is schematically shown in Figure 4. The solid line may be replaced by the dashed line for simplification of the analysis. Point A defines the transition point between regime 1 and regime 2, and point B the transition point between regime 2 and regime 3.

Figure 4:

Dependence of dynamical strength on strain rate in brittle materials (ε˙1≈100–102 s-1,ε˙s≈103s-1,ε˙2≈104s-1).

In regime 1, for the i-th structural scale level, the time for fracture of the elements at this level is

(28)ti=Δi/vsub (28)

With the consideration of Eq. (14), Eq. (28) may be rewritten as

(29)ti=δi/(μΔvsub) (29)

If we assume that a propagating crack in propagation process maintains its configuration, then the denominator in Eq. (29) μ_Δv_sub is simply the opening speed of cracks δ˙ and, thus,

(30)ti=δi/δ˙ (30)

On dividing the numerator and denominator of the right hand side term of Eq. (30) by Δ_i , we arrive at

(31)ti=(δi/Δi)/(δ˙/Δi)=εilim/ε˙i (31)

where εilim and ε˙i denote limit strain and strain rate, respectively. If we use Aleksandrov’s formula given by Eq. (22) as an expression for ε˙i, then we obtain Zhurkov’s formula given by Eq. (1). Thus, in regime I, Zhurkov’s formula or maximum strain may be used.

In regime 2, for many kinds of rock-like materials, we generally have [40]

(32)σdyn∼ε˙1/3 (32)

The incubation time is

(33)tinc=εdynε˙=σdynEε˙∼1ε˙2/3 (33)

Therefore, the impulse criterion given by Eq. (5) we have

(34)Jc=σc⋅tinc=σc⋅εdynε˙=σc⋅σdynEε˙∼ε˙-1/3 (34)

i.e. J_c will decrease with increasing strain rate. In order to obtain a rate-independent critical value in the right hand side of the failure criterion, we note that the combination of σ_dyn and t_inc should take the form Ic=σdyn2⋅tinc. Therefore, it may be argued that a proper failure criterion takes the following form

(35)∫0ti∗σ2(t)dt≥Ic (35)

i.e. in regime 2 where Eq. (32) holds, an energy based failure criterion is most appropriate to use.

In actuality, the criterion of Eq. (35) is the failure criterion at the scale level corresponding to the fragment size scale of the rock mass, since the strength at the activated by dynamic loading scale level (fragment size level) is

(36)σ(Dfrag)=σdyn(ε˙)∼ε˙1/3 (36)

where D_frag is the fragment size of the rock mass. The time to failure at this scale level is tinc=Dfragvc, and since according to Grady [41] Dfrag∼1/ε˙2/3, it follows that

(37)tinc=Dfrag/vc∼1/ε˙2/3 (37)

Eqs. (36) and (37) coincide with Eqs. (32) and (33).

To approximately evaluate the critical value of Ic=σdyn2⋅tinc, we use the values of σ_dyn and t_inc at the transition point A between regime 1 and regime 2. At this transition point, we have approximately σ_dyn≈σ_static, and tinc=εdyn/ε˙1≈εstatic/ε˙1≈μΔ/ε˙1. Therefore, the corresponding approximate value of I_c is

(38)Ic=σstatic2⋅μΔ/ε˙1 (38)

Eq. (38) may serve as a lower limit for the critical value of I_c . It is necessary to point out that Eqs. (33)–(38) are obtained based on Eq. (32).

In Regime 3, the loading is very intensive, well above the static strength of rock mass. The resistance of the material to external loading is mainly due to viscosity, and the dynamic strength of rock mass is approximately constant, i.e.

(39)σ=σdyn≈const (39)

It is noted that Eq. (39) is only a necessary condition for the failure of rock mass. In this regime, as shown in [42], the viscosity η of the rock at large strain rates is inversely proportional to the strain rate, i.e.

(40)η∼1ε˙ (40)

It is also known [43] that the viscosity η can be determined by the following equation

(41)η=G⋅τ (41)

where G is the shear modulus and τ is the relaxation time. From Eqs. (40) and (41) we obtain, ηε˙=G⋅τ⋅ε˙=Gεdyn=σdyn=const, i.e.

(42)ε=εdyn=const (42)

Therefore, the maximum dynamic strain criterion holds in this regime. The product of σ_dyn and ε_dyn gives a measure energy intensity at material failure, therefore Eq. (42) is equivalent to an energy intensity criterion.

5 Conclusions

To further improve the classical strength theories, it is necessary to introduce temporal-structural factors. To do this properly, we need to understand the interplay between internal times and internal scales and the effect that this has on microscopic deformation/fracture processes and macroscopic failure patterns. It is well known that classical elasticity and plasticity theories for cracks suffer from the stress/strain singularities at the crack tip. This prevents the direct use of stress or strain quantities “right at the crack tip” in order to formulate macroscopic fracture criteria for quasi-static and dynamic conditions. As mentioned earlier in Section 2 of the paper, nonlocal or gradient theories of elasticity (and plasticity) seem to dispense with such singularities and allow to develop failure criteria in terms of the maximum stress and strain(or a combination of the two) that occurs at a specific distance ahead of the crack tip. The location and value of this maximum stress/strain or energy quantity is determined by internal length/time parameters that such theories use for their formulation. For an account of recent development in the field of non-singular fracture mechanics, the reader may consult [44] and references quoted therein, especially those by Eringen (stress gradients) and the last author (strain gradients) of the present article. However, we have not used here this type of gradient theories, as the failure criteria that we discuss are by their nature of integral type (in time of space) and, thus, hereditary and nonlocal effects are already accounted for. Nevertheless, it is shown that such temporal-structural failure criteria are different expressions of internal kinetic structural changes under external action. Rocks have a complex internal structural hierarchy. Their failure is related to the development of cracks, the propagation velocity of which is finite. Thus, in order to describe properly deformation and fracture process in rock masses, it is necessary to consider both the internal structural hierarchy of the rock mass and the finiteness of crack propagation. In doing so, the concepts of limit failure strain, and different strain rate sensitivity regimes emerge and are comprehensively discussed. It is shown that for rock mass under intensive dynamic loading, the limit failure strain is not a constant, but increases with the strain rate. The incubation time in related impulse criteria accounts for the time that cracks need to go through the dynamically activated structural element of rock mass. It is also shown that in view of Eq. (32), energy failure criteria may serve as a rational failure criterion for rock masses under intensive dynamic loading.