PRSL

On the growth and decay of accelerationwaves in random media

By Martin Ostoja-Starzewski1 and Jerzy T r e bicki21Institute of Paper Science and Technology, and

Georgia Institute of Technology, Atlanta, GA 30318-5794, USA2Institute of Fundamental Technological Research,

Polish Academy of Sciences, 00-049 Warsaw, Poland

Received 9 October 1997; accepted 8 December 1998

We study the effects of material spatial randomness on the growth to shock or decayof acceleration waves. In the deterministic formulation, such waves are governed bya Bernoulli equation dα/dx = −µ(x)α + β(x)α2, in which the material coefficientsµ and β represent the dissipation and elastic nonlinearity, respectively. In the caseof a random microstructure, the wavefront sees the local details: it is a mesoscalewindow travelling through a random continuum. Upon a stochastic generalizationof the Bernoulli equation, both coefficients become stationary random processes,and the critical amplitude αc as well as the distance to form a shock x∞ becomerandom variables. We study the character of these variables, especially as comparedto the deterministic setting, for various cases of the random process: (i) one whitenoise; (ii) two independent white noises; (iii) two correlated Gaussian noises; and(iv) an Ornstein–Uhlenbeck process. Situations of fully positively, negatively or zerocorrelated noises in µ and β are investigated in detail. Particular attention is given tothe determination of the average critical amplitude 〈αc〉, equations for the evolutionof the moments of α, the probability of formation of a shock wave within a givendistance x, and the average distance to form a shock wave. Specific comparisons ofthese quantities are made with reference to a homogeneous medium defined by themean values of the µ, βx process.

Keywords: acceleration wave; random media; mesoscale; stochastic mechanics;wave propagation; Bernoulli equation

1. Introduction

(a) Microscale heterogeneity versus wavefront thickness

Acceleration waves are moving singular surfaces with a jump in particle acceleration(see, for example, Kosinski 1986). The behaviour of acceleration waves is governedby a Bernoulli equation

dαdx

= −µ(x)α+ β(x)α2, (1.1)

where x is position, α is the jump in particle acceleration, while the coefficients µ andβ represent the dissipation and elastic nonlinearity, respectively. Classical referenceson this subject include Bland (1969), Chen (1973), Coleman & Gurtin (1965), Menonet al . (1983), McCarthy (1975) and Christensen (1982). The resulting competition

Proc. R. Soc. Lond. A (1999) 455, 2577–2614Printed in Great Britain 2577

c© 1999 The Royal SocietyTEX Paper

2578 M. Ostoja-Starzewski and J. Trebicki

t

0x0

f t = 0

x0 + L

L L d L(a) (b) (c)

wavefront

p

ppp

x

Figure 1. Propagation of a wavefront in the x, t space-time. The wavefront is a zone of finitethickness L (between x0 and x0+L at time t = 0) propagating in the direction p (co-oriented withx) in a microstructure of characteristic grain size d. Three cases are distinguished: (a) L d,which shows the trend to a classical (deterministic) continuum limit, in which fluctuations die outto zero as L/d → ∞; (b) L finite relative to d, where spatial fluctuations render the wavefronta statistical mesoscale volume element; and (c) L < d, which leads to a piecewise-constantevolution.

between the two aforementioned effects leads to a possibility of shock formationin a finite distance x∞ (so-called distance to form a shock), providing the initialamplitude α0 exceeds a so-called critical amplitude αc. In the case of a homogeneousmedium, these are

x∞ = − 1µ

ln(

1− µ

βα0

), αc =

µ

β. (1.2)

Proc. R. Soc. Lond. A (1999)

Growth and decay of acceleration waves in random media 2579

For initial amplitudes α0 < αc, the dissipation wins over the nonlinearity, and thewave decays to zero exponentially. While a number of various cases of deterministicspatial dependence of µ and β had been studied, principally in the 1960s and 1970sin the rational mechanics literature, the analysis has to be conducted ab initio whenthe material displays spatial random fluctuations. Let us begin by considering aspace-time view of a wavefront of finite thickness L propagating in the x-positivedirection (the wavefront in figure 1). The field variable is f(x, t), and so we seethe initial profile f |t=0. The material has a spatially homogeneous (in terms of itsstatistics) ergodic microstructure with a single characteristic grain size d. Here, forthe sake of illustration of basic concepts, we use a two-dimensional Voronoi mosaic(based on a Poisson point process), which is an excellent generic model of a numberof heterogeneous materials. With reference to figure 1, there are, in general, threepossible cases as follows.

(a) L d: this is the classical (deterministic) continuum limit, in which fluctua-tions die out to zero as L/d→∞.

(b) L is finite relative to the grain size: here fluctuations are significant and notnegligible.

(c) L > d: the wave’s evolution is random and piecewise-constant. Locally, it is gov-erned by classical continuum mechanics, but the random properties of grains itencounters give it a stochastic character. In fact, one can distinguish two sub-cases: grain boundaries are either normal to the direction of wave propagation;or at arbitrary angles and include corners.

We recognize the wavefront of case (a) to be analogous to a representative volumeelement (RVE) of deterministic continuum mechanics; the RVE, i.e. L, is so largerelative to the grain size that the wavefront does not ‘see’ the local material disorder.However, as L decreases (case (b)), the wavefront becomes a statistical mesoscalewindow affected by random continuum type fluctuations of the microstructure. Incase (c), the grains are uniform continua and the wavefront evolves as a random jumpprocess. This requires consideration of backscattering of a pulse at all the grain–grainboundaries similar to what was done in Ostoja-Starzewski (1991).

In this paper we focus on case (b): the µ and β coefficients are now certain randomprocesses in x, so that α becomes a stochastic process driven by a µ, βx vectorprocess according to a stochastic Bernoulli equation (1.1), and the aforementionedcompetition becomes stochastic. This means that αc and x∞ become random vari-ables. In other words, there is a finite range of values of the initial amplitude, ratherthan just a single deterministic value (1.2)2, which may result in either a shock or adecay. Furthermore, the distance to form a shock, given a specific initial amplitude,is diffused over a certain range. This situation is exemplified in figure 2a, b with thehelp of a numerical simulation of a now random differential equation (1.1) driven bya white noise (see § 3).

(b) Outline of the paper

Given the fact that α is a stochastic process driven by the µ, βx vector process,our major objective is to statistically characterize αc and x∞. This stochastic problem



Figure 2. Simulation of ten exemplary evolutions of an acceleration wave α (a) and its inverseζ = 1/α (b) originating from the critical amplitude of a reference homogeneous deterministicmedium αc(det) = 〈µ〉〈β〉 as functions of distance x in a random medium described by onewhite noise (stochastic model of § 3). Observe that either a growth to ∞ or a decay to 0 occur.Parameters: 〈µ〉 = 1, 〈β〉 = 1, S1 = 0.2 and S2 = 0.35.

of αc was first studied by Ostoja-Starzewski (1993) in the setting of a white noisemedium; the analysis was carried out in terms of the inverse amplitude ζ = 1/α,which allowed a transformation of the original problem into a linear one. Later on,Ostoja-Starzewski (1995) considered the non-white-noise character of either one or



both components of the µ, βx process. The present investigation, first signalled inOstoja-Starzewski & Trebicki (1999), is more complete in that:

(i) the wavefront is examined as a zone of finite thickness relative to a microstruc-ture in which it propagates;

(ii) it deals with a fully nonlinear problem;

(iii) it assesses both αc and x∞; and

(iv) it considers a wider range of random media.

In particular, we study the following issues in this paper.

(a) Set-up of the Bernoulli equation in a random continuum due to a finite size ofthe wavefront.

(b) Bernoulli equation with one white noise, including:

(1) formulae for the average critical amplitude 〈αc〉, which yield conditionsfor 〈αc〉 to be greater or smaller than the critical amplitude αc(det) of thedeterministic problem, which is defined by the mean values of µ and β(see below);

(2) an exact solution for the inverse amplitude;(3) an approximation of the probability distribution of the inverse amplitude ζ

by a Winterstein method, and a resulting approximation of the amplitudeα.

(c) Bernoulli equation with two independent white noises, and analyses similar tothose listed in point (a) above.

(d) Bernoulli equation with one Wiener process, the case that is equivalent to aperturbation by two correlated Gaussian noises and also includes a discussionof the probability distributions of α and ζ.

(e) Equations for the moments of ζ, and explicit solutions in the special cases.

(f) Numerical comparisons of the first four moments of αc and ζc for various mod-els.

(g) The probability of formation of a shock wave as a function of the medium’srandomness.

(h) Bernoulli equation perturbed by an Ornstein–Uhlenbeck process.

Before we proceed with the analysis, we would like to note that the Bernoulliequation (1.1) describes the behaviour of acceleration waves in a wide variety ofmedia, both solids and fluids (see numerous references in the aforementioned rationalmechanics literature). Our analysis is, therefore, set in a general context of materialsdescribed by nonlinear functional constitutive relations with the presence of random-ness without any special reference to a particular material; we shall return to thispoint in the final section of the paper.



The basic class of random media we shall investigate is wide-sense stationary,ergodic processes µ(x) and β(x), understood as perturbations µ′(x) and β′(x) im-posed upon the constant mean values 〈µ〉 and 〈β〉. As noted in point (g) above, theOrnstein–Uhlenbeck process will also allow a study of a non-stationary material ran-domness. Furthermore, we shall always consider µ′(x) and β′(x) to be smaller thanthe means 〈µ〉 and 〈β〉. Indeed, these means define a reference homogeneous medium,which serves as a deterministic counterpart to all the stochastic problems. Compar-isons between the ensemble average solutions for αc and x∞ in the latter problem,and the solutions of the deterministic reference homogeneous medium problem, aremade throughout the paper.

2. Wavefront dynamics in random microstructures

(a) Classical continuum mechanics viewpoint

Let us first recall that a jump f(x, t) in the classical case (a) is defined by

[[f ]] = f2 − f1, (2.1)

where f1 and f2 are, respectively, the quantities immediately ahead of and behindthe wavefront. The discontinuity surface (propagating in the direction of positive X,material coordinate) is, here, of zero thickness, i.e. L→ 0, and located at X = Y (t).

It is well known from continuum mechanics that the convected derivative of thejump [[f ]] travelling at velocity c is governed by

ddt

[[f ]] =[[∂f

∂t

]]+ c

[[∂f

∂x

]]. (2.2)

In the special case, when f is continuous, we obtain the (Hadamard) kinematiccompatibility condition of the first order[[

∂f

∂t

]]= −c

[[∂f

∂x

]], (2.3)

which must hold, for instance, for the displacement field up to the point of materialfracture.

Next, we consider the dynamic condition of compatibility

[[σij ]]pj = −ρc[[ui]], (2.4)

where we recognize the left-hand side to be the jump in tractions across the dis-continuity surface. In the case of figure 1a, given the limit L/d → ∞, the tractionsand velocities on either side of the jump are practically uniform. This homogeniza-tion limit is conventionally taken for granted in continuum mechanics analyses whensubstituting the constitutive law into (2.4). However, in the case of figure 1b, thetraction and velocity fields are non-uniform, and, thus, what is implied here is avolume averaging in a direction perpendicular to p.

Let us scrutinize a constitutive law that has to come into this whole picture. Inthe classical case (figure 1a), assuming, for simplicity, an isotropic Hooke’s law (λand µ are the Lame constants),

σij = Cijklεkl, Cijkl = λδijδkl + µ(δjkδil + δjlδik), (2.5)



we have

[[σij ]] = λδij [[uk,k]] + µ([[ui,j ]] + [[uj,i]]). (2.6)

This is clearly interpreted as a very large window limit relative to d, i.e. the RVE ofcontinuum mechanics. Ostoja-Starzewski (1998) gives quantitative prescriptions ofthe approach of mesoscale moduli to unique macroscopic values as functions of thewindow size relative to the grain size for a number of specific microstructures (e.g.circular inclusions in a matrix, needles in a matrix, fibrous media).

The case (b), L finite relative to d, is a situation below the RVE limit: it corre-sponds to the wavefront moving as a statistical volume element through a microstruc-ture. It requires, in place of (2.5), a random response law so that

Cijkl(x, L/d, ω); ω ∈ Ωis a random stiffness tensor field; Ω is the sample space of all possible realizations.Thus, we should have

[[σij ]] = Cijkl(x, L/d, ω)([[uk,l]] + [[ul,k]]). (2.7)

We now proceed in the same fashion for a nonlinear elastic/dissipative law ex-pressed as a nonlinear functional of the entire deformation history

σij(X, t) = ψ∞s=0[F (t− s), F (t)], (2.8)

where F = ∂x/∂X is the deformation gradient with explicit dependence on time. Asshown for example in Christensen (1982), one derives in the classical case (a), theBernoulli equation (1.1), in which the coefficients µ and β representing the dissipationand elastic nonlinearity, respectively, are

µ = − 14Et

[4It + 2G′t(0) + 4EtF 1 − dEt

dt

], β = − Et

2Etc. (2.9)

We now define several symbols appearing in the above. First,

Et =d

dFψ∞s=0[F (t− s), F (t)],

Et =d2

dF 2ψ∞s=0[F (t− s), F (t)],

(2.10)

are called the instantaneous tangent modulus and the instantaneous second-ordertangent modulus. Next,

It =d

dFδψ∞s=0[ ]|X=Y (t), (2.11)

where δψ∞s=0[ ] is a Frechet derivative given by (r is a real constant)

δψ∞s=0[F (t− s), F (t) | f(t− s)] =ddrψ∞s=0[F (t− s) + rf(t− s), F (t)] |r=0 . (2.12)

Finally, we have

G′t(0) =ddtEt |t=0, F 1 =

∂F 1

∂t, (2.13)

and the velocity of the acceleration wave

c =√Et/ρR, (2.14)



where ρR is the position-independent mass density in the reference configuration.A consideration of case (b) for the acceleration wavefronts with L finite relative to

d implies randomness in all the parameters present in the µ and β coefficients. This,in turn, leads to a stochastic differential equation (of form (1.1)) of a prohibitivelyhigh degree of complexity, whose solution would be unwieldy. However, just as isoften done in continuum mechanics, we can consider a simpler specialized case of(2.9): that of acceleration waves entering a homogeneously deformed medium ofsome deformation gradient F0. In that case, for X > Y (t),

x(X, t) = F0X +X0, (2.15)

where the second term refers to a possible rigid body displacement. Then, ahead ofthe wave, F 1 = 0 and

Et = Gt(0) = G0, Et = E0, G′t(0) = G′0. (2.16)

Next, from (2.14), the velocity of the acceleration wave is constant,

c =√G0/ρR, (2.17)

while it follows from (2.15) that F (t − s) = 0 for X > Y (t). Finally, the Frechetderivative (2.12) vanishes, which in turn, by (2.14), implies It = 0. As a result, theµ and β coefficients become

µ = − G′02G0

, β = − E0

2G0c. (2.18)

Solution of (1.1) in the deterministic case (a) is given as

α(t) =µ

[(µ/α(0))− β]eµt/β + β. (2.19)

Physically realistic conditions are

G0 > 0, G′0 6 0, E0 6= 0, (2.20)

and one considers three cases as follows.

(i) If sgnα(0) = sgn E0 or if sgnα(0) = − sgn E0 and |α(0)| < |µ/β|, then we haveα(t)→ 0 monotonically as t→∞.

(ii) If α(0) = µ/β, then we have α(t) = α(0), a constant; recall the critical ampli-tude of (1.2)2.

(iii) If sgnα(0) = − sgn E0 and |α(0)| > |µ/β|, then we have |α(t)| → ∞ monoton-ically in a finite time given by x∞ of (1.2)1 divided by c.

(b) Stationarity and ergodicity of random media models

Clearly, everything is deterministic in the above because G0, G′0 and E0 are con-stants dependent on a given physical situation. However, if they are to be consideredrandom fields in the sense of case (b), analogous to the field of stiffness in (2.7), theevolution does not follow equation (2.19) any more, and has to be investigated anew.We can now consider two basic types of randomness:



(I) G′0 and E0—and, hence, µ and β—are uncorrelated random processes in posi-tion X;

(II) G′0 and E0—and, hence, µ and β—are correlated random processes in posi-tion X.

Consequently, we arrive at a stochastic version of the Bernoulli equation driven by avector process µ, βx. The resulting competition between the dissipation and elasticnonlinearity becomes stochastic, and the two quantities of interest, distance x∞and critical amplitude α0, become random variables. Consistent with the discussionleading to (2.7), we see that both coefficients µ and β are stochastic processes incases (b) and (c), and, in the following, as stated in § 1, we shall focus on (b).

We assume µ and β to be wide-sense stationary random processes

µ(X) = 〈µ〉+ µ′(X), 〈µ′〉= 0,

β(X) = 〈β〉+ β′(X), 〈β′〉= 0,

(2.21)

where 〈µ〉 and 〈β〉 are constant, and µ′ and β′ are zero-mean wide-sense stationaryrandom processes. Without much loss of generality, we can take µ′ and β′ as zero-mean wide-band Gaussian processes with correlation functions of the type

K(∆X) = σ2Xf(a,∆X), (2.22)

where ∆X = |X−X ′|, σ2X is the variance of a given process, a is a parameter inverse

to the correlation radius ρ, and function f satisfies conditions

f(a, 0) = 1, lim∆X→±∞

f(a,∆X) = 0. (2.23)

In particular, we may consider an Ornstein–Uhlenbeck process for which

K(∆X) = σ2Xe−a|∆X|. (2.24)

Since we focus on the effect of heterogeneities very small relative to the wavefrontthickness L, we may further consider situations where the correlation radii of bothµ′ and β′ are much smaller than the correlation radius ρα of the amplitude processα(X). It is reasonable, on the basis of theoretical analyses of random microstruc-tures (Ostoja-Starzewski 1994) and of experimental measurements of mechanicalproperties of paper (DiMillo & Ostoja-Starzewski 1998), to take ρ of µ′ and ρ ofβ′ to decrease with L increasing, and to increase with d increasing, i.e. ρ ∼ d/L.Consequently, for L/d 1, we see that

ρ ρα, (2.25)

which allows us to treat α(X,ω) as a diffusion Markov process driven by two zero-mean white noise processes ξ1(X,ω) and ξ2(X,ω) that are equivalent to processesµ′(X,ω) and β′(X,ω). That is, their correlation functions are of the form

Kξi(∆X) = 2Dξiδ(∆X), i = 1, 2, (2.26)

in which the intensities Dξi are determined by

Dξ1 =∫ ∞

0Kµ′(∆X) d∆X, Dξ2 =

∫ ∞0

Kβ′(∆X) d∆X. (2.27)



In our analysis, we assume that the physical processes µ′ and β′ satisfy (withprobability one) ergodic properties

limy→∞

1y

∫ y

0ξi(X,ω) dX = mξi = 〈ξi(X,ω)〉, i = 1, 2,

limy→∞

1y

∫ y

0(ξi(X,ω)ξi(X + ∆X,ω)) dX = Kξi(∆X) +m2

ξi .

(2.28)

In practice, the left- and right-hand sides of (2.28)1 would be replaced by a spa-tial (or volume) average from a finite number of sampling points n taken over onerealization ω

ξi(ω) =1N

N∑n=1

ξi(Xn, ω), i = 1, 2, (2.29)

and an ensemble average from a finite number of realizations ω taken at one samplingpoint

〈ξi(X)〉 =1N

N∑n=1

ξi(X,ωn), i = 1, 2. (2.30)

The ergodicity of these estimators, i.e. ξi(X,ω) = 〈ξi(X,ω)〉, is assured for sufficientlylarge N , by the property of the correlation function

lim|∆X|→∞

Kξi(∆X) = 0, i = 1, 2, (2.31)

which is satisfied by our Ornstein–Uhlenbeck as well as the white noise processmodels. Indeed, many material microstructures may even possess a stronger property,that of mixing. This, for instance, is the case with Voronoi mosaics based on Poissonpoint processes, both in two and three dimensions (Stoyan & Stoyan 1994); ourfigure 1 employs such a mosaic.

Finally, we note that the statistics (mean and correlation function) of mesoscaleproperties corresponding to any given L/d ratio, can be readily sampled experimen-tally. This would involve measuring two coefficients specified in (2.18) on appropriatesized aggregates of grains, i.e. corresponding to the mesoscale window of figure 1b.While any specific choice of the ratio L/d may suggest so-called local averaging ofstochastic processes ξi(X,ωn) (i = 1, 2), it is just a superficial analogy: the effectiveµ and β on a given scale L are not a result of spatial averaging over L. They couldformally be defined via a stochastic homogenization procedure, but, given the nonlin-ear/dissipative nature of all the grains, a quantitative solution would be prohibitivelycomplex.

3. Bernoulli equation with one white noise

(a) Basic stochastic formulation

The evolution of an acceleration wave is governed by a Bernoulli equation

dαdx

= −µ(x)α+ β(x)α2, ∀x (µ, β > 0). (3.1)



Here we use the position x as an independent variable, but the time may also play thisrole; interpretation in terms of x (henceforth representing the Lagrangian coordinateX) is preferable, because µ(x) and β(x) are material properties. We shall assume αtake non-negative values, and prescribe a deterministic initial condition

α(x0) = α0. (3.2)

It is, however, straightforward to generalize the ensuing analysis by admitting ran-domness in the initial condition.

In equation (3.1), µ(x) describes the attenuation of the wave, and β(x) its amplifi-cation due to a material’s nonlinear elasticity. Next, we consider both µ(x) and β(x)to be perturbed by the same standard zero-mean white noise ξ(x, ω), that is

µ(x, ω) = 〈µ〉+ S1ξ(x, ω), β(x, ω) = 〈β〉+ S2ξ(x, ω),S1, S2 > 0, S1 + S2 = S,

(3.3)

where 〈µ〉 and 〈β〉 are mean values of µ(x) and β(x), respectively, and 〈µ〉 S1,〈β〉 S2. Also, ω ∈ Ω, where (Ω,F, P ) is the basic probability space. For simplicity,in further exposition, the parameter ω will be omitted. Equation (3.3) means that thewhite noise ξ(x) appearing at the wavefront affects its dissipation with the intensityS1, and its nonlinear amplification with the intensity S2.

Introducing (3.3) into (3.1), we obtain a stochastic differential equation for α:

dαdx

= −〈µ〉α+ 〈β〉α2 + [S2α2 − S1α]ξ(x). (3.4)

We next set up a Stratonovich equation for equation (3.4):

dα = [−〈µ〉α+ 〈β〉α2] dx+ [S2α2 − S1α] dW (S)(x), α(x0) = α0. (3.5)

In (3.5), dW (S)(x) is a Stratonovich-type differential of the Wiener process. Wechoose the Stratonovich interpretation because the stochastic perturbations in equa-tion (3.4) are of a parametric type (see, for example, Schuss 1980).

For the sake of computing various functions of α, we will need the Ito formularelated to the Ito equation equivalent to the Stratonovich equation (3.5). To thisend, we define

A(α) = −〈µ〉α+ 〈β〉α2, B(α) = S2α2 − S1α, (3.6)

and proceed according to the general formula

dα =[A(α) + 1

2B(α)d

dαB(α)

]dx+B(α) dW (x), α(x0) = α0, (3.7)

to find the Ito equation, equivalent to (3.5), that is

dα = A(α) dx+ B(α) dW (x), α(x0) = α0, (3.8)

where

A(α) = (12S

21 − 〈µ〉)α+ (〈β〉 − 3

2S1S2)α2 + S22α

3 = A(α) + 12B(α)

∂

∂αB(α),

B(α) = B(α) = S2α2 − S1α.

(3.9)



Let us now consider a process y = g(α), which is a certain memoryless transfor-mation of the process α(x). The Ito equation for this new process is obtained fromthe following Ito formula:

dy =A(α)

dgdα

+ 12 B

2(α)d2g

dα2

dx+ B(α)

dgdα

dW (x). (3.10)

The first application of the above will concern the moments of α(x). To this end,we introduce a function

y = αk, k = 1, 2, . . . , (3.11)

and applying Ito’s formula (3.10), we find

dαk(x) = [kA(α)αk−1 + 12k(k − 1)B2(α)αk−2] dx+ kB(α)αk−1 dW (x). (3.12)

Upon averaging (3.12), we arrive at the following system of equations for the momentsof the process α(x)

ddx〈αk〉 = k〈A(α)αk−1〉+ 1

2k(k − 1)〈B2(α)αk−2〉,〈αk(x0)〉 = 〈αk0〉, k = 1, 2, . . . ,

(3.13)

whereby we note that the last term in (3.12) vanishes because 〈dW (x)〉 = 0 andα(x) is a non-anticipating random process. It is noteworthy that (3.13) is not aclosed system of equations.

The inverse amplitude, defined by

ζ(x) = 1/[α(x)], (3.14)

is one of the primary functions of interest in the ensuing analysis; it transforms theproblem of the blow-up of the wave amplitude α to infinity to the one of zero-crossing.We find from Ito’s formula (3.10) the following equation for ζ(x):

dζ(x) =[A(α)

(− 1α2

)+ B2 1

α3

]dx+ B(α)

(− 1α2

)dW (x). (3.15)

Noting (3.13), we get

dζ = [(S21 − 1

2S21 + 〈µ〉)ζ − (〈β〉 − 3

2S1S2 + 2S1S2)] dx+ (S1ζ − S2) dW (x), (3.16)

so that, finally,

dζ = [b1ζ + b2] dx+ (S1ζ − S2) dW (x),

b1 = 〈µ〉+ 12S

21 , b2 = −〈β〉 − 1

2S1S2.

(3.17)

Note that (3.17) is a linear inhomogeneous equation.Next of interest are moments of the inverse amplitude. These can also be obtained

from Ito’s formula with a substitution y = ζk. Thus,

ddx〈ζk〉 = [kb1 + 1

2k(k − 1)S21 ]〈ζk〉+ [kb2 + k(k − 1)S1S2]〈ζk−1〉

+ 12k(k − 1)S2

2〈ζk−2〉,〈ζk(x0)〉 = 〈ζk0 〉, k = 1, 2, . . .

(3.18)



For the first two moments we findd

dx〈ζ〉 = b1〈ζ〉+ b2,

ddx〈ζ2〉 = [2b1 + S2

1 ]〈ζ2〉+ [2b2 − S1S2]〈ζ〉+ S22 ,

(3.19)

with b1 and b2 being given by (3.17). Solving the above linear equations we get

〈ζ(x)〉 = −b2b1

+(ζ0 +

b2b1

)eb1(x−x0),

〈ζ2(x)〉 = −c1 − c2eb1(x−x0) + [ζ20 + c1 + c2]e(2b1+S2

1)(x−x0),

(3.20)

where

c1 =S2

2 − 2(b2/b1)(b2 − S1S2)2b1 + S2

1, c2 =

2(b2 − S1S2)(ζ0 + (b2/b1))b1 + S2

1. (3.21)

Taking x0 = 0 we get from (3.20)

〈ζ(x)〉 =〈β〉+ 1

2S1S2

〈µ〉+ 12S

21

+(ζ0 −

〈β〉+ 12S1S2

〈µ〉+ 12S

21

)e(〈µ〉+ 1

2S21)x. (3.22)

Since 〈µ〉+ 12S

21 > 0 and 〈β〉+ 1

2S1S2 > 0, then, assuming

ζ0 > −b2b1

=〈β〉+ 1

2S1S2

〈µ〉+ 12S

21, (3.23)

we see that 〈ζ〉 → ∞ as x → ∞, and we have no shock formation with respect tothe mean value.

On the other hand, if

ζ0 < −b2b1

=〈β〉+ 1

2S1S2

〈µ〉+ 12S

21, (3.24)

then 〈ζ〉 → −∞ as x → ∞, which implies crossing the zero-level and, thereby, anexplosion of the process α = ζ−1. It follows that the average critical inverse amplitude〈ζc〉 is given by

〈ζc〉 =〈β〉+ 1

2S1S2

〈µ〉+ 12S

21. (3.25)

Considering that the reference deterministic homogeneous medium is specified by 〈µ〉and 〈β〉, and that the corresponding critical amplitude is ζc(det) = 〈β〉/〈µ〉 (recall(3.3)), we establish that the assumption of a white noise perturbation of µ and βleads to the following relationships between 〈ζc〉 and ζc(det):

〈ζc〉 > ζc(det), forS2

S1>〈β〉〈µ〉 ,

〈ζc〉 < ζc(det), forS2

S1<〈β〉〈µ〉 .

(3.26)

Now, since 〈α〉 > 〈ζ〉−1, we find, in view of (3.25), the average critical amplitude〈αc〉 given in terms of the average inverse critical amplitude, that is

〈αc〉 =1〈ζc〉 =

〈µ〉+ 12S

21

〈β〉+ 12S1S2

. (3.27)



Figure 3. (a) Five exemplary evolutions of the inverse amplitude ζ = 1/α of an accelerationwave originating from the same initial value ζ0 = 0.8〈ζc〉, with 〈ζc〉 = 1.0, and demonstrating ablow-up to a shock wave. (b) The same evolutions in terms of α. Parameters: 〈µ〉 = 1, 〈β〉 = 1,S1 = 0.35 and S2 = 0.35.

It can now be shown from (3.26) that we can expect 〈αc〉 to be smaller (greater)than αc(det) = 〈µ〉/〈β〉 accordingly as the ratio S2/S1 is greater (smaller) than theratio 〈β〉/〈µ〉.

(b) Numerical simulations of trajectories

Evolutions of the deterministic Bernoulli equation are smooth functions tendingeither to zero or to infinity. In the case of randomness in the µ and β coefficients,



Figure 4. The mean ζ and standard deviation σζ of the inverse amplitude ζ as functions ofposition corresponding to the case of figure 3; all parameters are the same as in figure 3.

there is, as already pointed out, a stochastic competition between the dissipationeffect due to the −µ(x)α and the β(x)α2 terms. This competition means that

(i) the trajectories of the stochastic system are not smooth, but, rather ‘jagged’;

(ii) there is a range of different trajectories, decaying and exploding, that emanatefrom the same initial value; and

(iii) there might be a crossover from the behaviour where µ dominates to the onewhere β dominates.

In order to numerically simulate the system evolution, we adopt Milshtein’s method(Milshtein 1978; Sobczyk 1991).

In view of (3.8), this method leads to the following finite difference scheme for theαx process

αi+1 = αi + [A(αi)− 12 B

2(αi)]h+ B(αi)∆W (xi) + 12 B

2(αi)[∆W (xi)], (3.28)

while, considering (3.17), we obtain for the ζx process

ζi+1 = ζi + [(b1αi + b2)− 12(S1ζi − S2)2]h

+ (S1ζi − S2)2∆W (xi) + 12(S1ζi − S2)2[∆W (xi)], (3.29)

where h is a step on the x-axis.Using these schemes, in figure 3 we display five trajectories, generated in a Monte

Carlo sense, in terms of α and ζ, respectively. They correspond to the case 〈µ〉 =〈β〉 = 1.0, and S1 = S2 = 0.35, from which, given (3.26), we calculate the criticalamplitude 〈αc〉 = 1/〈ζc〉 = 1.0. In particular, in figure 3a, b we treat the case ofζ0 = 0.8 and α0 = 1/ζ0 = 1.25, respectively, which is the case of blow-up of thewave. These plots are complemented by the graphs of the average and the standard



Figure 5. (a) Five exemplary evolutions of an acceleration wave ζ = 1/α originating from thesame initial value ζ0 = 1.25 > 〈ζc〉, and demonstrating a decay; all other parameters are thesame as in figure 3. (b) The same evolutions in terms of α.

deviation of ζ in figure 4, computed from (3.20) and corresponding to this initialvalue case. Figure 4 is exemplary of a scenario of explosion of an acceleration waveto a shock, in which the mean 〈ζ〉 goes down to zero with x increasing, while theassociated standard deviation grows.

Next, in figure 5a, b, we present, for the same random medium, the case of decayof an acceleration wave by choosing ζ0 = 1.25 and α0 = 1/ζ0 = 0.8, respectively.



Figure 6. The mean 〈ζ〉 and standard deviation σζ of the inverse amplitude ζ as functions ofposition corresponding to the case of figure 4; all parameters are the same as in figure 2.

Figure 6 shows that the growth of 〈µ〉 is accompanied by an equally rapid growthof its standard deviation. This means, as may be confirmed by figure 5b, that thestandard deviation of the α process actually vanishes as x grows.

Finally, we mention an obvious feature of these simulations: as S1 and S2 go downto zero, the jaggedness in the trajectories vanishes and they become deterministicsolutions of (1.1).

(c) Exact solution for the inverse amplitude

Let us return to the linear equation

dζx = (Ax +Bxζx) dx+ (Cx +Dxζx) dWx. (3.30)

For completeness, after (Kallianpur 1980), its exact solution has the following form:

ζx = exp⌊∫ x

0(Bs − 1

2D2s) ds+

∫ x

0Dx dWs

⌋×ζ0 +

∫ x

0exp[−∫ s

0(Bu − 1

2D2u) du−

∫ s

0Du dWu

](As −DsCs) ds

+∫ x

0exp[−∫ s

0(Bu − 1

2D2u) du−

∫ s

0Du dWu

]Cs dWs

, (3.31)

wherein, considering (3.17), we identify

Ax = b2, Bx = b1,

Cx = −S2, Dx = S1.

(3.32)



(d) Approximation of the probability distribution of the inverse amplitude ζ

Let us consider the probability distribution of ζ. It could, in principle, be obtainedfrom (3.31). Another possibility, however, is to approximate it using the firstmoments; the point being that the system of equations for moments (3.18) is closed.For any fixed x, the process ζ(x) becomes a random variable ζ(ω). In order to approx-imate ζ(ω), the following Hermite polynomial transformation will be considered (cf.Ditlevsen & Madsen 1996)

ζ(ω) =N∑i=0

biHi(X(ω)), (3.33)

where the bi are constant coefficients, and X(ω) is a standardized normal randomvariable, i.e. its density is of the form

fX(x) = (1/2√

2π)e−x2/2. (3.34)

Also, Hi(x) = 1, H1(x) = x, H2(x) = x− 1, H3(x) = x3− 3x, H4(x) = x4− 6x2 + 3,etc., are the Hermite polynomials. Obviously, the formula (3.31) can be rewritten inthe form

ζ(ω) =N∑i=0

aiXi(ω). (3.35)

Now, in order to calculate the values of the N+1 parameters ai, the information onthe moments 〈ζk〉, k = 1, . . . , N + 1 is used. Taking into account the transformation(3.33), these parameters are determined from the system of nonlinear (with respectto ai) equations

〈ζk〉 =⟨( N∑

i=0

aiXi(ω)

)k ⟩, k = 1, 2, . . . , N + 1. (3.36)

The above equation can be rewritten in a more explicit form since we have

〈Xk〉 =

0, for n odd,1 · 3 · · · · · (n− 1), for n even.

(3.37)

Next, the system (3.36) is solved numerically, and, finally, the probability densityfunction of ζ(ω) is determined as

fζ(z) =∑j

|g−1j (z)|fX(g−1

j (z)), (3.38)

where g−1j is the jth branch of the inverse g−1 restricted to an appropriate interval

of monotonicity. We note here that in the special case of N = 4, the transformation(3.35) is called a Winterstein approximation (Ditlevsen & Madsen 1996).

(e) Approximation of the probability distribution of the amplitude α

With the approximate form of the distribution f(y) of the inverse amplitude ζ, wemay derive an approximate form of the distribution of the amplitude α itself. Thisis done by noting that

fαx(y) =∣∣∣∣− 1y2

∣∣∣∣fζx(1y

)=

1y2 fζx

(1y

), y > 0. (3.39)



Furthermore, it should be pointed out here that this formula can be used far enoughfrom the explosion time. The latter is a random variable x∞(ω) defined as follows:

x∞(ω) = infx; ζ(x, ω) > 0. (3.40)

Taking four moments calculated from

〈αkx〉 =∫ ∞

0ykfαx(y) dy =

∫ ∞0

(1y

)kfζx

(1y

)dy, (3.41)

we can use the Winterstein approximation for the probability density of α. In theparticular case when the distribution of ζ can be approximated by a Gaussian, thedistribution of α is simply obtained as a nonlinear transformation of the Gaussiandistribution. Finally, note that in the case of S1 = 0, S2 > 0, we find that the inverseamplitude is governed by the Ornstein–Uhlenbeck equation, implying that fζx isGaussian.

4. Bernoulli equation with two independent white noises

(a) Basic considerations

We now perturb the original Bernoulli equationdαdx

= −µ(x)α+ β(x)α2, (4.1)

by two zero-mean white noises ξ1 and ξ2

µ(x) = 〈µ〉+ ξ1(x), β(x) = 〈β〉+ ξ2(x), (4.2)

having a correlation matrix

Kξiξj (x1, x2) = gijδ(x2 − x1), gij =[g11 00 g22

]. (4.3)

From this we getdαdx

= −(〈µ〉+ ξ1)α+ (〈β〉+ ξ2)α2, (4.4)

which is interpreted in the Stratonovich sense as

dα = (−〈µ〉α+ 〈β〉α2) dx−√g11α dW (S)1 (x) +

√g22α

2 dW (S)2 (x), (4.5)

where dW (S)1 and dW (S)

2 are differentials of two independent Wiener processes in aStratonovich sense.

A more convenient way of writing the above is in a matrix form

dα = F (α) dx+ σ(α) dW (S)(x), (4.6)

whereF (α) = −〈µ〉α+ 〈β〉α2,

σ(α) = [σ11, σ12] = [−√g11α,√g22α

2],

dW (S) = [dW (S)1 , dW (S)

2 ]T.

(4.7)

An Ito equation, equivalent to (3.6), is

dα = F (α) dx+ σ(α) dW (S)(x), (4.8)



where

F (α) = F (α) + 12

dσdα

σT,

σ(α) = σ(α),

dW = [ dW1, dW2]T.

(4.9)

In the above, dW1 and dW2 are the Ito-sense differentials of two independent stan-dard Wiener processes. Taking (4.7) into account, we rewrite equation (4.6) as

dα = (−〈µ〉α+ 〈β〉α2) dx+ 12 [−√g11, 2α

√g22]

[−α√g11α2√g22

]dx

−√g11α dW1 +√g22α

2 dW2. (4.10)

Finally, the Ito equation for (4.1) takes the form

dα = [(12g11 − 〈µ〉)α+ 〈β〉α2 + g22α

3] dx−√g11α dW1 +√g22α

2 dW2. (4.11)

We next set up an Ito equation for the inverse amplitude. Thus, from the Itoformula in the two-dimensional case (see (7.12))

dh(α) = F (α)dhdα

+ 12

d2h

dα2 [σ211(α) + σ2

12(α)] + σ[dW ]Tdhdα

, (4.12)

we find

dζ = [(12g11 − 〈µ〉)α+ 〈β〉α2 + g22α

3](−1/α2) dx

+ (1/α3)[g11α3 + g22α

4] dx+ (−1/α2)[−√g11α dW1 +√g22α

2 dW2]= [(〈µ〉+ g11)ζ − 〈β〉] dx+

√g11ζ dW1 −√g22 dW2. (4.13)

This linear equation with a parametric noise dW1 and an additive noise dW2 can bewritten in a vector form as

dζ = [(〈µ〉+ g11)ζ − 〈β〉] dx+ [√g11ζ,

√g22]

[dW1dW2

]. (4.14)

(b) Moment equations for the inverse amplitude

We apply the Ito formula (4.13) to the function h(ζ) = ζk and then average toobtain

ddx〈ζk〉 = k〈(〈µ〉+ g11)ζk − 〈β〉ζk−1〉+ 1

2k(k − 1)〈g11ζk−1 + g22ζ

k−2〉. (4.15)

From this we find equations governing the first two momentsd

dx〈ζ〉 = (〈µ〉+ g11)〈ζ〉 − 〈β〉, 〈ζ0〉 = 1/α0,

ddx〈ζ2〉 = 2(〈µ〉+ g11)〈ζ2〉+ (g11 − 〈β〉)〈ζ〉+ g22, 〈ζ2

0 〉 = 1/α20,

(4.16)

where the initial conditions are being interpreted in the same sense as in § 2. It followsthat the solutions of (4.16) are

〈ζ〉 =〈β〉

〈µ〉+ g11+(〈ζ0〉 − 〈β〉

〈µ〉+ g11

)exp[(〈µ〉+ g11)x],

〈ζ2〉 = −c1 − c2e〈µ〉x + [〈ζ20 〉+ (c1 + c2)] exp[2〈µ〉+ g22]x,

(4.17)



in which

c1 =g22 − (g11/2〈µ〉)(g11 − 〈β〉)

2〈µ〉+ 12g11

, c2 =(g11 − 〈β〉)(〈ζ0〉+ (g11/2〈µ〉))

〈µ〉+ 12g11

. (4.18)

It is interesting that the equation governing the evolution of 〈ζ〉 is similar to theone of the previous section. However, in the present case, the Wiener process dW2enters additively into (4.14), which leads to 〈ζ〉 in (4.17) being independent of theintensity g22 of the β(x) process. From (4.17)1, we now find the decay and growthconditions for the acceleration wave

〈ζ〉 → ∞, for 〈ζ0〉 > 〈β〉〈µ〉+ g11

,

〈ζ〉 → 0, for 〈ζ0〉 < 〈β〉〈µ〉+ g11

.

(4.19)

The same result was obtained by a different method in Ostoja-Starzewski (1993).The analysis regarding the probability distributions of ζ and α is entirely analogous

to that conducted in § 3, and is, therefore, not repeated.

5. Bernoulli equation with one Wiener process equivalent to aperturbation by two correlated Gaussian noises


Again, let us begin with the stochastic Bernoulli equation

dαdx

= −(〈µ〉+ ξ1)α+ (〈β〉+ ξ2)α2, (5.1)

where ξ1 and ξ2 represent two zero-mean Gaussian noises having a correlation matrix

gij =[g11 g12g21 g22

], g12 = g21 = g, g11g22 > g2. (5.2)

Equation (5.1) is rewritten as

dαdx

= −〈µ〉α+ 〈β〉α2 + [−α, α2][ξ1ξ2

], (5.3)

which leads to

dα = F (α) + σ(α) dW,

σ(α) = [σ11, σ12] = [−α, α2],

F (α) = −〈µ〉α+ 〈β〉α2,

dW =[dW1dW2

], 〈dWi,dWj〉 = gij = Gδij .

(5.4)

Now, by interpreting equation (5.4) in the Stratonovich sense, we arrive at the equiv-alent Ito formula

dα = F (α) dx+ σ(α) dW, F (α) = F (α) + 12

dσdα

GσT. (5.5)



The solution of (5.5) is given by a Markov process with drift and diffusion coeffi-cients

A(α) = F (α), B(α) = σGσT. (5.6)

Each diffusion Markov process having such A(α) and B(α) is a solution of a stochasticdifferential equation; that is, there exists a certain Wiener process W (x) such thatα(x) is a solution of

dα = A(α) +B(α) dW. (5.7)

Thus, we can replace equation (5.5) with two correlated Wiener processes by theequation (5.7) with one Wiener process. Considering the expressions for α and G,we obtain

A(α) = −〈µ〉α+ 〈β〉α2 + 12 [−1, 2α]

[g11 gg g22

] [−αα2

]= (1

2g11 − 〈µ〉)α+ (〈β〉 − 32g)α2 + g22α

3, (5.8)

and

B(α) = [−α, α2][g11 gg g22

] [−αα2

]= g11α

2 − 12gα

3 + g22α4. (5.9)

Now, applying the Ito formula for a function ζ = h(α) to equation (4.7), we obtain

dζ = dh(α) = A(α)dhdα

+ 12 [√B(α)]2

d2h

dα2 +√B(α)

dhdα

dW, (5.10)

which leads to the following equation for the inverse amplitude ζ = 1/α:

dζ = dh(α) =−1α2 A(α) +

1α3B(α)− 1

α2

√B(α) dW,

A(α)(−1/α2) = (〈µ〉 − 12g11)α+ (3

2g − 〈β〉)α2,

−B(α)(1/α3) = g11ζ − 2g.

(5.11)

Thus, we finally arrive at

dζ = (〈µ〉+ 12g11)ζ − (〈β〉+ 1

2g)−√g11ζ2 − 2gζ + g22 dW. (5.12)


We are now ready to study the moments of the inverse amplitude. To this end, weuse the Ito formula (5.10) with respect to equation (5.12) to obtain

ddx〈ζk〉 = 〈kζk−1[ζ(〈µ〉+ 1

2g11)− (12g + 〈β〉)]〉+ 1

2〈k(k − 1)ζk−2[g11ζ2 − 2gζ + g22]〉

= [k(〈µ〉+ 12g11) + 1

2k(k − 1)g11]〈ζk〉+ [k(1

2g + 〈β〉) + k(k − 1)g]〈ζk−1〉+ 12k(k − 1)g22〈ζk−2〉. (5.13)

In the case k = 1 we findd

dx〈ζ〉 = (〈µ〉+ 1

2g11)〈ζ〉 − (12g + 〈β〉), (5.14)



which leads to

〈ζ〉 =〈β〉+ 1

2g

〈µ〉+ 12g11

+(ζ0 −

〈β〉+ 12g

〈µ〉+ 12g11

)e(〈µ〉+(g11/2))x. (5.15)

From this, we note immediately that the average critical inverse amplitude 〈ζc〉 isnow given by

〈ζc〉 =12S1S2 + 〈β〉

12S

21 + 〈µ〉 . (5.16)

Let us note here that for g = 0, the analysis reduces to the one conducted earlier in§ 4. On the other hand, for the case g → S1S2, g11 → S2

1 and g22 → S22 , the analysis

reduces to that of § 1.It will now be convenient to denote a kth order moment and its initial condition

as follows:

〈ζk〉 ≡ mk(x), 〈ζk0 〉 ≡ mk(x0). (5.17)

From the linear equation (5.13) we determine

mk(x) = ζk0 exp[∫ x

x0

r1(t) dt]

+∫ x

x0

(r2(t) exp

[−∫ t

x

r1(τ) dτ])

dt, (5.18)

wherer1(t) = k[b1 + 1

2(k − 1)g11],

r2(t) = 12k(k − 1)g22mk−2(t)− k[b2 + (k − 1)g]mk−1(t).

(5.19)

Taking x0 = 0, we find

mk(x) = er1xζk0 +

∫ x

0r2(t)e−r1t dt

. (5.20)

While we have already found m1(x) in (5.15), the second moment is given explicitlyas

m2(x) = exp[(2b1 + g11)x]2b21b2(ζ20 + 1) + b2g11(g22 − 2gζ0 + 3ζ2

0b1 − 2bζ0 + ζ20g11)

+ b1b2(g22 − 4b2ζ0 − 4gζ0) + 2gb21+ exp(b1y)2b1b2(2b2ζ0 − g11 − 2b1 + 2gζ0)− b1g(2b1 + g11)

+ 2b2g11ζ0(g + b2) + 2b1g11(g + g11) + b1b2(2b1 − g22) + b2g11g22 + 2b21g,(5.21)

where

b1 = 〈β〉+ 12g, b2 = 〈µ〉+ 1

2g11. (5.22)

6. Specification of random events at the explosion ofan acceleration wave

Consideration of the numerical simulations of trajectories in figures 2, 3, 5 and 6leads us to analyse several issues as follows.

(i) Explosion of an acceleration wave starting from a deterministic initial conditionis a random, rather than a deterministic, phenomenon, and thus, one can onlytalk of a probability to form a shock within a given finite distance.



Figure 7. The probability of formation of a shock wave within a distance x, P (ζ(x) < 0), forvarious types of correlation when ζ0 = 0.7: (1) full positive correlation; (2) zero correlation;(3) full negative correlation; other parameters are 〈µ〉 = 〈β〉 = 1, g11 = g22 = 0.01.

(ii) The mean distance to such an event (i.e. crossing the ζ = 0 line) is a certainfunction of random noises in the material.

(iii) The inverse amplitude is a certain random variable at the mean distance toform a shock.

(a) Probability of formation of a shock wave

It is clear from the foregoing analysis that when ζ0 = 1/α0 is sufficiently farfrom ζcr = b2/b1, then the probability distribution of ζ may be approximated by aGaussian distribution N(mζ(x), σζ(x)). The formula

P (ζ(x) < 0) = 1− P (ζ(x) > 0) = 1−∫ ∞

0fz(z;x) dz, (6.1)

gives the probability of formation of a shock wave within a distance x. For smallfluctuations of the white noises in µ and β and an appropriate choice of ζ0 = 1/α0,we can, therefore, work with

fζ(z;x) =1√

2πσζ(x)exp[−(z −mζ(x))2

2σ2ζ (x)

]. (6.2)

In the following, we plot (6.1), assuming (6.2), for various cases of the initial inverseamplitude ζ0, means 〈µ〉 and 〈β〉, noise strengths g11, g22, and cross-correlationg1 between both noises. Thus, in figure 7 we display the probability of formationof a shock wave within a distance x, P (ζ(x) < 0), for three types of correlationwhen ζ0 = 0.7: (1) full positive correlation; (2) zero correlation; (3) full negativecorrelation; other parameters are 〈µ〉 = 〈β〉 = 1, g11 = g22 = 0.01. It is seen that, asthe correlation between µ and β decreases, the range of distances over which a shock



Figure 8. The probability of formation of a shock wave within a distance x, P (ζ(x) < 0), forvarious types of correlation when ζ0 = 0.6; the meaning of the three curves and particularparameter values is the same as in figure 6.

Figure 9. A comparison of the probabilities P (ζ(x) < 0) at full positive correlation, for threecases: (1) ζ0 = 0.7; (2) ζ0 = 0.6; (3) ζ0 = 0.5; all other parameters are the same as in figure 6.

may form increases. While we used here ζ0 = 0.7, figure 8 is devoted to a situationwhen ζ0 = 0.6, all other parameters being the same. Clearly, the same trends areobserved here with all the curves being shifted to lower values of x.



Figure 10. A comparison of the probabilities P (ζ(x) < 0) at zero correlation, for three cases:(1) ζ0 = 0.7; (2) ζ0 = 0.6; (3) ζ0 = 0.5; all other parameters are the same as in figure 6.

Figure 11. A comparison of the probabilities P (ζ(x) < 0) at full negative correlation, for threecases: (1) ζ0 = 0.7; (2) ζ0 = 0.6; (3) ζ0 = 0.5; all other parameters are the same as in figure 6.

To put the shifting of P (ζ(x) < 0) in perspective, a comparison of the probabilitiesP (ζ(x) < 0) at full positive correlation is given in figure 9 for three cases of theinitial amplitude. This comparison is continued in figure 10 at zero correlation, andin figure 11 at full negative correlation.



Figure 12. A comparison of the average distance to form a shock, x∞ ≡ xc, and the standarddeviation of the inverse amplitude at the point of explosion, σζ(x∞), as a function of the initialamplitude ζ0 ≡ z0, for three cases: (1) full positive correlation; (2) zero correlation; and (3) fullnegative correlation; all other parameters are the same as in figure 6.

(b) Distance to form a shock wave

Using again (6.1), in figure 12 we give a comparison of the average distance toform a shock, x∞, as a function of the initial amplitude ζ0, for three cases of theµ, βx process: (1) full positive correlation; (2) zero correlation; and (3) full negativecorrelation; all other parameters are the same as in figure 7. We do not carry the plotsbeyond ζ0 = 0.9, as the distance x∞ then tends to go to infinity. In this figure wealso display the standard deviation of the inverse amplitude at the point of explosion,σζ(x∞).

Our next three figures continue such a comparison for other sets of parameters atfull positive, zero and negative correlation. In particular, we have here three cases: (1)g11 = g22 = 0.01; (2) g11 = 0.04 and g22 = 0.01; and (3) g11 = 0.01 and g22 = 0.04.Figures 13, 14 and 15 give the full positive, zero and negative correlation cases,respectively. Note that, with correlation between µ and β decreasing, the distancex∞ tends to go up. It is also interesting to note from these figures that the standarddeviation of the inverse amplitude σζ(x∞) tends to grow with ζ0 as well, and isstrongest when both noises in the material are fully negatively correlated.

(c) Character of the inverse amplitude at the point of shock formation

The final issue we address is that of the character of the probability distributionof the inverse amplitude at the point of shock formation x∞. We do it for a rangeof ζ0 values that make the explosion to shock possible. First, in figure 16, we showthe skewness γ1 and flatness γ2 of the inverse amplitude at the point of blow-up,



Figure 13. A comparison of the average distance to form a shock, x∞ ≡ xc, and the standarddeviation of the inverse amplitude at the point of explosion, σζ(x∞), as a function of the initialamplitude ζ0 ≡ z0, at full positive correlation, for various levels of noise. Notation: (1) forg11 = g22 = 0.01; (2) for g11 = 0.04 and g22 = 0.01; (3) g11 = 0.01 and g22 = 0.04.

x∞, as functions of ζ0. This is all meant with respect to the mean trajectory, thatis when m1(x∞) = 0. We use here 〈µ〉 = 1, 〈β〉 = 1 and g11 = g22 = 0.01, and,as before, study three cases: (1) full positive correlation; (2) zero correlation; and(3) full negative correlation between µ and β. It is interesting that the skewnessand flatness are practically non-existent in case (1), and they grow with decreasingcross-correlation. Also in this figure, in the inset we show the density function of theinverse amplitude at point x∞ for the initial amplitude ζ0 = 0.9.

Because the inverse amplitude process is fully positive, the boundary ζ = 0should, from a mathematical viewpoint, be considered as an absorbing boundaryand an appropriate condition for the inverse amplitude’s probability density (i.e.f(ζ, x)|ζ=0 = 0) should be taken into account. Nevertheless, in practice, and espe-cially when the diffusion effects are weak relative to the drift, it is more convenientto obtain the probability density of the inverse amplitude process in the whole spaceand, further, to consider only a part of the probability density for ζ > 0.

In the next three figures we continue this study. First, in figure 17, which is focusedon the full positive correlation, in addition to the above discussed parameter case wealso plot what happens as either g11 or g22 gets larger: cases (2) and (3), respectively;case (1) is a reference case from above. In case (2) the skewness and flatness increase,whereas in case (3) only the flatness increases and the skewness decreases. The changeof g11 or g22 also has the effect of broadening the density function of ζ at x∞.

In figure 18 we analyse the situation of zero correlation between µ and β as eitherg11 or g22 gets larger: cases (2) and (3), respectively; and case (1) is a reference casefrom figure 16. Here, the magnitude of changes in the skewness and flatness is much



Figure 14. A comparison of the average distance to form a shock, x∞ ≡ xc, and the standarddeviation of the inverse amplitude at the point of explosion, σζ(x∞), as a function of the initialamplitude ζ0 ≡ z0, at zero correlation, for various levels of noise. The meaning of (1), (2) and(3) is the same as in figure 12.

greater than before. However, as we go to the case of fully negative correlation infigure 19, these changes are more powerful by one more order of magnitude. Thethree cases investigated in figure 19 are entirely analogous to those analysed in thetwo previous figures. Finally, we note from the last insets that increasing either oneof the noises, g11 or g22, broadens and skews the density of ζ at point x∞.

These figures indicate that the analysis of a problem of growth to shock based onthe first moment of ζ is insufficient, and should incorporate either the probabilitydistribution of this quantity or its higher moments. This is done with the help offigures 16–19. They contain a wide range of information on the coefficients of skewnessand flatness of the inverse amplitude. In particular, we observe here a very non-Gaussian character of ζ at x∞, even for very small intensities of perturbations in µand β when their correlation is not fully positive.

7. Bernoulli equation perturbed by an Ornstein–Uhlenbeck process


We now perturb the original Bernoulli equation

dαdx

= −µ(x)α+ β(x)α2, α(x0) = α0, (7.1)

by two noises µ(x) and β(x)

µ(x) = 〈µ〉+ S1U(x), β(x) = 〈β〉+ S2U(x), S1, S2 > 0, (7.2)



Figure 15. A comparison of the average distance to form a shock, x∞ ≡ xc, and the standarddeviation of the inverse amplitude at the point of explosion, σζ(x∞), as a function of the initialamplitude ζ0 ≡ z0, at full negative correlation, for various levels of noise. The meaning of (1),(2) and (3) is the same as in figure 12.

Figure 16. Skewness γ1 (– – –) and flatness γ2 (——) of the inverse amplitude at the point x∞(m1(x∞) = 0), as functions of ζ0 at 〈µ〉 = 1, 〈β〉 = 1, and g11 = g22 = 0.01, for three cases: fullpositive correlation (1); zero correlation (2); and full negative correlation (3) between µ and β.The inset shows the density function of the inverse amplitude at point x∞ for ζ0 ≡ z0 = 0.9.



Figure 17. Skewness γ1 (– – –) and flatness γ2 (——) of the inverse amplitude at the point x∞(m2(xc) = 0), as functions of ζ0 at 〈µ〉 = 1, 〈β〉 = 1, at full positive correlation for three cases:(1) g11 = g22 = 0.01; (2) g11 = 0.04 and g22 = 0.01; and (3) g11 = 0.01 and g22 = 0.04. Theinset shows the density function of the inverse amplitude at point x∞ for ζ0 ≡ z0 = 0.9.

Figure 18. Skewness γ1 (– – –) and flatness γ2 (——) of the inverse amplitude at the pointx∞ (m2(xc) = 0), as functions of ζ0 at 〈µ〉 = 1, 〈β〉 = 1, at zero correlation for three cases:(1) g11 = g22 = 0.01; (2) g11 = 0.04 and g22 = 0.01; and (3) g11 = 0.01 and g22 = 0.04. Theinset shows the density function of the inverse amplitude at point x∞ for ζ0 ≡ z0 = 0.9.



0.6 0.7 0.8 0.9–10

30

70

110

150

(1)

(3)

(2)

z0

–6 –3 0 3 6 9

0.0

0.1

0.2

0.3

(1)

(2)

(3)

z0= 0.9

skew

ness

and

fla

tnes

s

density function

Figure 19. Skewness γ1 (– – –) and flatness γ2 (——) of the inverse amplitude at the point x∞(m2(xc) = 0), as functions of ζ0 at 〈µ〉 = 1, 〈β〉 = 1, at full negative correlation for three cases:(1) g11 = g22 = 0.01; (2) g11 = 0.04 and g22 = 0.01; and (3) g11 = 0.01 and g22 = 0.04. Theinset shows the density function of the inverse amplitude at point x∞ for ζ0 ≡ z0 = 0.9.

where, as before, 〈µ〉 and 〈β〉 are the mean values of stationary µ(x) and β(x) pro-cesses, but U(x) is an Ornstein–Uhlenbeck process. That is,

dU = −aU(x) dx+ ξ(x), a > 0, U(x0) = U0, (7.3)

and ξ(x) is a Gaussian white noise, such that

〈ξ〉 = 0, Kξ(x, x′) = 2Dδ(x− x′). (7.4)

The Ito equation for (7.3) is

dU = −aU(x) dx+√

2D dW (x), (7.5)

where W (x) is a standard Wiener process. For every x, U(x) has a normal distribu-tion with mean and variance (assuming initial condition U(x0) = 0)

〈U〉 = 0, 〈〈U〉〉 = σ2U [1− e−2ax], (7.6)

where σ2U = D/a. The correlation function of the process U(x) is

KU (x, x′) = σ2Ue−a|x−x

′|.

Let us also assume that for every x

〈µ〉 √〈〈S1U〉〉 = S1

√〈〈U〉〉, 〈β〉

√〈〈S2U〉〉 = S2

√〈〈U〉〉. (7.7)

We thus arrive at a stochastic differential equation for the vector y = [α,U ]T ≡[y1, y2]T

dy1 = [−〈µ〉y1 + 〈β〉y21 ] dx+ [−S1y1 + S2y

21 ]y2 dx,

dy2 = −ay2 dx+√

2D dW (x).

(7.8)



Figure 20. The influence of the correlation coefficient a of the process U(x) on the mean andstandard deviation of the inverse amplitude in the case when σU (x0) = 1. Values of otherparameters are: S1 = S2 = 0.15, 〈µ〉 = 〈β〉 = 1, 〈U〉 = 0.

We next introduce an inverse (vector) amplitude via a transformation of coordi-nates

ζ = [ζ1, ζ2] ≡ [f1(y1), f2(y2)] ≡ [(1/y2), y2] ≡ [(1/α), U ], (7.9)

so that (7.8) leads to

dζ1 = [〈µ〉ζ1 − 〈β〉] dx+ [S1ζ1 − S2]ζ2 dx,

dζ2 = −aζ2 dx+√

2D dW (x),

(7.10)

or, in a vector form,

dζ = A(ζ) dx+ σ(ζ) dW (x),

A(ζ) =[〈µ〉ζ1 − 〈β〉+ (S1ζ1 − S2)ζ2

−aζ2]

= [ai], i, j = 1, 2,

σ(ζ) =[0 00√

2D

]= [σij ].

(7.11)

We treat the above by considering the moments of ζ.


In order to obtain the moment equations for the inverse amplitude in the case ofa Bernoulli equation perturbed by the Ornstein–Uhlenbeck process, we begin with



Figure 21. A comparison of the influence of the correlation coefficient, a = 1.5, of the processU(x) on the mean and standard deviation of the inverse amplitude in the case when σU (x0) = 1(——) vis-a-vis σU (x0) = 0 (– – –). Values of other parameters are the same as in figure 19:S1 = S2 = 0.15, 〈µ〉 = 〈β〉 = 1.

an Ito formula in an n-dimensional case

dh(x1, x2, . . . , xn)

=[ n∑i=1

∂h

∂xlai + 1

2

n∑l=1

n∑m=1

∂2h

∂xl∂xm

r∑k=1

σlkσmk + 12

n∑l=1

n∑m=1

∂h

∂xlσlm dWm(x)

],

(7.12)

and apply it to

h(ζ1, ζ2) = ζi1ζj2 . (7.13)

Noting that, as before, x plays the role of the time parameter with the ζi being thecomponents of state vector ζ, we get

ddx〈ζi1ζj2〉 = 〈iζi−1

1 ζj2a1(ζ) + jζi1ζj−12 a1(ζ) +Dj(j − 1)ζi1ζ

j−22 〉,

i, j = 0, 1, 2, . . . ,M, 0 6= i+ j 6M. (7.14)

M is the highest order of a mixed moment to be considered. Upon carrying out theaveraging in (7.12), the last term vanishes, since the average value of a stochasticintegral of a non-anticipating function with respect to a Wiener process equals zero.



Now, assuming M = 2, and taking into account the vector A(ζ), we find thefollowing system of equations for the moments:

ddt〈ζ1〉 = 〈µ〉〈ζ1〉 − S2〈ζ2〉+ S1〈ζ1ζ2〉 − 〈β〉,

ddt〈ζ1ζ2〉 = (〈µ〉 − a)〈ζ1ζ2〉+ S1〈ζ1ζ2

2 〉 − S2〈ζ22 〉,

ddt〈ζ2

1 〉 = 2[〈µ〉〈ζ21 〉 − S2〈ζ1ζ2〉+ S1〈ζ2

1ζ2〉 − 〈β〉〈ζ1〉],ddt〈ζ2〉 = −a〈ζ2〉,

ddt〈ζ2

2 〉 = −2a〈ζ22 〉+ 2D.

(7.15)

Due to the presence of moments 〈ζ1ζ22 〉 and 〈ζ2

1ζ2〉 of order higher than M in (7.15),this system is not closed. According to the cumulant closure method (see, for exam-ple, Cramer 1946; Stratonovich 1963), the indicated moments can be approximatedas

〈ζ1ζ22 〉 = 〈ζ1〉〈ζ2

2 〉,〈ζ2

1ζ2〉 = 〈ζ1〉〈ζ1ζ2〉,

(7.16)

so that (7.15) becomes closed and linear.It is interesting to note that with the Ornstein–Uhlenbeck process, we can model

stationary as well as non-stationary material statistics. Thus, in the case when initialconditions for the process ζ2(x) are 〈ζ2(x0)〉 = 0 and 〈ζ2

2 (x0)〉 = σU , the disturbanceacting on µ and β has a constant variance in the whole material domain. For othercases of initial values of 〈ζ2

2 (x0)〉, the variance of the disturbance increases accordingto (7.6). These two cases are analysed numerically in figures 20 and 21.

In figure 20, we illustrate the dependence of the mean and standard deviation ofthe inverse amplitude on the correlation coefficient a of U(x), when σU (x0) = 1,which is the same as in the stationary state of the Ornstein–Uhlenbeck process.Clearly, as the correlation radius ρ = 1/a increases, so does the standard deviationof the inverse amplitude process. However, the mean remains practically unchanged.

In figure 21, we demonstrate the difference between wave evolutions in the sta-tionary versus non-stationary media at a = 1.5. This is done again in terms of themean and standard deviation of ζ1 = 1/α. The perturbation in the non-stationarymedium starts from zero and grows to the level typical of the stationary case.

Let us end this section with a note on the critical value ζc of the inverse amplitude.If the process ζ1, ζ2 starts from the following set of initial values

〈ζ1(x0)〉 = m10, 〈ζ21 (x0)〉 = m2

10,

〈ζ1(x0)〉 = 0, 〈ζ22 (x0)〉 = σ2

U , 〈ζ1(x0)(ζ2(x0))〉 = 0,

(7.17)

and a 0, which describes a stationary random medium, we obtain from (7.15)d

dx〈ζ1〉 = 〈µ〉〈ζ1〉 − 〈β〉. (7.18)

It follows from this thatζc = 〈β〉/〈µ〉, (7.19)

which is the same as for the reference homogeneous case.



8. Closure

As noted in § 1, the beauty of the Bernoulli equation (1.1) lies in its ability to describethe behaviour of acceleration waves, and, thus, the possibility of shock formation,in a wide variety of media, both solid and fluid. These wavefronts have convention-ally been treated as singular surfaces propagating in deterministic continua. Thepresence of a disordered material microstructure, however, has led us to considerthese wavefronts as zones of a thickness finite relative to the microscale of materialmicrostructure, such as the single grain size. Consequently, the wavefront is recog-nized as a mesoscale window (analogous to a statistical volume element) travellingthrough a sample realization of a random continuum. In the ensemble sense, we thusneed to treat the Bernoulli equation as a stochastic differential equation describ-ing a random evolution. As the wavefront thickness becomes very large relative tothe microscale, the variability of the effective mesoscale response over the wavefrontdomain tends to zero: this is the limit of the classical representative volume ele-ment (RVE) of continuum mechanics. This issue of mesoscale windows and of theRVE dilemma has also been studied, among others, in problems of linear elasticityand stochastic finite elements (Ostoja-Starzewski 1998; Ostoja-Starzewski & Wang1999), plasticity (Ostoja-Starzewski & Ilies 1996), and damage mechanics (Alzebdehet al . 1998).

By considering the spatial correlations of two material coefficients, we arrive atthe µ, βx process driving the stochastic Bernoulli equation, modelled as either awhite noise process, or an Ornstein–Uhlenbeck process. We then derive a range ofstochastic models that analyse relative effects of strength of noises of two underly-ing random processes that drive the stochastic competition of dissipation (µx) andmaterial elastic nonlinearity (βx). The principal conclusions of this investigation aresummarized as follows.

(1) It has been shown that, for small perturbations in µ(x) and β(x), and forthe initial value of ζ0 = 1/α0 sufficiently far away from the critical amplitudeζc = 1/αc, the probability density of ζ is approximately Gaussian. This allowsone to rapidly compute the probability of an acceleration wave explosion for afixed value of x.

(2) The method of Winterstein can be employed for the approximation of ζ as wellas α in the cases when the conditions required for the aforementioned Gaussianfit are not being met.

(3) The degree of correlatedness in the µ(x) and β(x) processes may have a signif-icant effect on the probabilistic characteristics of the inverse amplitude processζ(x).

(4) In the case of the random medium being modelled by the Ornstein–Uhlenbeckprocess, an increase in the correlation radius of this process results in thegrowth of the standard deviation of the inverse amplitude process, but leavesits mean practically unchanged.

(5) Many of our results indicate that the analysis of a problem of growth to shockbased on the first moment of ζ is insufficient and should incorporate either theprobability distribution of this quantity or its higher moments.



All these results were obtained for microinhomogeneous media described by wide-sense stationary processes. Thus, given the correlation radius of the µ(x) and β(x)processes being much smaller than that of the wave process, the medium couldclearly be taken as ergodic, and even mixing. However, in accord with the classicalinterpretation of random processes as a family of realizations over the Ω space, ouranalyses set the stage for materials not satisfying the ergodic property (e.g. globallyinhomogeneous media), where the averaging over an ensemble is more natural.

In conclusion, the results presented in this paper may become useful not onlyin the direct prediction of the behaviour of acceleration waves, but also in inverse-type problems, which could allow one to infer the statistical character of microscaleproperties from global observations of acceleration wave behaviours.

The constructive comments of a reviewer helped improve this paper. Support of this researchby the National Science Foundation under grants MSS-9202772 and CMS-9713764 is gratefullyacknowledged.

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