Flexure with damage - Geophysical Journal International

Geophys. J. Int. (2006) 166, 1368–1383 doi: 10.1111/j.1365-246X.2006.03067.xG

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Flexure with damage

David M. Manaker, Donald L. Turcotte and Louise H. KelloggDepartment of Geology, University of California, Davis, 1 Shields Ave., Davis, CA 95616, USA. E-mail: [email protected]

Accepted 2006 May 11. Received 2006 May 10; in original form 2005 July 20

S U M M A R YDuctile behaviour in rocks is often associated with plasticity due to dislocation motion ordiffusion under high pressures and temperatures. However, ductile behaviour can also occurin brittle materials. An example would be cataclastic flow associated with folding at shallowcrustal levels. Engineers utilize damage mechanics to model the continuum deformation ofbrittle materials. In this paper we utilize a modified form of damage mechanics that includes ayield stress. Here, damage represents a reduction in frictional strength. We use this empiricalapproach to simulate bending of the lithosphere through the problem of plate flexure.

We use numerical simulations to obtain quasi-static solutions to the Navier equations ofelasticity. We use the program GeoFEST v. 4.5 (Geophysical Finite Element Simulation Tool),developed by NASA Jet Propulsion Laboratory, to generate solutions for each time step. Whenthe von Mises stress exceeds the critical stress on an element we apply damage to reducethe shear modulus of the element. Damage is calculated at each time step by a power-lawrelationship of the ratio of the critical stress to the von Mises stress and the critical strain tothe von Mises strain. This results in the relaxation of the material due to increasing damage.To test our method, we apply our damage rheology to a semi-infinite plate deforming under itsown weight. Where the von Mises stress exceeds the critical stress, we simulate the formationof damage and observe the time-dependent relaxation of the stress and strain to near theyield strength. We simulate a wide range of behaviours from slow relaxation to instantaneousfailure, over timescales that span six orders of magnitude. Using this method, stress relaxationproduces perfectly plastic behaviour in cases where failure does not occur. For cases of failure,we observe a rapid increase in damage, analogous to the acceleration of microcrack formationand acoustic emissions prior to failure. Thus continuum damage mechanics can be used tosimulate the irreversible deformation of brittle materials.

Key words: crustal deformation, flexure of the lithosphere, geodynamics, rheology, shearmodulus.

1 I N T RO D U C T I O N

Solving problems in geodynamics requires an understanding of the

processes that govern deformation within the rigid lithosphere. How

the lithosphere deforms depends on the rheological properties of the

material. For the shallow lithosphere, elastic rheologies are appro-

priate and permanent deformation is in the form of brittle fracture.

These fractures span a wide range of spatial scales from microcracks

to regional joint systems and faults. At depths of greater than about

10 km (based on a typical continental geotherm of 30◦C km−1),

deformation changes from microfracturing to plastic flow as tem-

peratures reach 300◦–400◦C for quartz and 550◦–650◦C for feldspar

(Scholz 2002, p. 50). Plastic rheologies are appropriate below this

depth and permanent deformation is in the form of creep and duc-

tile behaviour. At high temperatures (T ∼ 1000◦C) creep processes

become dominant, ranging from diffusion creep with Newtonian

fluid behaviour to dislocation creep with non-Newtonian fluid be-

haviour. Plasticity is also observed at low temperatures if the confin-

ing pressure of the rock approaches the brittle strength (Turcotte &

Schubert 2002, p. 292). The transition from brittle to ductile be-

haviour occurs over a wide range of pressure and temperature con-

ditions, however microscopic brittle processes remain important

within the semi-brittle region (Scholz 2002, p. 43, 147).

The style of deformation of the rigid lithosphere can also depend

upon the timescale considered. On short timescales, from seconds

to hundreds of years, an elastic rheology is likely to be dominant.

For the longer timescales associated with plate deformation and

folding of rock layers, ductile rheologies maybe appropriate. Clearly,

both brittle and ductile rheologies play a role in the deformation of

the rigid lithosphere. However, there are instances in lithospheric

deformation where the appropriateness of a particular rheology is

not entirely clear. A specific example is the bending of the oceanic

lithosphere at a subduction zone. In some cases the bending is purely

elastic (Caldwell et al. 1976), but in other cases a ‘plastic’ hinge

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Flexure with damage 1369

develops. This behaviour has been shown to be consistent with a

perfectly plastic rheology (McAdoo et al. 1978). However, there is

extensive seismicity associated with this bending, indicating that

brittle deformation is occurring.

Another example is the large-scale crustal deformation associated

with mountain building and continental collisions. This large-scale

deformation has been treated by both block models and by con-

tinuum models. McKenzie & Jackson (1986), King et al. (1994),

Thatcher (1995) and Jackson (2002) have discussed the merits

of block (microplate) models. Continuum models have often uti-

lized a non-Newtonian viscous rheology. Examples include the vis-

cous thin-sheet models of England & McKenzie (1982), England

et al. (1985), England & Houseman (1986), Houseman & England

(1986a,b), Sonder & England (1986, 1989), Sonder et al. (1986)

and England & Molnar (1997).

Folding of the lithosphere has recently been recognized as another

phenomenon where the appropriateness of a particular rheology has

been challenged. Folding is often attributed to a viscous rheology

(Johnson & Fletcher 1994). At high temperatures and pressure, dis-

location and/or diffusion creep processes may give this behaviour. At

lower pressures and temperatures pressure solution processes give

a viscous rheology. Kink and chevron folds have tight hinge areas

separated by relatively straight limbs. This behaviour can be asso-

ciated with a perfectly plastic rheology within the hinges. Based on

these viscous rheologies, folding is generally considered to be a duc-

tile phenomenon. However, the importance of brittle mechanisms

in folding through elastic–plastic (Johnson 1980) and viscoelastic

processes (Schmalholz & Podladchikov 1999) have also been rec-

ognized. It has also been recognized that brittle processes can also

produce ductile behaviour (Hadizadeh & Rutter 1983), through cat-

aclasis (including fragmentation, fragment rotation) and cataclastic

flow (Sibson 1977; Paterson 1978; Hadizadeh & Rutter 1983). On

smaller scales, these brittle processes are responsible for catacla-

sis within fault zones. On a regional scale, cataclastic processes

are responsible for the brittle deformation of the crust. Cataclastic

flow can be the primary process responsible for the deformation in

folding. Hadizadeh & Rutter (1983) discuss numerous examples of

folding through cataclastic flow within the Umbrian mountain fold

belt in Italy, the Barrios quartzite in Spain and the Moine thrust belt

in Scotland. Ismat & Mitra (2001) attribute the folding responsible

for the formation of the Canyon Range syncline to cataclastic flow

associated with displacements on fractures and deformation zones.

Thus, brittle processes can lead to ductile behaviour without creep

and plasticity. The desire to model this behaviour while preserving

the brittle nature of the deformation forms the basis of our applica-

tion of continuum damage mechanics to crustal deformation.

Continuum damage mechanics is an empirically derived model

that is widely used in civil and mechanical engineering to quan-

tify the brittle deformation of solids associated with microcracking

(Kachanov 1986; Krajcinovic 1996; Skrzypek & Ganczarski 1999;

Voyiadjis & Kattan 1999). A number of authors have applied dam-

age mechanics to the brittle deformation of rock (Lyakhovsky et al.1997a, 2001, 2005; Ben-Zion & Lyakhovsky 2002; Turcotte et al.2003; Shcherbakov & Turcotte 2003; Hamiel et al. 2004; Katz &

Reches 2004). Shcherbakov & Turcotte (2004) and Shcherbakov

et al. (2005) have applied damage mechanics to explain the Omori’s

law decay of aftershock sequences. Using damage mechanics

Turcotte & Glasscoe (2004), Nanjo & Turcotte (2005) and Nanjo

et al. (2005) have derived a non-Newtonian viscous rheology for the

brittle deformation of solids. Shcherbakov et al. (2005) and Nanjo

et al. (2005) introduced the concept of a yield stress into damage

mechanics to describe deformation due to slip on faults. Where

the stress is less than the yield stress, brittle deformation in the

form of fault slip does not occur. Thus the yield stress is associated

with the dynamic coefficient of friction. If the stress is greater than

the yield stress, the brittle displacements on faults can result in a

non-Newtonian viscous rheology. This result is consistent with the

concept of a ‘schizosphere’ introduced by Scholz (2002, p. 3). At

stresses less than the yield stress the lithosphere acts as a stress guide.

At stresses greater than the yield stress, intraplate earthquakes can

cause deformation. Evidence for this behaviour comes from induced

seismicity, for example, due to the filling of a reservoir. The yield

stress of the lithosphere is certainly spatially variable, depending on

composition and crustal heat flow.

We express the irreversible damage of a material in terms of a

non-dimensional scalar damage variable α. As commonly applied

to brittle materials, this variable quantifies the effects of microfrac-

tures within a medium. The formation of microfractures reduces

the elastic strength of the material. As fractures initiate, grow and

coalesce with other fractures, the material becomes weaker. Rather

than attempt to describe each fracture individually, the degree of

microfracturing is described in terms of a state variable that relates

the amount of damage to the elastic moduli of the material. We

are considering deformation within the crust such as the bending

of the lithosphere or folding accommodated by cataclastic flow. We

are concerned with slip on faults (of varying scales) and catacla-

sis. Here, damage is associated with the reduction of the frictional

strength through wear of asperities, fracturing and comminution.

A primary objective of this paper is to show how damage me-

chanics, as a continuum model for a brittle rheology, can produce

behaviour that closely resembles a perfectly plastic rheology. We

demonstrate this by applying damage mechanics to a uniformly

loaded plate with end support. Plate flexure is a major aspect of

lithospheric deformation. Examples include: (1) the flexure of the

oceanic lithosphere at subduction zones, (2) the flexure of the litho-

sphere in response to volcanic loads (i.e. the Hawaiian islands) and

(3) the flexure of the lithosphere associated with sedimentary basins.

We apply the formulation of Shcherbakov et al. (2005) and Nanjo

et al. (2005) with minor modifications. This damage model includes

a yield stress σ y and at stresses σ < σ y damage does not occur. The

transient development of damage is obtained using a finite element

model. We present four different simulations of plate flexure, using

a simple plate that deforms under body forces with identical initial

and boundary conditions. The only difference among these models

is the yield stress. Regions in the model that exceed the prescribed

yield stress experience damage formation and inelastic deforma-

tion. Our simulations exhibit a range of time-dependent deforma-

tion from slow relaxation to catastrophic failure, demonstrating that

this methodology can be used to model a wide range of behaviour

in the brittle lithosphere, and demonstrating that brittle materials

can behave in a ductile manner approaching perfect plasticity.

2 A N A LY T I C A L F O R M U L AT I O N S

F O R DA M A G E D U E T O S H E A R

We first consider how damage occurs in a material that is subjected

to a constant shear stress. Under most conditions in the lithosphere,

Mode I (tensile) fractures are rare due to the confining lithostatic

pressure. As a result, shear fracture due to differential stress is the

dominant form of brittle deformation in the lithosphere. Addition-

ally, deformation in the lithosphere frequently occurs as slip on

pre-existing faults of all scales in response to tectonic stresses. As

previously mentioned, we apply the formulation of Shcherbakov

et al. (2005) and Nanjo et al. (2005). They use damage to describe

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1370 D. M. Manaker, D. L. Turcotte and L. H. Kellogg

the behaviour of the lithosphere in response to earthquakes and af-

tershock sequences, and their formulations consider the reduction

of Young’s modulus E as a result of increasing damage and strain.

Lyakhovsky et al. (1997a) found that in order to allow for healing

(closure of cracks) under isotropic compaction, a constant Lame pa-

rameter λ0 was required. We also assume a constant Lame parameter

λ0 and associate damage with the reduction of the shear modulus

G, and we modify the formulation by analogy, substituting G for E.

We now consider an infinite planar region subjected to shear. Our

approach follows Shcherbakov & Turcotte (2003, 2004), and the

sole difference is the modification of our damage formulation to

consider only changes in the shear modulus G. We consider two

different boundary conditions: (1) The application of a constant

shear strain at t = 0 and (2) the application of a constant shear stress

at t = 0. These two different initial conditions produce markedly

different behaviours.

First, we determine the behaviour of a planar region that is sub-

jected either to a uniform shear strain ε0 or to a uniform shear stress

σ 0. The planar region subject to damage has a thickness h, − h/2 <

y < h/2 defines this region. We assume that this region has a well-

defined yield stress σ y below which it behaves elastically, that is:

σxy = 2G0εxy, (1)

where ε xy is the shear strain and G0 is the shear modulus of the

undamaged material. We assume that eq. (1) is valid for 0 ≤ σ xy ≤σ y . From eq. (1) the corresponding yield strain ε y is given by

εy = σy

2G0

. (2)

In order to quantify the deviation from linear elasticity, we introduce

a damage variable α according to

(σxy − σy) = 2G0(1 − α)(εxy − εy), (3)

which is valid for σ xy > σ y . This is similar to that defined by

Shcherbakov et al. (2005).

We have 0 < α < 1, when α = 0 the material is undamaged and

when α = 1 either the stress relaxes to the yield stress or the material

fails. As α increases from 0 to 1 the material weakens and the strain

under a constant applied stress increases.

In order to complete the formulation of the problem, it is necessary

to prescribe the rate at which damage occurs. A detailed discussion

of the time evolution of damage based on thermodynamic consider-

ations has been given by Lyakhovsky et al. (1997a). We will utilize

the time evolution of the damage variable given by Shcherbakov

et al. (2005)

dα

dt= 0 for σxy ≤ σy, (4)

dα

dt= 1

td

[σxy

σy− 1

]ρ [εxy

εy− 1

]2

for σxy > σy, (5)

where td is a characteristic timescale for damage and ρ is a

power to be determined from experiments. Nanjo et al. (2005) and

Shcherbakov et al. (2005) found a value ρ = 3 to be consistent with

the rate of earthquake aftershock decay in California. We use this

value for the analytic and numerical solutions that follow throughout

this paper. Nanjo et al. (2005) demonstrated that the damage formu-

lations given by eqs (3) and (5) produce a non-Newtonian viscous

fluid rheology when stresses exceed the yield stress.

In our first example we will assume that a constant shear strain

ε0 greater than the yield strain ε y is applied at t = 0. We introduce

the non-dimensional variables

σxy

σy= σ ′

xy,ε0

εy= ε′

0,t

td= t ′, (6)

and substitute these quantities into eqs (2)–(5), with the result

σ ′xy = 1 + (1 − α)(ε′

0 − 1), (7)

dα

dt ′ = (σ ′xy − 1)ρ(ε′

0 − 1)2. (8)

Combining these equations and integrating with the initial condition

α = 0 at t ′ = 0 gives

α = 1 − 1

[1 + t ′(ρ − 1)(ε′0 − 1)ρ+2]

1ρ−1

, (9)

which describes the damage evolution in the material with a constant

applied strain that is greater than the yield strain, ε′0 > 1. Substitution

of eq. (9) into eq. (7) gives

σ ′xy = 1 + ε′

0 − 1[1 + t ′(ρ − 1)(ε′

0 − 1)ρ+2]1/ρ−1

. (10)

We see that α → 1 and σ ′xy → 1 as t ′ → ∞. The stress relaxes to

the yield stress. We define a non-dimensional relaxation time t ′r as

the time when α = 0.9. From eq. (9) we find that this time is given

by

t ′r = [10(ρ−1) − 1]

(ρ − 1)(ε′0 − 1)ρ+2

. (11)

The dependence of t ′r on ε′

0 is given in Fig. 1 for ρ = 3. We take

this value for ρ because Nanjo et al. (2005) found it to be applicable

to crustal deformation. It is seen in Fig. 1 that the non-dimensional

relaxation times are long for values of ε′0 slightly larger than unity

and that the relaxation time approaches infinity as a power law for

ε′0 → 1. For ε′

0 < 1 no stress relaxation occurs because the defor-

mation is governed by linear elasticity.

0.1

101

103

105

107

109

1011

1.0 1.2 1.4 1.6 1.8 2.0

Relaxation Time

Failure Time

Numerical Approx.

t'r, t'

f

'0, '

0

Figure 1. Comparisons of timescales for the behaviour of a plate subject

to shear stress. The dependence of the non-dimensional relaxation time t ′rdefined in eq. (11) on the non-dimensional constant applied strain ε′

0 (solid

line) and the dependence of the non-dimensional failure time t ′f defined in

eq. (17) on the non-dimensional constant applied shear stress σ ′0 (dashed

line). In both cases we have taken ρ = 3. Data points represent the estimated

failure times obtained through out numerical simulation. The agreement in

this benchmark test suggests that the numerical method used is sufficiently

accurate.

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As a second example we will assume that a constant shear stress

σ 0 greater than the yield stress σ y is applied at t = 0. In this case

we introduce the non-dimensional variables

σ0

σy= σ ′

0,εxy

εy= ε′

xy,t

td= t ′ (12)

Substitution of these variables into eqs (2)–(5) gives

ε′xy = 1 + (σ ′

0 − 1)

(1 − α), (13)

dα

dt ′ = (σ ′0 − 1)ρ(ε′

xy − 1)2. (14)

Combining these equations and integrating with the initial condi-

tions α = 0 at t ′ = 0 gives

α = 1 − [1 − 3t ′(σ ′

0 − 1)ρ+2] 1

3 , (15)

which describes the damage evolution in the material with a constant

applied stress that is greater than the yield stress,σ ′0 >1. Substitution


ε′xy = 1 + σ ′

0 − 1[1 − 3t ′(σ ′

0 − 1)ρ+2]1/3

. (16)

Failure occurs at the non-dimensional time t ′f when α = 1 (ε′

xy →∞), from eq. (15) we obtain

t ′f = 1

3(σ ′0 − 1)ρ+2

. (17)

The dependence of t ′f on σ ′

0 is also given in Fig. 1 for ρ = 3. For

equal values of ε′0 and σ ′

0, the relaxation times t ′r are about two

orders of magnitude larger than the failure times t ′f . Again, the time

to failure approaches infinity as a power law as σ ′0 → 1. Substitution


α = 1 −(

1 − t ′

t ′f

) 13

, (18)

which was previously given by Ben-Zion & Lyakhovsky (2002) and

Shcherbakov & Turcotte (2003). The corresponding time depen-

dence of strain is given by

ε′xy = 1 + σ ′

0 − 1(1 − t ′

t ′f

)1/3. (19)

The approach to failure is in the form of a power law. It is important

to note from eq. (17) that the time to failure is well defined for the

case of constant stress σ ′0 > 1, whereas the application of constant

strain ε′0 > 1 results in the relaxation of the stress σ ′

xy to 1 at infinite

time.

3 DA M A G E R E S U LT I N G F RO M

P L AT E F L E X U R E

We now turn our attention to plate bending. We begin with a homo-

geneous, isotropic elastic plate subject to the load of its own weight.

The problem simplifies to two-dimensional plane strain for an infi-

nite plate supported by a pivot at the base of one end and a roller on

the other end as illustrated in Fig. 2(a). The initial elastic response

of the plate has an analytic solution:

w(x) = −qx

24D(L3 − 2Lx2 + x3), (20)

where w is the plate deflection, q is the distributed load per unit

length, D is the modulus of rigidity, L is the length of the plate and

x is the position along the plate from 0 to L (Turcotte & Schubert

2002, pp. 114–115). For a plate, the modulus of rigidity is given by:

D = Eh3

12(1 − ν2), (21)

where E is Young’s modulus, h is the plate thickness and ν is Pois-

son’s ratio. The plate fibre stresses are given by:

σxx = 6q

h2

(y

h− 1

2

)(Lx − x2), (22)

σzz = 6qν

h2

(y

h− 1

2

)(Lx − x2), (23)

and the shear stress is:

σxy = 6q

h3

[y(h − y)

(L

2− x

)], (24)

where y is the vertical profile position from 0 to h within the plate.

We again introduce a set of non-dimensional variables defined by

x ′ = x

L, y′ = y

h, w′ = wD

q L4, σ ′ = σ

q

(h

L

)2

. (25)

Substitution of these variables into eqs (20)–(24) gives

w′ = x ′

24(1 − 2x ′2 + x ′3), (26)

σ ′xx = 6x ′(1 − x ′)

(y′ − 1

2

), (27)

σ ′zz = 6νx ′(1 − x ′)

(y′ − 1

2

), (28)

σ ′xy = 6h

L

(1

2− x ′

)(1 − y′)y′. (29)

This is the non-dimensional analytic solution for the elastic bending

of an infinite plate. It should be noted that this solution is strictly

valid only in the limit h/L → 0. This solution is always valid at t =0 and is valid at all times if the maximum stresses at x ′ = 0.5 and

y′ = 0 and 1 are less than the yield criteria.

Plate bending with damage was previously considered by

Krajcinovic (1979) but without time evolution. To solve the plate

bending problem for the time evolution of damage requires a nu-

merical solution. We again assume a homogeneous, isotropic elas-

tic plate bending under its own weight in two-dimensional (plane)

strain. However, we will solve the full two-dimensional problem

using Hooke’s law for an isotropic material:

σi j = λδi jεkk + Gεi j . (30)

When stresses within the plate are less than the yield stress, the

deformation is purely elastic and satisfies eqs (20)–(24). However,

where stresses within the plate exceed the yield stress, damage will

occur. Under crustal conditions, lithostatic pressure plays a signifi-

cant role as confining pressure and the formation of shear fractures

or slip on pre-existing faults due to the differential stress is the

dominant mechanism of brittle deformation. Therefore, we apply

the yield criterion for damage by considering the differential stress.

We calculate the differential stress by using the von Mises stress,

which is commonly used in determining the onset of plasticity. How-

ever, we are not using the differential stress to evaluate the onset of

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-0.012

-0.008

-0.004

0

0 0.2 0.4 0.6 0.8 1

Analytic Solution

FEM Solution

w'

x'

-0.8 -0.4 0 0.4 0.80

0.2

0.4

0.6

0.8

1

Analytic Solution

FEM Solution

'xx

y'

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08

Analytic Solution

FEM Solution

y'

'xy

q = constant

L

h

y

x

A. B.

C. D.

Figure 2. Plate bending problem due to constant load. (a) Schematic diagram of the problem geometry. The plate has a width L in the x-direction, a thickness

h in the y-direction, and extends an infinite length in the z-direction. The uniform load q per unit length in the x- and z-directions is due to the weight of the

plate and acts in the—y-direction. (a) Comparison of the distribution of the non-dimensional fibre stresses σ ′xx across the plate at its centre x ′ = 0.5 for the

finite element computation with the analytic solution. (b) Comparison of the distribution of the non-dimensional shear stresses σ ′xy across the plate at its centre

x ′ = 0.5 for the finite element computation with the analytic solution. (c) Comparison of the non-dimensional plate deflection w′ of the plate at its centre

x ′ = 0.5 for the finite element computation with the analytic solution. Solutions are based on an elastic plate with L ′ = 10 and ν = 0.25. The data points for

the stresses are for the elements and the data points for the deflection are for the nodes of the finite element computation. In each case the data points are from

the finite element computation and the solid lines are the analytic solutions.

plasticity, but the onset of brittle damage. The von Mises stress is

given by:

σvm =√

1

2

[(σ1 − σ2)2 + (σ1 − σ3)2 + (σ2 − σ3)2

], (31)

where σ 1, σ 2 and σ 3 are the principal stresses (Turcotte & Schubert

2002, p. 334). This allows us to consider the contributions of the

entire stress tensor. As an example, we will determine the maximum

non-dimensional von Mises stress for the elastic bending of a plate

using the analytical results obtained using eqs (27) to (29). We will

consider h/L to be small and can neglect the shear stress σ ′xy com-

pared to σ ′xx and σ ′

zz. The maximum von Mises stress is at x ′ = 0.5

and y′ = 0, 1. From eq. (27) we find σ ′1 = σ ′

xx = ±0.75 and from

eq. (28) we find σ ′3 = σ ′

zz = ±0.75ν. With σ ′2 = 0 and ν = 0.25,

we find that the maximum non-dimensional von Mises stress from

eq. (31) is (σ ′vm)max = 0.676. Where the von Mises stress exceeds

the yield stress, the material behaves inelastically and damage oc-

curs. However, the damage we consider here is not plasticity, but

the inelastic behaviour of brittle materials. For the plate considered

above the bending will be elastic at all times if the non-dimensional

von Mises yield stress is greater than σ ′y = 0.676.

Additionally, we introduce a von Mises strain, since the ratio of

the strains to a yield strain is also taken into account in our damage-

rate formulation and the relationship between damage and the shear

modulus. The expression for the von Mises strain is:

εvm =√

1

2

[(ε1 − ε2)2 + (ε1 − ε3)2 + (ε2 − ε3)2

]. (32)

We generalize the damage eqs (3) and (4) to the form

dα

dt= 0 for σvm ≤ σy, (33)

dα

dt= 1

td

[σvm

σy− 1

]ρ [εvm

εy− 1

]2

for σvm > σy . (34)

Following (Lyakhovsky et al. 1997a) we obtain solutions using a

shear modulus G which is reduced by damage and a constant Lame

parameter λ0. Thus we have, based on eq. (3), that:

G = G0

[(1 − εy

εvm

)(1 − α) + εy

εvm

], (35)

λ = λ0, (36)

for εvm > ε y and dεvm/dt > 0. Where dεvm/dt ≤ 0 and εvm > ε y ,

eq. (35) becomes:

G = G0

[(1 − εy

εvm,max

)(1 − α) + εy

εvm,max

]. (37)

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Thus, in cases when strain is reduced from above the yield strain to

below the yield strain, the reduction in the shear modulus remains.

To simplify, following eqs (6) and (25), we once again introduce the

non-dimensional variables

x ′ = x

L, y′ = y

h, L ′ = L

h, w′ = wD0

q L4,

σ ′ = σ

q

(h

L

)2

, t ′ = t

td. (38)

This reduces the parameters that we need to specify to ρ, L ′, σ ′y and

ν 0 in order to obtain solutions for the evolution of the plate rheology

due to damage.

3.1 The finite element model

We use the finite element modelling program GeoFEST v. 4.5

(Geophysical Finite Element Simulation Tool), developed by the Jet

Propulsion Laboratory (NASA-JPL) to compute quasi-elastostatic

solutions for our problem. Although GeoFEST v. 4.5 allows for

viscoelastic rheology and both Newtonian and non-Newtonian be-

haviour, we utilize only the elastic capabilities of the code. Using

an infinite plate model the problem is simplified to plane strain and

requires only a 2-D finite element model. The cross-sectional profile

of the plate has a 10:1 aspect ratio (that is L ′ = 10), and is represented

by a 600 × 60 element grid. This grid was adequate to determine

the structure of the boundary-layer features of the solutions. The

stability of the solutions was demonstrated by grid refinement. The

36 000 bilinear quadrilateral (square) elements are constructed from

36 661 nodes. The body force is equally distributed throughout the

plate by giving each element an equal body force, which remains

unchanged. The plate configuration is illustrated in Fig. 2. We con-

sider body forces and deflections to be negative. We also define

compressive stresses to be negative, consistent with the engineering

convention.

Each element is given identical elastic parameters for the initial

undamaged condition; however, the shear modulus for each element

is allowed to evolve over time with increasing damage based on

eq. (35). Each element has a unique shear modulus, and therefore is

locally homogeneous and isotropic, so Hooke’s law can be applied

for each element. We apply boundary conditions to the finite element

model to duplicate the initial analytic solution described earlier.

Each model starts at t = 0 with elastic flexure and is allowed to

evolve.

Before describing the damage evolution, we compare the results

of the initial finite element model solution to the steady-state elastic

plate flexure under a body force taking ν = 0.25. Figs 2(b)–(d) shows

a comparison of the results to the analytic solution. In Fig. 2(b) the

distribution of non-dimensional fibre stresses σ ′xx across the plate

at its centre point x ′ = 0.5 is given. The solid line is the analytic

solution from eq. (27) and the data points are from the elements in

the finite element computation. In Fig. 2(c) the distribution of non-

dimensional shear stresses σ ′xy across the plate at its centre point

x ′ = 0.5 is given. Again, the solid line is the analytic solution from

eq. (29) and the data points are from the finite element computation.

In both cases the finite elements solutions are in close agreement

with the analytic solutions. In Fig. 2(d) the non-dimensional plate

deflections w′ are given. In this case the deflections from the finite

element computation are about 2 per cent greater than the analytic

displacements from eq. (26). We attribute this small difference to

the finite aspect ratio (L ′ = 10) of our model rather than an infinite

aspect ratio. We also find the maximum non-dimensional von Mises

0.0130

0.0135

0.0140

0.0145

0.0150

0.0 5.0 x 104 1.0 x 105 1.5 x 105 2.0 x 105

| w' |

t'

A.

B.

C.

-0.8 -0.4 0.0 0.4 0.80

0.2

0.4

0.6

0.8

1

Initial

t' = 200

t' = 1200

t' = 10000

t' = 100000

t' = 200000

'xx

y'

0.0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1

t' = 10

t' = 50

t' = 400

t' = 10000

t' = 200000

y'

Figure 3. Evolution of damage in the plate for a non-dimensional von Mises

yield stress σ ′y = 0.5309. (a) Distribution of the non-dimensional fibre

stresses σ ′xx across the plate at its centre x ′ = 0.5 for various non-dimensional

times t′. (b) Distribution of the damage α across the plate at its centre

x ′ = 0.5 for various non-dimensional times t′. (c) The non-dimensional

plate deflection w′ of the plate at its centre x ′ = 0.5 as a function of

time.

stress at x ′ = 0.5 and y′ = 0 and 1. From our numerical solution, we

find (σ ′vm)max = 0.666, which compares with the value (σ ′

vm)max =0.676 obtained above for our analytic solution. Again we attribute

this difference to the finite aspect ratio of our numerical model.

3.2 Numerical approximation of time evolution of damage

Following the initial elastic response, we apply the damage relations

given in eqs (33) and (34). Where the von Mises stress exceeds the

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0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0.0 0.2 0.4 0.6 0.8 1.0

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

t' = 10

t' = 1200

t' = 10000

t' = 100000

t' = 200000

A.

B.

C.

D.

E.

Figure 4. Distribution of damage throughout the plate for a non-dimensional von Mises yield stress σ ′y = 0.5309 at various non-dimensional times t′. (a) t ′ =

10; (b) t ′ = 1200; (c) t ′ = 10 000; (d) t ′ = 100 000; and (e) t ′ = 200 000.

yield stress, the material experiences damage. Obtaining the value

of damage at a particular place in space and time is not possible an-

alytically. The rate of damage is a non-linear function of the stresses

and strains, which vary in response to the changing rheology due

to damage. We calculate the value of the damage parameter us-

ing the stresses and strains from quasi-static solutions produced

by the finite element model, allowing for spatially varying elastic

properties.

We calculate the change in the damage parameter using a forward

Euler approximation to numerically integrate the time derivative of

damage over a specified time interval. The damage αji at any element

i and time step j is obtained from the damage αj−1i at the previous

time step j − 1 using the relation:

αji = α

j−1i + dα

j−1i

dt ′ t ′, (39)

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where t ′ is the non-dimensional time step. For each time step, we

apply the relationship between damage and the shear modulus given

in eq. (24) to modify the rheology for each element. Through this

iterative approach, we achieve a spatial and temporal history for

damage and stress-strain in the plate.

Although the forward Euler approximation is simple to imple-

ment, it is not without drawbacks. It is best suited for nearly linear

functions. Since we do not allow for material healing, damage will

either be constant or increase through time, ensuring a degree of sta-

bility. However, in cases where a rapid increase in damage occurs,

significant errors can be introduced to the damage approximation.

For a particular time step, the error for the forward Euler approxi-

mation is:

e ji = 1

2

d2αj−1i

dt ′2 (t ′)2, (40)

and the global error for a particular element is given by:

ei = N

2

d2αj−1i

dt ′2 (t ′), (41)

where N is the total number of time steps. Therefore, unless the

time-derivative function for the element is nearly linear or small,

the error will be dominated by the size of the time step. In order to

verify the accuracy of our numerical approximation, we first calcu-

late a time-series for damage (and the resulting evolving stress and

strain field and material rheology) using the forward Euler approx-

imation with a trial time step. The trial time step is selected so the

maximum value of the damage parameter α is small for the first time

step (less than 0.1). We then compute a second time-series using a

time step that is reduced by a factor of 10. We compare these two

time-series at various time intervals to verify that the solutions are

identical (or nearly so), indicating that these time-series produce the

same solution. If the solutions are not in agreement, we discard the

solution for the trial time step, and the second time-series becomes

the new trial time step. We further reduce the time step by a factor

of 10 and the process is repeated until the time-series are essentially

identical.

To test the numerical method for the approximation of the time

evolution of damage, we apply our methodology to the analytic so-

lutions previously presented in Section 2. Specifically, we address

the problem of a planar region of thickness h subjected to a con-

stant shear stress that leads to failure. The analytic solution gives

the progression to failure with a uniform distribution of damage

over the volume. We represent this in our finite element model as

a single element subjected to a shear stress. If the applied shear

stress exceeds the yield stress, damage will accumulate within the

element. We apply eqs (1)–(5) and (39) to the elastostatic solution

obtained using GeoFEST to calculate the change in the damage and

elastic parameters for each time step. Fig. 1 shows the comparison

between the analytic solution for failure times and eight numerical

determinations of failure times for values of σ ′0 ranging from 1.1

to 1.8. There is excellent agreement between our numerical deter-

minations and the analytic solution for failure times. The error in

failure times was calculated from

e =∣∣t ′

f,analytic − t ′f,approx

∣∣t ′

f,analytic

. (42)

For the initial selection of time step size, errors ranged from 0.35

to 3.5 per cent, with an average error for the eight estimates for

failure time of 1.6 per cent for the original time step used. A further

reduction of the time-step size reduced the estimation error. This

suggests that our use of the forward Euler approximation for the

0.0120

0.0160

0.0200

0.0240

0.0 2.0 x 104 4.0 x 104 6.0 x 104

| w' |

t'

-0.8 -0.4 0 0.4 0.80

0.2

0.4

0.6

0.8

1

Initial

t' = 200

t' = 600

t' = 2000

t' = 60000

'xx

y'

0.0 0.2 0.4 0.6 0.8 1.0

0

0.2

0.4

0.6

0.8

1

t' = 2

t' = 200

t' = 600

t' = 2000

t' = 60000

y'

A.

B.

C.




times t′. (b) Distribution of the damage α across the plate at its centre


plate deflection w′ of the plate at its centre x ′ = 0.5 as a function of time.

time evolution of damage is acceptable for this problem. The reason

for this behaviour is that dα/dt is typically small with the exception

of regions where stresses are very high relative to the yield stress.

This typically occurs in regions close to failure, where rapid dam-

age formation is occurring. The material will fail regardless of the

size of time step due to the high strain. Additionally, there is an

upper limit to damage (α = 1) that prevents the approximation from

increasing without bound, which ensures a degree of stability and

convergence.

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0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0.0 0.2 0.4 0.6 0.8 1.0

t' = 2

t' = 200

t' = 2000

t' = 10000

t' = 60000

A.

B.

C.

D.

E.

Figure 6. Distribution of damage throughout the plate for a non-dimensional von Mises yield stress σ ′y = 0.4409 at various non-dimensional times t′. (a) t ′ =

2; (b) t ′ = 200; (c) t ′ = 2000; (d) t ′ = 10 000; and (e) t ′ = 60 000.

4 N U M E R I C A L R E S U LT S F O R

DA M A G E E V O L U T I O N D U E

T O P L AT E F L E X U R E

We now extend our analyses to the numerical modelling of the dam-

age for a plate bending under a constant load. Our model config-

uration was described in Section 3.1, and we take the reference

(undamaged) Poisson’s ratio ν 0 = 0.25. We model the damage evo-

lution and the deformation of the plate for a range of yield stresses.

In order for damage to occur, the assumed von Mises yield stress

must be less than the value obtained for the initial elastic solution,

that is σ ′y < 0.666. Because of the non-homogeneous distribution

of stresses, damage will occur only in regions where stresses exceed

the yield stress, resulting in strain localization. The increase in dam-

age can associated with the nucleation, growth and coalescence of

microfractures. This can occur in a both a stable and unstable fash-

ion. When stable fracturing occurs, damage formation and strain

localization will relax the stresses and dissipate strain energy. When

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unstable fracturing occurs, the relaxation of stress and dissipation of

strain energy is exceeded by the formation of damage, and the end

result is plate failure. There is a critical yield stress which represents

the threshold between plate failure (unstable fracturing) and stress

relaxation (stable fracturing). Damage can also be associated with

slip on faults, cataclasis, or the reduction of frictional strength.

We examine the time evolution of damage for plate flexure re-

sulting from four different yield stresses to observe the spatial and

temporal changes in stress, strain and damage. The non-dimensional

yield stresses we consider are σ ′y = 0.5309, 0.4409, 0.3779 and

0.2520. In terms of real rock materials, these non-dimensional yield

stresses correspond to differential yield stresses of 950, 790, 675

and 450 MPa, respectively, for a granodiorite plate 300 m long and

20 m thick, or 280, 230, 200 and 130 MPa, respectively, for a plate

of low-porosity calcite limestone that is 200 m long and 20 m thick.

These values are within the ranges of experimentally determined

differential yield stress of rocks given by Paterson (1978, pp. 24–

25). Because the deformation is driven by the weight of the plate

itself, the initial elastic stresses are identical for each case. However,

the spatial distributions of initial stresses that exceed the yield stress

within the plate differ greatly due to the differing yield stresses. The

resulting evolution of damage, stresses and plate deformation for

these cases span a wide range of behaviours.

4.1 Case 1: σ′y = 0.5309

For our first case, the ratio of the non-dimensional von Mises yield

stress σ ′y = 0.5309 to the maximum initial non-dimensional von

Mises yield stress σ ′vm = 0.666 is 0.797. Profiles of σ ′

xx across the

plate at the centre x ′ = 0.5 are given in Fig. 3(a) at several times t′.Profiles of the damage variable across the plate at the centre x ′ = 0.5

are given in Fig. 3(b) at several times t′. The deflection of the plate

w′ at its centre x ′ = 0.5 as a function of time t′ is given in Fig. 3(c).

Distribution of damage α across the entire plate is given in Fig. 4

at several times. Simulations using t ′ = 2.0 and 0.2 were found

to be essentially identical. The simulation was run to t ′ = 2.0 ×105, where the changes in the plate deflection (Fig. 3c) and stress

profile (Fig. 3a) suggest near complete relaxation. The initial stress

profile at t ′ = 0 given in Fig. 3(a) is identical to the elastic solution

given in Fig. 2(b). For the longest time calculated, t ′ = 2 × 105,

almost complete stress relaxation has occurred. The stress has been

reduced to the yield value (σ ′xx)y in boundary layers with an elastic

core. The damaged boundary layers constitute about 40 per cent of

the plate and the elastic core about 60 per cent of the plate. Since the

plate must carry the same bending moment at all times, the stress

in the elastic core increases as the stresses in the damaged region

decrease. The long-term behaviour of the plate is identical to a plate

with a perfectly plastic rheology.

From Fig. 3(c) we see that the initial deflection of the plate w′ =0.0133 is identical to the centre deflection of the elastic beam given

in Fig. 2(d). The final deflection of the plate at large times corre-

sponding to the perfect-plastic state is w′ = 0.0149. Thus damage

results in an 11 per cent increase in deflection at its centre. The pro-

files of the damage variable α at various times t′ given in Fig. 3(b)

show the evolution of damage with time. Initially, at t ′ = 0, we have

no damage and α = 0. At large times, α = 0 in the elastic core and

α = 1 in the damaged boundary layers. We define a non-dimensional

relaxation time t ′r to be the time required for 90 per cent of the re-

laxation displacement, that is

w′(t ′r ) − w′(0)

w′(∞) − w′(0)= 0.9. (43)

0

0.05

0.1

0.15

0.2

0 10 20 30 40

| w' |

t'

A.

B.

C.

-1.0 -0.5 0.0 0.5 1.00

0.2

0.4

0.6

0.8

1

t' = 0

t' = 4

t' = 20

t' = 30

t' = 36

t' = 37

'xx

y'

0.0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1

t' = 2

t' = 4

t' = 20

t' = 30

t' = 36

t' = 37

y'




times t′. (b) Distribution of the damage α g cross the plate at its centre



From the values given above we have w′(t ′r ) = 0.01474 and from

Fig. 3(c) we have t ′r = 3.75 × 104.

The time evolution of damage throughout the entire plate is illus-

trated in Fig. 4. It is seen that damage is confined to a thin boundary

layers at the top and bottom (y′ = 0 and 1) and near the plate cen-

tre (x ′ = 0.5). A significant elastic core remains through the plate

centre, with over 60 per cent of the material remaining undamaged

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t' = 1

t' = 10

t' = 30

t' = 38

t' = 39

A.

B.

C.

D.

E.

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0.0 0.2 0.4 0.6 0.8 1.0

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

Figure 8. Distribution of damage throughout the plate for a non-dimensional von Mises yield stress σ ′y = 0.3779 at various non-dimensional times t′.

(a) t ′ = 1; (b) t ′ = 10; (c) t ′ = 30; (d) t ′ = 38; and (e) t ′ = 39.

in the vertical profile through the centre. This preserves the stiff-

ness of the plate and prevents a significant increase in deflection.

The initial damage rates and accompanying stress change rates are

relatively small and the material relaxes slowly, on the order of 106

characteristic time units.

Although a perfect-plastic rheology gives the final state of our

plate, it does not give the temporal evolution to that state. Damage

mechanics provides the time-dependent relaxation solution from the

initial elastic solution to the final perfectly plastic solution. This is

accomplished through solely through a brittle mechanism.

4.2 Case 2: σ′y = 0.4409

For our second case, the ratio of the non-dimensional von Mises yield

stress stress σ ′y = 0.4409 to the maximum initial non-dimensional

von Mises yield stress σ ′vm = 0.666 is 0.662. The resulting stress

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0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

| w' |

t'

A.

B.

C.

0.0 0.2 0.4 0.6 0.8 1.0

0

0.2

0.4

0.6

0.8

1

t' = 0.02

t' = 0.04

t' = 0.12

t' = 0.20

t' = 0.24

t' = 0.247

y'

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.00

0.2

0.4

0.6

0.8

1

t' = 0

t' = 0.04

t' = 0.12

t' = 0.20

t' = 0.24

t' = 0.247

'xx

y'




times t′. (b) Distribution of the damage α g cross the plate at its centre



profiles, damage profiles and beam deflections are given in Fig. 5,

and the distributions of damage throughout the cross section of the

plate at several times t′ are given in Fig. 6. We used t ′ = 0.2 for

this simulation, which was run to t ′ = 6.0 × 104. At this time, plate

deflection and stress profiles suggest near complete relaxation. The

behaviour of this solution is similar to the first case considered above.

Once again the initial elastic solution relaxes to a ‘perfectly plastic’

solution with damaged boundary layers transmitting the yield stress

(σ ′xx)y = ±0.57 and an undamaged elastic core. Because of the

reduced yield stress, damaged boundary layers constitute about 80

per cent of the plate and the elastic core 20 per cent of the plate.

From Fig. 6(c) we see that the central non-dimensional deflection

of the plate relaxes from the initial elastic deflection w′ = 0.0133 to

w′ = 0.024. In this case, damage results in an 80 per cent increase

in deflection compared to 11 per cent in our first example. From

eq. (41) we have w′ (t ′r ) = 0.02293 and from Fig. 5(c) we have

t ′r = 2.3 × 104. A comparison of Fig. 6 with Fig. 4 shows the much

larger damage zone in this case.

Case 2 differs from case 1 in that the damage occurs rapidly at

first, resulting in rapid changes of stress within the plate. The rapid

damage and deformation shifts the highest stresses toward the centre

of the plate. At one point, shown in Fig. 5(a), the plate appears to

be approaching failure at t ′ = 2000. However, the stress relaxation

outpaces damage formation toward the plate core, and a thin elastic

core is preserved (Fig. 6). By t ′ = 5000 most of the damage has

occurred and the stress relaxation begins to slow. Between t ′ =10 000 and 60 000 there is very little additional damage formation

or stress relaxation. Most of the stress relaxation has occurred by

t ′ ∼ 104.

4.3 Case 3: σ′y = 0.3779

For our third case, the ratio of the non-dimensional von Mises yield

stress stress σ ′y = 0.3779 to the maximum initial non-dimensional

von Mises yield stress σ ′vm = 0.666 is 0.567. The resulting stress pro-

files, damage profiles and beam deflections are given in Fig. 7, and

the distributions of damage throughout the plate at several times

t′ are given in Fig. 8. We used t ′ = 0.02 for this simulation.

Case 3 shows a gradual increase in the damage towards the plate

core that ultimately outpaces the stress relaxation. This can be seen

by the increase in the maximum fibre stress towards the core after

t ′ = 16 (Fig. 7a). Damage is concentrated in the centre, penetrat-

ing downward through the core (Fig. 8). The plate deflection rapidly

increases after t ′ = 30, indicating impending failure (Fig. 7c). Catas-

trophic failure of the plate begins at t ′ = 37, where the plate deflec-

tion rapidly increases. Complete damage (α = 1) through the plate

core exists at t ′ = 39, indicating failure. The transition from stress

relaxation in case 2 to catastrophic failure in case 3 is expected.

The maximum bending moment that a plate with a perfect plas-

tic rheology can carry is at the yield stress throughout (Turcotte

& Schubert 2002, p. 335). By t ′ = 39 the von Mises stress is re-

laxed to the yield stress, indicating perfect plasticity through the

core.

4.4 Case 4: σ′y = 0.2520

For our fourth and final case, the ratio of the final non-dimensional

von Mises stress σ ′y = 0.2520 to the maximum initial non-

dimensional von Mises yield stress σ ′vm = 0.666 is 0.378. The

resulting stress profiles, damage profiles, and plate deflections

are given in Fig. 9, and the distributions of damage throughout the

plate at several times t′ are given in Fig. 10. We use t ′ = 0.0002

for the simulation, decreasing the time step to 0.0001 at t ′ ≈ 0.24

to provide better accuracy as the plate approaches failure. Case 4

shows almost immediate failure, showing similar behaviour to case

3 over a time period that is two orders of magnitude smaller. Damage

formation outpaces stress relaxation and the maximum fibre stress

increases towards the core by t ′ = 0.12 (Fig. 9a). The plate deflection

shows the catastrophic failure beginning at t ′ = 0.24 (Fig. 9c), and

the stresses rapidly increase between t ′ = 0.24 and 0.247 (Fig. 9a).

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0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0.0 0.2 0.4 0.6 0.8 1.0

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

0 0.60.2 0.8 1.0

0.5

1.0

x'

y'

0.40

t' = 0.01

t' = 0.10

t' = 0.20

t' = 0.246

t' = 0.248

A.

B.

C.

D.

E.

Figure 10. Distribution of damage throughout the plate for a non-dimensional von Mises yield stress σ ′y = 0.2520 at various non-dimensional times t′.

(a) t ′ = 0.01; (b) t ′ = 0.10; (c) t ′ = 0.20; (d) t ′ = 0.246 and (e) t ′ = 0.248.

Complete damage through the plate midpoint profile exists at this

time (Figs 9c and 10). Like case 3, the von Mises stress through the

plate centre indicates perfect plasticity by t ′ = 0.25.

An important point of interest is determining the critical yield

stress that marks the transition between relaxation of stresses and

failure. Fig. 11 gives the non-dimensional failure time t ′f as a func-

tion of the non-dimensional yield stress σ ′y from a series of numeri-

cal simulations. We estimate a non-dimensional critical yield stress

of approximately 0.42 based on the asymptotic behaviour of the fail-

ure time curve. When the yield stress is less than this critical yield

stress, the plate will fail. Conversely, where the yield stress is greater

than the critical yield stress, stresses within the plate will relax to the

yield stress and damage formation will cease. The non-dimensional

critical yield stress of 0.422 did not produce failure by t ′ = 106. At

this time, the α per time step was 5 orders of magnitude less than

the amount needed to increase α to 1, with α decreasing with each

time step as stresses relaxed closer to the yield value. Therefore, we

consider this estimate to be reasonable.

C© 2006 The Authors, GJI, 166, 1368–1383


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0.1

1

101

102

103

0.2 0.3 0.4

t'f

'y

Figure 11. Dependence of the non-dimensional failure time t ′f defined in

eq. (38) on the non-dimensional yield stress σ ′y for the plate flexure problem.

Failure times determined by numerical simulations obtained from the finite

element modelling.

5 D I S C U S S I O N A N D C O N C L U S I O N S

The four plate flexure cases show a wide range of behaviour, from

slow relaxation to rapid failure. Since the simulations begin with the

same distributions of strain and stress, as well as the same elastic

parameters, the differences in their behaviour are due solely to the

different yield stresses and the subsequent damage evolution. The

ability to simulate a wide range of behaviour over time periods that

differ by many orders of magnitude demonstrates the versatility of

this method.

Our cases 3 and 4 simulate failure, but on timescales that differ

by 2 orders of magnitude. As the material weakens, the deformation

grows and stresses increase towards the plate core as the material

becomes unstable. Failure accelerates for both cases when the plate

deflection w′ [given by eq. (26) for the undamaged case] reaches a

value in the 0.02 to 0.025 range. Although case 2 reaches a deflection

in the same range, it preserves an elastic core and is able to relax the

stresses to the yield stress. Therefore the rate of stress relaxation due

to damage and the accompanying strain rate have a direct impact on

whether the material will fail or not.

An inspection of the von Mises stress profiles through the mid-

line of the plate gives a clear indication of whether the plate will

relax or fail (Fig. 12). In case 2 illustrated in Fig. 12(a), the max-

imum value of the von Mises stress continually decreases to the

yield stress. Although the stresses may increase at any particular

point the maximum von Mises stress in the plate profile contin-

ues to decrease. Stress relaxation occurs through damage. If we

interpret the increase in the damage parameter as the amount of

microfracturing, this process involves energy dissipation in sur-

face energy and kinetic energy as acoustic emissions (Scholz 2002,

pp. 29–30). In case 3 illustrated in Fig. 12(b), the maximum von

Mises stress in the plate profile decreases at first, but after reaching

a minimum the maximum stress in the profile begins to increase

toward the centre. This change in the slope of the maximum von

Mises stress is a precursor to the impending failure of the material.

Turcotte & Shcherbakov (2005) found that the value of the char-

acteristic time for damage td consistent with the aftershock decay

(M > 2.5) following the M = 7.3 Landers (California) earthquake

0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

t' = 0t' = 2t' = 12t' = 20t' = 30t' = 38

'vm

y'

0.0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

t' = 0t' = 200t' = 1000t' = 5000t' = 10000t' = 60000

'vm

y'

A.

B.

To failure

To relaxation

Figure 12. Evolution of the von Mises stress at its centre x ′ = 0.5 for

various non-dimensional times t′. (a) A case of stress relaxation for a non-

dimensional von Mises yield stress σ ′y = 0.4409. (b) A case of failure for a

non-dimensional von Mises yield stress σ ′y = 0.3779.

(June 28, 1992) is td = 4 s. For cases 1 and 2 we find typical non-

dimensional relaxation times t ′r to be 104 to 105. Taking td = 4 s,

we find the corresponding relaxation times tr to be 10 to 100 hr. For

cases 3 and 4 we find the non-dimensional failure times t ′f to be 39

and 0.248. Again, taking td = 4 s we find the corresponding failure

times to be 156 and 1 s.

We use the von Mises stress criterion, which is commonly applied

to the onset of plasticity, for the onset of inelastic behaviour. This

criterion takes into account only the magnitude of the deviatoric

stress, and not whether overall differential stress is compressive

or tensional. It is well known that rocks are stronger under com-

pression than tension especially under low confining pressures. The

strength of most materials under uniaxial loading is 10–20 times

greater in compression than extension. However, in triaxial tests the

differential stresses for shear fracture under extension are of a simi-

lar order of magnitude to those under compression (Paterson 1978,

p. 22). Lyakhovsky et al. (1997b) examined the results of 4-point

C© 2006 The Authors, GJI, 166, 1368–1383


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beam tests of Indiana limestone and observed that under a confining

pressure of 20 MPa, the differences in the values for Young’s mod-

ulus and the shear modulus at the transition from shortening to

extension in the tensile side of the beam were only ∼14 per cent.

This confining pressure corresponds to a lithostatic pressure of less

than 1 km depth for typical rocks of the continental crust. Addition-

ally, deformation in the lithosphere is frequently accommodated by

slip on existing faults. Therefore, under most geological conditions

where lithospheric deformation is taking place, the difference in

strength under extension and compression is not significant. Un-

der such conditions of confining pressure, all components are in

compression and deformation is dominated by shear fracture and

slip. Based on this observation and for simplicity, we assume no

difference in yield stress.

This modelling of brittle flow can be applied to deformation

within the brittle lithosphere in response to deviatoric stresses.

Damage in this case is largely associated with the slip on faults

and cataclastic flow. The plate bending that we describe can be

applied to flexure of the lithosphere at subduction zones and sub-

sequent seismicity in the hinge region. It can also be applied to

folding in the elastico-frictional regime. We achieve a wide range of

behaviour in our simulations in response to body forces with identi-

cal initial conditions, with the only difference being the yield stress.

This methodology can be readily adapted to include more complex

models of lithospheric deformation.

A C K N O W L E D G M E N T S

We wish to acknowledge the valuable input from Greg Lyzenga in

helping with GeoFEST v. 4.5. We also extend our gratitude to our

reviewers, Vladimir Lyakhovsky and Boris Kaus, for their insightful

critique and comments that helped improve this manuscript. This

research was supported by the National Science Foundation under

Grant ATM 0327571.

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