Cohesive Crack Propagation in a Random Elastic Medium

8/13/2019 Cohesive Crack Propagation in a Random Elastic Medium

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Probabilistic Engineering Mechanics 23 (2008) 2335

www.elsevier.com/locate/probengmech

Cohesive crack propagation in a random elastic medium

M. Bruggia, S. Casciatib, L. Faravellia,

aDepartment of Structural Mechanics, University of Pavia, via Ferrata 1, 27100 Pavia, ItalybASTRA Department, School of Architecture, University of Catania, via Maestranze 99, 96100 Siracusa, Italy

Received 2 April 2007; received in revised form 28 September 2007; accepted 1 October 2007

Available online 12 October 2007

Abstract

The issue of generating non-Gaussian, multivariate and correlated random fields, while preserving the internal auto-correlation structure of each

single-parameter field, is discussed with reference to the problem of cohesive crack propagation. Three different fields are introduced to model

the spatial variability of the Young modulus, the tensile strength of the material, and the fracture energy, respectively. Within a finite-element

context, the crack-propagation phenomenon is analyzed by coupling a Monte Carlo simulation scheme with an iterative solution algorithm based

on a truly-mixed variational formulation which is derived from the HellingerReissner principle. The selected approach presents the advantage

of exploiting the finite-element technology without the need to introduce additional modes to model the displacement discontinuity along the

crack boundaries. Furthermore, the accuracy of the stress estimate pursued by the truly-mixed approach is highly desirable, the direction of crack

propagation being determined on the basis of the principal-stress criterion. The numerical example of a plain concrete beam with initial crack

under a three-point bending test is considered. The statistics of the response is analyzed in terms of peak load and loadmid-deflection curves, in

order to investigate the effects of the uncertainties on both the carrying capacity and the post-peak behaviour. A sensitivity analysis is preliminarily

performed and its results emphasize the negative effects of not accounting for the auto-correlation structure of each random field. A probabilistic

method is then applied to enforce the auto-correlation without significantly altering the target marginal distributions. The novelty of the proposed

approach with respect to other methods found in the literature consists of not requiring the a priori knowledge of the global correlation structure

of the multivariate random field.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Multivariate non-Gaussian random fields; Auto-correlation; Cohesive crack propagation; Truly-mixed finite-element method; Monte Carlo simulations

1. Introduction

The cohesive crack propagation problem is considered

as a suitable example of having to generate non-Gaussian

correlated random fields when considering the uncertainties

of the physical parameters. The issue arises from observing

that the simulation of non-Gaussian, multivariate random fields

with a cross-correlation structure cannot be conceived but inan approximated manner [1]. Indeed, the task of matching

the target marginal distributions conflicts with the one of

preserving the spatial auto-correlation of each single-parameter

random field. A probabilistic method for the generation

of the non-Gaussian random fields is developed starting

from the availability of traditional Gaussian random field

Corresponding author.E-mail address: [email protected](L. Faravelli).

realizations for each physical parameter, initially considered

as independent of the others. In contrast to other methods

found in the literature [24], the proposed procedure avoids

the computational burden of directly considering the global

correlation structure of the multivariate random field. In

particular, the Gaussian realizations obtained by assigning

each spectral density function are used to statistically estimate

the covariance matrix of each corresponding random field.

Thence, the auto-correlation structure of each random field is

obtained in an already discretized manner, as an alternative

to the common practice of first assigning an auto-correlation

function of exponential type and then discretizing it [2]. The

eigenvectors of each covariance matrix are then applied to

the cross-correlated, non-Gaussian entries resulting from the

inverse Nataf transform, in order to restore the auto-correlation

structure of each field. Although the latter eigenvector mapping

slightly alters the marginal distributions of the random

0266-8920/$ - see front matter c

2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.probengmech.2007.10.001
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24 M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335

variables, the results of a sensitivity analysis show that

accounting for the internal auto-correlation is fundamental in

order to obtain reliable results in terms of statistics of the

response.

The developed probabilistic method is first validated using

a simple numerical example (whose results are reported in the

Appendix), and is then applied to the problem of cohesive crackpropagation. Three different fields are generated in order to

model the spatial variability of the Young modulus, the tensile

strength of the material, and the fracture energy, respectively,

associated with the crack development. It is worth noting that

a scalar representation of the Young modulus at any point of

the body implicitly introduces an isotropy assumption, which

is in conflict with the inherent anisotropic nature of a random

medium. The finite-element discretization allows, however, to

conceive an anisotropic medium as the assemblage of isotropic

finite elements. Being a full simulation of all anisotropic elastic

and failure parameters beyond the scope of this study, a scalar

definition of the Young modulus is assigned at each point for

the sake of convenience.

The numerical study of a crack-propagation phenomenon

requires a mechanical model able to follow the crack-path

evolution, which is a priori unknown and not aligned with

the body discretization of the initial un-cracked domain.

In [5], an adaptive remeshing strategy was proposed. Further

studies aimed to limit and possibly avoid the computationally

expensive remeshing phase. The procedures developed from

these studies usually rely on either one of two alternative

strategies: the XFEM (extended finite-element method), or

the meshless approach. The XFEM method is based on a

continuous displacement formulation which needs to be locally

enriched with discontinuous modes in order to be able to cross

the existing mesh by exploiting the partition-of-unit property

of the shape functions [6]. The meshless strategy [7] seems

to be ideally tailored to handle crack-propagation problems,

but must overcome some numerical difficulties, such as the

quadrature formulas and the boundary conditions assignment,

which are easily solved by a finite-element approach.

The approach adopted in the present paper cannot be

grouped in any of the two afore-mentioned categories, since

it is based on an extension of the truly-mixed variational

formulation developed in [8] from the HellingerReissner

principle. The associated solution algorithm, which is able

to follow the a priori unknown crack path in both the pre-and post-peak regimes, was proposed and verified in [9].

This approach is chosen to be coupled with a Monte Carlo

simulation scheme because it presents several advantages.

By using equilibrated stress fields, with square-integrable

divergence and inherently discontinuous displacements, the

stress-flux continuity is imposed in an exact manner at each

load step. Furthermore, the potentially active discontinuity of

the displacements at each crack-interface element allows the

direct inclusion of a cohesive law. Within this framework,

the stress element of Johnson and Mercier [10] is selected, it

being one of the very few elements able to pass the infsup

condition required for the convergence of the method.

The problem of cohesive crack propagation in elastic media

is investigated by coupling the above-mentioned mechanical

model based on the truly-mixed formulation, and the newly

proposed probabilistic method for the generation of 2D and

multiparameter cross-correlated random fields of non-Gaussian

nature. In the following, the theoretical backgrounds of the

random field generation procedure and the truly-mixed finite-element formulation are discussed in separate sections, and

are then jointly applied to a numerical example. Within this

example, Monte Carlo simulations with finite elements are

carried out to determine the statistics of the response of a

plain concrete beam undergoing a three-point bending test.

A sensitivity analysis is performed on a limited number

of samples in order to preliminarily check the influence of

different probabilistic assumptions on the results. Finally, the

effects of the uncertainties on both the carrying capacity and the

post-peak behaviour are quantified by considering a significant

number of random field realizations.

The methodology developed in this work can be applied

for further developments within fracture mechanics of quasi-brittle materials. Indeed, both the energetic size effect (of

a deterministic nature) and the randomness in the material

properties affect the maximum load-carrying capacity of a

structural component. According to Ref. [11], for a certain class

of structures, the first factor governs the deterministic mean

of the nominal strength, while the second is responsible for

the higher order moments. As such, a probabilistic approach

is needed in order to evaluate the probability density function

of the response, in view, for example, of an estimate of the

reliability of the structure [12]. Furthermore, when considering

the microscopic origin of the crack formation, homogenization

techniques are usually applied to model a standard continuumwhich behaves like the originally micro-cracked body

[13,14]. It is, therefore, of interest to check the hypothesis of

a homogenized random field by evaluating the influence of the

spatial variability of the material properties on the response.

2. The probabilistic approach

2.1. Framing the problem

In the literature, random fields were first introduced as a 2D

natural extension of stochastic processes [1517]. Moreover,

the simulation of their realizations provided the support for the

development of stochastic finite elements [1821]. Advancedtopics include the generation of spatial-temporal wind velocity

fields[22,23]and the discretization issues in crack-propagation

analysis[24]. An extended state-of-the-art report can be found

in Ref. [25]. Refs. [14] are devoted to the development

of simulation methods for non-Gaussian processes. Very

few authors [2,4] discuss the simulation of multivariate and

cross-correlated non-Gaussian random fields and the existing

approaches are all, to some extent, approximate. For this

reason, the issue of providing a non-Gaussian nature to a cross-

correlated, multiparameter random field while preserving its

internal auto-correlation structure, is still considered an open

area of research.


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M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335 25

The selective list of references mentioned above does

not aim to entirely capture the broad spectrum of results

provided by the research activities in the related areas, but

only to represent those that directly influenced the authors

in developing the probabilistic approach followed in this

paper. In particular, one assumes that a commercial software

for the simulation of Gaussian random fields with assignedspectral density function is readily available. The task is then

to investigate how it can be conveniently exploited when

simulating cross-correlated random fields of non-Gaussian

nature. The answer found in this work consists of being able

to statistically estimate, from several independently generated

Gaussian random fields, the corresponding auto-correlation

matrices whose eigenvectors are then used to restore the

spatial auto-correlation destroyed when applying a non-linear

transformation, such as the inverse Nataf transform. This

approach avoids the computational burden of building the

global auto-correlation structure of the multidimensional and

multivariate random field. Furthermore, the auto-correlation

of each considered random field is assigned in an alreadydiscretized manner, thus avoiding the following of the standard

procedure of first assigning an auto-correlation function of

exponential type, and then discretizing it. The steps that must

be taken in order to first match the given marginal probability

distributions and cross-correlation structure, and then enforce

the spatial auto-correlation are discussed in the following sub-

section.

2.2. The proposed algorithm for the simulation of cross-

correlated non-Gaussian random fields

A multivariate (m= 3) and multidimensional (n= 2) non-Gaussian stochastic field is defined over a rectangular domainH(x), x Rn . (1)The domain is discretized into an rby s grid consistent with the

finite-element mesh, so that N= r s is the number of points towhich each random field sample of size m is assigned.

The first step is to generate a discrete sample of uncorrelated

standard Gaussian vectors. For computational convenience, the

random field vectors are organized in a matrix ofNrows andm

columns. Therefore, each row vector, U(x i ), is the uncorrelated

multiparameter Gaussian sample associated to the i th node of

the grid,i=

1, . . . ,N.

A mapping of each vector U(x i )to the correlated Gaussian

space is performed as follows

z= L U (2)L being the Cholesky decomposition of the correlation matrix

obtained by applying the regression formulas in Ref.[26] to the

values of the correlation coefficients originally prescribed for

H(x) in the non-Gaussian space. The resulting sample, z, of

correlated Gaussian variables is then transformed into a non-

Gaussian vector, by applying the inverse Nataf transformation

to each scalar component of the random field

Hh(x i )=F1

H h[(zi )] (3)

for h = 1, . . . ,m and i = 1, . . . ,N. Due to the non-linearity of the transformation, the single Hh(x)vector (i.e., thevalues assumed by each parameter in different nodes) is made

of uncorrelated random variables. Hence, a transformation

restoring the internal correlation must be performed, and it is

given by

Hh(x)=N

i=1

iHh(x i ) (4)

where iare the eigenvectors of the target auto-correlation

matrix of the single h th discretized random field. This matrix

can be either obtained by discretization of the auto-correlation

function, which is usually assumed to be of exponential type

with given correlation length, or by assigning the spectral

density function of each single-parameter random field. In the

latter manner, one can ignore the global structure of the cross-spectral density matrix and consider only each single spectral

density function, together with the correlation coefficients of

the vector of sizem assigned to each node by Eq.(2).It is worth noting that, in the above summarized procedure,

the order in which the operations in Eqs. (3) and (4)

are performed is fundamental. Indeed, if the eigenvector

mapping takes place before the inverse Nataf transformation,

the non-linearity of the latter transformation destroys the

correlated results expected from the first operation. As a

consequence, non-Gaussian cross-correlated fields made of

auto-uncorrelated entries would be generated. An iterative

procedure could be pursued by altering the auto-correlationstructure, but the inverse Nataf transformation would still

produce internally uncorrelated variables. Instead, when

the inverse Nataf transformation precedes the eigenvector

mapping, the second transformation reaches its expected goalof correlating the internal variables, whereas the marginal

distributions pursued by the Nataf transformation are just

slightly altered. This alteration is minimal if the starting point

is made of internally independent realizations. In other words,

the sequence eigenvectorNatafeigenvector would produce

less accurate results than the simplest path Natafeigenvector

does. The above statements are supported by the results of the

numerical analyses carried out in theAppendix.

3. The mechanical model

3.1. Truly-mixed finite-element formulation for cracked media

The crack-propagation algorithm used in this paper was

originally developed in [9], and it is intimately tied to the

truly-mixed finite-element formulation which is here only

briefly recalled. In the following, the governing equations are

presented in a mixed weak continuous form, which is soon after

discretized by resorting to the JohnsonMercier stress element.Adopting a classical notation, one defines, within the domain

R2, the square-integrable vector of the body loads, g , andindicates with and the unknown stress and strain tensors,

respectively.The compatibility equation (written in terms of stresses

through the elastic constitutive law, =

C

1, with C the


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fourth-order tensor of the elastic constants) is tested for a virtual

stress field, . The equilibrium equation is tested by means of

a virtual displacement field, v. The resulting weak variational

formulation reads: find( , u)S(div;)22sym[L2()]2 suchthat:

C

1

: dx+ div udx

[[u]] ( n)dx=0, S(div;),

div vdx=

g vdx= 0, v [L2()]2(5)

where one defines S(div;)22sym = { : i j = ji L2(),div [L2()]2}, andL 2()is the space of the functions thatare square integrable on the domain .

When a macro-crack propagates through the medium, the

line integral in the first row of Eq. (5) accounts for the

(cohesive) energy dissipated across the fracture, it being derived

from the GaussGreen theorem

u dx=

div udx+

u ( n)d (6)

where = 1 2, with 1,2 the two opposite sides of thecrack, and( n)is the stress flux acting on them.

In Eq.(5), the operator[[]]denotes the strong discontinuityof the quantity to which it is applied, i.e., the displacement jump

between the two opposite edges of the crack, which, under the

small displacements assumption, can be written as

u ( n)d=1

u ( n)d2

u ( n)d

=

[[u]] ( n)d. (7)

By introducing the rate independent and piece-wise linear

idealization in Fig. 1 as a cohesive law, the displacement

jumps, [[u]], can be expressed as functions of the tractionstress fluxes, ( n). Three regions (labeled A, B and C,respectively, in Fig. 1) need to be distinguished in order to

model the global behaviour of the cohesive interface. The first

(zone A) is representative of a regime where the medium is

un-cracked and the resistance, t , is larger than the currentnormal traction. After the crack initiation (zone B), an energy

release takes place driving the current normal traction to a value

that is smaller than the resistance, t. However, as long as thedisplacement jump,[[u]]n , in the normal direction with respectto the crack, is smaller than a critical value, [[u]]n , there is still aresidual cohesion between the two sides of the crack. Once this

threshold value is reached (zone C), no cohesion between the

two sides of the crack is experienced.

When deploying the described truly-mixed approach within

a finite-element discretization, the main difficulty to cope

with is due to the symmetry of the stress tensors. The two

interpolation fields of stresses and displacements must satisfy

the so-called infsup condition in order for the method to be

globally convergent. Among the few available approaches able

to pass the infsup condition in a truly-mixed setting, the

Fig. 1. Pure mode I cohesive law.

Fig. 2. The JM triangular element.

Fig. 3. Stress (circles) and displacement (squares) degrees of freedom.

Fig. 4. Deterministic geometry assumed for the numerical example of a three-

point bending test on a plain concrete beam with initial crack (dimensions inmm).

Table 1

Defining the marginal distributions of the three-variate random field

Random variable Distribution

type

Mean Coefficient of variation

Young modulus, E Lognormal

(L)

36.5

(GPa)

0.2

Tensile strength, f Weibull (W) 3.19

(MPa)

0.2

Specific fracture

energy,G

Weibull (W) 100

(N/m)

0.2


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Fig. 5. Statistics of the 1000 realizations in the first node of the grid, before

forcing the internal correlation.

JM element (by Johnson and Mercier) is selected. It consists

of a composite triangular element, which is made of three

sub-triangles (Fig. 2). Stress shape functions with complete

first-order polynomial bases are defined in each sub-triangle,

while shape functions of the same polynomial type model the

globally discontinuous displacements over the whole triangle.

This discretization results into a number of degrees of freedom

per element equal to six for the displacements, and equal to

fifteen for the stresses (Fig. 3).

The main advantage of this approach consists of not

requiring the introduction of any extra mode or shape function

in order to handle the energy dissipation across the fracture.

In fact, the displacement field is discontinuous per se and it is

Fig. 6. Statistics of the1000 realizationsin thefirst node of thegrid,accounting

for the internal correlation.

sufficient to evaluate the line integral of Eq.(5)across the crack

to take into account the entire phenomenon.

3.2. The solution algorithm

The fracture path, , being a priori unknown, the problem

is inherently non-linear. In the present paper, the solution

process is handled by an iterative algorithm, which works on a

linearization of the original problem at each load step. First, the

current mixed matrix is computed and the following linearized

matrix equationA() Bu

Bu 0

u

=

0

g

, (8)

is solved, with the blockA()evaluated by updating the line

integral in Eq.(5).


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Fig. 7. Example of one realization of the three-parameter random field: (a) Elastic modulusE(GPa), (b) Tensile strength f(MPa), (c) Fracture energy G (N/m).

A check on the elements in the regimes [B] or [C] ofFig. 1

is then performed. For each edge element on the crack, thedisplacement jump is evaluated as the difference between the

displacements of the nodes facing each other on the opposite

sides of the crack itself. If the jump is less than the critical value,

an update of type [B] is added to the mixed governing matrix.

Conversely, if the threshold jump value is exceeded, the relevant

node is added to the tail of the crack where no residual cohesion

is detected, and the two sides of the crack are independent of

each other.

This procedure must be repeated at fixed load until

convergence is achieved, meaning that the cohesive constitutive

law in Fig. 1 is exactly imposed over the entire crack. Once

achieved, a (positive) load increment is further applied to thestructure, until a proper stress average on a small contour

centered at the crack tip reaches the limit stress value, t. Whenthis occurs, the principal-tensile-stress criterion is adopted, and

the crack is allowed to propagate in the direction normal to the

maximum tensile stress.

The presented steps are repeated until no equilibrium

configurations are found. If this is the case, the maximum

load sustainable by the specimen is reached and negative

loading increments (decrements) are applied to the structure so

that it exploits its post-peak softening regime. The procedure

is stopped when failure occurs, i.e. when the residual load-

carrying capacity of the structure is null.

4. Numerical example

The classical problem of a three-point bending test on a plain

concrete beam with initial crack (Fig. 4) is considered. This ex-

ample was deterministically analyzed in [27]and probabilisti-

cally approached in [2]. In the latter reference, a non-Gaussian,

multiparameter random field was generated by building an ex-

tended covariance matrix and it was coupled with a meshless

strategy to run the Monte Carlo simulations. In the present

work, a random field generation method that avoids consider-

ing the global correlation structure is instead preferred, as em-

phasized in Section2. Furthermore, the Monte Carlo simula-

tions are performed within a finite-element scheme exploiting

the truly-mixed variational formulation described in Section3.Using the JM element inFig. 2,the values of each random

field sample are assigned to the vertices of the squares formed

by four elements. The random fields are accordingly discretized

into a grid of 10 by 40 nodes. At each node, the realizations

of three random variables corresponding to the material Young

modulus, its tensile strength and the specific fracture energy,

respectively, are specified.

Table 1 collects the statistics of the random variables

considered in the problem. In particular, the same coefficient of

variation, equal to 0.2, and the same correlation coefficient of

0.8 are assumed for all the random variables having, however,

different distribution types and mean values. The latter ones


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Fig. 8. Local remeshing for crack propagation: (a) crack path at the specimen collapse and (b) relevant stress map for stress normal to the edge (MPa).

Fig. 9. Load-relative mid-deflection statistical and deterministic curves computed over the 100 MC samples obtained from the probabilistic assumptions of cases:

(a) (a) and (b) (b) ofTable 2.

Table 2

Sensitivity analysis of the statistical assumptions, on the basis of only 100 realizations of the multiparameter random field

Case Distribution types Coefficients of variation Auto-correlation Correlation coefficients Peak load statistics

E f G E f G E f E G f G Mean (kN) Variance(kN2)

(a) L W W 0.20 0.20 0.20 Yes 0.80 0.80 0.80 62.802 36.5276

(b) L W W 0.20 0.20 0.20 No 0.80 0.80 0.80 62.6317 14.8563

(c) L W W 0.15 0.18 0.20 No 0.80 0.80 0.80 62.5745 12.8776

(d) L W W 0.20 0.20 0.20 No 0.70 0.50 0.90 62.6624 17.0197

(e) L W W 0.15 0.18 0.20 No 0.70 0.50 0.90 62.6088 14.8543

(f) L L L 0.20 0.20 0.20 No 0.80 0.80 0.80 62.6337 15.1436

(g) L L L 0.20 0.20 0.20 No 0.00 0.00 0.00 62.2682 12.1846

(h) L L L 0.20 0.20 0.20 Yes 0.00 0.00 0.00 61.4451 24.7527

reflect the physical quantities used for the deterministic study

in [27], while the second-order moments are selected as in [2]

to allow for a comparison of the probabilistic results. In [2] this

choice was motivated by a higher computational simplicity. It

is worth noting that the probabilistic approach proposed in the

present work does not have these computational limitations,

because it avoids building the global covariance matrix of

the multivariate random field. Furthermore, Ref. [2] assumes

all the variables to be lognormally distributed, while here

a Weibull distribution assumption is introduced for both the

tensile strength and the fracture energy, as suggested by most

of the general literature. The sensitivity of the results to

different probabilistic assumptions is verified in the following

Section4.2.

In order to generate each random field, the corresponding

spectral density matrix is given. The covariance matrix and the


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spectral density matrix are then handled by simply assigning

one function to each variable, rather than considering a matrix

of functions. Following the reasoning in [19,28], the spectral

density function is assumed to be of the form

G(kx , ky)

=2

dx dy

4 exp

kx dx

2

2

+ ky dy

2

2

(9)where kx and ky are the wave numbers defined in the

interval (,+), and is the standard deviation. The twoparametersdx anddy must be selected in such a way that they

are consistent with the finite-element mesh. For this purpose,

one first estimates the minimal lag of the mesh in the two

directions [28], then the Nyquist cut-off values, kxu and kyu ,

and thence

dx=

2

kxu /3; dy=

2

kyu/3. (10)

For the specific example under investigation, Eq.(10)leads to avalue of 20.257 mm in both directions, so that the associated

exponential auto-correlation function shows a length which

covers nearly three elements. The target auto-correlation matrix

of each non-Gaussian random field could then be obtained

from the discretization of the afore-mentioned auto-correlation

function. As an alternative, the following strategy is instead

adopted: 1000 realizations of the standard Gaussian process

are simulated following the spectral density scheme in Eq. (9),

and their auto-correlation matrix is afterwards estimated on a

statistical basis. This operation provides the 400 by 400 matrix

whose eigenvectors are used in Eq. (4), to give an internal

correlation to the realizations achieved after the inverse Nataf

transformation of Eq.(3).

By applying the stochastic modelling procedure proposed

in Section 2.2, a total of 1000 realizations are generated for

each random field.Figs. 5and6provide a comparison between

the assigned marginal distributions of each parameter and

those estimated from the corresponding 1000 realizations, as

achieved before and after forcing the spatial auto-correlation

by means of the eigenvectors mapping in Eq. (4), respectively.

In particular,Fig. 5provides a synthesis of the statistics of the

three variables after the inverse Nataf transformation of Eq.

(3),but before forcing the internal correlation of each random

field by Eq.(4). Instead,Fig. 6provides the final statistics of

the input parameters, thus accounting for their spatial auto-correlation. By comparing the two figures, one can observe how

the last step of the stochastic modelling procedure slightly alters

the marginal probability distributions of the parameters, but the

entity of this effect is negligible. The sensitivity analysis carried

out in the following Section 4.2 emphasizes the importance

of accounting for the auto-correlation, whose absence leads

to unreliable results in terms of statistics of the response.

Within an approximated framework, the slight alteration of the

marginal distributions is, therefore, acceptable with respect to

the advantage of accounting for the spatial auto-correlation.

Finally, the realizations generated by accounting for the spatial

auto-correlation are assigned to the corresponding nodes of

Fig. 10. Mean values of peak loads calculated on increasing number of MC

realizations compared to the deterministic value (dotted line).

Fig. 11. Standard deviation of the peak loads calculated for an increasing

number of MC realizations.

the finite-element mesh. Fig. 7 shows, as an example, one

realization of the resulting multiparameter random field.

4.1. Remark on the solution algorithm

In principle, the algorithm outlined in Section2.2can follow

any crack-path geometry, by choosing the crack-propagation

direction on the basis of the maximum tensile-stress criterion.

For this purpose, the accuracy of the stress estimate provided by

the mixed approach is higher than the other methods. However,

the computation of the line integral in Eq. (5) calls for a

slight local remeshing to align the evolving crack path with

the boundaries of the adjacent finite elements involved in thefracture. This procedure is automatic, but it slows down the

structural analysis. In view of the high computational effort

always demanded when adopting a Monte Carlo simulation

scheme, the assumption that the system is symmetric with

respect to both the geometry and the load is made.

Sufficient checks are performed to ensure the validity of

this assumption. For this purpose, the algorithm in its general

formulation, able to represent any crack trajectory, is initially

applied to random field samples of reduced size (just 100

replicates). As an example of the achieved results, Fig. 8(a)

shows, at the end of a simulation, a very small deviation of

the crack path from the straight vertical line. Fig. 8(b) plots


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Fig. 12. Load-relative mid-deflection diagrams obtained over the 1000 MC samples referring to case (a) inTable 2: (a) statistical and deterministic curves; (b)

envelope.

Fig. 13. Histogram and fitting normal probability distribution for peak loads

calculated on 1000 MC samples.

the relevant stress map at the same load step. Similar results

are obtained when using different random field realizations as

input. It can be concluded that the removal of the symmetry

assumption does not significantly improve the accuracy of the

method, but only augments its computational time. Therefore,

in order to rely on a faster algorithm, all the following

computations are carried out by a priori approximating the

crack path with a straight line. It is worth noting that, under this

assumption, the mixed approach still presents advantages with

respect to the XFEM method, since the inherent displacement

discontinuity prevents us from having to introduce extra

discontinuous modes in order to allow the crack propagation.

4.2. Preliminary studies of the uncertainty propagation

A sensitivity analysis is performed with respect to the

statistical assumptions made for the random variables involved

in the problem under consideration. The effects of changing the

data as reported inTable 2are investigated by calculating, for

each considered case, the statistics of the response. In order

to cover all the cases envisioned in Table 2 with a moderate

computational effort, the analyses are performed on only 100

realizations of each parameter random field.

The first row ofTable 2 denotes the original assumptions

inTable 1 as case (a). Case (b) differs only in not having an

auto-correlation structure. Cases (c) through (e), together with

not having an auto-correlation structure, also consider different

values of the coefficients of variation (case (c)), the correlation

coefficients (case (d)), or both (case (e)). Finally, cases (f)through (h) assume a lognormal distribution for all the variables

which are first considered as mutually correlated but without

any auto-correlation (case (e)), as independent and not auto-

correlated (case (f)), and lastly the introduction of the auto-

correlation in the independent case is evaluated (case (h)).

The results in terms of peak load statistics are reported in

the last two columns of Table 2. It can be noted that, while

the mean value is not significantly affected by the changes of

the initial assumptions, the higher order moments seem to be

more sensitive to the actual statistics of the random variables.

In particular, when the internal auto-correlation within the

realizations of each random field is not considered, a drop inthe variance of the response is observed.

Fig. 9(a) and (b) illustrate the deterministic and statistical

(mean and mean standard deviation) load-relative mid-deflection curves obtained from the cases (a) and (b) ofTable 2,

respectively. The difference between the values of the peak

load variance in the two cases is evident along the entire

curves. Moreover, the curve of the mean values resulting

from neglecting the auto-correlation structure (case (b)) does

not match the deterministic curve as good as it does when

accounting for the auto-correlation (case (a)), especially in the

softening branch.

In conclusion, the results of the sensitivity analyses justify

a certain freedom in selecting the statistical properties of the

input parameters (e.g., same coefficients of variation and same

correlation coefficients), but also emphasize the importance of

simulating each random field with an internal structure that is

in agreement with an assigned auto-correlation matrix.

4.3. Statistical analysis of the response

Figs. 10and11provide evidence that the results obtained

in the last two columns of Table 2 by considering only 100

replicates are not representative estimates of the mean and

variance of the response, respectively. Indeed, to reach a

convergence in the response statistical properties, at least 500


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Fig. A.1. Statistic elaboration over a sample of size 1000: (a) after the Nataf transformation, and (b) after the Natafeigenvector sequence.

realizations of each parameter random field must be considered.

For this reason, the analyses are now repeated for samples of

1000 realizations. These realizations are generated according

to the probabilistic assumptions of case (a) in Table 2, thus

accounting for the spatial auto-correlation structure of each

parameter random field.The Monte Carlo finite-element analyses are carried out by

running the solution algorithm described in Section 3.2 for

the 1000 realizations obtained from the simulation strategy of

Section2.2,with the marginal distributions as given in Table 1

and the spectral density function of Eq.(9).At the end of each analysis, the loaddisplacement curve

is calculated. Its statistics (mean and standard deviation)

are then computed over its 1000 realizations. In Fig. 12(a),

the resulting statistical loaddisplacement curves (mean and

mean standard deviation) are plotted together with thedeterministic diagram. A good agreement between the curve

of the mean values and the deterministic one is obtained. In

Fig. 12(b), the envelope of the curves obtained from each Monte

Carlo simulation shows that the uncertainties have remarkable

effects in the zone next to the load-carrying capacity and in the

softening branch.

The histogram in Fig. 13 gives the probability density

function (pdf) of the peak load. Among the known distribution

models, a Normal pdf with a mean of 62.50 kN and a standard

deviation of 6.81 kN is also drawn in Fig. 13as the curve that

best fits the histogram.


11/13


Fig. A.2. Influence on the statistics of the realizations of the starting auto-correlation: (a) wished (i.e., assigned as inTable A.1), (b) fully correlated, and (c)

uncorrelated.

Table A.1

Assigning the three-parameters, 2D field of theAppendixexample: (a) marginal distribution and central moments, (b) local cross-correlation coefficients, and (c)auto-correlation matrix

Random variable Marginal distribution Mean Standard deviation

(a)

X1 Lognormal 36.5 7.3

X2 Weibull 3.19 0.638

X3 Weibull 100 20

(b)

Cross-correlation coefficients, i j X1 X2 X3X1 1 0.8 0.8

X2 0.8 1 0.8

X3 0.8 0.8 1

(c)

Ra=

1 0.8 0.5 0.2 0 0.5 0.4 0.25 0.1 0 0 0 0 0 00.8 1 0.8 0.5 0.2 0.4 0.5 0.4 0.25 0.1 0 0 0 0 0

0.5 0.8 1 0.8 0.5 0.25 0.4 0.5 0.4 0.25 0 0 0 0 0

0.2 0.5 0.8 1 0.8 0.1 0.25 0.4 0.5 0.4 0 0 0 0 0

0 0.2 0.5 0.8 1 0 0.1 0.25 0.4 0.5 0 0 0 0 0

0.5 0.4 0.25 0.1 0 1 0.8 0.5 0.2 0 0.5 0.4 0.25 0.1 0

0.4 0.5 0.4 0.25 0.1 0.8 1 0.8 0.5 0.2 0.4 0.5 0.4 0.25 0.1

0.25 0.4 0.5 0.4 0.25 0.5 0.8 1 0.8 0.5 0.25 0.4 0.5 0.4 0.25

0.1 0.25 0.4 0.5 0.4 0.2 0.5 0.8 1 0.8 0.1 0.25 0.4 0.5 0.4

0 0.1 0.25 0.4 0.5 0 0.2 0.5 0.8 1 0 0.1 0.25 0.4 0.5

0 0 0 0 0 0.5 0.4 0.25 0.1 0 1 0.8 0.5 0.2 0

0 0 0 0 0 0.4 0.5 0.4 0.25 0.1 0.8 1 0.8 0.5 0.2

0 0 0 0 0 0.25 0.4 0.5 0.4 0.25 0.5 0.8 1 0.8 0

0 0 0 0 0 0.1 0.25 0.4 0.5 0.4 0.2 0.5 0.8 1 0

0 0 0 0 0 0 0.1 0.25 0.4 0.5 0 0.2 0.5 0.8 1


12/13


5. Conclusions

The problem of stochastic cohesive crack propagation is

numerically investigated by coupling a truly-mixed finite-

element approach with a Monte Carlo simulation scheme.

The formulation manages the fracture problem thanks to the

peculiar nature of the adopted discretizing fields. Discontinuous

displacements and continuous stress fluxes directly allow the

simulation of crack propagation along element boundaries.

Furthermore, a multiparameter stochastic field is introduced to

model the material properties. For this purpose, a methodology

that pursues to assign an auto-correlation structure to the

generated non-Gaussian, cross-correlated fields is developed,

without having to consider the global structure of the

multiparameter covariance matrix. A classical three-point

bending specimen made of concrete is used to perform

numerical tests on the proposed methodology. Results from

Monte Carlo analyses are shown to determine the statistics

of the response, with peculiar attention to load-crack mouth

opening diagrams and load-carrying capacity.

Acknowledgements

The authors acknowledge the grants received from the

Athenaeum research funds of the University of Catania and the

University of Pavia.

Appendix

In order to support the development of the probabilistic

approach proposed in Section 2 for the generation of cross-

correlated, non-Gaussian random fields, a simplified example

is used here to test the consequences of adopting different

options. The objective is to simulate a three-parameter, 2D

random field, with the marginal distributions, the local cross-

correlation structure, and the spatial auto-correlation assigned

inTable A.1.In particular, the auto-correlation is specified by a

15 15 matrix, corresponding to a 5 3 nodal discretization ofthe field.

The simplest way to generate the realizations of the

multiparameter random field consists of simulating a sequence

of three times 5 3, independent standardized Gaussiannumbers, which can be regarded as a realization of three

independent Gaussian fields. They must then be transformed

into non-Gaussian cross-correlated quantities, and subsequentlyinto auto-correlated fields. The two operations are performed

by the Nataf transform and by the classical mapping using the

eigenvectors of the auto-correlation matrix, respectively. The

first transformation is summarized in Eq.(6);the eigenvectors

mapping in Eq.(7).The only selectable option is in the order of

the operations.

(1) Eigenvector mapping first and then Nataf: the second

transformation destroys the expected result of the first

operation; as a consequence, one has three non-Gaussian

cross-correlated fields made of uncorrelated entries.

Since iterations are often introduced when dealing with

non-linear transformations, the auto-correlation structure

is altered to check for the consequences of a stronger

correlation. Once again the Nataf transformation results

into internally uncorrelated variables, thus showing that the

iteration path cannot be pursued within this framework.

(2) Nataf first and then the eigenvectors mapping. Fig. A.1

shows the statistic elaborations over a sample of size

1000, when only the Nataf transformation is applied (a)and after the Natafeigenvector sequence (b). The second

transformation reaches its expected goal, whereas the

marginal distributions pursued by the Nataf transformations

are just slightly altered. This alteration is minimal if

the starting point is made of internally independent

realizations. Fig. A.2 shows the influence of the starting

auto-correlation on the statistics of the realizations. It is

evident that the sequence eigenvectorsNatafeigenvectors

would produce less accurate results than the simplest path

Natafeigenvectors do.

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Cohesive Crack Propagation in a Random Elastic Medium