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
http://www.elsevier.com/locate/probengmechmailto:[email protected]://dx.doi.org/10.1016/j.probengmech.2007.10.001http://dx.doi.org/10.1016/j.probengmech.2007.10.001mailto:[email protected]://www.elsevier.com/locate/probengmech -
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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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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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M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335 27
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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M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335 31
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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32 M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335
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.
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M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335 33
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
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34 M. Bruggi et al. / Probabilistic Engineering Mechanics 23 (2008) 2335
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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