Reliability analysis of fracture in piezoelectric components with a random microstructure

Reliability analysis of fracture in piezoelectric components with

a random microstructure

Citation for published version (APA):

Verhoosel, C. V., & Gutiérrez, M. A. (2009). Reliability analysis of fracture in piezoelectric components with a random microstructure. In X. Furuta, X. Frangopol, & X. Shinozuka (Eds.), Proceedings of the 10th International Conference on Structural Safety and Reliability (ICOSSAR10), 13-17 September 2009, Osaka, Japan (pp. 1518-1525). Taylor and Francis Ltd..

Document status and date: Published: 01/01/2009

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Reliability Analysis of Fracture in Piezoelectric Components with a Random

Microstructure

C.V. Verhoosel

& M.A. Guti´errez

1_{Fac. of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands}

2_{Fac. of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Delft, The Netherlands}

Keywords: Piezoelectricity, Micro electromechanical systems (MEMS), Reliability analysis

ABSTRACT: The pronounced piezoelectric effect on small length scales is a driving force for the miniaturi-sation of electromechanical components. With the decrease of component sizes, controllability of production processes becomes increasingly difficult. As a consequence the reliability of these components is significantly influenced by microscopic imperfections. In this contribution, microscopic imperfections in lead zirconate titanate (PZT) are studied. Numerical homogenisation is used to derive random fields for bulk and elastic properties of the piezoelectric material. Stochastic finite element simulations are performed to gain insight in the reliability of a prototypical micro component.

1 INTRODUCTION

Piezoelectric ceramics are materials that exhibit rel-atively large deformations through application of an electric field and vice versa. This electromechani-cal coupling makes piezoelectrics suitable for many applications, such as micro electromechanical sys-tems (MEMS). The pronounced piezoelectric effect on small scales is one of the driving forces of minia-turisation of piezoelectric components. Since con-trollability of production processes of (piezoelectric) components on this small scale is more difficult than on larger scales, the structural properties of MEMS are subject to relatively large uncertainties. These un-certainties can have a significant influence on both the performance and reliability of the components. Nu-merical tools for assesing these issues are important for the further development of piezoelectric microsys-tems.

In this contribution the fracture process of a thin PZT (lead zirconate titanate) specimen is studied. In section 3, the microstructure of the considered spec-imen is characterised using scanning electron mi-croscopy (SEM). On the basis of the experimen-tally observed microscale characteristics, finite ele-ment models that locally represent the microstructure are constructed in section 4. Numerical homogenisa-tion is then performed in order to derive bulk and co-hesive properties of the macroscale specimen. After

having constructed the random fields for the bulk and cohesive properties of the specimen, a macroscopic finite element analysis is performed to determine the probability of failure of the specimen. The stochas-tic finite element analysis required for this reliabil-ity analysis is discussed in section 5. The results of numerical experiments are presented in section 6 and some conclusions are drawn in section 7.

2 PROBLEM STATEMENT AND MODELLING APPROACH

The specimen considered in this contribution is a film of PZT (Figure 1), with a mixture corresponding to the morphologic phase bound (MPB), where the strongest piezoelectric effect is observed. The speci-men is not poled, hence no bulk piezoelectric effect is present. Since the poling process has no intrinsic ef-fect on the microstructural geometry, the absence of a global piezoelectric effect is assumed to be of minor importance for the study of microstructural imperfec-tions. The specimen has a width and height of approx-imately 4.5 mm and 7.0 mm and is produced using mi-cro moulding (Rosqvist and Johansson 2002). This is a relatively cheap technique compared to other exist-ing methods and is recently under investigation for application to MEMS components.

For the specimen considered, significant spatial variations in the microstructure are observed

experi-Safety, Reliability and Risk of Structures, Infrastructures and Engineering Systems – Furuta, Frangopol & Shinozuka (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-47557-0

18µm

Substrate PZT sample 4.5 mm

7.0 mm

Figure 1. Schematic representation of the considered spec-imen.

mentally. The most important variations observed are: 1) a variation in average grain size; 2) a variation in porosity. Production techniques exist that offer bet-ter controllability of the microstructural imperfections than micro moulding. Although the influence of the microscale imperfections on the reliability of piezo-electric components will then be smaller, their influ-ence is likely not to be negligible. The framework considered here is therefore also useful for other pro-duction techniques.

Incorporation of the microscopic imperfections in a numerical model can be accomplished in several ways. One approach is to take into account the full microstructural geometry. This approach is, however, impractical due to the computational effort that would be required. Alternatively, the microstructural vari-ations can be incorporated in a macroscopic model by means of numerical homogenisation (Kouznetsova et al. 2002). Such an approach is commonly applied in a deterministic setting.

In this contribution the benefits of homogenisation are exploited in order to obtain information regard-ing the reliability of the specimen. The macroscopic component is modelled using a plane stress finite ele-ment model, requiring relatively few degrees of free-dom. The microstructural complexity is described by means of random bulk and cohesive properties, which are derived from the microscale.

3 CHARACTERISATION OF MICROSTRUC-TURAL GEOMETRY

3.1 Microscopic observations

The microstructure of the considered specimen is characterised on the basis of SEM images. The aver-age grain size is studied on the basis of the top-view of the specimen, hence assuming this surface to be char-acteristic for the internal geometry. As shown in Fig-ure 2, the top view is studied using 20× 20µm win-dows. The number of grains in each window, ntop(x),

is taken as an indirect measure of the local average grain size. Matlab’s image processing toolbox is used to pronounce the grain boundaries in the SEM im-ages. Grains surpassing a single image boundary are counted as a half, images surpassing two boundaries are considered to only have a quarter contribution.

20µm

105µm

Figure 2. Schematic representation of the SEM images used to characterise the microstructural geometry.

The porosity of the specimen is observed by SEM images of the cross-section shown in Figure 2. The cross-sectional images have a width of approximately 105 microns. Using various image manipulations, these SEM images are transformed into black-white images. From the latter images the cross-sectional pore ratio, pcs(x), is directly obtained.

3.2 Generation of random fields

It should be emphasised that the microstructural ge-ometry as observed above is fully dependent on the specimen considered. Production of another speci-men and generating the same images would yield completely different images. These differences are a consequence of the microstructural randomness present in the problem. A realistic description of the microstructural geometry therefore requires random fields. The number of grains in the top-view windows, for example, can be described by a random field

ntop= ntop(x, ˜θ), (1)

with ˜θ being an arbitrary random variable used to in-dicate the random nature of the field. Discretisation of this random field requires the measurement of some statistical properties of the specimen. Under the as-sumption of ergodicity (i.e. the statistical mean is ap-proximated by the spatial mean), the mean is directly

obtained by µntop = E  ntop(x, ˜θ)  ≈ 1 Ntop Ntop

∑

with Ntop being the total number of top view

im-ages (40 in this case). In this expression, θ0 is one

realisation of the random variable ˜θ, indicating that the observations are carried out on a single spec-imen. The covariance is obtained by assuming the random field to be homogenous and isotropic (with di j= xj− xi ) as Cntop(x1, x2) = E  ntop(x1, ˜θ)ntop(x2, ˜θ)  −µ_n2_top = 1 Ntop Ntop

∑

∑

ntop(xi,θ0)ntop(xj,θ0) −µn2top.

(3)

Obviously, the finite number of measurement points (images) yields a finite number of inter point dis-tances, di j. Moreover, the covariance can only be

de-termined accurately if a significant number of point couples with the same distance is present. As a con-sequence, the covariance can only be determined for a finite number of distances as indicated in Figure 3. The statistical properties are given in Table 1.

As can be seen in Figure 3, the correlation length associated with the number of grains in the top view is of the same order of magnitude as the used win-dow size. In the remainder of this work, the number of grains is therefore described by a stationary spatially uncorrelated random field with a Poisson distribution, with coefficient equal to the mean. As a consequence the standard deviation is slightly overestimated.

The correlation length associated with the poros-ity is considerably larger than the window size. The porosity is therefore described by a spatially corre-lated lognormal random field. In order to determine the discretisation of this random field, an autocorrela-tion funcautocorrela-tion of the form

ρpcs(kx2− x1k) = exp  −kx2− x1k lc  (4)

is fitted to the experimental data in Figure 3. The cor-relation length is given in Table 1. With the experi-mentally obtained mean and covariance, the random field (1) is discretised by rewriting the lognormal ran-dom field as

pcs(x, ˜θ) = exp G(x, ˜θ)

, (5)

with G(x, ˜θ) being an underlying Gaussian random

field. The mean and covariance of this underlying random field are analytically related to these of the

Mean Std. Dev. Char. Len. ntop 72 7.8

pcs 4.20 % 1.97 % 614 µm

Table 1. Statistical properties of the random fields describ-ing the micro structural geometry.

ρpcs (fitted) ρntop (measured) ρpcs (measured) kx2− x1k (µm) C o rr el at io n 2000 1500 1000 500 0 1 0.8 0.6 0.4 0.2 0

Figure 3. Autocorrelation data (function) as observed from the micro structure SEM images.

lognormal field (Der Kiureghian and Liu 1986). The Karhunen-Loeve expansion of this underlying field

G(x, ˜z) =µG+ m

∑

is obtained by numerically solving the Fredholm equation (Guti´errez and Krenk 2004)

Z

ΩCG(x1, x2)gG,i(x2)dx2=λG,igG,i(x1). (7)

The approximation of the random field (1) is then ob-tained by application of (5) to yield

pcs(x, ˜z) = qµpcs 1+V2 pcs m

∏

with{˜zi} being a set of standard normal random

vari-ables.

3.3 Local representation of the microstructure To derive random fields for the elastic and bulk prop-erties by means of numerical homogenisation, it is re-quired to locally reproduce the microstructure, such that microscale finite element simulations can be per-formed. The microscale model considered for ho-mogenisation is a three-dimensional specimen with the dimensions of the considered window and the thickness of the specimen, as schematically shown in Figure 6. The geometry is assumed to be periodic in the in-plane directions.

The granular microstructure of the specimen is re-produced by means of a Voronoi tesselation (Okabe

Figure 4. Schematic representation of the micro structure representative volume element (RVE), used for numerical homogenisation.

et al. 1992). Such a tesselation mimics isotropic grain growth. The in-plane geometrical periodicity is guar-anteed by the usage of a geometrically periodic set of nucleation points.

To locally match the number of grains in the top view, nucleation points for the Voronoi tesselation are sequentially added. The number of grains in the top view are counted after the addition of a nucle-ation point. Once the number of grains in the top-view equals the required value, the addition of nucleation points is stopped.

The porosity is reproduced by randomly removing some of the grains. The number of grains to be re-moved is determined by a Poisson distribution with parameter, λ = floorpcs(x, z) · ngr



, with ngr being

the total number of grains in the Voronoi tesselation. The grains to be removed are selected randomly. Once a set of grains is removed, the average cross-sectional porosity is computed by consideration of many differ-ent cross-sections and computing the average poros-ity. If this average cross-sectional porosity matches the target value with a specified tolerance, the re-moved grains are assumed to appropriately represent the local micro structure. If the porosity is not rep-resented correctly, the procedure is repeated by the removal of a new set of grains.

As an example, the microstructure corresponding to the mean realisation with ntop= 72 and pcs= 4.2 %

is shown in Figure 4. Some of the cross-sections used to compute the pore ratio are shown as well.

4 CONSTRUCTION OF THE RANDOM FIELDS OF THE BULK AND COHESIVE PROPERTIES 4.1 Numerical homogenisation

Numerical homogenisation is used to determine the homogenised bulk and cohesive properties of the microstructural polycrystals. The polycrystals are loaded in the x-direction, while the top and bottom surface are assumed to be traction free, according to the macroscopic plane stress condition (see Figure 6). In order to remove rigid body translations and rota-tions, control point 1 is fully fixed, point 2 is

con-E 82.3 GPa

ν 0.36

tult 80.0 MPa

G_{c 2.34 N/m}

Table 2. Parameters used for microscale simulations.

u7(µm) F (N ) 0.2 0.15 0.1 0.05 0 0.012 0.009 0.006 0.003 0

Figure 5. Force-disaplcement curves for two realisations of the microstructure.

straint in the y- and direction, point 4 in x- and z-direction and point 5 in x- and y-z-direction. Moreover, periodicity of the displacement field is assumed for the in-plane surfaces (e.g. u7 = u2+ u5+ u4, with u= (u, v, w) being the displacement vector).

The quasi-static behaviour of the microstructure in the absence of body forces is governed by the partial differential equation

div(σσσm) = 0, (9)

withσσσmdenoting the Cauchy stress tensor on the

mi-croscale. The finite element method is used to solve this partial differential equation in combination with constitutive laws for the bulk material as well as for intergranular cracks (Verhoosel and Guti´errez 2009). Interface elements are used to model these cracks. Note that piezoelectric ceramics normally require the solution of a coupled problem, in which the electric field is solved for as well. Since an unpoled piezo-electric component is considered, a purely mechani-cal solution suffices in this case. The most important parameters used for the simulations are given in Ta-ble 2. The force-displacement curves for two different polycrystal realisations are shown in Figure 5.

The average strain in x- and z-direction are defined as εxx= Z Ωrve εxxdΩ= u7 brve and εzz= w7 hrve , (10)

respectively. Following from the Hill energy condi-tion (Hill 1963) (i.e. the virtual work performed by the homogenised quantities matches that of the mi-croscopic field quantities), the corresponding expres-sion for the average stress in x-, y- and z-direction are

brve x z y hrv e brve F F 6 3 7 4 8 2

Figure 6. Schematic representation of the microscale model used for homogenisation.

obtained as

σxx=

F brvehrve

, σyy= 0 and σzz= 0. (11)

The effective modulus of elasticity and Poisson ratio then follow from

E=∂σxx ∂εxx = 1 brve ∂F ∂u7 and ν = −∂w7 ∂u7 , (12)

evaluated in the undeformed state. Similarly, a ho-mogenised fracture strength and fracture toughness are obtained by evaluation of

tult= Fmax brvehrve and Gc= 1 brvehrve ∞ Z 0 F du7. (13)

Given a force-displacement curve as shown in Figure 5, the homogenised bulk and cohesive properties can be computed using (12) and (13), respectively.

4.2 Construction of random fields for the macro-scopic properties

In the previous sections it has been outlined how the experimentally observed random microstructure can be represented. Moreover, the homogenisation of bulk and cohesive properties was discussed. As mentioned in section 2, the goal of the homogenisation proce-dure is to find random fields for these properties. Although the material properties could not be mea-sured directly, an indirect measurement is obtained by means of the outlined homogenisation procedure.

Derivation of the random fields for the material properties requires computation of the mean and co-variance of these properties. As for the construction of the geometry random fields this is done using a moving window technique. The homogenised bulk and cohesive properties are therefore computed on a square grid using the outlined homogenisation proce-dure. Once the data is available, the mean and covari-ance are computed as in (2) and (3). The results are given in Table 3 and Figure 7.

Once the mean and covariance function of the bulk and cohesive random fields are obtained, a dis-cretisation of these fields is obtained by means of a

Mean Std. Dev. Char. Len. E 27.4 GPa 1.56 GPa 291 µm

ν 0.189 0.007 1658 µm t_ult 43.8 MPa 8.18 MPa 428 µm

G_{c 1.2 N/m 0.35 N/m 328} µ_m

Table 3. Statistical properties of the random fields describ-ing the macroscale bulk and cohesive properties, as ob-tained by numerical homogenisation.

ρG_c (fitted) ρtult (fitted) ρν (fitted) ρE (fitted) ρG_c (measured) ρtult (measured) ρν (measured) ρE (measured) kx2− x1k (µm) C o rr el at io n 3000 2500 2000 1500 1000 500 0 1 0.8 0.6 0.4 0.2 0

Figure 7. Autocorrelation data (function) as observed from the micro structure SEM images.

Karhunen-Loeve expansion. As for the geometric ran-dom fields, a stationary lognormal distribution is as-sumed. The Karhunen-Loeve expansion is then found by first determining the field of the underlying Gaus-sian process, to finally and up with

E(x, ˜z) = qµE 1+V2 E m

∏

for the modulus of elasticity. Similarly, fields are de-rived for the Poisson ratio, the fracture strength and the fracture toughness. The parameters for these fields are given in Table 3.

5 MACROSCOPIC RELIABILITY ANALYSIS 5.1 Macroscopic equilibrium

As described in section 2, a macroscopic reliability analysis is performed using the previously obtained random fields. The macroscale is modelled by means of a partition of unity-based finite element model (Babuska and Melenk 1997). The key feature of this model is that the partial differential equation

div(σσσ) = 0 (15)

is discretised using C0-continuous elements, en-hanced with Heaviside functions to mimic discrete cracks. A plane stress isotropic elastic law is used to represent the bulk constitutive behaviour. The mod-ulus of elasticity and Poisson ratio are represented

by the random fields discussed. The cohesive be-haviour of the cracks is governed by a commonly used traction-opening law (Wells and Sluys 2001). The fracture strength and fracture toughness required for this law, are modelled by the random fields dis-cussed previously.

The partition of unity framework yields a nonlinear system of the form

fint(a, z) =λˆf, (16)

with fint being the discrete internal force vector that

depends on the nodal displacement vector a and on a realisation of the random vector ˜z. The parameterλ denotes the loadscale and ˆf is a load vector represent-ing the loadrepresent-ing conditions of the system.

In order to determine the equilibrium path(λ, a) of

the nonlinear system, an additional constraint equa-tion is required. In this contribuequa-tion, dissipaequa-tion-based solution control (Verhoosel et al. 2008) is applied by means of the constraint equation

g= 1

2ˆf

T_{(λ∆a−∆λa) −∆τ. (17)}

Using this constraint, the complete equilibrium path can be determined in a stepwise fashion, resulting in a set of equilibrium positions{(λ, a)i}.

A typical quantity of interest is the maximum load. In this work, the maximum load is defined as the largest loadscale in the set of solution points

λmax= max [{λi}] . (18)

The stochastic finite element methods considered here rely on the computation of the sensitivity of this max-imum load with respect to the material parameters.

5.2 Solution sensitivity

The sensitivity of the equilibrium solution can be ob-tained by differentiation of equation (16) and (17) to the random variables zi to yield a linear system of

equations (Guti´errez and De Borst 1999). Solving this system yields the sensitivities of the solution points to the random variables.

Computation of the solution point sensitivities re-quires the assembly of the derivative of the internal force vector to the random variables, zi. This requires

the computation of the sensitivities of bulk stresses and interface tractions in all integration points. Since analytical laws are considered on that level, analytical expressions for these derivatives are obtained. In ad-dition, the sensitivities of history parameters (used to distinguish between the cases of loading and unload-ing) need to be incorporated. Another contribution to the sensitivity of the internal force vector is caused by the sensitivity of the crack path. Finite difference

simulations show that the influence of the crack path on the sensitivity of the internal force vector is small. Hence, in this work the sensitivity of the crack path is not incorporated.

The sensitivity of the maximum load follows di-rectly from (18) as ∂λmax ∂zi = ∂λi ∂zi , with i satisfying (18). (19)

Strictly speaking this means that the maximum al-ways occurs at the same amount of dissipation, which is not fully correct. Finite difference computations, however, demonstrate that the additional sensitivity due to the shifting of the maximum load is limited in the cases considered. In cases where this shift turns out to be important, a better definition of the ultimate load is required. This definition can, for example, be based on a quadratic interpolation of the ultimate load (Guti´errez 1999).

5.3 Stochastic finite element analysis

In this contribution sensitivity-based methods for un-certainty and reliability analysis are considered.

A first-order approximation of the mean and stan-dard deviation of the maximum load is obtained using the perturbation method (Guti´errez and Krenk 2004)

µ_λmax=λmax and σ_λ2_max =

m

∑

It should be noted that for the coefficients of varia-tion considered, the approximavaria-tion of the mean will be inaccurate. The standard deviation can, however, be used to get a good indication of the relative impor-tance of the various sources of randomness.

Reliability analysis is in this work performed using the first-order reliability method (FORM). The prob-ability of the occurrence of a maximum load below

λ∗

max is approximated by determination of the

small-est (in magnitude) point z∗on the limit state function g(z∗) =λmax(z∗) −λmax∗ = 0. (21)

This point is generally referred to as the design point and is found using the sensitivity-based HL-RF algo-rithm (Liu and Der Kiureghian 1991). In the case that a unique design point is found, a first-order approxi-mation of the probability of occurrence of a maximum load smaller thanλ_max∗ is given by

Pr[λmax(˜z) <λmax∗ ] ≈Φ(− kz∗k), (22)

with Φ being the cumulative density function of the standard normal distribution.

It is not obvious that a unique design point exists. For this reason, the HL-RF algorithm is used with var-ious initial values for the random vector ˜z. The points where the axes of the random space cross the limit state are used as initial points.

λ_Fˆ

∆

Figure 8. Specimen of 2× 2 mm with an initial notch (with

radius 20µm), loaded in tension. The opening of the initial notch is denoted by∆.

6 NUMERICAL EXPERIMENTS

In order to limit the amount of Karhunen-Loeve terms required for an accurate representation of the ran-dom fields, numerical experiments are performed on a specimen that is smaller than the one used for the construction of these fields. The number of Karhunen-Loeve modes, m, is determined by truncation the Karhunen-Loeve eigenvalues (the λ’s in equation (14)) at 10 percent of the maximum eigenvalue. Since the same set of random variables is considered for all random fields, the field with the smallest correlation length (in this case the modulus of elasticity) deter-mines the number of terms required. From this re-quirement it follows that 18 Karhunen-Loeve modes are used.

A square specimen of 2× 2 mm with a 20µm ra-dius notch is loaded in tension (Figure 8). The spec-imen is discretised using 6606 elements, leading to a system of 6844 degrees of freedom. All points on the top edge are moved upward with the same dis-placment by the external force λF, with ˆˆ F = 4 N.

The opening of the notch, ∆, is used to construct a loadscale-displacement diagram (see Figure 9). From this figure, the deterministic maximum loadscale is obtained asλmax= 8.682 (the maximum load is

there-fore 34.73 N).

To quantify the relative importance of the various sources of randomness, the coefficients of variation of the ultimate load as a consequence of each of the random fields are computed using the first-order per-turbation method. The results are given in Table 4. As can be seen, the coefficient of variation of the ultimate load is approximately 9.9 percent. From the individ-ual contributions to this uncertainty, it is observed that the randomness in the fracture strength has the most significant influence.

The probability of occurrence of an ultimate load,

∆(µm) λ 0.1 0.08 0.06 0.04 0.02 0 10 8 6 4 2 0

Figure 9. Loadscale versus the opening of the notch for the deterministic problem.

Source of randomness V_λmax Modulus of elasticity 0.021 Poisson ration 0.002 Fracture strength 0.085 Fracture toughness 0.037 Combined 0.099

Table 4. Coefficient of variation of the ultimate load for various sources of randomness. The results are obtained using the first-order perturbation method.

λmax, below λmax∗ = 6.5 is determined using the

first-order reliability method. Using the deterministic problem (z= 0) as a starting point for the HL-RF

al-gorithm, the design point is found askz∗k = 2.28. An

approximation of the failure probability is then ob-tained by evaluation of (22). The probability of failure is then found as 0.011.

In order to determine if there is only a single design point, the HL-RF algorithm is also started with differ-ent initial points. The point where the z1-axis crosses

the limit state surface (at z1= −5.1) is used as one

initial point. In that case, the HL-RF algorithm con-verges to the same design point as found above. With using the intersection points of the other axes with the limit state surface as initial points, the HL-RF algo-rithm also converges to the same design point. From this it can be concluded that for the problem con-sidered a unique design point exists. This is primar-ily caused by the presence of the initial notch, which forces the fracture pattern to be independent of the randomness.

7 CONCLUDING REMARKS

A homogenisation framework is proposed to study the influence of microscopic imperfection on the reliabil-ity of miniaturised components. The main idea of the work is to describe the material properties of the spec-imen by random fields, such that stochastic finite ele-ment methods (SFEM) can be applied to gain insight

in the effect of uncertainties. The random fields for the bulk and cohesive properties are constructed us-ing models for the microscale.

The framework is applied to an unpoled lead zir-conate titanate (PZT) specimen, produced using mi-cro moulding. For this specimen it is shown that the randomness in the microstructure has a signifi-cant influence. A relatively strong influence of crostructural imperfections is also expected for mi-croscale components using other materials (produced using other production processes). The framework presented here can also be applied to these different cases.

The accuracy of the results presented in this work strongly depends on the quality of the experimen-tal observations. The results presented in this paper can be further improved by adding the information of more SEM images.

In this paper a fixed value is taken for the size of the microstructures used for homogenisation. This size is taken such that the homogenised response de-pends on the local value of the porosity and grain size, but is independent of the realisation of the mi-crostructure. A problem here is that, strictly speaking, the required size depends on the porosity and grain size. Since these two quantities are random, the mi-crostructure size preferred for homogenisation is ran-dom as well. Incorporation of this ranran-domness is part of future work.

The numerical simulations presented in this paper demonstrate that stochastic finite element methods can be used to gain insight in the effect of uncertain-ties. For the considered numerical experiment, only a single fracture mode is found. The proposed methods can, however, also be applied to cases in which multi-ple fracture modes are present. The determination of appropriate starting points for the HL-RF algorithm then becomes important.

ACKNOWLEDGEMENTS

The MicroNed programme (part of the BSIK pro-gramme of the Dutch government) is acknowledged for supporting the research of Clemens Verhoosel. Joost van Bennekom is acknowledged for providing the SEM images.

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