Probabilistic Engineering Mechanics

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1 Probabilistic Engineering Mechanics 6 (11) Contents lists available at ScienceDirect Probabilistic Engineering Mechanics journal homepage: Variability response unctions or eective material properties Sanjay R. Arwade a,, George Deodatis b a Department o Civil & Environmental Engineering, University o Massachusetts, Amherst, 13, USA b Department o Civil Engineering and Engineering Mechanics, Columbia University, 17, USA a r t i c l e i n o abstract Article history: Received 15 July 1 Accepted 8 November 1 Available online 18 November 1 Keywords: Homogenization Material properties Uncertainty quantiication Random ields The variability response unction (VRF) is a well-established concept or eicient evaluation o the variance and sensitivity o the response o stochastic systems where properties are modeled by random ields that circumvents the need or computationally expensive Monte Carlo (MC) simulations. Homogenization o material properties is an important procedure in the analysis o structural mechanics problems in which the material properties luctuate randomly, yet no method other than MC simulation exists or evaluating the variability o the eective material properties. The concept o a VRF or eective material properties is introduced in this paper based on the equivalence o elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists or the eective material properties o statically determinate structures. The VRF or eective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance o the response displacement at two points in a statically determinate beam with randomly luctuating material properties modeled using random ields. Two numerical examples are presented that demonstrate the character o the VRF or eective material properties, the method o calculation, and results that can be obtained rom it. 1 Elsevier td. All rights reserved. 1. Introduction In many solid and structural mechanics problems the material properties exhibit random, non-periodic, spatial luctuations. It is common in such problems to replace the randomly luctuating material properties with a deterministic, homogenized set o material properties. The homogenized properties can be determined by direct averaging o the randomly luctuating properties or by constraining the response o the homogenized system to be equivalent to that o the heterogeneous system in some sense such as having equivalent strain energy. Homogenization is particularly important in applying numerical techniques such as the inite element method to problems with randomly luctuating material properties since such problems typically require very reined meshes to achieve high solution accuracy i the local material property luctuations are to be represented. Homogenization is also a key part o upscaling material properties in the context o multi-scale analysis o solid mechanics problems (See, or one example, [1]) A key concept in homogenization is that o the representative volume element (RVE). The concept o the RVE is that, i material properties are homogenized over a suiciently large volume, Corresponding address: Department o Civil & Environmental Engineering, University o Massachusetts, Amherst, 13 Natural Resources Rd. Amherst, MA 13, USA. address: arwade@ecs.umass.edu (S.R. Arwade). the resulting homogenized properties approach deterministic values. In other words, the variance o the estimate o the homogenized properties approaches zero as the homogenization volume increases. Practically speaking, o course, the volume o material in any solid mechanics problem is inite, and urthermore it may be desirable in some problems to retain medium to long range random luctuations in the material properties by homogenizing over intermediate length scales. An example o the latter is evident in the approach o the moving window generalized method o cells which generates a smoothed version o random material property luctuations to ease numerical analysis []. In homogenization problems in which the material volume is smaller than an RVE, it is critical to recognize that the homogeneous problem is itsel stochastic, despite neglect o the spatial luctuations o the material properties. This stochasticity arises rom homogenizing over a inite volume so that the homogenized material properties are themselves random. An important issue, thereore, is the variance o the homogeneous, or eective, material properties which in turn deines the uncertainty in the response o the homogeneous system. Note that i homogenization occurs over an RVE this variance becomes zero and the problem is deterministic. I, on the other hand, the averaging volume is not an RVE, then the original stochastic problem that contains random spatially luctuating material properties is replaced by a stochastic problem in which the material properties are spatially constant but random. Currently, the only practical way to evaluate the variance 66-89/$ see ront matter 1 Elsevier td. All rights reserved. doi:1.116/j.probengmech

2 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) o the eective properties is through Monte Carlo (MC) simulation, which can be computationally very expensive. A method is introduced here or evaluating the variance o the eective properties that circumvents the need or brute orce MC simulation by introducing a variability response unction (VRF) or the eective properties that connects the spectral density o the randomly luctuating material properties (that are modeled as homogeneous random ields) to the variance o the eective material properties. A variability response unction [3,4] describes the inluence o the spectral content o a random material property ield on the response o a structural system, usually the displacement. In cases where a VRF exists, the variance o the structural response can be calculated by a straightorward integration o the product o the spectral density o the material property ield and the VRF, completely obviating the need or MC simulation or any other approximate technique such as perturbation or expansion methods. This results in very rapid evaluation o the response variance, and, perhaps more importantly, the possibility o evaluating the sensitivity o the response variability to the spectral content o the underlying material property ield. The body o this paper contains: (1) A general deinition o the concept o the VRF or eective material properties and the proo that such a VRF, i it exists, depends on the VRF or the displacement variance and covariance; () proo o the existence o a VRF or the eective material properties o a statically determinate structure with a single applied point load and a corresponding numerical example; (3) proo o the existence o a VRF or the eective material properties o a statically determinate structure with an applied uniorm distributed load and a corresponding numerical example. Item (3) requires the proo o the existence o a VRF or the covariance o the response displacement ield o a statically determinate structure, which is also included.. Variability o eective properties Consider a domain Ω R 3 with volume V Ω occupied by an isotropic elastic continuum characterized by a second order constitutive matrix C(x) and its inverse D(x) = C 1 (x). These matrices deine the standard Hooke s law or three dimensional isotropic elastic solids σ(x) = C(x)ɛ(x) (1) where σ = [σ 11,σ,σ 33,σ 1,σ 13,σ 3 ] T and ɛ = [ɛ 11,ɛ, ɛ 33,ɛ 1,ɛ 13,ɛ 3 ] T are the stress and strain vectors commonly used in engineering notation. In this section we consider only C(x) noting that C(x) and D(x) can be treated interchangeably since they completely deine each other. I the material occupying Ω is heterogeneous, C(x) is a unction o position x, and i the heterogeneity is random, then C(x) can be considered a matrix random ield with mean matrix µ that is independent o position and matrix o spectral density unctions S C (κ 1,κ,κ 3 ) that depend on three wave numbers κ 1, κ, and κ 3 i C(x) is stationary. Analysis o continuum mechanics problems in which the material properties are heterogeneous poses substantial challenges, and it is thereore oten desirable to replace the heterogeneous material with a homogeneous material that is, in some sense, equivalent. This homogenization corresponds to replacing the matrix random ield C(x) with the eective constitutive matrix C = (C(x), Ω, boundary conditions) () which is a unction not only o C(x) but also o the problem domain Ω and boundary conditions. The ollowing discussion is restricted to luctuations o the elastic modulus E(x) = C 11 (x) = C (x) = C 33 (x), noting that it could equivalently be applied to the shear modulus G(x) = C 44 (x) = C 55 (x) = C 66 (x). It should be pointed out that the elastic modulus cannot luctuate independently o the shear modulus and Poisson s ratio i the material is to remain locally isotropic. The simplest deinitions o the eective elastic modulus Ē are based on harmonic or arithmetic averaging o E(x). These deinitions give the Reuss and Voigt bounds on the eective modulus deined by Ē r Ē Ē v Ē r = E(x) 1 Ω = 1 Ē v = E(x) Ω = 1 V Ω V Ω [ Ω E(x) 1 dv ] 1 E(x)dV. (3) Ω The Voigt and Reuss bounds on the eective elastic modulus do not depend on the boundary conditions o the problem, and thereore the variances var[ē v ] and var[ē r ] depend directly on integrals o E(x) that can be calculated using stochastic calculus [5] or estimated using MC simulation. An alternative deinition or the eective elastic modulus ensures that the elastic strain energy in the homogenized version o the problem is equivalent to that in the heterogeneous version. To use this deinition the problem must be deined such that the solution o the homogeneous problem depends only on a single material constant, in this case assumed to be Ē. Common examples include a uniaxial state o stress, pure shear, or a beam bending problem. For a set o traction boundary conditions t(x) that satisy the condition that the response o the homogeneous problem depends only on a single elastic constant, and that result in a displacement ield u(x) in the heterogeneous body, the equivalence o strain energy can be expressed as strain energy o homogeneous problem = strain energy o heterogeneous problem g(ē, Ω, boundary conditions) = ɛ(x)c(x)ɛ(x)dv Ω = t(x)u(x)ds. (4) The boundary conditions (bcs) and Ω are usually chosen so that an exact expression or g(ē, Ω, bcs) is available in closed orm, in which case Eq. (4) can be solved directly or Ē. For example, i the boundary conditions are chosen to generate a uniaxial state o stress σ 11 = σ, g(ē, Ω, bcs) = Ω σ /Ē and i the problem is a cantilever beam with a point load applied at the ree end, g(ē, Ω, bcs) = P 3 /3ĒI with P being the applied load, the length o the beam, and I the moment o inertia. Assuming that g(ē, Ω, bcs) is known and Eq. (4) can be solved or Ē, the eective elastic modulus can be expressed as ) Ē = g (Ω, bcs, t(x)u(x)ds. (5) In many cases, the eect o Ω and o the boundary conditions will appear as a deterministic coeicient C 1, so that the variance o the eective elastic modulus can be expressed as [ ] var[ē] =C 1 var t(x)u(x)ds. (6) This expression suggests that the variance o the eective elastic modulus depends on the variance o an integral o the displacement ield. Furthermore, it is known that in certain cases the variance o the displacement depends on a variability response unction through the ollowing integral expression var[u i (x)] = S (κ 1,κ,κ 3 )VRF ui (x,κ 1,κ,κ 3 )dκ 1 dκ dκ 3 (7)

3 176 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) where S (κ 1,κ,κ 3 ) is the spectral density unction describing the spatially random luctuations o the material property ield and VRF ui (x,κ 1,κ,κ 3 ) is the deterministic variability response unction. Note that this relationship has been proven to exist only or statically determinate problems in one dimension [3,4], although it is written here in the more general three dimensional orm. Note also that it has been shown to be approximately valid or certain statically indeterminate problems in one and two dimensions [6 9]. The existence o a VRF or the displacement suggests that a VRF may exist or the eective elastic modulus such that var[ē] = S (κ 1,κ,κ 3 )VRFĒ(κ 1,κ,κ 3 )dκ 1 dκ dκ 3 (8) and that it may be possible to express VRF Ē(κ 1,κ,κ 3 ) in terms o VRF u (x,κ 1,κ,κ 3 ). Veriying these last two statements, that a VRF exists or the eective material properties, and that this VRF depends on the VRF o the response displacements, are the two main objectives o this paper, and they are now investigated or some speciic structural mechanics problems. 3. Eective lexibility o beams For structural beams it is convenient to consider uncertainty in the lexibility in the orm D(x) = 1 EI(x) = 1 EI (1 + (x)) (9) where EI is the nominal stiness and (x) is a zero mean homogeneous random ield with spectral density unction S (κ) that is a unction o the wave number κ. The variance o the response transverse displacement u(x) o such beams can be expressed in the ollowing integral orm involving a VRF var[u(x)] = VRF u (x, κ)s (κ)dκ. (1) Note that this expression is exact or statically determinate beams but only approximate or statically indeterminate beams. The equivalent homogeneous beam has the same geometry and boundary conditions, but has D(x) replaced with the eective lexibility D (constant along the length o the beam) which plays the role o Ē in the previous discussions. 4. Statically determinate beams with a single point load Consider a statically determinate beam o length in which the lexibility varies in the way described in Eq. (9) and which is loaded with a single point load P applied at position x P. Eq. (4) expressed or such a beam is g( D,, bcs) = C1 P D = Pu(xP ) = t(x)u(x)ds (11) which can be solved or D = 1 C 1 P u(x P) (1) where u(x P ) is the transverse displacement o the randomly heterogeneous beam at location x P and C 1 is a deterministic constant depending on the length and the boundary conditions. The variance o the eective lexibility can then be expressed as ( ) 1 var[ D] = var[u(x P )] C 1 P ( ) 1 = S (κ) VRF u (x P, κ)dκ C 1 P = S (κ)vrf D(κ)dκ (13) a b Fig. 1. Cantilever beams with point and distributed loads. with ( ) 1 VRF D(κ) = VRF u (x P, κ). (14) C 1 P Consequently, or a statically determinate beam with a single applied point load the VRF o the eective lexibility has the same unctional orm as the VRF o the displacement, but is multiplied by a coeicient that depends on the applied point load, the geometry o the beam, and the boundary conditions Example The beam shown in Fig. 1(a) is a statically determinate cantilever with a single point load P applied at the ree end x P =. For this particular beam P g( D, 3 D, bcs) = 3 as C 1 = 3 /3. Using Eq. (1), the expression or D is (15) D = 3u(). (16) P 3 It should be noted that as u() is the tip delection o the randomly heterogeneous beam, u() is a random variable and consequently D is also a random variable. Eq. (14) leads to the ollowing expression relating the VRF o the displacement to the VRF o the eective lexibility ( ) 3 VRF D(κ) = VRF P 3 u (, κ). (17) et P = 16, N, = 16 m and EI = Nm. The variability response unction VRF u (, κ) resulting rom these numerical parameters is shown in Fig.. It has been estimated by Fast Monte Carlo (FMC) simulation [1,11] though it can be calculated rom a closed orm expression ollowing the procedures in [3,4,1]. The corresponding variability response unction VRF D(κ) is shown in Fig. 3, has the same shape as VRF u (κ) and diers only by the constant ( 3/P 3) = For demonstration purposes, consider the two spectral density unctions σ S,1 (κ) = β π(κ + β ) {.33 κ 3 S, (κ) = otherwise (18) with σ =.1 and β =.5 that deine the homogeneous random ield (x). Both spectral densities correspond to the same variance σ =.1. Fig. 4 displays a ew samples o D(x) resulting rom

4 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) Fig.. VRF or cantilever tip displacement (example in Section 4.1). Fig. 5. VRF or eective lexibility shown with spectral densities o lexibility ield (example in Section 4.1). Fig. 3. VRF or eective lexibility (example in Section 4.1). each o the spectra under a truncated Gaussian assumption or the marginal PDF. It should be noted that the results are invariant to the choice o a marginal PDF, however, the marginal PDF should respect physical restrictions in the numerical values allowable or the lexibility (consequently the Gaussian is not acceptable as a marginal PDF). Fig. 5 shows the two spectra on the same plot with VRF D(κ) in order to visualize the corresponding overlaps. Eq. (13) provides the ollowing estimates or the variance o D or the two spectral densities: var[ D] = or S,1 (κ) and var[ D] = or S, (κ). These values agree well with results o direct MC simulations o the beam response with 5 samples which yield var[ D] = and var[ D] = , respectively. Note that the VRF s use here or the eective lexibility are exact, and any discrepancy between the prediction o the VRF approach (Eq. (13)) and the results o MC simulations stems rom three potential sources o numerical error: (1) estimation error associated with the inite number o samples used in the MC simulations; () estimation error associated with the inite number o samples used in the estimation o the VRF; (3) error associated with the numerical integration o the product o the VRF and the spectral density to predict the eective property variance (Eq. (13)). Each o these errors could be reduced to achieve an arbitrary degree o accuracy provided suicient time and computational resources were available. The eective lexibility as deined in this work diers rom the arithmetic and harmonic averages o D(x). For the same Fig. 4. Samples o D(x) generated using spectral densities S,1 (κ) and S, (κ), both with the same marginal PDF (truncated Gaussian) and variance σ =.1.

5 178 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) simulations described above, the variance o the arithmetic mean o D(x) (Reuss) and o the harmonic mean (Voigt) are and or S,1 (κ) and and or S, (κ). Note that since lexibility is considered in this case, the Voigt average corresponds to a harmonic average o D(x) whereas the Reuss average corresponds to an arithmetic average o D(x). In certain cases the variance o the Voigt and Reuss averages diers substantially rom the variance o the strain energy based eective lexibility. 5. Statically determinate beams with uniorm load Consider now a statically determinate beam identical to that introduced in the previous section but with a deterministic uniorm distributed load q applied along the entire length o the beam. For this case g( D,, bcs) = qū(x)dx = q C D (19) where ū(x) denotes the transverse displacement o the homogeneous beam and the constant C represents the inluence o the beam s geometry and boundary conditions, as beore. The righthand side o Eq. (4) is written as t(x)u(x)ds = qu(x)dx () where u(x) denotes the transverse displacement o the randomly heterogenous beam. Combining now Eqs. (4), (19) and () leads to the ollowing expression or D D = 1 qc u(x)dx. (1) The variance o D can be written as ( ) [ 1 ] var[ D] = var u(x)dx qc ( ) 1 = c(u(x 1 ), u(x ))dx 1 dx () qc where c(u(x 1 ), u(x )) is the covariance unction o u(x) (The second part o Eq. () is derived rom the irst part by direct calculations ollowing equations K.1 and K. in [1]). The expression in Eq. () suggests that i a VRF exists such that c(u(x 1 ), u(x )) = S (κ)vrf u1 u (x 1, x, κ)dκ (3) with S (κ) being the spectral density o random ield (x), then var[ D] ( ) 1 = S (κ)vrf u1 u qc (x 1, x, κ)dκdx 1 dx [ ( ) ] 1 = S (κ) VRF u1 u qc (x 1, x, κ)dx 1 dx dκ = S (κ)vrf D(κ)dκ (4) with [ ( ) ] 1 VRF D(κ) = VRF u1 u qc (x 1, x, κ)dx 1 dx (5) What diers signiicantly rom the previous case involving a point load is that VRF D(κ) is now a unction o integrals o VRFs. The existence o VRF u1 u (x 1, x, κ) will now be established. This in turn proves the existence o VRF D(κ) or statically determinate beams loaded with a uniorm distributed load. The displacement ield can be expressed as x u(x) = D h(x, ξ)m(ξ)[1 + (x)]dξ (6) where D is the nominal value o the lexibility (Eq. (9)), h(x, ξ) is the deterministic Green s unction and M(ξ) is the bending moment, which is independent o the lexibility and deterministic since the beam is statically determinate. The correlation unction r(u(x), u(y)) is easily computed using Eq. (6), r(u(x 1 ), u(x )) = E[u(x 1 )u(x )] = D x1 x h(x 1,ξ 1 )h(x,ξ )M(ξ 1 )M(ξ ) [ 1 + R (ξ 1 ξ ) ] dξ 1 dξ (7) where R (ξ 1 ξ ) is the autocorrelation o the homogeneous, zero mean random ield (x). The covariance unction c(u(x 1 )u(x )) is thereore written as c(u(x 1 ), u(x )) = D x1 x h(x 1,ξ 1 )h(x,ξ )M(ξ 1 ) M(ξ )R (ξ 1 ξ )dξ 1 dξ. (8) I the spectral density S (κ) is substituted or R (ξ 1 ξ ) through the Wiener Khintchine transorm, the covariance unction o u(x) becomes c(u(x 1 ), u(x )) = D x1 x M(ξ ) h(x 1,ξ 1 )h(x,ξ )M(ξ 1 ) S (κ)e iκ(ξ 1 ξ ) dξ1 dξ. (9) Changing the order o integration yields c(u(x 1 ), u(x )) = VRF u1 u (x 1, x, κ)s (κ)dκ. (3) where the VRF is deined as VRF u1 u (x 1, x, κ) = D x1 x h(x 1,ξ 1 )h(x,ξ ) M(ξ 1 )M(ξ )e iκ(ξ 1 ξ ) dξ1 dξ. (31) Eqs. (3) and (31) are o interest or several reasons. First, they show that a VRF exists or the covariance unction o the non-homogeneous, random ield u(x). Second, the derivation o this VRF is independent o the marginal distribution o (x). Third VRF u1 u (x 1, x, κ) is deined on the two-dimensional domain (x 1, x ). In principal VRF u1,u (x 1, x, κ) can be calculated exactly using closed orm expressions or the Green s and moment unctions h(x, ξ) and M(ξ), but in practice it is oten more convenient to estimate the VRF using the very eicient Fast Monte Carlo (FMC) method [1,11]: 1. Select the variance σ o the lexibility luctuations.. Select the number o simulations N sim used to estimate each value o the VRF. 3. Fix values o x 1 and x. 4. Fix the value o κ. 5. Generate N sim realizations o (x) with j = 1,,..., N sim : (x) = σ ( cos κx + θ (j)) ( θ (j) = j 1 )( ) π (3) N sim 6. For each realization o (x) compute u (j) (x 1 ) and u (j) (x ). 7. Estimate the covariances using

6 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) Fig. 6. VRF u1,u (x 1, x, κ) shown as surace and contour plots or ixed values o κ. Fig. 7. VRF u1,u (x 1, x, κ) shown as surace and contour plots or ixed values o x 1 (example in Section 5.1). c(u(x 1 ), u(x )) 1 N sim ( u (j) (x 1 ) û(x 1 ) )( u (j) (x ) û(x ) ) N sim j=1 û(x 1 ) 1 N sim u (j) (x 1 ) N sim j=1 û(x ) 1 N sim u (j) (x ) (33) N sim 8. Calculate j=1 VRF u1,u (x 1, x, κ) = c(u(x 1), u(x )) σ (34)

7 18 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) Fig. 8. VRF u1,u (x 1, x, κ) shown or ixed values o x 1 and x (example in Section 5.1). 9. Repeat steps 4 8 or a number o equidistant values o κ on the wave number domain rom zero to an appropriately selected upper cuto wave number κ u beyond which the values o the VRF become negligibly small. 1. Repeat steps 3 9 or a number o equidistant values o x 1 and x on a two-dimensional grid in the x 1, x domain. Once VRF u1,u (x 1, x, κ) is estimated by this FMC procedure, VRF D(κ) is readily obtained by numerical integration ollowing Eq. (5), or, alternatively, VRF D(κ) can be estimated directly by an FMC algorithm similar to that described above Example Consider now the statically determinate cantilever beam in Fig. 1(b) with a uniorm load q applied over the entire length o the beam. For the homogeneous beam, the displacement is given by ū(x) = q D ( x x + 3 4) 4 (35) which yields, according to Eq. (19), C = 5 /. Eq. () is then written as var[ D] = ( q 5 and Eq. (5) as VRF D(κ) [ ( ) = q 5 ) [ ] var u(x)dx VRF u1,u (x 1, x, κ)dx 1 dx ] (36). (37) et q = 1 N/m, = 16 m and EI = Nm, and consider again the two spectral density unctions deined by Eq. (18). Using the FMC algorithm described above, VRF u1,u (x 1, x, κ) has been estimated using N sim = 5. FMC simulations were carried out or 5 5 values o x 1 and x evenly spaced in the two-dimensional interval [, 16], and or n κ = 5 values o κ evenly spaced in the interval [, 3]. Consequently, the complete procedure requires the generation o n κ N sim = 5 samples o D(x) according to Eqs. (9) and (3) and the subsequent solution or u(x) or each one o these samples. Using the inite element method to calculate u(x) with 1 Euler Bernoulli beam elements, the entire FMC estimation requires approximately 75 seconds when executed in MATAB on a MacBook Pro laptop computer with a.6 GHz Intel Core Duo processor and 4 GB o RAM. The FMC algorithm provides an estimate o VRF u1,u (x 1, x, κ) which depends on three input arguments, making it diicult to visualize. Fig. 6 shows VRF u1,u (x 1, x, κ) as surace and contour Fig. 9. VRF D(κ) shown with spectral densities S,1 (κ) and S, (κ) (example in Section 5.1). plots or κ ={.86, 1.5, 3.}. This igure shows that the general shape o the VRF is similar at dierent values o κ, with the peak value always occurring at x 1 = x =, corresponding to the tip o the beam. As κ increases towards its maximum value o 3., the magnitude o the VRF decreases, which is in agreement with the general orm o VRF u (κ) (Fig. ) which has a maximum at κ = and approaches zero as κ increases. In Fig. 7, sections through VRF u1,u (x 1, x, κ) at ixed values o x 1 are shown. Sections through these suraces at constant x are shown in Fig. 8, and show a shape similar to that in Fig., with the magnitude o the VRF decreasing as x 1 x increases. Fig. 9 shows VRF D(κ), calculated using Eq. (37), together with the two spectral density unctions o Eq. (18). Numerical integration o the third line o Eq. (4) with S,1 (κ) gives var[ D] = and direct MC simulations o the beam s response with 1 samples gives , a reasonably good agreement. For S, (κ) the agreement is also good, with the VRF approach giving and direct MC simulations yielding Note again that the VRF s used here or the eective lexibility are exact, and any dierence between the VRF-based predictions or the variance o D and the corresponding MC results is merely an artiact o various estimation and numerical integration errors that can be reduced to arbitrarily small values given suicient computational resources. 6. Conclusion A variability response unction (VRF) has been introduced or the eective elastic material properties o a heterogeneous body. This VRF depends directly on the boundary conditions and geometry o the body, as well as on the VRF or the displacement response, when the homogenization o the medium is deined based on equivalence o elastic strain energy in the heterogeneous and

8 S.R. Arwade, G. Deodatis / Probabilistic Engineering Mechanics 6 (11) homogeneous bodies. For the case o a statically determinate beam subject to either a single point load or a uniormly distributed load the VRF or the eective lexibility exists and can be calculated by scaling the displacement VRF (point load case) or scaling and integrating the VRF or the covariance o displacements along the length o the beam (uniorm load case). In order to demonstrate the second result, the existence o a VRF or the covariance o displacements at points along a statically determinate beam has been proven. Two numerical examples demonstrate the eicacy o the VRF approach or predicting the variance o the eective lexibility. The VRF or eective material properties introduced here is useul or eiciently evaluating the variance o eective material properties averaged over inite material volumes and also or conducting studies on the sensitivity o the eective property variability to the spectral contents o the underlying material property random ield. The calculation o the VRF is done using a very eicient Fast Monte Carlo algorithm. Once the VRF is calculated, the evaluation o eective material property variability and sensitivity can be accomplished without recourse to any additional MC simulation, requiring only simple numerical integration o a one dimensional integral in the wave number domain. Acknowledgements This work was supported by the United States National Science Foundation under Grant No. CMMI-9819 with Dr. Mahendra P. Singh as Program Director and Grant No. CMMI-8665 with Dr. awrence Bank as Program Director. Reerences [1] Koutsourelakis S. Stochastic upscaling in solid mechanics: an exercise in machine learning. Journal o Computational Physics 7;6:31 5. [] Graham, Baxter S. Simulation o local material properties based on movingwindow gmc. Probabilistic Engineering Mechanics 1;16: [3] Shinozuka M. Structural response variability. Journal o Engineering Mechanics 1987;113:85 4. [4] Papadopoulos V, Deodatis G, Papadrakakis M. Flexibility based upper bounds on the response variability o simple beams. Computer Methods in Applied Mechanics and Engineering 5;194: [5] Grigoriu M. Stochastic Calculus. Boston: Birkhauser;. [6] Wall J, Deodatis G. Variability response unctions o stochastic plane stress/strain problems. Journal o Engineering Mechancis 1994;1: [7] Graham, Deodatis G. Variability response unctions or stochastic plate bending problems. Structural Saety 1998;: [8] Graham, Deodatis G. Response and eigenvalue analysis o stochastic inite element systems with multiple correlated material and geometric properties. Probabilistic Engineering Mechanics 1;16:11 9. [9] Miranda M, Deodatis G. On the response variability o beams with large stochastic variations o system parameters. In: 1th International conerence on structural saety and reliability, IASSAR, Osaka, Japan; 9. [1] Miranda M. On the response variability o beam structures with stochastic variations o parameters. Ph.D. Thesis, Columbia University. [11] Deodatis G, Shinozuka M. Bounds on response variability o stochastic systems. Journal o Engineering Mechanics 1989;115: [1] Grigoriu M. Applied non-gaussian Processes. Englewood Clis, NJ: Prentice Hall; 1995.