A Stochastic Finite Element Approach on Creep of Rock Salt · 2016-08-30 · Keywords : Stochastic...

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Page 1: A Stochastic Finite Element Approach on Creep of Rock Salt · 2016-08-30 · Keywords : Stochastic Finite Element Method, Polynomial Chaos, Creep est,T Viscoelastic, Rock Salt 1 Introduction

Proceedings of XLIV International Summer School � Conference APM 2016

A Stochastic Finite Element Approach on Creep ofRock Salt

M. Grehn, A. Fau, U. Nackenhorst

[email protected]

Abstract

A robust computational material model for rock salt with a stochasticapproach for Young's modulus is developed to predict rock salt behavior duringa creep test. The uncertain input data as well as the unknown displacement�elds are described by Hermite polynomial chaos. The numerical problemis solved using Stochastic Finite Element Procedure. The material model isable to reproduce the experimental results during the two �rst creep periodesincluding the scattering of experimental creep curves.

Keywords: Stochastic Finite Element Method, Polynomial Chaos, Creep Test,Viscoelastic, Rock Salt

1 Introduction

Old salt domes are used as reservoirs for toxic and nuclear waste in Germany. Nu-clear waste has a long half-life time, so the mechanical stability of salt domes hasto be guaranteed for a long period. In the last decades, a large expertise in the me-chanical behavior of rock salt was gained. This led to a variety of material models,which may be based on a macroscopic phenomenological approach [1], derived fromBurger's modi�ed model [2], or from micromechanics using the composite dilatancymodel [3, 4, 5, 6] for examples.However, from experimental results obtained during creep tests for rock salt fromAsse mine in Germany [7] reproduced in Figure 1 it can be seen that the materialproperties are subject to large variation as also observed by [4, 8]. The goal of thispaper is to reproduce this creep test including the scattering of experimental databy introducing a stochastic material model for rock salt.

2 Origin of uncertainties

To analyze numerically the behaviour of rock salt, we refer to a material modelbased on di�erent established works from the literature [7, 9, 5, 6].

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A Stochastic Finite Element Approach on Creep of Rock Salt

Figure 1: Experimental uniaxial creep test for three samples of Asse rock salt re-produced after [7]

2.1 Material model

The analysis of the creep test can be split into three parts as illustrated in Fig-ure 2 [10]. During the primary creep period (I), the dislocations move in the latticestructure of the crystal salt. The strain rate is relatively high, but it slows down byincreasing time. In the secondary or steady-state creep period (II) the dislocationdensity increases leading to an increasing resistance. The strain rate is nearly con-stant. The tertiary creep period (III) is characterized by an exponentially increasingof the strain rate. We focus here only on the two �rst creep periods to reproducethe experimental results previously presented.

Figure 2: Analysis of the creep behavior of rock salt

The material model is based on an additive split of the strain tensor ε:

ε = εel + εv + εcr. (1)

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Proceedings of XLIV International Summer School � Conference APM 2016

The elastic strain tensor εel and the viscous strain tensor εv describe the primarycreep period. The steady-state creep period is described by the creep strain tensorεcr. We can reproduce the viscoelastic part for one-dimensional systems by springsof sti�nesses Ei and dampers of coe�cients ηi using a generalized Maxwell modeland the following free energy function [9]:

ψ(ε, qi) = W ◦(ε) +

NMW∑i=1

qi · ε+ Ξ

(NMW∑i=1

qi

). (2)

The function W ◦ is the initial stored energy function. The internal variables qicharacterize the viscoelastic response and ε = ε − 1

3tr(ε) is the deviatoric strain.

NMW is the number of Maxwell elements. We consider one spring of sti�ness E∞connected in parallel with two Maxwell elements whose sti�nesses are respectivelyE1 and E2. The total Young modulus E is computed as a summation of all singleYoung's modulus:

E = E∞ +2∑i=0

Ei. (3)

The creep behavior of rock salt for the deviatoric part of the creep strain tensor isdescribed by Norton's law:

˙εcr = A exp

(−QRT

)·(σvM

)N, (4)

where N is a material parameter and σvM denotes the von Mises stress. The tem-perature dependency of the creep curve is described by Arrhenius equation, whichis linked multiplicative with the creep rates. The pre-exponential factor A is deter-mined by experiments. R denotes the universal gas constant and T the absolutetemperature. The macroscopic activation energy Q is provided by [5].

Experiments [7, 6] show that for rock salt, the volumetric deformation dependson the state of stress in comparison with the dilatancy boundary. This materialbehavior is described by the functional rv introduced in [7]:

rv =

3[τokt−τDσokt

]2

for τokt > τD

0 otherwise.(5)

The function τD represents the dilatancy boundary equation [7]. τokt is the octahe-dral shear stress and σokt is the octahedral normal stress. When the state of stressis below the dilatancy boundary, the deformation is at constant volume, otherwisearises the volumetric part of the creep strain tensor de�ned as:

εcr,vol = rv ˙εcr. (6)

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A Stochastic Finite Element Approach on Creep of Rock Salt

In loosened rock salt the moisture has a big in�uence on the creep rate. Thisbehavior is modeled by the following equations [7]:

fΦ = cΦ1 sinh (cΦ2 Φ) ,

fc =

[1 +

(σ3σu

)cfc1 τokt]−( cfc2

1+( τoktσu )

)2

for σ3 ≥ 0

1 otherwise,

Fh = 1 + fΦ fc.

(7)

The factor Fh, which is linked multiplicative with the creep rates, sets together twoe�ects. The �rst part fΦ describes the in�uence of relative humidity Φ on the creeprate. The second one fc takes into account the in�uence of minimum principal stressσ3 and the octahedral shear stress τokt. The other components in this equation arematerial parameters de�ned by [6].

To predict the primary and the secondary creep behavior of rock salt over long timethe evolution equations must be solved e�ciently by numerical techniques. We ap-ply �nite element technique to an axisymmetric geometrical model to reduce thecomputational e�orts. We use for the primary creep periode the algorithm intro-duced in [9] and for the secondary creep periode the one proposed in [11].

2.2 Identifying the random parameters

The three uniaxial creep tests performed by [7] have the same set up with an axialstress equal to 11 MPa, see Figure 1. We investigate which material parameters haveto be considered di�erent for the three rock salt samples to reproduce consistentlythe scattering of the experimental curves.Based on Equation 4 we can see that a scattering of the material parameters A or Nor of the relative humidity Φ changes the slope of the secondary creep period, whichis contradictory with the experimental results. The sti�ness of the �rst spring E∞in the generalized Maxwell model characterizes the asymptotic solution εv(t = ∞)for the second creep period. For this reason the free energy function in Equation 2will be stochastically distributed assuming an uncertain Young's modulus E∞. Thestochastic approaches chosen to describe E∞ as a random �eld and to solve thestochastic �nite element problem are introduced in the following section.

3 Stochastic Finite Element Approach

For solving the boundary value problem, we use the stochastic �nite element method[12] with the aid of polynomial chaos expansions to describe the input random �eldE∞(θ) as well as the displacement random �eld u(θ). Each random quantity ischaracterized by θ.

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Proceedings of XLIV International Summer School � Conference APM 2016

3.1 Representation of the random input

The Young modulus E∞(θ) is approximated as a Hermite polynomial chaos (PC)series expansion:

E∞(θ) ≈p∑i=0

aiHi (ξ(θ)) , (8)

where p is the order of Hermite polynomials Hi de�ned as in [13]:

Hi (x) = (−1)i1

ϕ (E∞)

diϕ (x)

dxi, (9)

ϕ (x) = 1√2πe−

x2

2 is the standard normal probability density function (PDF). Tocompute the coe�cients ai in Equation 8, we use the projection method [13]. TheHermite polynomials are orthogonal with respect to the Gaussian measure. For aprobability space with dimension M equals to 1 one obtains

E[E∞Hi (ξ(θ))] = ai E[H2i (ξ(θ))], (10)

where E is the mean operator and E[H2i (ξ(θ))] = i!. With help of the transformation

to the standard normal space E∞ → ξ : Fx(E∞) = Φ(ξ(θ)), we can write the randomvariable as following:

E∞ (ξ(θ)) = F−1x (Φ(ξ(θ))). (11)

The coe�cients {ai, i = 0, ...,∞} are computed as:

ai =1

i!E[[E∞ (ξ(θ))Hi (ξ(θ))] =

1

i!

∫RF−1x (Φ(t))Hi(t)ϕ(t)dt. (12)

Therefore, for the speci�c cases of normal or log-normal distributions, we get:

if E∞(θ) ≡ N (µE∞ , σE∞), a0 = µE∞ , a1 = σE∞ , ai = 0 for i ≥ 0

if E∞(θ) ≡ LN (λE∞ , ζE∞), ai =ζ iE∞i!

exp

[λE∞ +

1

2ζ2E∞

]for i ≥ 0,

(13)

where the index i goes from 0 to P the dimension of PC basis. The parametersλE∞ , ζE∞ for log-normal distributions depend on the mean and the standard devia-tion see [13]. The size P of the polynomial chaos basis is given by:

P =(M + p)!

M !p!. (14)

Based on the independence of the random variable ξ(θ) inside the PC expansion,we can compute the multi-dimensional basis:

ψα =M∏i=1

Hαi , (15)

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A Stochastic Finite Element Approach on Creep of Rock Salt

Figure 3: Two realizations of Young's modulus [GPa] �eld for a rock salt sample.

with αi ≥ 0 and∑M

i=1 αi ≤ p as a product of one-dimensional basis. An illustrativetransmission of an urn problem is introduced in [13]. Thus the PC can be writtenas:

E∞ =P−1∑i=0

E∞iψi (ξ(θ)) , (16)

where E∞ is the deterministic PC-coe�cient.The considered creep test results are not statistically representative to have atrustable representation of material uncertainties, a large number of experimen-tal data would be required. We assume for E∞ a normal (N) or a lognormal (LN)distribution. The mean of the Young modulus µE∞ is estimated as the parameterreproducing the mean of the three experimental results, i.e. µE∞ = 2, 140 GPa.The standard deviation σE∞ is computed such that at least about 60 % of the prob-abilistic creep test results are between the two extreme experimental curves i.e.σE∞ = 0, 535 GPa.In Figure 3 two realizations of the Young modulus random �eld (µE = 37, 8 GPaand σE = 0, 535 GPa) are shown over the cross section x = 30mm for a polynomialchaos basis up to order 2. For all the Gauss points, Young's modulus values arescattered around the mean value, which is represented by a dashed line.

3.2 Estimation of the random displacement �eld

The linearization of the deterministic discrete FEM for static problems yields alinear system of size Nd ×Nd where Nd is the number of degrees of freedom in the

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Proceedings of XLIV International Summer School � Conference APM 2016

FEM geometry:

K∆u = r − rint, (17)

K is the total sti�ness matrix, u is the displacement �eld, r and rint are respectivelythe vectors of external and internal forces.In the stochastic �nite element model we assume the loading and the geometryas certain. We introduce input random variables for material properties as a PCexpansion:

K(θ)∆u(θ) = r − rint(θ), (18)

where K(θ) is the stochastic sti�ness matrix. Details to compute the total sti�nessmatrix can be found in [12, 13, 14]. By expanding each component of the unknowndisplacement �eld as:

u(θ) ≈P−1∑i=0

uiψi (ξ(θ)) , (19)

we get the following discretized problem:

P−1∑i=0

Kiψi (ξ)P−1∑j=0

∆ujψj (ξ) =P−1∑j=0

rjψj (ξ)−P−1∑j=0

rintj ψj (ξ) (20)

where · represents the nodal discretization. The resulting linear system: K0,0 · · · K0,P−1...

. . ....

K0−1,0 · · · KP−1,P−1

· ∆u0

...∆uP−1

=

r1...

rP−1

− r

int1...

rintP−1

(21)

has a size (Nd ∗ P ) × (Nd ∗ P ). The mean value contribution is contained in themain diagonal elements [13]. Considering a probability space of dimension 1, thesize of the linear system is (Nd ∗ (p+1)!

p!)× (Nd ∗ (p+1)!

p!). The size of the linear system

is largely increased considering random input data. For a case using 156 nodes, thesize of the deterministic sti�ness matrix is 312×312, whereas for a polynomial chaosof order 1, the size of the stochastic sti�ness matrix is 624×624. But, these matricesare symmetric and sparse matrices.To solve the large system of equation, we can choose to use a direct solver or aniterative scheme as for example the Gauss-Seidel PC solver introduced in [15, 16].The advantage of the direct solver is that the equation system must be solved onlyone time but it may face memory problems for very large sti�ness matrix. Here,as we tackle an axisymmetric case with a unique random �eld, we can use a directsolver for this calculation.

4 Numerical results for the creep test

We reproduce the creep test using the parameters listed in Table 8:

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A Stochastic Finite Element Approach on Creep of Rock Salt

Parameter Value UnitµE 2.140 GPaσE 0.535 GPaP 3 -

Distribution log-normal -Realizations 300 -

Table 8: Stochastic parameters

The results of the stochastic �nite element calculation of the creep test are illus-trated in Figure 4. The robust stochastic numerical model for rock salt allows toreproduce the test for a time period of 900 days. The dashed line represents themean of the deformation of the rock salt sample. With the assumed distribution(see Section 3.1), 64, 2 % of the stochastic deformation lies between the two extremeexperimental curves reproduced by the black lines.

Figure 4: Distribution of strain �eld

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Proceedings of XLIV International Summer School � Conference APM 2016

4.1 Convergence of the expected value

We investigate the convergence of our computations with respect to the number ofrealizations. The calculation parameters are summarized in Table 9:

Parameter Value UnitµE∞ 2.140 GPaσE∞ 0.535 GPar 2 -

Realizations [3,...,2187] -

Table 9: Stochastic parameters

Figures 5a and 5b show the means of Young's modulus E∞ in [GPa] and of thedeformation in [%] at 900 days. The convergence behavior for the input and out-put random �elds are quite similar. We consider they reach an acceptable errorfor 200 realizations. By increasing the number of realizations, the computationaltime becomes larger. From three to 2000 realizations, the computational times hasbeen multiplied by nine. For further computations we consider a number of 300realizations as a good compromise between accuracy and computing time. As we

Figure 5: Convergence of the expected value for the input parameter and the defor-mation.

assume here an axisymmetric geometry, we impose a priori a speci�c structure forthe random �eld. So, we also investigate the results considering the Young modulus

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REFERENCES

as a random variable and a homogeneous �eld. We see that this assumption doesnot in�uence the results largely. Homogeneous �eld using a unique random variableconvergences by around 500 realizations whereas the computations for random �eldconverge by around 200 realizations to the expected value. For this simple creeptest, a homogeneous �eld allows to describe correctly the behavior. The develop-ment of the random �eld has been done as a preparation to compute damage forthe third creep period.

5 Conclusions

In this paper, we have presented a numerically robust stochastic material model topredict the creep behavior for rock salt. We can reproduce the creep tests shown in[7] and [6].Here, we have considered homogeneous random �elds. We could extend our com-putations to heterogeneous random �elds to include for example some localizedimpurities. Further work is to simulate the third creep period of the experimentaltests and to predict the damage in real salt domes within a stochastic framework.

References

[1] Z. Hou, R. Wolters, and U. D�usterloh. Die Modellierung des mechanis-chen Verhaltens von Steinsalz: Vergleich aktueller Sto�gesetze und Vorge-hensweisen. BMBF-Verbundvorhaben, Ergebnisbericht zu Teilvorhaben 1,F�orderkennzeichen 02 C 1004, 2007.

[2] R. G�unther, W. Minkley, and K. Salzer. Die Modellierung des mechanis-chen Verhaltens von Steinsalz: Vergleich aktueller Sto�gesetze und Vorge-hensweisen. BMBF-Verbundvorhaben, Ergebnisbericht zu Teilvorhaben 1,F�orderkennzeichen 02 C 1004, 2007.

[3] A. Hampel, U. Hunsche, P. Weidinger, and W. Blum. Description of the Creepof Rock Salt with the Composite Model - II. Steady-State Creep. Series on soiland rock mechanics, 1996.

[4] U. Hunsche. Visco-Plastic Behaviour of Geomaterials, chapter Uniaxial and Tri-axial Creep and Failure Tests on Rock: Experimental Technique and Interpre-tation, pages 1-53. Springer Vienna, Vienna, 1994.

[5] P. Weidinger, A. Hampel, W. Blum, and U. Hunsche. Creep behaviour of naturalrock salt and its description with the composite model. Materials Science andEngineering: A, 234236:646 - 648, 1997.

[6] A. Hampel. Die Modellierung des mechanischen Verhaltens von Steinsalz: Ver-gleich aktueller Sto�gesetze und Vorgehensweisen. BMBF-Verbundvorhaben,Ergebnisbericht zu Teilvorhaben 1, F�orderkennzeichen 02 C 1004, 2006.

197

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REFERENCES

[7] U. Heemann and O. Schulze. Die Modellierung des mechanischen Verhaltensvon Steinsalz: Vergleich aktueller Sto�gesetze und Vorgehensweisen. BMBFVer-bundvorhaben, Ergebnisbericht zu Teilvorhaben 1, F�orderkennzeichen 02 C 1004,2007.

[8] G. Wang, L. Zhang, Y. Zhang, and G. Ding. Experimental investigations ofthe creepdamagerupture behaviour of rock salt. International Journal of RockMechanics and Mining Sciences, 66:181 - 187, 2014.

[9] J. C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer,Berlin,1998.

[10] M.L. Jeremic. Rock Mechanics in Salt Mining. Taylor and Francis, 1994.

[11] H. L. Schreyer. On time integration of viscoplastic constitutive models suitablefor creep. Int. J. Numer. Meth. Engng., 2001.

[12] O.C. Zienkiewicz and R.L. Taylor. Finite Element Method (5th Edition) Volume1 - The Basis. Elsevier, 2000.

[13] B. Sudret, M. Berveiller, and M. Lemaire. A stochastic �nite element proce-dure for moment and reliability analysis. European Journal of ComputationalMechanics, 2012.

[14] B. Sudret and A. Kiureghian. Stochastic Finite Element Methods and Reliabil-ity A State-of-the-Art Report. Department of civil and environmental engineer-ing university of california, 2000.

[15] Y. Saad. Iterative Methods for Sparse Linear Systems: Second Edition. Engi-neeringPro collection. Society for Industrial and Applied Mathematics (SIAM,3600 Market Street, Floor 6, Philadelphia, PA 19104), 2003.

[16] D.M. Young and W. Rheinboldt. Iterative Solution of Large Linear Systems.Computer science and applied mathematics. Elsevier Science, 2014.

M. Grehn, Leibniz Universit�at Hannover, Institute of Mechanics and Computational Me-

chanics, Appelstr. 9a, 30167 Hannover

A. Fau, Leibniz Universit�at Hannover, Institute of Mechanics and Computational Mechan-

ics, Appelstr. 9a, 30167 Hannover

U. Nackenhorst, Leibniz Universit�at Hannover, Institute of Mechanics and Computational

Mechanics, Appelstr. 9a, 30167 Hannover

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