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Mechanics of Time-DependentMaterialsAn International Journal Devoted to theTime-Dependent Behaviour of Materialsand Structures ISSN 1385-2000Volume 18Number 1 Mech Time-Depend Mater (2014)18:81-96DOI 10.1007/s11043-013-9215-3

Simulation of cooling and solidification ofthree-dimensional bulk borosilicate glass:effect of structural relaxations

N. Barth, D. George, S. Ahzi, Y. Rémond,N. Joulaee, M. A. Khaleel & F. Bouyer

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Mech Time-Depend Mater (2014) 18:81–96DOI 10.1007/s11043-013-9215-3

Simulation of cooling and solidificationof three-dimensional bulk borosilicate glass:effect of structural relaxations

N. Barth · D. George · S. Ahzi · Y. Rémond · N. Joulaee ·M.A. Khaleel · F. Bouyer

Received: 20 November 2012 / Accepted: 21 March 2013 / Published online: 20 April 2013© Springer Science+Business Media Dordrecht 2013

Abstract The modeling of the viscoelastic stress evolution and specific volume relaxationof a bulky glass cast is presented in this article and is applied to the experimental coolingprocess of an inactive nuclear waste vitrification process. The concerned borosilicate glassis solidified and cooled down to ambient temperature in a stainless steel canister, and thethermomechanical response of the package is simulated. There exists a deviant compressionof the liquid core due to the large glass package compared to standard tempered glass plates.The stress load development of the glass cast is finally studied for different thermal loadscenarios, where the cooling process parameters or the final cooldown rates were changed,and we found a great influence of the studied cooldown rates on the maximum stress build-up at ambient temperature.

Keywords Cooling process · Nuclear glass cast · FEM simulation · Viscoelasticity ·Structural relaxation

1 Introduction

The vitrification of high-level radioactive waste (HLW) is used to dispose of nuclear fissionproducts for periods of thousands of years. Processing of the waste and the glassy matrix,

Electronic supplementary material The online version of this article(doi:10.1007/s11043-013-9215-3) contains supplementary material, which is available to authorizedusers.

N. Barth · D. George (�) · S. Ahzi · Y. Rémond · N. JoulaeeICube, Université de Strasbourg, CNRS, 2 rue Boussingault, 67000 Strasbourg, Francee-mail: george@unistra.fr

M.A. KhaleelFundamental and Computational Science Directorate, Pacific Northwest National Laboratory, Richland,WA 99352, USA

F. BouyerCEA Centre de Marcoule DTCD/SECM, 30207 Bagnols-Sur-Cèze Cedex, France

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82 Mech Time-Depend Mater (2014) 18:81–96

which makes up the major part of the containment, is currently in operation at MarcouleVitrification Plant, France (Bonniaud et al. 1980) among other places in the world. At theend of this process, the glass melt is generally poured at temperatures of around 1373 K intostainless steel canisters. The solidification of the glass during the cooling process then takesplace in these HLW canisters. High temperature gradients arise within the glass during thecooling of the package, leading to a point at which glass cracking is likely to occur, as statedby the observations of Kamizono and Niwa (1984) and Faletti and Ethridge (1988). This me-chanical behavior is the consequence of the thermal loads that themselves greatly vary withthe different HLW process protocols used; see, for example, the thermal studies of Falettiand Ethridge (1988) and of Kahl et al. (1991). Faletti and Ethridge (1988) reviewed exper-imental cracking surface areas that are well different as a function of the thermal gradientsobtained during such various vitrification processes.

The glass cracking provides additional surfaces that can interact with water. This phe-nomenon was demonstrated by Perez and Westsik (1981) by nuclear glass leaching exper-iments. As for the chemical and leaching behaviors of the borosilicate glass studied in thiswork, the nuclear glass “SON68”, they were reviewed by Frugier et al. (2008) and the per-meability of such cracking network was recently studied by Ougier-Simonin et al. (2011a).

The thermal history of the glass package is determined by the manufacturing processprotocols (see, for example, Barth et al. 2012) and storage conditions (see, for example, Liet al. 2011). The thermal gradients induce a mechanical load of the package which can beaddressed, in a first approach, by viscoelasticity theory as studied in the current work. Then,beyond the scope of the current study, it can be coupled to a damage behavior as reportedby Dubé et al. (2010) and by Doquet et al. (2013) aiming at predicting failure in mode I, orin mode II (Ougier-Simonin et al. 2011b).

The determination of the thermomechanical response of an HLW package involves tak-ing into account relaxation phenomena that are time and temperature dependent. There-fore, the solidification process requires the precise knowledge of the time and temperaturedependence of each material point within the changing phase process between liquid andsolid state (since it is not instantaneous and varies in a given temperature range). For this,Tool (Tool and Eichlin 1931; Tool 1946, 1948) introduced the fictive temperature (Tf ) as ameans of quantifying these phenomena during the liquid–solid glass transition range. Later,Narayanaswamy (1971) generalized a structural relaxation model of the glass propertiesthrough shift functions and master relaxation curves that are built to predict these relaxationphenomena. Moynihan (DeBolt et al. 1976; Moynihan et al. 1976) improved the expressionof the Tool–Narayanaswamy structural relaxation model thus completing the “TNM” for-malism. In parallel to the development of the TNM model, Kovacs et al. (1979) developedthe “KAHR” model (Kovacs–Aklonis–Hutchinson–Ramos). The specificities of these twomodels were compared in the equilibrium state, for example, by Hodge (1994), especiallyconcerning the “x” parameter (between 0 and 1) defining the degree of nonlinearity in bothmodels.

The temperature jumps are the principal means of studying the glassy transition (Tg) onvarious materials among which are also polymers (Kovacs 1958). The merged TNM-KAHRmodel was also extended to other kinds of structural solicitations that are considered in thesame manner as temperature, such as with relative humidity jumps (Zheng and McKenna2003) or carbon dioxide jumps (Alcoutlabi et al. 2011). Kovacs (1964) addressed also thecase of pressure solicitations and the glassy transition pressure Pg . The defined nonlineari-ties in the TNM-KAHR model showed limitations towards the above-mentioned experimen-tal solicitations regarding structural recovery. This is discussed in details by Alcoutlabi et al.(2011).

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Soules et al. (1987) computed these nonlinear behaviors within the TNM frameworkusing a finite element method (FEM) and detailed in their algorithms the two relevant relax-ation laws, namely the structural or specific volume change relaxation (through structuralrecovery) MV and the viscoelastic or stress relaxation (shear modulus) MS . The fictive tem-perature is calculated given the thermal history for the MV relaxation process, thus evaluat-ing the structural state of the glass at any time. Thereafter, the very same fictive temperature,nonlinearity parameter and activation energy are employed to compute MS , but with differ-ent relaxation times (different master curves). Relatively to the assumption of identical ac-tivation energy between MV and MS , it was found in the albite–anorthide–diopside ternarysystem that 70 silicate melts compositions have similar shear and volume relaxation energyterms (Webb and Knoche 1996).

In the current work, we used, for the viscoelasticity of glass, a stress relaxation MS thatonly concerns the shear modulus G(T , t) and assumed a time-independent bulk modulusK(T ), as in the works by Soules et al. (1987), Ganghoffer (2000), Jain and Yi (2005) andKong et al. (2007). We should note that the time-dependent bulk modulus K(T , t) was em-ployed for soda–lime silicate glasses by Daudeville and Carré (1998), Nielsen et al. (2010)and Siedow et al. (2005). As for polymer glasses, Alcoutlabi et al. (2003) review and studythoroughly the time-dependence of the bulk modulus (in particular) and its direct role overthe isotropic residual stresses. Yet, keeping a constant bulk modulus before and after glasstransition showed to be a realistic approximation as presented by Ganghoffer (2000).

In the current study, we investigate the mechanical response of the SON68 (Frugier et al.2008) inactive glass block at a scale 1:1 following the experimental vitrification processwithout damage. The mentioned inactivity of the studied borosilicate glass is relative to ra-dioactivity and to the radioactive decay energy acting as a thermal source (see Barth et al.2012). This thermal power source and the radioactivity as glass network modifier (by pro-ducing ionization rays and following the chemistry of radioactive decay) are not taken intoaccount in the present work, but may have at first a significant effect from the mechanicalpoint of view over the short-term range cooling (the thermal load being incomplete since thepackage is not cooled down to ambient temperature).

2 Thermomechanical analysis

For the simulation of the HLW package, we used a decoupled thermal/mechanical FEManalysis where the thermal process of the glass casting is computed and used as input thermalmaps into the mechanical analysis (assuming that the mechanical behavior produces thennegligible thermal flux during the cooling and solidification process).

Based on the work of Markovsky and Soules (1984) and Soules et al. (1987) encompass-ing the TNM formalism, the thermal strain increment is given by:

�εth = αg(T + �T/2) · �T + [αl(Tf ) − αg(Tf )

] · �Tf (1)

with αg and αl the glassy (solid) and liquid linear thermal expansion coefficients, respec-tively; and where the fictive temperature Tf for the corresponding property (specific volume)reflects the structural relaxation state through the usual response function MV in reducedtime ξ . For practical reasons, MV is decomposed in the following exponential or Pronyseries forms:

ξ(t) =∫ t

0

τVref

τV (t ′)dt ′,

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MV

(ξ(t)

) ≈ exp[−(

ξ(t)/τVref

)β] ≈nV∑

i=1

cVi exp

[−ξ(t)/τVi,ref

], (2)

Tf (t) = T (t) −∫ t

0MV

(ξ(t) − ξ

(t ′))dT (t ′)

dt ′dt ′

where τVref is the relaxation time at an arbitrary reference temperature Tref; τV (t) is the re-

laxation time at t ; the stretched exponential parameter β is between 0 and 1 and typicallynearby 0.5; nV is the number of terms (superscript V for the series of exponential functionsdetermining the spectrum of MV ); cV

i is the corresponding weighting coefficient for the ithterm whose sum is normalized to 1; and τV

i,ref is the ith relaxation time at Tref.For a given experiment/simulation, τV (t) can be well described for temperature jumps

(whatever the current temperature or pressure, thermal history, internal state) using a robustset of parameters from the structural model given by Narayanaswamy (1971). The nonlinearbehavior is being introduced by the previously discussed x parameter, see Eq. (3) below.

Numerically, Tf is determined at any increment of time ending at t + �t (with t theinitial time in the increment and t = 0 s for the first increment):

Tf (t + �t) =nV∑

i=1

cVi Tf i(t + �t)

with Tf i(t + �t) =Tf i(t) + �t · T (t+�t)

τVi

1 + �t

τVi

,

τ Vi (t + �t)

τVi,ref

= exp

[−�h∗

R

(1

Tref− x

T (t + �t)− 1 − x

Tf (t)

)]

Tf i(0) = T0. (3)

Here, T0 is the initial temperature; τVi is the current ith relaxation time in the Prony series

used to discretize MV in Eq. (2); �h∗/R is the ratio between the total activation energyand the ideal gas constant; the non-linear parameter x is the ratio between glassy and total(glassy + structural) activation energies that is typically around 0.5 (for additional modelingdetails, see the TNM-KAHR works (Narayanaswamy 1971; DeBolt et al. 1976; Moynihanet al. 1976)).

Tf , Tref, �h∗, and x are chosen to be common in the TNM formalism for the responsefunctions MV and the viscoelastic response function MS (given below by Eq. (5)). This sim-plification is driven by the fact that our present work is a first approximation of the overallstructural relaxation of the borosilicate glass SON68. For these reasons we also use relax-ation parameters that are optimized for a soda–silica–lime glass (for which both responsefunction MV and MS were available in Soules et al. (1987) and are then coherent for asimilar inorganic glass).

The linear hereditary viscoelastic law employed for the stress relaxation MS in Souleset al. (1987) is then applied, without taking into account their partial stresses. These partialstresses were calculated in the case of Soules et al. (1987) only through an implicit non-ageing hypothesis that is invalid in our case (also denoted as time translational invariance;see, for example, Fielding et al. 2000).

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The stress–strain constitutive relation has been then discretized into the following incre-mental formulation from Soules et al. (1987):

�σ = Ke(T (t + �t/2), t + �t/2

) : (�ε − �εth1)

(4)

where σ and ε are the Cauchy stress and strain tensor, respectively, 1 is the second orderidentity tensor and Ke the elastic stiffness tensor, determined by both bulk and shear modu-lus (K(T ) and G(T , t)). The stress relaxation MS controls the time dependence of the shearmodulus relaxation in Ke such that

G(T (t + �t/2), t + �t/2

) = G(T (t + �t/2),0

) · MS(t + �t/2)

with MS(t + �t/2) =nS∑

k=1

cSk exp

[−(t + �t/2)/τ Sk (t + �t/2)

],

τ Sk (t + �t/2)

τ Sk,ref

= exp

[−�h∗

R

(1

Tref− x

T (t + �t)− 1 − x

Tf (t + �t)

)]. (5)

Here, nS is the number of terms (superscript S for the series of exponential relaxation func-tions MS ), cS

k the coefficient to the kth term, τ Sk the current kth relaxation time, τ S

k,ref thekth relaxation time at Tref (for additional modeling details, see the article of Soules et al.1987).

During the analysis, the time increment needs to be adjusted carefully in order to avoidthe errors associated with numerical integration, such as in Eq. (2). A good discussion on theerror accumulation is reported by Chambers and Becker (1986). However, from an empiricalpoint of view, the followed rule proposed by Soules et al. (1987) was set for a time increment�t such that

�t

τSk(t)

≤ 0.4. (6)

This rule showed to have a reasonable precision and stability during the transition region.Hodge (1994) discussed also these numerical evaluations and proposed fictive temperatureincrements less than 1 °C. Moreover, during the numerical cooling process of the package,we paid attention to avoid the numerical artifacts (i.e., discontinuities) of the Tf map overthe whole structure.

3 Modeling of the cooling

The finite element code Abaqus (Abaqus FEA 2009) was used to compute these equationswithin a 3D model of a half-canister totally filled with glass (with 21500 hexahedral el-ements, 24000 nodes and 73000 degrees of freedom). The half-canister is modeled as ahalf-cylinder, cut longitudinally, defined by a vertical axis of 1.1 m and a diameter of 0.4 m(for the whole canister, approximating the geometry of the glass block). The upper face ofthe cylinder is identified with the top free surface of the glass. The geometry and mesh areidentical to the ones used in our previous work (Barth et al. 2012).

At first, we compute the evolution of the package temperatures during the whole pro-cess as described in details in our previous work (Barth et al. 2012). The glass transitionwhich varies locally as function of the thermal history is then analyzed from these results,and fictive temperatures are calculated during the process as a function of time. We chose to

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Table 1 Glass relaxation properties from Soules et al. (1987); in this work, the MS parameter set was ex-tracted from the work of Kurkjian (1963); we normalized the cV

iterms (note that these were not correctly

normalized in the work of Soules et al. 1987)

MV MS

First term of the series cV1 = 0.05640 τV

1 = 27070 s cS1 = 0.067 τS

1 = 10.75 s

Second term cV2 = 0.51010 τV

2 = 121300 s cS2 = 0.053 τS

2 = 155 s

Third term cV3 = 0.21745 τV

3 = 329700 s cS3 = 0.086 τS

3 = 1406 s

Fourth term cV4 = 0.13270 τV

4 = 896300 s cS4 = 0.230 τS

4 = 10150 s

Fifth term cV5 = 0.04101 τV

5 = 2436000 s cS5 = 0.340 τS

5 = 46080 s

Last term (nV = nS = 6) cV6 = 0.04234 τV

6 = 10920000 s cS6 = 0.224 τS

6 = 107500 s

For both relaxations Tref = 746.15 K�h∗/R = 76200 Kx = 0.5

start the fictive temperatures computation at the moment when the glass free surface (whichis the coldest point at the top of the glass cast in the simulation) is at a temperature arbi-trarily set around the Littleton temperature (900 K for the glass SON68; above which glassis considered to be in a softening behavior). After this, the thermomechanical analysis isconducted locally as soon as the glass solidifies in a given element. Once the liquid glassin the package core is entirely surrounded by the solid (this moment is defined thereafteras “solidification start”), we compute the mechanical response of the entire package, in-cluding the liquid core. In addition, the liquid glass depression is restrained in the liquidstate analysis since it is considered to be incompressible (then, cavitation occurs instead ofdepression). The glass constitutive laws presented in the previous section are coded into aFortran subroutine (UMAT in Abaqus FEA 2009).

3.1 Cooling process and glass properties

The manufacturing of a full scale package during an experimental vitrification process al-lowed us to collect a set of temperature boundary conditions using thermocouples. A cylin-drical canister (1.3 m high, with a diameter of 0.43 m) was filled with a nominal volume ofthe inactive glass SON68, and then was cooled down to ambient temperature.

The details of the manufacturing process and the thermal properties of the SON68 glasscast were reported by Barth et al. (2012). The thermal results studied there were directlyused in the current study through temperatures interpolations on an identical time scale. Atthe end of all simulations, the glass is at a uniform temperature of 300 K (±5 K).

The glass relaxation properties were taken from Soules et al. (1987) for a soda–lime glassand are reported in Table 1. The thermomechanical properties were taken from Dubé et al.(2010) for the same borosilicate glass as the one used in our experiments and are reportedin Table 2. In addition, the Young modulus and Poisson ratio were calculated assuming anidentical bulk modulus before and after the glass transition. The liquid phase shear moduluswas fixed at a value recommended by Ref. (UMAT in Abaqus FEA 2009) for incompressiblematerials.

The solid and liquid coefficients of thermal expansion were obtained experimentallyfrom dilatometric tests (experiments were carried out at CEA Centre de Marcoule usingan Adamel Lhomargy dilatometer). Evaluation was made from graphical interpretation (in-ternal CEA technical note). The experimental tests were made from room temperature up

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Table 2 SON68 glass thermomechanical properties, for the solid from Dubé et al. (2010)

Solid (glassy) properties Liquid properties

Temperatures [K] 293.15 373.15 473.15 573.15 673.15 773.15 Any T

Young modulus [GPa] 92.8 89.3 88.3 87.2 86.1 39.8 6.62510−4

Poisson ratio 0.22 0.245 0.21 0.215 0.24 0.2 0.499995

αg or αl [10−6 K−1] 8.3 8.3 8.3 8.3 9.1 9.1 80

to 1330 K at an identical heating rate of 5 °C per minute. From these tests, the averagesolid coefficient of thermal expansion was evaluated for a range of temperature between310 K and 730 K corresponding to the linear variation of the curve. It was measured tobe around αg = 9 × 10−6K−1. For the average liquid coefficient of thermal expansion, itwas measured between 790 and 850 K and found to be around αl = 80 × 10−6 K−1. Theglass transition temperature Tg (located at the change of slope in the dilatometric curve)was evaluated graphically from the dilatometric curve and also using a thermal differentialmicroanalysis with a Setaram M5 microanalyzer (internal CEA technical note). From bothof these analyses, Tg was evaluated at the same value around 775 K.

3.2 Boundary conditions

The stainless steel canister has a thickness of 5 mm. In the primary thermal simulations of theglass cast, the canister has no influence and was not taken into account. In the mechanicalresponse, however, its role is less negligible, as it is expected to increase the local stressload near the edge of the glass (at the glass/steel interface and due to its thermal expansioncoefficient mismatch). An example where the stainless steel/glass interface is the principalfactor to the damaging behavior is treated, for example, by Chambers et al. (1989) (there, thedamage of the glass comes from the interface and must be avoided). Nevertheless, this effecthad a minor influence in all our simulation and for the sake of simplicity, we chose to neglectthis role (we rely only on the thermal gradients of the bulky glass cast to initiate damage)and decided to conserve free displacement boundary conditions (BCs) at any glass–canisterinterface. The results presented hereafter will concentrate only on the effect of the viscousrelaxation of the bulk glass.

As the cylinder is modeled only by a half representation, we apply a symmetry bound-ary condition on the longitudinally cut surface. We define a fixed point at the bottom ofthe cylinder axis and a displacement-constrained point at the bottom face (displacement isimposed along the radius of the canister).

For the thermal boundary conditions (BCs) we have computed five cases (see Barth et al.2012):

Case I—Experimental BCs following an experimental manufacturing process.Cases II and III—Start at an imposed homogenous temperature in the entire package and

then follow the experimental BCs for the remaining time. For Case II (H high T), thehomogeneous temperature is 1373 K (pouring temperature). For Case III (H low T), thehomogeneous temperature is 856 K (a highly viscous liquid state for the glass).

Cases IV and V—We first follow the experimental BCs until we reach an external temper-ature of 856 K in the final cooling down period and then we change the cooling rate. ForCase IV (linear BC), the cooling rate is switched to very slow, and constant down to ambi-ent temperature (the rate is slower than the experimental one until a critical time). For Case

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V, the cooling rate is changed to a high value that corresponds to a quenching at ambienttemperature (quench BC).We note that the thermal results from our previous study (Barth et al. 2012) did not takeinto account any relaxation phenomenon (only a “setting temperature” for the transition asdenoted by Soules et al. 1987).

4 Results and discussion

The simulations of the thermomechanical models are reported and discussed in this section.Each simulation corresponds to one of the five thermal BC sets described above. The glassrelaxation results are reported firstly for a thermal history following the experimental re-sults (Case I), and then by the four other thermal BCs for which the vitrification process isdifferent.

4.1 Effect of each glass relaxation during the experimental case (Case I)

Through the thermo-viscoelastic algorithm, the phase transition region is computed in thisreference simulation and varies over temperatures between 778 and 799 K. These are thefinal fictive temperatures at the end of the cooling period which are determined over thepackage for the specific volume and shear modulus relaxations (MV , MS ).

Once the fictive temperatures as a function of time are known, we obtain the thermalstrain during the cooling down period from Eq. (1). The strain driven by the thermal historythrough MV leads to a mechanical response of the entire package using the constitutiveEq. (4) for which the elastic properties of the liquid and solid glass are then modulated bythe relaxation MS in Eq. (5).

The vitrification process computed with both of these relaxations (MV &MS or denotedhere as Case I-a) leads, at the end of the cooling, to a stress state equivalent to temperedglass: compression of the external surface and by equilibrium tensile stresses of the packagecore. It is used as the reference case hereafter.

Two other simulations using the same thermal history (Case I) and the same solidificationstart were undertaken and reported in this section in order to illustrate the effect of MS andMV individually in our model, as it was done in the work of Ganghoffer (2000). In the firstsimulation (denoted by “MV –MS” or here by Case I-b), we only compute MV relaxationassuming MS = 1 at any time, and the fictive temperatures defined by MV enable us toattribute to the glass the corresponding elastic properties (either of the liquid or the solidglass, depending on the temperature difference with Tf during the increment). Then, ina second simulation (denoted as “setting T ” or Case I-c), we assume the uniform fixedtransition temperature at 775 K, the linear expansion coefficient and elastic properties beingaccordingly defined as in either the liquid or the solid glass.

The MV –MS (Case I-b) simulation allows us to compare it with the reference MV &MS

(Case I-a) simulation in order to evaluate the influence of MS relaxation on the stress state.The second “setting T ” simulation where there is neither MV nor MS (Case I-c) enables usto compare it with the MV –MS simulation (Case I-b) in order to evaluate the influence ofthe Tf map (and through MV ) on the stress state.

4.1.1 Pressure of the liquid glass

Observations in solid glass of “critical” transient tensile stresses (for glass fracture nucle-ation), at the beginning of the solidification process, have not been confirmed in our mod-eling of the experimental vitrification process (Case I-a, b or c). This phenomenon could be

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Fig. 1 Liquid glass pressure during the solidifications of the experimental cooling simulations (Case I):MV &MS simulation (MV with MS , Case I-a); MV –MS simulation (MV without MS , Case I-b); and settingtemperature simulation (neither MV nor MS relaxation, Case I-c)

due to the glass surface vicinity that contracts more rapidly than the inner part (Daudevilleand Carré 1998) and thus generates a compression state of the core and transient tensilestresses at the edges. For the “non-critical” transient tensile stresses, we present this effecthere through the compression state of liquid glass during the solidification given the fact thatthe liquid and the solid are in a mechanical equilibrium (the pressure computations presentedin our study assumes a zero atmospheric pressure outside the package).

Figure 1 shows the liquid core pressure for the different Cases I-a, b and c. No com-pression state (conventionally, positive pressure) was observed for the reference simulation(Case I-a with both relaxations MV &MS ).

In Fig. 1, the liquid glass shows a rather limited compression load at the very beginningof solidification only when MV is present (MV –MS simulation, Case I-b). The compressionof the core of the package is initiated there only when the complex transition region mapis taken into account (due to thermal history–dependence of MV , in comparison with CaseI-c). On the time axis of Case I-c, the setting temperature of 775 K makes the solid glassenclose the liquid glass later than for the case of the MV –MS simulation (Case I-b), but withan equivalent solidification end. For the Case I-c, this period of time is then shorter (lesstime to build-up), and with slower cooling rates, inducing a less important pressure build-up in the liquid phase in comparison to Case I-b. As for the reference Case I-a (MV &MS

simulation), the elastic behavior of the solid glass is there with a smoother transition fromthe incompressible behavior to the elastic one of the solid glass, compared to the MV –MS

case (I-b). This can explain why the liquid core tends to be compressively most reactive inthe MV –MS simulation (Case I-b).

The rather limited amount of pressure in the three relaxation cases (I-a, b and c) rep-resented in Fig. 1 leads toward a stress state of the liquid glass that could be considered apressure free state during this experimental thermal load (Case I).

It should be noticed that the ability to have a liquid in depression has been omitted, sothat at the end of the solidification period this behavior is restored for the solid glass. Themechanical load tending to be similar towards a tensile stress state for Case I, it is shown bythe final vertical asymptotes represented in Fig. 1.

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Table 3 Evaluation of the stress build-up at the end of the cooling down period with experimental BC(Case I)

Simulation Max. principal stress(center) [MPa]

Max. Tresca stress(edge) [MPa]

Reference model (MV &MS ) Case I-a 193 233

MV without MS(MV –MS) Case I-b 258 415

Neither MV nor MS (setting T ) Case I-c 185 206

4.1.2 Residual and transient stresses

MV –MS (Case I-b) and setting T (Case I-c) simulations showed a stress state qualitativelysimilar to the principal stresses map of the reference simulation (MV &MS , Case I-a). As forthe quantitative characterizations, we chose to compare the results from these simulations inTable 3 through the maximum values in the entire glass block of the shear stress (localizedat the edges) and the maximum principal stress (localized along the centerline of the glassblock). The Tresca (maximum shear stress) value enables us to compare it with a shearfailure criterion (mode II of failure), and the maximum principal stress with a tensile failurecriterion (mode I). These results show that the MS relaxation (Case I-a versus Case I-b) andthe MV relaxation (Case I-b versus Case I-c) have opposite influences on the final stress loadfor this experimental thermal BC case.

For Case I-a, the stress build-up increases until the cooling ceases at ambient tempera-ture, as represented in Fig. 2 for cross-sections in the glass block taken at the height of thewarmest point.

For the principal stresses evolution shown in Fig. 3, the glass tends to be loaded in biaxialcompression on the edge and in triaxial tension at the center. Plots of the pressure as afunction of time after solidification starts are also shown in Fig. 3 for the cylinder centerand for an edge point (the radial projection of the center on the edge). It can be comparedto similar plots for tempered glass plates as in the work of Nielsen et al. (2010, see theirFig. 10) except for the tension state of the surface at the beginning of the cooling that isabsent in our case (it is not represented in Fig. 3).

4.2 The different thermal boundary conditions

For the relaxations investigated, the transition region is highly dependent on the coolingrates, which is illustrated for the thermal BCs in Fig. 4 (Cases I to V). The higher the coolingrate differences between the center and the edge, the larger the range of Tf will be in thepackage. Moreover, the lowest Tf are localized in the center of the package for all thethermal BCs, since the cooling rate of the edge is greater than at the center during the phasetransitions.

4.2.1 Liquid–solid equilibrium

Theses simulations were also studied during solidification for each thermal load scenario.The liquid pressure computations are presented in Fig. 5 (MV &MS : Case I to V-a). For eachthermal BC, the solidification end is marked by the same asymptotes towards depressionafter a duration corresponding to the solidification (see also Fig. 4(a)).

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Fig. 2 Pressure of the Cauchy stress tensor, along diameters, at arbitrarily chosen times of the referencemodel (Case I-a: experimental thermal BC, MV &MS simulation) taking into account the relaxations MS andMV ; liquid glass is present at 0 min on the whole diameter, at 70 min between R = ±0.15 m, and finallyat 175 min between R = ±0.1 m; the diameters are chosen between the height of the second upper castingcenter and the height of the cylinder center to follow the warmest point in the core

Fig. 3 Principal stresses of the Cauchy stress tensor at the half-height of the package, and correspondingpressure, during the solidification and cooling process of the reference MV &MS simulation (Case I-a) takinginto account the relaxations MS and MV ; the period from 0 to 1000 s, not represented here, does not showany significant stress values; the end of the time axis approximates here the package at ambient temperature

H high T (Case II-a) and Linear BC (Case IV-a) simulations show close behaviors whencompared to the reference model. H low T (Case III-a) and Quench BC (Case V-a) simula-tions present a different behavior yielding an important compression of liquid glass during

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Fig. 4 Solidification transition region for the relaxation models in the entire package (two fictive tempera-tures that are near the extreme values are represented in two different locations of the glass block for eachcase); the two curves obtained are for each case on the center of the glass free surface and the last volumesolidified (this latter position is not the same on the cylinder axis for all cases); the symbols on each curve de-limit the liquid–solid transition at the surface or in the core of the package through the same lag of Tf behindT ; (a) The timescale is in the same referential as in the results from the thermal model without relaxation(Barth et al. 2012); in the purely thermal study, the maximum thermal gradients were calculated and ordered(1)–(5); Here, they correspond respectively to values of 9460, 4472, 4297, 3239, and 1303 K/m; (b) The Tf

are now plotted as a function of temperature during the cooling, the lag of Tf behind T greater than 2 °C(marked by the symbols) is the beginning of the departure from the equilibrium straight line Tf = T

solidification (and corresponding tensile stresses in the solid by equilibrium) as in the case ofthe tempered glass plate simulations (Daudeville and Carré 1998). A qualitative correlationbetween these load types and thermal results is presented in Online Resource 1.

The transient tensile stresses at the surface described, for instance, by Daudeville andCarré (1998), through the liquid–solid equilibrium, makes it possible to induce damage ofthe glass in mode I only for the quench BC case (V-a), whose liquid pressure is the highestas reported in Fig. 5. It should be highlighted that these transient tensile stresses in theglassy state are only critical at the upper and lower extremities, which is in a rather limited

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Fig. 5 Liquid glass pressureduring solidification of thedifferent cooling BCs with bothof the relaxations MV and MS

(e.g., reference model is then theMV &MS simulation Case I-a);(a) the global view; (b) the zoomof (a) for lower pressure values

Table 4 Evaluation of the stress build-up at the end of the cooling down period for the different thermal BCs(modeled with MV and MS )

BC set name Linear BCCase IV-a

H low TCase III-a

Referencemodel Case I-a

H high TCase II-a

Quench BCCase V-a

Maximumprincipal stress(center) [MPa]

112 176 193 192 294

Maximum Trescastress (edge)[MPa]

136 206 233 235 449

volume in the glass block. The assumption made in the thermal study of the liquid glass ina pressure free referential (Barth et al. 2012) was therefore valid for the other simulations(Cases I to IV).

4.2.2 Residual stresses

At the end of the cooling process (at a uniform ambient temperature), all simulations under-taken in this study present a similar load of the package, characterized by a triaxial tensileload in the center and by a biaxial compression at the edge of the package.

For the sake of simplicity, we chose the usual maximum stress criterion as shown inTable 4 (maximum shear stress and principal stress) to draw comparisons between the dif-ferent stress loads. To avoid edge effects, the criterion was not applied on the 1/16 heightextremities of the package.

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The stress states reported are in good agreement with the maximum thermal gradientsindicated in the caption of Fig. 4, concerning their sorting (1 to 5). There is an exceptionwith the differences of these maximum stresses that are considered as negligible between thereference simulation (Case I-a) and H high T simulation (Case II-a). These results underlinethe fact that the cooldown rates (linear (Case IV-a) and quench BC (Case V-a) simulations)have a greater impact on the stress build-up. From these stress states, we would expect ourborosilicate glass to fail in mode I in the center, and eventually in mode II at the edge (whereno mode I failure occurs).

5 Conclusion

We modeled the structural relaxation during the cooling of glass in a nuclear waste vitrifi-cation process using viscoelasticity without damage. Our results, based on simulations ofan experimental vitrification process, showed significant differences from commonly stud-ied tempered glass plate processes. Indeed, the thermomechanical load of liquid glass (inmechanical equilibrium with the surrounding glassy state) did not show significant transientcompressive stress values for our experimental vitrification process. This result was obtainedalthough we have observed a final thermal stress load in the solidifying and cooling glasspackage that is similar to the tempered glass processes (residual stresses).

We separated and analyzed the contribution of the two modeled relaxations (stress relax-ation and/or specific volume relaxation) to the stress load and found them to have oppositeinfluence on the residual stress state.

We have also studied four other thermal boundary conditions. Their build-up stress loadswere found to qualitatively correspond to those that may arise from thermal gradients ob-tained through a pure thermal study. These results highlight the fact that the external coolingrates are more important than the internal state of the package for the simulated thermalboundary conditions.

The pressure load of liquid glass has highlighted some critical influence (through tran-sient tensile stresses) on the mechanical response of the package for the case of a quenchsimulation (Case V-a). In this particular case, the failure criterion is met in a very limitedvolume of the glass block which contributes to a tensile transient stress failure scenario thatis well known for tempered glass.

Apart from this quench case process, the failure of the glass in the whole package, forthe simulated cooling processes, is expected to be in mode I at the center of the package andin mode II at the edge. It is reasonable to think that the amount of damage will follow theamplitude of the stress build-up inside the package.

The stress relaxation and specific volume relaxation were applied together to the differentcases of thermal boundary conditions (simulating different vitrification processes). Eachrelaxation (stress relaxation or specific volume relaxation) showed a major effect on thestress state building up during the simulated thermal load, either on the residual stresses atthe end of the cooling down period, or on the transient stresses observed in our study throughthe compression of the liquid glass core inside the package.

Acknowledgements The authors wish to acknowledge the CEA (LCLT) for its financial support as wellas providing the experimental data. The ANDRA (French National Radioactive Waste Management Agency)and AREVA NC are also gratefully acknowledged for supporting this study. Finally, thanks are due to V. Do-quet for fruitful discussions (Laboratoire de Mécanique des Solides, CNRS–École Polytechnique, Palaiseau,France).

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References

Abaqus FEA: Abaqus/Standard version 6.9-1. Theory manual 2.1.1 and 4.1.1, user subroutine. manual 1.1.36(2009). SIMULIA/Dassault Systèmes

Alcoutlabi, M., McKenna, G.B., Simon, S.L.: Analysis of the development of isotropic residual stresses ina bismaleimide/spiro orthocarbonate thermosetting resin for composite materials. J. Appl. Polym. Sci.88(1), 227–244 (2003). doi:10.1002/app.11649

Alcoutlabi, M., Banda, L., Kollengodu-Subramanian, S., Zhao, J., McKenna, G.B.: Environmental effectson the structural recovery responses of an epoxy resin after carbon dioxide pressure jumps: intrinsicisopiestics, asymmetry of approach, and memory effect. Macromolecules 44(10), 3828–3839 (2011).doi:10.1021/ma1027577

Barth, N., George, D., Ahzi, S., Rémond, Y., Doquet, V., Bouyer, F.: Modeling and simulation of the coolingprocess of borosilicate glass. J. Eng. Mater. Technol. 134(4), 041001 (2012). doi:10.1115/1.4006132

Bonniaud, R., Jouan, A., Sombret, C.: Large scale production of glass for high level radioactive waste. Nucl.Chem. Waste Manag. 1(1), 3–16 (1980). doi:10.1016/0191-815X(80)90024-8

Chambers, R.S., Becker, E.B.: Integration error controls for a finite element viscoelastic analysis. Comput.Struct. 24(4), 537–544 (1986). doi:10.1016/0045-7949(86)90192-6

Chambers, R.S., Gerstle, F.P. Jr., Monroe, S.L.: Viscoelastic effects in a phosphate glass-metal seal. J. Am.Ceram. Soc. 72(6), 929–932 (1989). doi:10.1111/j.1151-2916.1989.tb06246.x

Daudeville, L., Carré, H.: Thermal tempering simulation of glass plates: inner and edge residual stresses. J.Therm. Stresses 21(6), 667–689 (1998). doi:10.1080/01495739808956168

DeBolt, M.A., Easteal, A.J., Macedo, P.B., Moynihan, C.T.: Analysis of structural relaxation in glass usingrate heating data. J. Am. Ceram. Soc. 59(1–2), 16–21 (1976). doi:10.1111/j.1151-2916.1976.tb09377.x

Doquet, V., Ben Ali, N., Constantinescu, A., Boutillon, X.: Fracture of a borosilicate glass under triaxialtension. Mech. Mater. 57, 15–29 (2013). doi:10.1016/j.mechmat.2012.10.008

Dubé, M., Doquet, V., Constantinescu, A., George, D., Rémond, Y., Ahzi, S.: Modeling of ther-mal shock-induced damage in a borosilicate glass. Mech. Mater. 42(9), 863–872 (2010).doi:10.1016/j.mechmat.2010.07.002

Faletti, D., Ethridge, L.: A method for predicting cracking in waste glass canisters. Nucl. Chem. Waste Manag.8(2), 123–133 (1988). doi:10.1016/0191-815X(88)90071-X

Fielding, S.M., Sollich, P., Cates, M.E.: Aging and rheology in soft materials. J. Rheol. 44(2), 323–369(2000). doi:10.1122/1.551088

Frugier, P., Gin, S., Minet, Y., Chave, T., Bonin, B., Godon, N., Lartigue, J.-E., Jollivet, P., Ayral, A., DeWindt, L., Santarini, G.: SON68 nuclear glass dissolution kinetics: current state of knowledge and basisof the new GRAAL model. J. Nucl. Mater. 380(1–3), 8–21 (2008). doi:10.1016/j.jnucmat.2008.06.044

Ganghoffer, J.-F.: Calculation of thermal stresses in glass-ceramic composites. Mech. Time-Depend. Mater.4(4), 359–379 (2000). doi:10.1023/A:1026581906549

Hodge, I.: Enthalpy relaxation and recovery in amorphous materials. J. Non-Cryst. Solids 169(3), 211–266(1994). doi:10.1016/0022-3093(94)90321-2

Jain, A., Yi, A.Y.: Numerical modeling of viscoelastic stress relaxation during glass lens forming process. J.Am. Ceram. Soc. 88(3), 530–535 (2005). doi:10.1111/j.1551-2916.2005.00114.x

Kahl, L., Kroebel, R., Storch, W., Holton, L., Dierks, R.: Investigation of the thermal behavior of high levelwaste glass in steel canisters during cooling and at equilibrium. Nucl. Eng. Des. 130(1), 77–88 (1991).doi:10.1016/0029-5493(91)90195-N

Kamizono, H., Niwa, K.: An estimation of the thermal shock resistance of simulated nuclear waste glassunder water quenching conditions. J. Mater. Sci. Lett. 3(7), 588–590 (1984). doi:10.1007/BF00719619

Kong, J., Kim, J.H., Chung, K.: Residual stress analysis with improved numerical methods for tem-pered plate glasses based on structural relaxation model. Met. Mater. Int. 13(1), 67–75 (2007).doi:10.1007/BF03027825

Kovacs, A.J.: La contraction isotherme du volume des polymères amorphes. J. Polym. Sci. 30(121), 1097–1162 (1958). doi:10.1002/pol.1958.1203012111

Kovacs, A.J.: Transition vitreuse dans les polymères amorphes—étude phénoménologique. In: Adv. Polym.Sci, vol. 3, pp. 394–507. Springer, Berlin (1964). doi:10.1007/BFb0050366

Kovacs, A.J., Aklonis, J.J., Hutchinson, J.M., Ramos, A.R.: Isobaric volume and enthalpy recovery ofglasses—II. A transparent multiparameter theory. J. Polym. Sci., Polym. Phys. Ed. 17(7), 1097–1162(1979). doi:10.1002/pol.1979.180170701

Kurkjian, C.R.: Relaxation of torsional stress in the transformation range of a soda–lime–silica glass. Phys.Chem. Glasses 4(4), 128–136 (1963)

Li, J., Yim, M.-S., McNelis, D.: A simplified methodology for nuclear waste repository thermal analysis.Ann. Nucl. Energy 38(2–3), 243–253 (2011). doi:10.1016/j.anucene.2010.11.002

Author's personal copy

96 Mech Time-Depend Mater (2014) 18:81–96

Markovsky, A., Soules, T.F.: An efficient and stable algorithm for calculating fictive temperatures. J. Am.Ceram. Soc. 67(4), c56–c57 (1984). doi:10.1111/j.1151-2916.1984.tb18826.x

Moynihan, C.T., Easteal, A.J., Tran, D.C., Wilder, J.A., Donovan, E.P.: Heat capacity and struc-tural relaxation of mixed-alkali glasses. J. Am. Ceram. Soc. 59(3–4), 137–140 (1976).doi:10.1111/j.1151-2916.1976.tb09450.x

Narayanaswamy, O.S.: A model of structural relaxation in glass. J. Am. Ceram. Soc. 54(10), 491–498 (1971).doi:10.1111/j.1151-2916.1971.tb12186.x

Nielsen, J.H., Olesen, J.F., Poulsen, P.N., Stang, H.: Finite element implementation of aglass tempering model in three dimensions. Comput. Struct. 88(17–18), 963–972 (2010).doi:10.1016/j.compstruc.2010.05.004

Ougier-Simonin, A., Guéguen, Y., Fortin, J., Schubnel, A., Bouyer, F.: Permeability and elastic prop-erties of cracked glass under pressure. J. Geophys. Res. B 116(7), B07203.1–B07203.12 (2011a).doi:10.1029/2010JB008077

Ougier-Simonin, A., Fortin, J., Guéguen, Y., Schubnel, A., Bouyer, F.: Cracks in glass under triaxial condi-tions. Int. J. Eng. Sci. 49(1), 105–121 (2011b). doi:10.1016/j.ijengsci.2010.06.026

Perez, J.M. Jr., Westsik, J.H. Jr.: Effects of cracks on glass leaching. Nucl. Chem. Waste Manag. 2(2), 165–168 (1981). doi:10.1016/0191-815X(81)90034-6

Siedow, N., Grosan, T., Lochegnies, D., Romero, E.: Application of a new method for radia-tive heat transfer to flat glass tempering. J. Am. Ceram. Soc. 88(8), 2181–2187 (2005).doi:10.1111/j.1551-2916.2005.00402.x

Soules, T.F., Busbey, R.F., Rekhson, S.M., Markovsky, A., Burke, M.A.: Finite-element calculation ofstresses in glass parts undergoing viscous relaxation. J. Am. Ceram. Soc. 70(2), 90–95 (1987).doi:10.1111/j.1151-2916.1987.tb04935.x

Tool, A.Q., Eichlin, C.G.: Variations caused in the heating curves of glass by heat treatment. J. Am. Ceram.Soc. 14(4), 276–308 (1931). doi:10.1111/j.1151-2916.1931.tb16602.x

Tool, A.Q.: Relation between inelastic deformability and thermal expansion of glass in its annealing range. J.Am. Ceram. Soc. 29(9), 240–253 (1946). doi:10.1111/j.1151-2916.1946.tb11592.x

Tool, A.Q.: Effect of heat-treatment on the density and constitution of high-silica glasses of the borosilicatetype. J. Am. Ceram. Soc. 31(7), 177–186 (1948). doi:10.1111/j.1151-2916.1948.tb14287.x

Webb, S., Knoche, R.: The glass-transition, structural relaxation and shear viscosity of silicate melts. Chem.Geol. 128(1–4), 165–183 (1996). doi:10.1016/0009-2541(95)00171-9

Zheng, Y., McKenna, G.B.: Structural recovery in a model epoxy: comparison of responses after temperatureand relative humidity jumps. Macromolecules 36(7), 2387–2396 (2003). doi:10.1021/ma025930c

Author's personal copy