Numerical modelling of moisture related mechanical stress in wooden cylindrical objects using COMSOL: a comparative benchmark

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1 erpt from the Proeedings of the COMSOL Conferene 2 Paris Numerial modelling of moisture related mehanial stress in wooden lindrial objets using COMSOL: a omparative benhmark H.L. (Henk) Shellen (PhD), A.W.M. (Jos) van Shijndel (PhD) indhoven Universit of Tehnolog, Department of Building and Arhiteture, VRT 6.29; P.O. Bo 53; 56 MB indhoven; The Netherlands; H.L.Shellen@tue.nl Abstrat: For preservation of artefats in a museum the indoor limate is often restrited to a ver narrow interval for temperature, but most of all for relative humidit (,7). In old buildings the museum onditions of artefats, e.g. near old walls, mostl are not in line with museum reommendations. To have an impression of indoor museum limates in old buildings, a large number of ase studies were arried out in several Duth museums. For at least one ear, temperature and relative humidit were reorded in different rooms and at different eternal wall surfaes of the museums. The results of this measurement ampaign reveal that there were a large number of indoor limate onditions that did not satisf the originall formulated restrited limate (). To have an impression of the impat of the measured indoor limates on the onservation properties of objets, a numerial simulation model was needed. The model should be able to determine the temperature and moisture ontent gradients in organi objets and the related mehanial stress due to hanging environmental indoor limate onditions. For this purpose a three dimensional numerial simulation model was onstruted in COMSOL. The partial differential equations for heat and moisture transport in objets were implemented in COMSOL and oupled to the mehanial stress module of COMSOL. The results were temperature and moisture profiles in three dimensional objets due to dnamiall hanging (measured) environmental onditions. The resultant mehanial stresses were alulated and ompared to maimum allowable threshold stresses to sta in the safe elasti strain regime. Kewords: Museum indoor limate, modeling, onservation, museum objets, stress, moisture indued damage.. Introdution The indoor limate onditions in a number of about 25 different museums in The Netherlands have been monitored (4). One of the ke questions was to determine the possible risk on damage to objets in relation to the varing indoor limate. From literature it is known that varing limate onditions, espeiall flutuating relative humidit (RH) onditions, result in varing moisture ontents in hgrosopi materials like wood. These varing moisture ontents indue deformations of the wooden materials. When these deformations sta in the elasti range of the wooden materials no damage is epeted. When a maimum stress level is reahed, the response of the materials, however, is no longer reversible. The inelasti region is reahed and the deformations will result in an irreversible response. S. Jakiela et al. (2) developed a numerial model of moisture movement and related stress in a lime wood linder. The model was developed to determine maimum stress levels in a wooden linder, representing objets of art eposed to hanging limate onditions. The objet of this work was to develop a similar model in COMSOL to determine the stress levels in a similar objet under measured boundar onditions in the different museum environments. In this wa it was possible to ompare different museum indoor limates regarding the damage the ma indue in objets of value. 2. COMSOL model The COMSOL model is based on a model of a lime wooden linder with diameter.3m. The oordinate sstem is lindrial and the material model is assumed to be orthotropi. 2. Hgro thermal equations The COMSOL model is based on the following equations: Thermal transport: T p ( k T ) () t = Densit [kg/m 3 ]

2 p = Constant pressure speifi heat k = Thermal ondutivit [J/kg.K] [W/m.K] wood an be onsidered as an orthotropi material. Due to the strength in the aial diretion the displaements in this diretion an be negleted. Water vapor transport b diffusion: P ( D( P) P) (2) t P = Vapor pressure [Pa] D(P)= Moisture diffusion oeffiient [m 2 /s] 2.2 Boundar onditions Thermal: q h T o T ) (3) ( s q = Heat flu at surfae [W/m 2 ] h = Heat transfer oeffiient [W/m 2.K] h = 7.7 W/m 2.K Hgri: g P o P ) (4) ( s g = Vapor flu at surfae [kg/m 2 s] = Vapor transfer oeffiient [s/m] =.5-6 s/m P o = RH o *P sat (T o ) [Pa] Indies: o : environmental s : surfae sat : saturated nvironmental onditions t=: T o =2 o C RH o = 7%RH t= : T o =2 o C RH o = 3%RH 2.3 Mehanial stress and strain In a first approimation the deformation of wood is onsidered to be elasti and orthotropi. Partial differential equations (PD) for the linear stress and strain desription b Hooke s Law will be oupled to PD s for heat and moisture desription. For a 3-D volume element in a ontinuous bod in equilibrium the differential equations of mehanial equilibrium in Cartesian oordinates are desribed b (3). In a first approimation Figure : eneral state of stresses (Hibbeler 997) The omponents of stresses in a ontinuous plane in equilibrium under the ation of surfae (without eternal bod fores) satisf two differential equations of equilibrium. These equations epressed in Cartesian oordinates have the form: where σ, = the normal omponents of stress [N/m 2 ] τ, τ = the shear omponents of stress assoiated with two aes [N/m 2 ] The relation between stress and strain is desribed b the generalized Hooke s law and for an anisotropi material it an be written as: where ε, ε = the normal strain omponents [-] γ = the shear strain omponent assoiated with two aes [-] υ, υ = Poisson s ratio [-], = moduli of elastiit or Young s moduli [N/m 2 ] w

3 = the shear modulus [N/m 2 ] α, α = the linear thermal epansivit [m/m.k] Δθ = a temperature inrement [ K] w = moisture ontent [kg/m 3 ] κ, κ = the linear relative deformation (shrinkage or swelling) due to moisture ontent hanging [m/m(kg/m 3 )] In ase of small displaements of a ontinuous bod, we an write u v v u, and Furthermore the matri must be smmetrial, therefore For the elasti desription of an orthotropi material, 4 independent elasti material properties must be known:,, and one of υ or υ. In fat, the material was treated as an anisotropi ontinuous bod, with orthotropi mehanial and deformation material properties. Inverting the two-dimensional matri and ombining the relations above, we obtain the displaement equations, whih an be written: ( u) Where is a rank four tensor, whih an be written as four 2-b-2 matries, 2, 2 and 22 : Where, the shear modulus, is defined b 2( And μ in turn is defined b v Material properties ) The material properties of lime wood were taken from Jakiel (2): The mean dr densit of the speimens was 53 kg/m 3. Poisson s ratio υ = υ =.4. The equilibrium moisture ontent and the moisture diffusion oeffiients were taken from graphs presented in (2) and were implemented in COMSOL in interpolation tables, as is presented in table. Table : Moisture diffusion oeffiient D and equilibrium moisture ontent MC of lime wood as a funtion of equilibrium relative humidit RH RH [%RH] D(RH) [m 2 /s] MC(RH) [m%] The modulus of elastiit properties of lime wood were opied from a table presented b Jakiel (2) into an interpolation table in COMSOL: Table 2: Modulus of elastiit of lime wood in tangential diretion (T) and radial diretion (R) RH [%RH] T [MPa] R [MPa] The dimensional hanges of lime wood in the radial and tangential diretion aross the entire range of MC were α R =.3 and α T =.28. These values were related to the hange in RH.

4 3. Verifiation The results of the COMSOL model were ompared with the results from Jakiel (2). These results are presented in figures and 2: The orresponding figures from the COMSOL model are presented below: Figure 2: Change in the distribution of moisture ontent at seleted distanes from the eternal surfae of a wooden linder for a step RH variation from 7 to 3% (2) Figure 4: COMSOL alulation of the hange in the distribution of moisture ontent at seleted distanes from the eternal surfae of a wooden linder for a step RH variation from 7 to 3% Figure 3: Tangential stress developing in wood as a result of the gradient of moisture ontent shown in figure (2) Figure 2 represents the hanging moisture ontent of the linder of wood on different depths from the surfae as a funtion of time. The initial equilibrium moisture ontent MC at a starting relative humidit of 7 %RH is 4%. The step variation in RH is from 7 to 3%RH. Figure 3 shows the tangential stress in the wooden linder due to the hanging moisture ontent. The largest stress is developing at the surfae, due to the shrinking of wood at the surfae, indued b the dring at the surfae. Figure 4 and 5 an be ompared to figure 2 and 3. The inrement of the urves depth is mm, starting from the surfae. Figure 5: COMSOL alulation of the tangential stress developing in wood as a result of the gradient of moisture ontent shown in figure 3 The equilibrium moisture ontent at the starting relative humidit of 7%RH slightl differs in the COMSOL alulations. This is the result of the interpolation tables, whih were taken from the urves from (2). Figures 6 and 7 show moisture gradient and stress gradient b Jakiel (2) after 24 h (a) and das (b). The lightest tone orresponds to the initial moisture ontent of 4% and lak of stress. The darkest tone orresponds to the final moisture ontent of 6 % and the stress level of 5.75 MPa. Figures 8 and 9 show COMSOL results.

5 Figure 6: Moisture ontent gradient (2) Figure 8: COMSOL results for moisture gradient Figure 7: valuation of stress gradient Figure 9: COMSOL results for stress gradient

6 4. Conlusion The implementation of a numerial model in COMSOL for alulating moisture indued mehanial stress in wooden objets seems to be promising. There was a lear equivalene between the results, presented b Jakiela and the results from the COMSOL model. In future the COMSOL model will be used to alulate mehanial stress in objets, indued b a a varing indoor limate, as has been measured in several museums. 5. Aknowledgement This work was supported b uropean Commission funding through the U Climate for Culture projet within FP7-NV Referenes. ASHRA. Museums, Libraries and arhives, in: 27 ASHRA Handbook,: Heating, ventilating, and air-onditioning appliations, SI edition, Amerian Soiet of Heating, Refrigerating and Air-Conditioning ngineers, In., 27, Chapter 2, p.p (27) 2. Jakiela, S., Bratasz, L., Kozlowski, R.: Numerial modeling of moisture movement and related stress field in lime wood subjeted to hanging limate onditions. Wood Si Tehnol (28) 42: 2-37 (27) 3. Lekhnitskii, S.. Theor of lastiit of an Anisotropi lasti Bod. Holden-Da, In. San Franiso (963) 4. Martens, M.H.J., Shellen, H.L. A sound indoor limate for a museum in a monumental building. Thermal performane of the building eterior envelopes of whole buildings XI International Conferene. Florida. (2) 5. Meklenburg, M.F., Tumosa, C.S., Temperature and relative humidit effets on the mehanial and hemial stabilit of olletions, in: ASHRA Journal, p.p (999) 6. Shellen, H.L. Heating monumental hurhes. Indoor limate and preservation of ultural heritage. PhD thesis. indhoven Universit of Tehnolog. ISBN: (22) 7. Thomson,. The Museum nvironment, st edition, Butterworths, in assoiation with the International Institute for Conservation of Histori and Artisti Works, London, United Kingdom, 978, p.p. 4. (978)