NUMERICAL SIMULATION OF DESICCATION CRACKING PROCESS BY WEAK COUPLING OF DESICCATION AND FRACTURE

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1 Geotec., Cont. Mat. & Env., ISSN:86-990, Japan, DOI: NUMERICAL SIMULATION OF DESICCATION CRACKING PROCESS BY WEAK COUPLING OF DESICCATION AND FRACTURE *Sayako Hirobe and Keni Oguni, Department of Sytem Deign Engineering, Keio Univerity, Japan *Correponding Author, Received: 09 March 06, Revied: 7 July 06, Accepted: 9 Nov. 06 ABSTRACT: The prediction of the poibilitie for the deiccation cracking i important for the building contruction becaue they could caue damage to the foundation tructure. While the experimental and numerical reearche were performed with variou material and condition, the mechanim of the deiccation cracking i till not clear. In thi reearch, the deiccation cracking i modeled by the coupling of the deiccation governed by the diffuion equation, the deformation, and the fracture. We perform the weak coupling analyi of the finite element analyi for the deiccation and the analyi of Particle Dicretization Scheme Finite Element Method (PDS-FEM) for the deformation and the fracture. The imulation i carried out with the different thickne of the deiccation layer under variou boundary condition. The imulation reult how the atifactory agreement with the experimental obervation in term of the crack pattern with net-like tructure, pattern formation proce, and the change in ize of the cell framed by the crack depending on the thickne of the deiccation layer. Thi agreement between the imulation reult and the experimental obervation indicate that the coupling of deiccation, deformation, and fracture i a fundamental mechanim of the deiccation cracking. Keyword: Deiccation Crack, Pattern Formation, Coupled Problem, PDS-FEM. INTRODUCTION The prediction of the poibilitie for the deiccation cracking i important for the building contruction becaue they could caue damage to the foundation tructure. While the experimental and numerical reearche were performed with variou material and condition, the mechanim of the deiccation cracking i till not clear. The reult of the previou experimental reearche how that the deiccation cracking ha geometric feature conerved in variou material and condition []-[4]. For intance, the deiccation cracking ha a net-like tructure and form polygonal cell framed by the crack. The ize and the hape of the cell change depending on the contraint condition and the thickne of the pecimen. In the previou numerical approache, ome model and numerical method are propoed to reproduce thee geometric feature of the deiccation cracking [5]-[8]. While thee model and method can reproduce the net-like crack pattern, they cannot reproduce the increae of the cell ize depending on the thickne of the pecimen or the change in crack pattern depending on the contraint condition. In thi paper, the deiccation cracking i modeled by the coupling of the deiccation governed by the diffuion equation, the deformation, and the fracture. Thi model can reflect the inhomogeneou water ditribution due to deiccation on the problem of the deformation and the fracture. We perform the weak coupling analyi of the finite element analyi for the deiccation and the analyi of Particle Dicretization Scheme Finite Element Method (PDS-FEM) [9], [0] for the deformation and the fracture. The imulation reult are compared with the experimental reult qualitatively.. MATHEMATICAL MODEL OF DESICCATION CRACKING The deiccation proce in the mixture of the powder and the water i decribed by the diffuion equation in term of the volumetric water content θ when the moiture diffuion coefficient D i aumed a contant and the gravitational effect i neglected. Conider a permeable and linearly elatic body Ω with external boundary Γ. When the water evaporate from the boundary Γ, the water ditribution in Ω i given by the next initial boundary value problem: θ = D t θ = Q n θ x Ω ( θ ) x on Γ (a) D (b) 8

2 ( x, 0) = θ x Ω θ (c) where Q (θ) i a water flux due to the evaporation from the boundary Γ and θ i a function of the poition x and time t. Here, only the liquid water movement i conidered. For the coupling of the deiccation and the fracture, the effect of the crack hould be embedded in the deiccation problem. In thi reearch, the crack urface Γ i conidered a a newly created evaporation urface and a hield for the permeable flow. The evaporation from the crack urface Γ can be introduced a an additional Neumann boundary condition of the initial boundary value problem for the deiccation proce Eq. (): θ = n ( θ ) x on Γ D Q () where Q (θ) i a water flux from the crack urface Γ due to the evaporation. The hield for the permeable flow can be expreed a the elimination of the water flux normal to the crack urface Γ. The Darcy low in the orthonormal coordinate ytem {e i} i J (3) = D θ where J i a water flux vector in the coordinate ytem {e i}. We define the orthonormal coordinate ytem {e i } with e 3 in the normal direction of the crack urface Γ. The proection of J on Γ (denoted a J c ) in the coordinate ytem {e i} i expreed a J = T P T J (4) c i i k kl l where the coordinate tranform matrix T i and the proection matrix which eliminate the water flux normal to Γ are = e i e (5) if i = =, P i = 0 otherwize. (6) Ti The introduction of J c in the place of J correpond to the introduction of the aniotropic moiture diffuion coefficient in the initial boundary value problem Eq. () and the Neumann boundary condition Eq. () on the crack urface. We olve thi initial boundary value problem for the water movement Eq. () with the Neumann boundary condition Eq. () on the crack urface by the ordinary FEM with the linear tetrahedral element. For the coupling of the deiccation and the deformation, the volume hrinkage correponding to the change in the volumetric water content θ hould be embedded in the deformation problem. The relationhip between the change in volumetric water content θ and the volumetric drying hrinkage train ε v i v ρ w ε ( x, t) = { θ ( x,0) θ ( x, t)} (7) ρ d where i a moiture hrinkage coefficient of the powder, ρ w i a ma denity of the water, and ρ d i a dry bulk denity of the powder. When Ω i homogeneou and iotropic, the drying hrinkage train ε i i v ε if i = ε = 3 (8) i 0 if i. In the cae of the deiccation crack phenomenon, the total train ε i can be divided into the elatic train ε i e and the drying hrinkage train ε i. The hrinkage train ε i doe not contribute to the generation of the tre and the train energy. Therefore, the tre-train relationhip and the train energy I for the deformation problem become σ ( ) i = cikl ε kl ε kl (9) I = Ω ( ε ) c ( kl kl )dv i ε i ikl ε ε (0) where σ i i a tre tenor and c ikl i an elatic tenor. In thi paper, the analyi of the deformation and the fracture i performed by PDS-FEM. For the evaluation of the functional I in Eq. (0), PDS- FEM applie the particle dicretization for the field variable with a pair of the conugate geometrie; Voronoi teellation {Φ } and Delaunay teellation {Ψ β }. The Delaunay teellation i a tetrahedron on the three-dimenional problem. The detailed dicretization cheme i hown in Oguni et al. [0]. Then, the dicretized train energy I i Î β β β β β ( ε ) c ( ε ε ) β ε Ψ () = M i i ikl kl kl β = where M i the number of Delaunay block and Ψ β i the volume of the β-th Delaunay block. The 9

3 diplacement u i minimizing the dicretized train energy Eq. () i atifying the equation of the force equilibrium: N γ γ K ik uk = γ = where K f i γ = M β β βγ ik B cikl Bl β = Ψ N β β β ε ( i = B ui + Bi u = f β β β β ( c B ) Ψ. = M β k i ikl l β = ) () (3) (4) ε (5) Here, N i the number of Volonoi block. Once the traction on the boundary of Voronoi block reache to the tenile trength, the interaction between the Voronoi block i lot. Thi lo of the interaction i introduced by changing B i β (and thu tiffne matrix K i βγ ). 3. NUMERICAL ANALYSIS We perform the weak coupling analyi of the finite element analyi for the deiccation and the analyi of PDS-FEM for the deformation and the fracture. The FEM analyi for the deiccation proce i carried out with a contant time tep t=0. hour. Then, the analyi for the deformation and the fracture i performed by PDS- FEM at each time tep. When the maximum traction among all element reache to the 97% of the tenile trength, the time tep i reduced to t=0.0 hour to capture the effect of the fracture urface on the deiccation and the deformation promptly. urface and the ide of the model. The initial volumetric water content i 7.% (contant) and the deiccation proceed until the averaged volumetric water content reache to the 3.4% (the averaged volumetric water content at which the crack propagation terminated in the deiccation tet of Peron et al. [4]). The nodal diplacement of the bottom urface of the model i contrained in the long ide direction and the vertical direction. We prepared the finite element model with the untructured meh (the number of the element i 56,597 and the number of node i,8). Table The model ize and the parameter for the imulation of one-dimenional crack pattern Model ize Soil dry denityρ d Evaporation peed on Γ Evaporation peed on Γ mm kg/m hour m/hour m/hour Moiture hrinkage coefficient 0.64 Moiture diffuion coefficient D m /hour Poion ratio 0.3 Young modulu Tenile trength 5.0 MPa 0.45 MPa 3. One-dimenional Crack Pattern In the cae of the deiccation tet on bar in Peron et al. [4], the mixture of water and clayey ilt i haped in a thin rectangular bar and the bar i dried on the plate. The bottom urface of the bar i contrained in the long ide direction only by the notche on the plate. In thi tet, the crack formed on the top urface are parallel to each other and normal to the long ide. In thi paper, we perform the imulation to reproduce thi experimental reult. The model ize and the parameter for the imulation are determined from the experiment of Peron et al. [4]; ee Table. The water evaporate from the top Fig. Simulation reult of the one-dimenional crack pattern The imulation reult (Fig. ) how the final crack pattern formed in the analyi model. The all crack are parallel to each other and normal to the long ide. The geometric feature of the crack and the number of crack formed on the top urface of the analyi model coincide with the experimental obervation of Peron et al. [4]. 0

4 3. Two-dimenional Crack Pattern 3.. Drying tet for the comparion with imulation reult of two-dimenional crack pattern We perform the drying tet of calcium carbonate lurry to oberve the crack pattern correponding to the thickne of the pecimen and to meaure the parameter for the imulation of two-dimenional crack pattern. The calcium carbonate lurry wa prepared at volumetric water content 7%. The lurry wa poured into the rectangular acrylic container ( mm). The thickne of the pecimen wa et a 5 mm, 0 mm, 0 mm, and 30 mm. The lurry wa dried in the air (0 C temperature and at 50 % relative humidity) until the pecimen dried out completely. Fig.3 Crack propagation proce of the drying tet in the cae of 0 mm thickne 3.. The imulation of two-dimenional crack pattern Fig. Final crack pattern on the top urface of the pecimen formed in the drying tet. (a) 5 mm, (b) 0 mm, (c) 0 mm, and (d) 30 mm The exceive water layer diappeared at the volumetric water content 56.0% and the crack initiated at the volumetric water content.4%. The crack propagation terminated at the volumetric water content 0.4%. Figure how the final crack pattern on the top urface of the pecimen with different thickne. The crack with net-like tructure form the polygonal cell framed by the crack and the averaged ize of the cell increae with the increae of the pecimen thickne. A hown in Fig. 3, in the crack pattern formation proce, ome long crack initiate on the edge of the pecimen and travere the pecimen at the initial tage of the deiccation cracking. Then, relatively hort crack appear to teellate the lager cell. Thee crack often branch and terminate when they meet the exiting crack. We perform the imulation to reproduce the two-dimenional crack pattern oberved in the drying tet of calcium carbonate lurry. The width and the height of the model i et a 00 mm and the depth wa et a 5 mm, 0 mm, 0 mm, and 30 mm. The parameter for the imulation of twodimenional crack pattern are hown in Table. The water evaporate from the top urface of the model and the nodal diplacement on the ide and bottom urface of the model i contrained. The initial volumetric water content i 56.0% (contant) and the deiccation proceed until the averaged volumetric water content reache to the 0.4%. We prepared the finite element model with the untructured meh; the meh ize are hown in Table 3. The final crack pattern on the top urface model with different thickne i hown in Fig.4. The crack have a net-like tructure and form polygonal cell. The cell ize are almot contant on each thickne and the averaged cell ize increae with the increae of the model thickne. Thee geometric feature of the crack pattern and the increaing tendency of the averaged cell ize qualitatively coincide with the obervation of the drying experiment of the calcium carbonate lurry hown in Fig.. In the crack pattern formation proce, ome long crack extend travering the top urface. Then, relatively hort crack propagate to teellate the lager cell. Thi hierarchical equence of the cell formation can be alo oberved in the drying tet of calcium carbonate lurry.

5 Table The parameter for the imulation of twodimenional crack pattern Soil dry denity ρ d Evaporation peed on Γ Evaporation peed on Γ 800 kg/m hour m/hour m/hour Moiture hrinkage coefficient 0.69 Moiture diffuion coefficient D m /hour Poion ratio 0.3 Young modulu Tenile trength 5.0 MPa.6 MPa Fig.5 The crack propagation proce of the drying tet in the cae of 0 mm thickne Table 3 The meh ize for the imulation of twodimenional crack pattern model ize number of number of [mm] node element ,930 50, ,337 5, , ,55 55,304 6,46 4. CONCLUSION In thi paper, the coupling model of deiccation, deformation, and fracture i propoed. The imulation for the one-dimenional crack pattern and two-dimenional crack pattern i performed baed on thi model and the reult of the imulation of the two-dimenional crack pattern are compared with the reult of the drying tet of calcium carbonate lurry. The imulation reult how the atifactory agreement with the experimental obervation in term of the crack pattern with net-like tructure, pattern formation proce, and the change in the ize of the cell framed by the crack depending on the thickne of the deiccation layer. Thi agreement between the imulation reult and the experimental obervation indicate that the coupling of deiccation, deformation, and fracture i a fundamental mechanim of the deiccation cracking. 5. REFERENCES Fig.4 The final crack pattern on the top urface of the pecimen formed in the imulation. (a) 5mm, (b) 0mm, (c) 0mm, and (d) 30mm [] A. Groiman and E. Kaplan, An experimental tudy of cracking induced by deiccation, Europhyic. Letter. Vol. 5, 006, pp [] H. Nahlawi and J.K. Kodikara, Laboratory experiment on deiccation cracking of thin oil layer, Geotechnical and Geological Engineering. Vol. 4, 006, pp [3] H.J. Vogel, H. Hoffmann, A. Leopold and K. Roth, Studie of crack dynamic in clay oil II. A phyically baed model for crack

6 formation, Geoderma. Vol.5, 005, pp [4] H. Peron, T. Hueckel, L. Laloui and L.B. Hu, Fundamental of deiccation cracking of fine-grained oil: experimental characteriation and mechanim identification, Canadian Geotechnical Journal. Vol. 46, 009, pp [5] G. Muielak and T. Śliwa, Fracturing of clay during drying: modelling and numerical imulation, Tranp Porou Med. Vol. 95, 0, pp [6] J.Sima, M. Jiang and C. Zhou, Numerical imulation of deiccation cracking in thin clayey layer uing 3D dicrete element modelling, Computer and Geotechnic. Vol. 56, 04, pp [7] R. Rodríguez, M. Sánchez, A. Ledeman and A. Lloret, Experimental and numerical analyi of deiccation of mining wate, Canadian Geotechnical Journal. Vol. 44, 007, pp [8] H. Peron, J.Y. Delenne, L. Laloui and M.S. El Yououfi, Dicrete element modelling of drying hrinkage and cracking of oil, Computer and Geotechnic. Vol. 36, 008, pp [9] M. L. L. Wierathne, K. Oguni and M. Hori, Numerical analyi of growing crack problem uing particle dicretization cheme, International Journal for Numerical Method in Engineering, Vol. 80, 009, pp [0] K. Oguni, M.L.L. Wierathne, T. Okinaka and M. Hori, Crack propagation analyi uing PDS-FEM and comparion with fracture experiment, Mechanic of Material. Vol. 4, 009, pp Copyright Int. J. of GEOMATE. All right reerved, including the making of copie unle permiion i obtained from the copyright proprietor. 3