Harris, P.J.

Hattori, G.

Trevelyan, J.

English

Hattori, G

Trevelyan, J

CUED - Cambridge University Engineering Department

A new enriched dual boundary element method for fracture in anisotropic materials

Durham Research Online

Proeedings of the 10th UK Conferene on Boundary Integral Methods,University of Brighton, UK, 13-14 July 20150 A new enrihed dual boundary element methodfor frature in anisotropi materialsG. Hattori, J. TrevelyanShool of Engineering & Computing Sienes, Durham University, UKgabriel.hattoridurham.a.uk, jon.trevelyandurham.a.uk0.1 IntrodutionFrature mehanis has been studied for over than 50 years, but it is still re-eiving great attention from the sienti ommunity. One of the reasons is theappearane of a range of new materials, suh as omposite, piezoeletri, magne-toeletroelasti materials, just to ite a few examples. All these materials have aommon harateristi: they present anisotropi behaviour, so material responseto a given exitation is also onditioned to the rystal arrangements of this ma-terial (for more about rystal arrangements see [1℄ for instane). Anisotropimaterials have been widely employed in the aerospae and automobile indus-tries, wind power generators, sporting goods, and also as piezoeletri sensorsand atuators.Analytial formulations to deal with anisotropi material in frature me-hanis problems are very limited [2, 3℄. Hene, frature mehanis problemsare solved using numerial methods, suh as the Finite Element Method (FEM)[4℄ and the Boundary Element Method (BEM) [5℄. It has been shown thatBEM is more aurate and eetive than FEM. However, in the last 15 years, anew paradigm has been established for the FEM, where the partition of unity[6℄ was applied to desribe the disontinuity due to the presene of the rak,where the rak is then modelled through additional degrees of freedom in theelements ontaining the rak. This approah was named Extended Finite El-ement Method (X-FEM) [7℄ and it has been shown that it an provide similarauray to the one found in BEM models [8℄.In this work we develop an alternative BEM formulation for anisotropifrature problems using the partition of unity method (PUM). This idea hasalready been disussed by [9℄ for isotropi materials, and our objetive is togeneralise this approah for anisotropi materials. To this end, the PUM isimplemented in a dual BEM ontext. The enrihment funtions are derived interms of the Stroh formalism [10℄ and further details on the implementation ofthe X-BEM are disussed.Furthermore, we ompare the results of the X-BEM with those obtainedby alternative numerial tehniques: dual BEM implemented in ombinationwith disontinuous quarter-point elements [11, 12℄ and X-FEM with anisotropienrihment funtions [8℄.10.2 Constitutive equationsConsider an anisotropi elasti domain Ω, then the stati equilibrium equationsin the presene of body fores b are dened asσij,j + bi = 0 (0.1)Symmetry holds for the stress and strain tensors σ and ε, respetively,σij = σji (0.2)εij = εji, (0.3)whereεij =12(ui,j + uj,i) (0.4)and ui represent the displaements.The linear onstitutive equations are given by the generalized Hooke's lawσij = Cijklεkl (0.5)where Cijkl dene the material onstants tensor, satisfying the following sym-metry relationsCijkl = Cjikl = Cijlk = Cklij (0.6)0.3 The Dual Boundary Element Method (DBEM)The boundary element method (BEM) has been established as a referene nu-merial method when dealing with linear elasti frature mehanis problems[5℄. BEM is known to be more aurate and robust than domain disretisationmethods suh as the more popular nite element method (FEM).The dual BEM (DBEM) is the usual hoie when dealing with fraturemehanis problems. Hong and Chen [13℄ presented the idea of the ombineduse of a BIE and its derivative. This was used for the rst time in a fraturemehanis ontext when [14℄ presented the DBEM. It an be summarised by adisplaement boundary integral equation (DBIE)cij(ξ)uj(ξ) +∫Γp∗ij(x, ξ)uj(x)dΓ(x) =∫Γu∗ij(x, ξ)pj(x)dΓ(x) (0.7)and a tration boundary integral equation (TBIE), obtained by the dierentia-tion of (0.7) and further substitution in (0.5)cij(ξ)pj(ξ) +Nr∫Γs∗rij(x, ξ)uj(x)dΓ(x) = Nr∫Γd∗rij(x, ξ)pj(x)dΓ(x) (0.8)where Γ represents all the boundaries (inluding rak boundaries) of domainΩ; Nr is the outward unit normal to the boundary at the olloation point ξ;cij is the free term deriving from the Cauhy Prinipal Value integration of2the strongly singular kernels p∗ij ; u∗ij and p∗ij are the displaement and trationfundamental solutions; d∗rij and s∗rij follow from dierentiation and substitutioninto the generalised Hooke's law of u∗ij and p∗ij , respetively. The kernels u∗ijand p∗ij are given byu∗ij = −1πℜ{AjmQmi ln(zxm − zξm)}(0.9)p∗ij =1πℜ{BjmQmiµmn1 − n2zxm − zξm}(0.10)where n = (n1, n2) is the unit normal at the observation point; zxm = x1 +µmx2, zξm = ξ1 + µmξ2 are evaluated at the observation and olloation points,respetively; ℜ denotes the real part ; Q = A−1(L−1+L−1)−1with L = iAB−1and L represents the omplex onjugate of L.The matries A, B and onstant µm are parameters from the Stroh formal-ism and an be obtained through the following eigenvalue problem( −C2ij2−1C2ij1 −C2ij2−1C1ij1 −C2ij1TC2ij2−1C2ij1 −C2ij1TC2ij2−1)(AmBm)= µm(AmBm)(0.11)where there is no summation on the m index.0.3.1 Extended Boundary Element Method (X-BEM)The extended boundary element method (X-BEM) was rst proposed by Simp-son and Trevelyan [9℄ for frature mehanis problems in isotropi materials.The main idea is the same as for the X-FEM, to model the asymptoti be-haviour of the displaements around the rak tips by introduing new degreesof freedom. The displaements uh(x) are thus redened asuh(x) =∑i∈NNi(x)ui +∑k∈NCTNk(x)∑αFα(x)aαk (0.12)where N and N CT are the sets of all nodes and the enrihed nodes, respetively,Ni is the standard Lagrangian shape funtion assoiated with node i, ui is thevetor of nodal degrees of freedom, and aαk represents the amplitudes of theenrihed basis funtions whih apture the asymptoti behaviour around therak tips. In elasti materials, aαk is an 8-omponent vetor for two-dimensionalproblems, sine only two nodal variables (u1, u2) and four enrihment funtionsare needed to desribe all the possible deformation states in the viinity of therak-tip [8℄.In this work, we use the anisotropi enrihment funtions obtained by [8℄ forthe X-FEM:Fl(r, θ) =√rℜ{A11B−111 β1 +A12B−121 β2}ℜ{A11B−112 β1 +A12B−122 β2}ℜ{A21B−111 β1 +A22B−121 β2}ℜ{A21B−112 β1 +A22B−122 β2}(0.13)3where βm =√cos θ + µm sin θ, r is the distane between the rak tip andan arbitrary position, θ is the orientation measured from a oordinate systementred at the rak tip.Let us emphasise that the anisotropi enrihment funtions an also be usedfor isotropi materials, sine this is a degenerated ase from anisotropi materi-als. For more details please refer to referene [8℄.The X-BEM formulation is similar to the one used by Simpson and Trevelyan[9℄ for isotropi materials. The extended DBIE and the TBIE an be restatedascij(ξ)uj(ξ) +∫Γp∗ij(x, ξ)uj(x)dΓ(x) +∫Γcp∗ij(x, ξ)Fα(x)aαkdΓ =∫Γu∗ij(x, ξ)pj(x)dΓ(x)(0.14)cij(ξ)pj(ξ) +Nr∫Γs∗rij(x, ξ)uj(x)dΓ(x) +Nr∫Γcs∗rij(x, ξ)Fα(x)aαkdΓ =Nr∫Γd∗rij(x, ξ)pj(x)dΓ(x)(0.15)where Γc = Γ+∪Γ− stands for the rak surfaes Γ+ and Γ−. In this work, onlythe element ontaining the rak tip reeives the enrihment funtion. Let usreall that strongly singular and hypersingular terms arise from the integrationof the p∗ij , d∗rij and s∗rij kernels and they are regularised in the same way asshown in [15℄.The enrihment funtions add new degrees of freedom in the BEM formu-lation, whih auses the linear system of equations to beome indeterminate.An easy form to overome this issue is to inlude additional olloation points.Eah new olloation point will provide 2 new equations (in x1 and x2 dire-tions). In this work, 3 nodes are enrihed in eah rak fae, whih will add3× 4× 2× 2 = 48 new degrees of freedom to the problem. Therefore, 24 extraolloation points are neessary to solve the linear system of equations.0.4 Numerial resultsIn this setion we show the rak opening displaement (COD) for anisotropimaterials in two frature mehanis examples. The numerial results are om-pared with the results from the extended BEM, the dual BEM with quarter-point elements [11℄ and the X-FEM.The BEM formulation presented in [11℄ has the partiularity of modellingonly one rak surfae, thus obtaining the COD straight from the solution ofthe linear system of equations.40.4.1 Double edge rakFigure 0.1 illustrates a square plate (h/w = 1) with two edge raks of lengtha under a uniform loading σ. The size of the rak is dened by a/w = 0.25.The material is a four graphite-epoxy laminate, where the elasti properties are:E1 = 144.8 GPa, E2 = 11.7 GPa, G12 = 9.66 GPa and ν12 = 0.21. The CODwas obtained for two distint bre orientation, φ = 75◦ and φ = 90◦. Resultsfor the X-FEM are obtained using topologial and geometrial enrihment, witha xed area of re/a = 0.2 and a mesh of 4525 elements. For more informationabout both adopted enrihment types please refer to [8℄ for instane.wha aσσE2E1φFigure 0.1: Double edge rak problem.For the dual BEM and the X-BEM a 10 ontinuous quadrati element perside mesh was used. The rak is disretised with 10 disontinuous elements.The dual BEM presents a quarter-point in the element at the rak tip, whilethe X-BEM presents an enrihed element.Results are given in Figures 0.2 and 0.3. The COD of the X-FEM models issmaller than the ones of the BEM models. The X-BEM COD is higher at theupper rak fae than its equivalent using quarter point. In this ase, the X-FEMresults an be underestimated ompared to the BEM results. Nevertheless, thedisplaements around the rak tip have the same√r behaviour.0.4.2 Slanted entred rakA square plate h/w = 1 with a slanted entred rak of length 2a under auniform loading is represented in Figure 0.4, where β = 30◦. The size of therak is a/w = 0.5. The material is a glass-epoxy omposite with the following50 0.2 0.4 0.6 0.8 11.851.91.9522.052.12.152.22.25Position − xPosition − y  X−BEMBEM QPX−FEM topologicX−FEM geometricFigure 0.2: COD for the double edge rak - θ = 75◦.elasti properties: E1 = 48.26 GPa, E2 = 17.24 GPa, G12 = 6.89 GPa andν12 = 0.29. The bre orientation angle φ assumes values of 90◦and 120◦ inthis example. The BEM meshes present 6 quadrati elements per side, and 10disontinuous elements at the rak. The X-FEM mesh ontain 2025 elementsand the geometrial enrihment uses a xed area of re/a = 0.2.The results are shown in Figures 0.5 and 0.5 for φ = 90 and 120◦, respe-tively. Here we present the results in terms of the u2 displaement only, sine itis diult to get an aurate COD for the X-FEM model, whih an be seen bythe deformed shape of the COD. Both CODs from the BEM models result insymmetrial relative displaements. The COD from the extended BEM is on-siderably larger than the other referene solutions. One of the possible reasonsis that the extended BEM model is underestimating the stiness of the plate,resulting in a larger COD.0.5 SummaryAn enrihed boundary element method was proposed in this work. This methoduses the benets of the partition of unit to model in a more general way theasymptoti displaements around the rak tip. Existing anisotropi enrihmentfuntions for the X-FEM were employed at the proposed method. Some numer-ial examples were evaluated, and the results ompared to a well establishedBEM formulation and the X-FEM . The rak opening displaement was om-60 0.2 0.4 0.6 0.8 11.81.851.91.9522.052.12.152.2Position − xPosition − y  X−BEMBEM QPX−FEM topologicX−FEM geometricFigure 0.3: COD for the double edge rak - θ = 90◦.2w2h2aσσE2E1φβFigure 0.4: Slanted entred rak problem.pared with the quarter point BEM and the X-FEM. For a at horizontal edgerak, both BEM models have the same behaviour around the rak tip, and the73 3.5 4 4.5 5 5.5−0.1−0.0500.050.1Position − xDisplacement − y  X−BEMBEM QPX−FEM topologicX−FEM geometricFigure 0.5: COD for the slanted rak problem - φ = 90◦.3 3.5 4 4.5 5 5.5−0.1−0.0500.050.1Position − xDisplacement − y  X−BEMBEM QPX−FEM topologicX−FEM geometricFigure 0.6: COD for the slanted rak problem - φ = 120◦.X-BEM onsiders a larger COD at the end of the rak. In a slanted rak, theCOD in the X-BEM model is larger than both BEM and X-FEM. The X-BEMould be used an alternative to the dual BEM with quarter-point elements.80.6 AknowledgmentsThe rst author aknowledges the Faulty of Siene, Durham University forhis Postdotoral Researh Assoiate funding.Referenes[1℄ M. J. P. Musgrave. Crystals Aoustis: Introdution to to the study ofelasti waves and vibrations in rystals. Holden-Day, 1970.[2℄ Carloni C. Nobile, L. Frature analysis for orthotropi raked plates.Composite Strutures, 68(3):285293, 2005.[3℄ G. C. Sih, P. C. Paris, and G. R. Irwin. On raks in retilinearly anisotropibodies. International Journal of Frature, 1:189203, 1965.[4℄ T. J. Boone, P. A. Wawrzynek, and A. R. Ingraea. Finite element mod-elling of frature propagation in orthotropi materials. Engineering Fra-ture Mehanis, 26:185201, 1987.[5℄ YJ Liu, S Mukherjee, N Nishimura, M Shanz, W Ye, A Sutradhar, E Pan,NA Dumont, A Frangi, and A Saez. Reent advanes and emerging ap-pliations of the boundary element method. Applied Mehanis Reviews,64(3):030802, 2011.[6℄ I. Babuska and J. M. Melenk. The partition of unity method. Interna-tional Journal for Numerial Methods in Engineering, 4:607632, 1997.[7℄ Niolas Moes, John Dolbow, and T Belytshko. A nite element method forrak growth without remeshing. Int. J. Numer. Meth. Engng, 46(1):131150, 1999.[8℄ G Hattori, R Rojas-Díaz, A Sáez, N Sukumar, and F Garía-Sánhez. Newanisotropi rak-tip enrihment funtions for the extended nite elementmethod. Computational Mehanis, 50(5):591601, 2012.[9℄ Robert Simpson and Jon Trevelyan. A partition of unity enrihed dualboundary element method for aurate omputations in frature mehanis.Computer Methods in Applied Mehanis and Engineering, 200(1):110,2011.[10℄ A. N. Stroh. Disloation and raks in anisotropi elastiity. Philosophialmagazine, 3:625646, 1958.[11℄ F Gara, A Sáez, and J Domnguez. Tration boundary elements forraks in anisotropi solids. Engineering analysis with boundary elements,28(6):667676, 2004.9[12℄ A. Sáez, R. Gallego, and J. Domínguez. Hypersingular quarter-pointboundary elements for rak problems. International Journal for NumerialMethods in Engineering, 38:16811701, 1995.[13℄ Hong-Ki Hong and Jeng-Tzong Chen. Derivations of integral equations ofelastiity. Journal of Engineering Mehanis, 114(6):10281044, 1988.[14℄ A. Portela, M. H. Aliabadi, and D. P. Rooke. The dual bem: eetiveimplementation for rak problems. International Journal for NumerialMethods in Engineering, 33:12691287, 1992.[15℄ F Garia-Sanhez, Andrés Sáez, and J Dominguez. Anisotropi and piezo-eletri materials frature analysis by bem. Computers & Strutures,83(10):804820, 2005.10

A new enriched dual boundary element method for fracture in anisotropic materials

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