3D XFEM Modelling of Imperfect Interfaces

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1 3D XFEM Modelling of Imperfect Interfaces Elena Benvenuti 1, Giulio Ventura 2, Antonio Tralli 1, Nicola Ponara 1 1 Dipartimento di Ingegneria, Universit di Ferrara, Italy elena.benvenuti@unife.it, tra@unife.it, nicola.ponara@unife.it 2 Dipartimento di Ingegneria Strutturale, Edile e Geotecnica Politecnico di Torino, Italy giulio.ventura@polito.it Keywords: XFEM, 3D, interface, coating. SUMMARY. In this paper, recent contributions to the modelling of coated inclusions by means of an extended Finite Element Method are presented. The matrix particle interface is modelled as a finite thickness, imperfect interface. Two approaches are considered: a variational approach inspired to Suquet s work on asymptotic analysis of thin layers, and an approach based on Eshelby s equivalent eigenstrain approach. The former approach is asymptotically consistent with imperfect interfaces, while the latter approach holds for thin and thick coating layers, whose behavior can be assimilated to that of imperfect interfaces. 1 INTRODUCTION New developments on the use of the extended Finite Element Method (XFEM) for the threedimensional modelling of imperfect interfaces have been proposed by the authors. The aim is modelling particle reinforced composites, where spherical and cylindrical coated inclusions are embedded within a matrix. Particle reinforced composites are employed for instance, in the automotive industry and in electronic products. Mechanical properties strongly depend on interfacial bonding quality. Hence the determination of the maximum radial stress at the particle surface as a function of the applied load and the adhesion parameters is of great interest [13][23]. The matrix particle interface can be modelled by means of cohesive zone models. However, the results depend on the geometry and the size of the finite element mesh [21]. Therefore, we adopt a different approach. Based on the Partition of Unity Property of the finite element shape functions [15], XFEM is a broad spectrum technique for dealing with cohesive embedded interfaces [17, 19, 1]. In the last years, we have developed a variant, called Regularized XFEM approach [4]. A two-dimensional application to the delamination problem of a FRP strip glued to a concrete block has shown an excellent comparison with experimental results [6]. The model implementation in the three-dimensional case has been also discussed in [5]. In the present work, imperfect interfaces are studied where the displacements field is discontinuous across the interface, while the traction vector is continuous. As usual in the XFEM approach, the surface of separation is implicitly defined via a level set function. We briefly present the main aspects of two original approaches, that we have recently developed [5, 7, 8]. The first approach is suitable for modelling thin interfaces, namely coatings with vanishing thickness. It is based on the variational formulation proposed by Suquet [20] and asymptotically convergent to a spring-like imperfect interface. The second approach is based on the use of some Eshelby s results concerning the use of eigenstrain in matrix-inclusion problems [10, 11]. The latter approach is suitable for both thin and thick coatings. 1

2 2 THE MATRIX-INCLUSION PROBLEM WITH IMPERFECT INTERFACE Isotropic materials and a small displacement regime are assumed. We consider a body V made of a material of elastic moduli C (1), i, j, h, k = 1, 2, 3. Let V be loaded on the surface V s by a surface loading f s. The displacement field is u 0. An inclusion made of material of elastic isotropic moduli C (2), is subsequently inserted. The inclusion is surrounded by a coating V c of thickness h and made of a material of elastic moduli C (3). We consider a zero-thickness interface across which the interfacial traction is continuous, while opening and sliding displacement discontinuities can occur. Let σ ij be the ij component of the stress tensor characterizing the mechanical behavior of the spring representative of the interface [2]. For a zero-thickness interface, the equations governing the mechanical behavior of the interface are σ ij n j = 0, u i n i n h = αn h, u i (δ ih n i n h ) = βt h, (1a) (1b) (1c) where N i = σ kj n k n j n i and T i = σ kj n j (δ ik n i n k ) denote the normal and shear traction at the interface. Moreover, α and β are the compliances in the normal and sliding directions, respectively. Eqs. (1b) and (1c) define the spring-like interface [14][2]. 3 REGULARIZED KINEMATICS AND PROPOSED APPROACHES Let S be the surface to which the coating reduces for vanishing thickness, and s(x) be the signed distance function from S [16]. Let φ be a sufficiently smoothed function of s(x). More precisely, φ is a regularized Heaviside function, converging to the Heaviside function H(s(x)), H(s(x)) = 1 if s(x) 0, and H(s(x)) = 1 otherwise, for vanishing ρ, namely when the coating thickness asymptotically vanishes. In particular, we adopt the smoothed Heaviside function [4] φ = 1 s(x) α ρ (ξ) dξ, (2) V ρ 0 where α ρ denotes any weighting function decaying to zero for increasing distances from S, and V ρ is a normalization volume [4]. Here ρ denotes a regularization length. The displacement field is assumed as u i (x) = u 0 i (x) + φ(s(x))b(x), (3) where b i (x) is the generic components of the vector field modulating the displacement opening at the inclusion-matrix interface with finite thickness. The compatible strain is ε ij = ε 0 ij + φ(s(x))β ij (x) δ φ(b i n j + b j n i ), (4) where δ φ = dφ ds. (5) 3.1 The reference problem Lets us adopt the scheme displayed in Fig. 1, namely the matrix occupies the volume V (1) \ V c +, while the inclusion the volume V (1) \ V c, V c + and V c being the two portions in which V c is 2

3 subdivided by the interface S. In general, the constitutive operators of the model problem are C (1) C a in V (1) \ V+ c, = C (2) in V (2) \ V c, C c in V c, where C c are the elastic moduli of the reference material, that have to be assumed in the coating layer in order for the formulation to be consistent with the imperfect interface model described in Sec Asymptotic approach for vanishing ρ In general, C c do not coincide with C(3), the elastic moduli of the true physical coating layer. Indeed, C c depend on the type of interfacial mechanical behavior. Suquet [20] proved that the constitutive moduli of the coating layer in the reference problem have to be assumed in the form C c = 1 tδ φ C (3) in V c, (7) where C (3) denotes the constitutive moduli of the coating interphase, and t a length parameter, which is introduced for dimensional consistency; t is set equal to a unit length and omitted in the following. The same relationship was independently derived by Benvenuti [4]. Then after assuming the displacement field (3), the solution of the imperfect interface problem can be found by minimizing the functional Π = 1 ε ij C (1) 2 ε hk dv + 1 ε ij C ε hk dv + 1 ε ij C V (1) \V+ c 2 V (2) \V c 2 ε c hk dv L ext, V c (8) where C c is defined according to Eq. (7). We have formulated in [7] the discrete version of the minimization problem associated with energy (8). 3.3 Eigenstrain based approach for finite thickness interface An alternative approach for solving the reference problem, called Problem (a), has been recently developed in [8]. We introduce a simplified problem, called system (b), in which the same body considered in (a) is taken into consideration. Systems (a) Fig.3a and (b) Fig.3b are in equilibrium under the same surface forces f s on V s. While the shape and size of the bodies in both systems is the same, different inhomogeneous constitutive constants C a and Cb are considered. In the simplified Problem (b), the coating layer is no longer physically present. The constitutive moduli are assumed as { C b C (1) in V (1), = C (2) (9) in V (2). The reference and the simplified problems are sketched in Fig. 3. Extending Eshelby s approach to eigenstrains [10], we set ε ij = 1 2 δ φ(b i n j + b j n i ). (10) The stress and the strain in the, simplified, Problem (b) are assumed as σ b ij = C (ε ij ε ij), ε b ij = ε ij ε ij. (6) (11a) (11b) 3

4 (a) (b) z f n z V c + S x y x 2 h y V c 1 Figure 1: Circular inclusion made of material (2) in a matrix (1) surrounded by an interface S of thickness h and normal n The mechanical internal work associated with Problem (b) becomes W b = σij(ε b ij ε ij) dv. (12) The internal virtual work of Problem (a) is written as V W a = W b + W ab (13) After appropriate calculations, it can be shown that the interaction energy W ab writes [8] W ab = σijε c ij dv. (14) V c where V c is the coating volume, and σij c is defined by following equation σ c ij = C c ε hk. (15) Note that in this way σij c = C(3) 1 2 (b hn k + b k n h ). Finally, the internal work (13) is [9] W a = σij b δε b ij dv + σij c δε ij dv. (16) V V c Replacement of the constitutive relationships in the principle of virtual work leads to Cε b b hkδε b ij dv + C c ε hk δε ij dv = f si δu 0 i ds (17) V V c V s for any virtual variations δe ij, δε ij, and δu0 i. The corresponding set of equilibrium and constitutive 4

5 Cε C(ε ε ) Figure 2: Eshelby s approach: The body is subjected to the stress C(ε ε ) where ε is the eigenstrain localized in the coating equations is [10] [9] σ b ij,j = 0 in V, (18a) σ b ij n j = f si on V s, (18b) σ b ij = C (1) εb hk in V (1), (18c) σ b ij = C (2) εb hk in V (2), (18d) σij b = σij c in V c. (18e) (a) (b) The system of equations (18) governs the simplified problem we want to study. f s f s σ a ij, εa ij σ b ij, εb ij Figure 3: The reference Problem (left) and the simplified problem (right) 5

6 p p Figure 4: Cylinder with a spherical inclusion subjected to tensile loading 4 THE DISCRETE PROBLEM In this section, a vector notation is used. The stiffness matrix corresponding to the variational formulation described in Sec. 3.2 has been derived in [7]. In the present contribution, we present the discrete problem associated with Eq.(17). The domain is discretized into N e finite elements. We interpolate the displacement u across the I N nodal degrees of freedom û0 I, and ˆb I. According to the partition of unity concept, the displacement field is approximated as [17] û(x) = N I (x)û 0 I + φ(s(x))n I (x)ˆb I. (19) I N I N h In Eq. (19), N I are the standard Finite Element shape functions, and N h are the degrees of freedom corresponding to the elements crossed by the enriched layer. We next omit the spatial dependence. The approximated strain field ˆε = ê + ˆε (20) is computed, in which ê = Bû 0 + φ Bˆb, ˆε = δ φ B ˆb. (21) In Eq. (21), vectors û and ˆb collect the nodal degrees of freedom, while B and B are the compatibility matrices. By replacing the finite element approximation (20) in the variational formulation (17), we get B t CB(û 0 + φˆb) (δû 0 + φ δˆb) dv + V δ 2 φ B T C c B ˆb δˆb dv = V c FNδû 0 ds, V s (22) for any δû 0 and δˆb, where N and F collect the N I shape functions and the nodal forces, respectively. Note that in Eq. (22), C = C (1) in V (1) while C = C (2) in V (2). The symmetric stiffness matrix K is obtained as ( ) Kuu K K = ub K bb + K, (23) bb K T ub 6

7 u u x x Figure 5: One-dimensional homogeneous bar with constant stress and imperfect interface at L/3; on the left the displacement obtained by means of the variational approach of Sec. 3.2, on the right that obtained by means of the eigenstrain approach of Sec. 3.3 where K uu = B t CBdV, K ub = V φ B t CB dv, V c K bb = φ 2 B t CB dv, V c K bb = δ 2 φ B t C c B dv, V c (24a) The proposed variational formulation has led to a great simplification of the solving equations, since the eigenstrain contribution to the stiffness matrix is reduced to a single additional term, K bb. Spurious energy couplings deriving from the combination of multiple enrichment types within the XFEM format were highlighted in [12]. Hence material and displacement discontinuities have been treated separately here. 5 NUMERICAL RESULTS 5.1 1D Comparison of the approaches We consider a one-dimensional bar of length L clamped at the left end and with assigned displacement at the right end. A discontinuity is placed at L/3 from the left end. The case of vanishing regularization length is considered. In Fig. 5, we compare the displacement profiles obtained with the variational approach described in Sec. 3.2 and with the eigenstrain approach described in Sec The displacement profiles are almost the same in the two cases for the asymptotic case of interface with vanishing thickness D Example Let us consider a cylinder with a spherical inclusion. The cylinder is subjected to remote uniaxial loading along the axis direction z, as shown in Fig. 4. The mesh made of tetrahedral elements shown in Fig.6 was adopted. The regularization zone coincides with the width l ρ of the (truncated) support of φ and δ φ. In order to recover the asymptotic behavior, the support of φ has been assumed equal to 20ρ in order to contain the truncation error, as shown in a previous study [5]. However, in general, the regularization length, namely the support of φ, and the mesh size h are assumed independent of each other. This implies that one or more layers of finite elements can fall within the enriched zone. 7

8 Figure 6: Cylinder with a spherical inclusion: geometry Figure 7: Enriched finite elements involving 1 (left) or 3 (right) layers of finite elements Figure 8: Soft inclusion: Stress σ zz obtained for perfect interface (left) and soft interface (right) This aspect is peculiar of the proposed approach and emphasizes the difference with respect to other regularized XFEM approaches, where the enrichment is confined within one layer of finite elements. The only requirement is that the width of the regularization zone has to be equal or larger to h in order for the stiffness terms related to the enriched zone to be accurately computed. In particular, the regularization zone was accurately resolved by assuming, in the following examples, an odd number of finite elements, typically one or three. The enrichment involves a shell finite elements of finite thickness interface as shown in Figure 8

9 (7), where the cases of thin and thick coating are shown. In Fig. 8, the σ zz stress component for the case of soft inclusion is shown when a perfect interface (left) ad a soft interface (right) are considered. 6 CONCLUSIONS We have briefly presented the main aspects of two original approaches that we have recently developed for modelling the mechanical response of matrix-inclusion systems under remote loading. The approaches are suitable for coatings whose mechanical behavior can be described through a spring-like, or imperfect, interface model. References [1] Belytschko, T., Gracie, R., Ventura, G.: A review of the extended/generalized finite element methods for material modelling. Model. Simul. Mat. Sci. Engng. 17 (2009). DOI / /17/4/ [2] Benveniste, Y., T.Miloh: The effective mechanical behavior of composite materials with imperfect contact between the constituents. Mech. Mat. 33, (2001) [3] Benvenuti, E.: Mesh-size-objective xfem for regularized continuousdiscontinuous transition. Finite Elements in Analysis and Design 47, (2011) [4] Benvenuti, E., Tralli, A., Ventura, G.: A regularized XFEM model for the transition from continuous to discontinuous displacements. Int. J. Num. Meth. Engng 197, (2008) [5] Benvenuti, E., Ventura, G., Ponara, N.: Finite element quadrature of regularized discontinuous and singular level set functions in 3d problems. Algorithms 5, (2012) [6] Benvenuti, E., Vitarelli, O., Tralli, A.: Delamination of frp-reinforced concrete by means of an extended finite element formulation. Comp.B 43, (2012) [7] Benvenuti, E., Ventura, G., Ponara, N., Tralli A.: Variationally consistent extended FE model for 3D planar and curved imperfect interfaces, submitted [8] Benvenuti, E.: XFEM with equivalent eigenstrain for matrix-inclusion interfaces, submitted [9] Dvorak, G.: Micromechanics of Composite Materials, vol. Solid Mechanics and Its Applications 186. Springer Verlag, Wien New York (2013) [10] Eshelby, J.: The force on elastic singularity. Philosophical Transactions of the Royal Society of London A: Mathematical and Physical Sciences 244, (1951) [11] Eshelby, J.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 241, (1957) [12] Fries, TP, Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns, Int.J. for Numerical Methods in Engineering 68, (2006) 9

10 [13] Fu, S.Y., Feng, X.Q., Lauke, B., Mai, Y.W.: Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulatepolymer composites. Composites Part B: Engineering 39, (2008) [14] Hashin, Z.: Thermoelastic properties of fiber composites with imperfect interface. Mech. Mat. 8, (1990) [15] Melenk, J., Babuska, I.: The partition of unity finite element method: Basic theory and applications. Comp. Meth. Appl. Mech. Engng. 139, (1996) [16] Moës, N., Cloirec, M., Cartraud, P., Remacle, J.: A computational approach to handle complex microsctructures geometries. Comp. Meth. Appl. Mech. Engng. 192, (2003) [17] Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Num. Meth. Engng 46, (1999) [18] Mura, T.: Micromechanics of Defects in Solids. (1982) [19] Sukumar, N., Chopp, D., Moës, N., Belytschko, T.: Modeling holes and inclusions by level sets in the extended finite-element method. Comp. Meth. Appl. Mech. Engng. 190, (2001) [20] Suquet, P.: Discontinuities and Plasticity. Springer Verlag, Wien New York (1988) [21] Turon, A., C.G. Dàvila, Camanho, P., Costa, J.: An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Engng. Fract. Mech. 74, (2007) [22] Ventura, G., Gracie, R., Belytschko, T.: Fast integration and weight function blending in the extended finite element method. Int. J. Num. Meth. Engng 77, 1 29 (2009) [23] Vörös, G., Pukánszky, B.: Effect of a soft interlayer with changing properties on the stress distribution around inclusions and yielding of composites. Composites Part A: Applied Science and Manufacturing 32, (2001) 10