Accepted Manuscript Thermal-mechanical crack propagation in orthotropic composite materials by the extended four-node consecutive-interpolation element (XCQ4) M.N. Nguyen, N.T. Nguyen, T.T. Truong, T.Q. Bui PII: DOI: Reference:
S0013-7944(18)31022-1 https://doi.org/10.1016/j.engfracmech.2018.11.036 EFM 6249
To appear in:
Engineering Fracture Mechanics
Received Date: Accepted Date:
18 September 2018 19 November 2018
Please cite this article as: Nguyen, M.N., Nguyen, N.T., Truong, T.T., Bui, T.Q., Thermal-mechanical crack propagation in orthotropic composite materials by the extended four-node consecutive-interpolation element (XCQ4), Engineering Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.engfracmech.2018.11.036
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Thermal-mechanical crack propagation in orthotropic composite materials by the extended four-node consecutive-interpolation element (XCQ4) M. N. Nguyena , N. T. Nguyena , T. T. Truonga , T. Q. Buib,c,∗ a
Department of Engineering Mechanics, Faculty of Applied Sciences, Ho Chi Minh City University of Technology - VNU, Vietnam b Institute for Research and Development, Duy Tan University, Da Nang City, Vietnam c Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo 152-8552, Japan
Abstract A numerical study of thermal-mechanical crack growth in orthotropic composite materials by the recently developed extended nodal gradient finite element method is presented. The extended four-node consecutive-interpolation quadrilateral element (XCQ4) used in this work is the conventional four-node quadrilateral element, but it is enhanced by additional terms of averaged nodal gradients to smoothen the derivative fields and increase the accuracy of solution, while keeping the same total number of degrees of freedom (DOFs) as that of the traditional element. Inspired by enrichment partition-of-unity method, the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack are mathematically described through known enrichment functions. Once the stress fields are obtained, stress intensity factors (SIFs) can thus be evaluated by interaction integral technique. The SIFs are then used to predict crack growth direction. Unlike isotropic media, material orientation plays a critical role and that has to be considered when modeling the evolution of crack in orthotropic composite materials under thermo-mechanical loading condition. In addition, we introduce for the first time an improved way for the enriched approximation of discontinuous temperature field in cracked orthotropic media by taking into account the effect of material orientation. Several numerical examples for thermo-mechanical fracture problems in orthotropic composites with both isothermal and adiabatic loading conditions are analyzed to show the accuracy and performance of the present XCQ4. The computed results of the SIFs are compared with reference solutions derived from experiments, the standard extended four-node quadrilateral element (XQ4), and other numerical methods available in literatures. Keywords: fracture; thermo-elasticity; finite element; orthotropic materials; crack growth
1. Introduction Composite materials in general or orthotropic composite ones in particular can be found both in nature (e.g., woods and bones) or in industry (e.g., fiber reinforced composite). In practice, orthotropic composite materials are of interest and importance due to, for instance, their higher strength per unit ∗ Corresponding author: Tinh Quoc Bui, Associate Professor, Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo 152-8552, Japan Email address: [email protected]
; [email protected]
(T. Q. Bui)
Preprint submitted to Elsevier
November 14, 2018
weight, compared with conventional materials. Thanks to that advantage, they have been applied in various structures from buildings or aircraft to daily products such as bicycles and tennis rackets. However, they are still susceptible or prone to defects like crack. Accordingly to the growth in popularities of orthotropic materials, studying fracture problems in such media is necessary. Although analytical solutions are available for some special problems, e.g., see Refs. [1, 2, 3, 4], it is quite limited to certain simple problems due to many aspects. For engineering applications which often contain complicated geometries and loadings and boundary conditions, advanced numerical methods are usually more suitable options. Many advanced numerical approaches for analysis of fracture problems in composite and orthotropic media have been developed including finite element method (FEM), boundary element method (BEM), and meshfree methods. The FEM has been shown to be the most popular one due to its simplicity and acceptable accuracy. In FEM, the problem domain is discretized into non-overlapping sub-domains, which are often named as elements. Any functions defined within the problem domain can be approximated based on values at nodes, which are the points interconnected by elements. In cases where geometry of problem domain has to be updated such as the evolution of crack or moving boundary, the mesh of elements has to be updated as well, which is unfortunately not a trivial task as the initial mesh has to be re-meshing. The approaches using enrichment technique, e.g., see [5, 6] and references therein, show an effective means that are able to completely eliminate such a burdensome task of re-meshing. The underlying and innovative idea is the utilization of mathematical functions (i.e., enrichments) to capture the geometrical jump of displacement and the singularity of stress fields in the vicinity of cracks. Although the technique was initially proposed in the framework of finite element analysis, known as extended finite element method (XFEM), the enrichment functions can be employed in other Galerkin-based methods such as isogeometric analysis [7, 8, 9] or meshfree methods [10, 11, 12] Based on its success in modeling linear elastic fracture mechanic (LEFM) problems, the enrichment technique was later extended to consider cracked structures under thermo-mechanical load [13, 14, 15, 16], taking into account the fact that local increment of thermally induced stress may cause severe degradation or eventually lead to failure of structures or components. Appropriate enriched terms must be selected for the field of temperature due to existence of cracks . For adiabatic condition, there is a jump in temperature across the crack and hence an enriched scheme similar to displacement field in LEFM has to be adopted. For isothermal condition where the temperature on crack surface is prescribed, it is not the temperature field but the thermal gradients are discontinuous instead. Further contributions on fatigue crack growth under thermal-mechanical loads in three-dimensional domain can be attributed to Pathak et al. [17, 18, 19, 20]. These works are fundamental for future investigation on thermo-mechanical crack growth due to fatigue phenomenon. Recently, an interesting study on experimental and numerical estimation of fatigue crack growth has been reported for Ni-based super alloy . The presented numerical approach is based on XFEM as well as von Mises yield criterion for crack tip plasticity and Paris law to predict crack propagation due to fatigue. Unlike modeling fracture in isotropic materials, the material orientation is critical and that has to be considered in orthotropic media. Specially when modeling the evolution of crack, the material orientation largely affects the crack paths. The experimental investigation of mixed-mode fracture in Norway spruce conducted by Jernkvist  shows that the original notch orientation and the degree of mode mixity has 2
little effect on direction of crack growth, as the crack always propagates along the wood fibers. Similar phenomenon is observed by Cahill et al.  for uni-directional fibre-reinforced composite laminae. However, such observation is not recorded in the tests conducted by Ke et al. . Hence, modeling of crack growth in orthotropic media is more complicated and needs further investigation. Numerical models of cracked bodies made of orthotropic materials under assumption of LEFM have been reported by various authors [25, 26, 27, 28, 29]. Nevertheless, works focusing on analysis of thermoelastic fracture problems in orthotropic media [30, 31, 32] are still limited in literatures, especially in the context of crack propagation. In recent years, the consecutive-interpolation procedure (CIP) has been introduced as an extension for conventional FEM, in which the nodal gradients are taken into the interpolation. The extra terms related to nodal gradients allow smoother and more accurate description of gradient fields (e.g., stress and strain fields). Interestingly, no additional DOFs are introduced into formulation, given the same mesh with standard FEM. The CIP was initially proposed for 2D linear elastic problems using three-node triangular elements  and four-node quadrilateral elements [34, 35]. Each element type has to be formulated differently using CIP. The bottleneck is resolved by a general formulation for a wide range of elements from 1D to 3D, which was recently proposed by Nguyen et al. [36, 37]. As mentioned above, stress components are essential in determination of SIFs, which in turn are key to properly predict crack path. In fact, the stress components obtained by the CIP are smoothed, gaining higher accuracy of the SIFs, which is desirable in fracture mechanics. This issue was previously explored by the present authors and reported in [38, 39] for linear elastic fracture problems and in  for linear thermo-mechanical fracture problems of cracks in isotropic materials. Inspired by the promising improvement brought by CIP, the extended four-node consecutive-interpolation element (XCQ4) is further developed in this paper for numerical analysis of thermal-mechanical crack propagation in orthotropic composite materials in the context of linear thermoelastic fracture mechanics. Our objective is to show the accuracy and performance of our XCQ4 element in modeling thermalmechanical crack propagation in composite materials. It should be stressed out that the modeling of thermal-mechanical crack propagation in orthotropic composite materials is more challenging in comparison with that in isotropic materials . Here, in this work, we particularly pay attention on how the thermo-mechanical loads, crack inclination angles, material orientation, mesh size, etc., affect the static SIFs and crack propagation path in 2D composite structures, which will be investigated and explored in detail through various numerical examples from simple to complex configurations. The accuracy of calculated results for SIFs and crack growth paths obtained by the present method is demonstrated by comparing with reference solutions available in literatures and the standard XFEM. In particular, we explore the accuracy of the static SIFs through four numerical examples of orthotropic composite materials and analyze the crack propagation in such media with two numerical examples. Another important issue once modeling thermal-mechanical crack propagation in orthotropic composite material is the enriched approximation of the temperature field. This is different from isotropic materials, where material properties are the same in any direction, and enriched approximation of temperature is hence straightforward. Duflot  proposed the employment of only one branch function to express the singularity of temperature gradient in isotropic materials. In the contrary, for orthotropic 3
media, the material orientation plays a crucial role and thus has to be considered. However, in some previous works about thermo-elastic fracture in orthotropic media, [32, 31], the issue were surprisingly not mentioned, such that the same enriched formulation as proposed by Duflot  for isotropic materials is utilized. Therefore, in this paper, we suggest that a scheme of two enriched function, which take the effect of material orientation into account, should be used in the approximation of temperature. The organization of the present study is as follows. Section 2 is a brief review on the consecutiveinterpolation procedure, highlighting major features of the CIP based elements. In Section 3, the formulation of the XCQ4 for linear thermo-mechanical fracture problems of orthotropic media is derived. Details on the problem definition, the enriched schemes for approximation of displacements and temperature field variables for both adiabatic and isothermal conditions, the calculation of static SIFs based on evaluation of interaction integral and the crack growth criterion are presented. In Section 4, numerical evaluation for static SIFs and quasi-static crack propagation of thermoelastic fracture problems are presented, by which the accuracy and performance of the proposed method are demonstrated and discussed. Major conclusions drawn from the study are given in Section 5. 2. A brief on consecutive-interpolation 4-node quadrilateral element (CQ4) The development and derivation of CQ4 element have been reported in detail in our previous studies [34, 35]. A more generalized formulation of the consecutive-interpolation procedure (CIP) applicable for a wide range of element types was also developed by the same authors . For the sake of completeness, we will hence briefly present the fundamentals of the CIP in this section. Let us consider an elastic body in the domain Ω bounded by Γ which is discretized into finite elements. Using CIP, the approximation of an arbitrary function u(x) defined in Ω can be written as a linear combination: n X u(x) ≈ u ˜(x) = RI (x)uI = Ru, (1) I=1
where uI and RI are respectively the nodal value of function u and the CIP-based shape function associated with node I (global index), while n is the number of nodes. The vector of CIP shape functions, R, is calculated based on the conventional Lagrange shape functions as described in  R=
¯ [I] + φIy N ¯ [I] . φI N[I] + φIx N ,x ,y
Here N[I] = [N1 (xI ), N2 (xI ), ..., Nn (xI )] denotes the vector of Lagrange shape functions evaluated at ¯ [I] ¯ [I] node I. N ,x and N,y are respectively the weighted average of the first-order derivative of Lagrange shape functions with respect to the x and y direction X [I][e] ¯ [I] = N w N , (3) e ,x ,x e∈SI
¯ [I] = N ,y
we N[I][e] . ,y
At each node I, the set SI containing all elements interconnected at I is determined. Then the derivative [I][e] of N[I] evaluated at each element e within SI is determined and stored in vector N,x . The weight 4
coefficient we is defined simply in a way that is formed by the ratio between area of element e with the total area of all elements in the set SI we = P
, e ∈ SI ,
The auxiliary functions φI , φIx and φIy are actually dependent on the types of element being considered. It means that they must be developed particularly for each type of element, as already reported in [33, 34, 36, 37]. The bottleneck is however resolved by a general formulation, which was proposed by the present authors . The development of our general formulation is an important step forwards the efficiency of this class of elements. In overall, the general formulation can be applied to determine auxiliary functions for a wide range of finite elements from 1D to 3D. By following our previous work, , the auxiliary functions are defined locally in each element e as follows: φi = Ni + Ni2 (Σ1 − Ni ) − Ni Σ2 − Ni2 , n X 1 2 φix = (xj − xi ) Ni Nj + Ni Nj (Σ1 − Ni − Nj ) , 2
Here, the subscript ith denotes the local ith (i = 1, 2, ..., ne) of element e, and ne is the total number of nodes within the element e. The terms Σ1 , Σ2 are determined by Σ1 = Σ2 =
ne X i=1 ne X
An illustration of the CIP scheme tailored to 4-node quadrilateral element is schematically depicted in Figure 1  for the sake of clarity. Denote i, j, k, and m as the four nodes of the element containing the point of interest x, the Lagrange shape functions associated with those nodes are calculated using standard finite element procedure. The four sets Si , Sj , Sk , and Sm are established by finding adjacent elements that share the node i, j, k, and m, respectively. The weighted average of first-order derivatives of Lagrange shape functions are calculated using Eq. (3) and (4) and subsequently, the CIP-shape function is obtained by Eq. (2). It is obvious that approximating the value of an arbitrary function u at the point of interest x requires all the nodes within the four sets of neighboring elements Si , Sj , Sk , and Sm , while in conventional FEM, only the four nodes i, j, k, and m are required. Therefore, compared to conventional FEM, the supporting domain for point x is larger in CIP-based elements, which is followed by higher computational effort to establish the stiffness matrix. Nevertheless, the efficiency of CIP-based elements, in particular CQ4 element, has been investigated and reported in detail by . In that study, the log-log graphs of relative errors in both displacement and energy norm versus computational time (in seconds) evidently exhibits that less time is required for CIP-based elements than standard FEM counterparts to achieve the same accuracy. For curious readers, a summary of seven major desirable features of elements enhanced by the CIP is provided in [16, 35]. 5
Finite element mesh m
k x j
Point of interest x Supporting nodes for the point x
Figure 1: Sketch of the CIP approach on a 4-node quadrilateral element in an irregular mesh .
3. Formulation of XCQ4 element for thermo-mechanical fracture analysis in orthotropic composite media The XCQ4 element was initially introduced by Kang et al.[38, 39, 40] for linear elastic fracture mechanics (LEFM), which was formed by integrating the enrichment functions  into the CQ4 element . The XCQ4 is found to perform better in evaluation of stress intensity factor (SIF) than its traditional extended four-node quadrilateral element (XQ4). The main reason behind that advantage is due to the fact that the stress fields obtained by XCQ4 are smoother and higher accurate than those by its opponent, XQ4, while stress fields are the key ingredients to compute SIFs. The XCQ4 element was extended to analyze linear thermoelastic fracture problems in isotropic materials by the present authors , which aimed to consider the effect of thermal stress and thermal shock on SIFs, and positive observations were also reported. In this paper, the XCQ4 element is further developed for thermal-mechanical crack propagation in orthotropic composite media to take into account the anisotropy of material properties. Details on the XCQ4 including its numerical integration have already been thoroughly presented in [38, 39, 40, 16] and therefore are not necessary to repeat here for brevity. Instead, the rest of this section focuses on the extension in methodology due to the anisotropy introduced into the materials. 3.1. Governing equations The governing equations for a static linear thermoelastic problem defined in a domain Ω bounded by Γ with the assumption of small displacements and small strains are given by −qi,i + Q = 0,
σij,j + bi = 0.
In Eq. (10), Q denotes the heat/sink source; and qi = −kij T,j is the heat flux; T stands for temperature; and kij is the coefficients of thermal conductivity. In Eq. (11), bi is the body force vector and σij is the Cauchy stress tensor σij = Cijkl εkl − Cijkl αkl (T − Tref ), (12) 6
in which Cijkl stands for the elastic stiffness tensor; αij are coefficients of thermal expansion measured at reference temperature Tref ; and εij = 12 (ui,j + uj,i ) is the strain tensor calculated from displacement ui . Repeated indices imply summation and differentiation is denoted by a comma. The associated boundary conditions are given as follows T
= T¯ on ΓT : prescribed temperature,
q · n = q¯ on Γq : prescribed heat flux, ¯ on Γu : prescribed displacement, u = u σ · n = ¯t on Γt : prescribed traction,
(13) (14) (15) (16)
with ΓT ∪ Γq = Γu ∪ Γt = Ω, and ΓT ∩ Γq = Γu ∩ Γt = ∅. A crack existing in Ω is denoted by Γc , where traction free condition (Γc ⊂ Γt , t¯ = 0 on Γc ) is assumed. 3.2. Asymptotic fields For orthotropic materials, the in-plane stress-strain relation can be written in the material principal axes 1 − 2 as follows: ε=S:σ (17) where strain tensor ε, stress tensor σ and compliance tensor S are written following Voigt notation as ε = (ε11 , ε22 , 2ε12 )T , T
σ = (σ11 , σ22 , σ12 ) , S11 S12 S16 S = S21 S22 S26 , S16 S26 S66
(18) (19) (20)
In general, the 1 − 2 axes are not aligned with the global X − Y axes and not aligned with the local x − y axes defined at the crack tip as well, see Figure 2. Here θ denotes the inclined angle of crack segment with the horizontal axis, while β stands for the angle between horizontal axis and the 1-axis. The characteristic equation of an orthotropic material is given by Lekhnitskii  S11 z 4 − 2S16 z 4 + (2z12 + S66 ) z 2 − 2S26 z + S22 = 0,
whose roots can only be either complex or purely imaginary, i.e. zk = zkx + izky (k = 1, 2), and occur in conjugate pairs as z1 , z¯1 and z2 , z¯2 . The asymptotic solutions for displacements and stress fields in the vicinity of crack tip under mixedmode loadings have been derived by  in terms of the roots of the characteristic equation and the local
Figure 2: Sketch of coordinate systems: X − Y are the global axes, while x − y are the local axes originated at the crack tip; and 1-2 are the material principal axes.
polar coordinates (r, θ) defined at the crack tip r r 2r 1 2r 1 ux = KI Re (z1 p2 ψ2 − z2 p1 ψ1 ) + KII Re (p2 ψ2 − p1 ψ1 ) (22) π z1 − z2 π z1 − z2 r r 2r 1 2r 1 uy = KII Re (z1 q2 ψ2 − z2 q1 ψ1 ) + KII Re (q2 ψ2 − q1 ψ1 ) (23) π z1 − z2 π z1 − z 2 2 KI z 1 z2 z2 z1 KII 1 z2 z12 σxx = √ Re − +√ Re − (24) z1 − z2 ψ2 ψ1 z1 − z2 ψ2 ψ1 2πr 2πr KI 1 z1 z2 KII 1 1 1 σyy = √ Re − +√ Re − (25) z1 − z2 ψ2 ψ1 z1 − z2 ψ2 ψ1 2πr 2πr KI z 1 z2 1 1 KII 1 z1 z2 σxy = √ Re − +√ Re − . (26) z1 − z2 ψ1 ψ2 z1 − z2 ψ1 ψ2 2πr 2πr Here, the operator Re returns the real part of the statement; KI and KII are the SIFs corresponding to mode-I and mode-II, respectively. The quantities pk , qk and ψk (k = 1, 2) are defined as follows: pk = S11 zk2 + S12 − S16 zk S22 qk = S12 zk2 + − S26 zk p ψk = cos θ + zk sin θ
(27) (28) (29)
Recent investigations on crack tip behavior have suggested the employment of higher-order terms in the asymptotic solution and the introduction of T -stress [43, 44, 45, 46]. Consideration of higher-order terms would result in higher accuracy of stress fields, computational cost however should be arisen as well. The use of eigenfunction expansion facilitates an analytical and efficient way to approximate both asymptotic displacement and stress fields up to desired higher order terms. Unfortunately, application of the symplectic analytical solution into extrinsic enriched element approach is not straightforward, as partition of unity is required. Nevertheless, in the current work, we focus on the performance of XCQ4 element, in comparison with its standard XFEM element, i.e., XQ4 8
element, in modeling behavior of fracture in orthotropic media. Therefore the formulation remains as close to standard XFEM as possible, to highlight the efficiency of CIP-enhanced element, i.e., XCQ4. In fact, as the results suggest, due to presence of CIP, the XCQ4 element is able to provide highly accurate estimation of SIFs. Extension of the current XCQ4 element by incorporation of higher order terms into asymptotic solutions has been scheduled in our future works. 3.3. Approximation of the displacement By employing the enrichment approach  and the shifting technique , the displacement fields with discontinuity are approximated in terms of XCQ4 element can be expressed as  X X uh (x) = Ri (x)ui + Rj (x) (H(x) − H(xj )) aj i∈I s
(F (x) − F
(xk )) bαk ,
Being originally proposed by , the shifting technique allows the enrichment function to be non-zero only in elements that contain the discontinuity. In other elements, the enriched term vanishes. This simple technique is considered as an effective treatment for blending elements, in which just some of the nodes are enriched . The technique was later employed by several authors [49, 50, 51] due to its convenience. An alternative approach is the floating node method [52, 53] which is quite suitable to study delamination in composite laminates. Geometrically, an arbitrary element e in the discretization can fall into one of the following three categories: (i) e is not cut by the crack; (ii) e is completely cut by the crack; (iii) e is partially cut by the crack and thus contains the crack tip. The nodes that belong to elements of category (iii) is defined as set K tip . The nodes that belong to elements of category (ii) and is not included in K tip is stored as set J split . Set I s denotes all the nodes in finite element mesh. The first term of Eq. (30) is the usual approximation when there is no discontinuity as in Eq. (1). However, the second term mathematically describes the jump in displacement by the aid of a step function H(x) = sign(φ(x)), which returns the sign of the signed distance φ(x) from an arbitrary point x to the crack surface. The third term of Eq. (30) also plays an important role as it is capable of capturing the singularity of stress fields in the vicinity of crack tip. Based on knowledge of asymptotic stress fields, four branch functions F α (α = 1, ..., 4) for isotropic materials were determined by  √ √ θ √ θ √ θ θ α [F , α = 1, . . . , 4] = r sin , r cos , r sin sin (θ) , r cos sin (θ) . (31) 2 2 2 2 For orthotropic composite materials, the four branch functions are essentially different from the isotropic
materials, and they are derived by  F
F2 = F3 = F4 =
θ1 p r sin g1 (θ), 2 √ θ1 p r cos g1 (θ), 2 √ θ2 p r sin g2 (θ), 2 √ θ2 p r cos g2 (θ), 2
(32) (33) (34) (35)
in which the functions gk and θk takes into account the material orientation and are given as follows q gk = (cos θ + zkx sin θ)2 + (zky sin θ)2 , (36) zky sin θ θk = arctan (37) cos θ + zkx sin θ The vector of nodal displacement at node i is written as ui while aj and bk are the additional DOFs due to the enriched terms. 3.4. Approximation of the temperature 3.4.1. Adiabatic crack As discussed in the previous section, the temperature is discontinuous at the crack surface in adiabatic crack problems. Analogously to the displacement field, the jump in the temperature field can be modeled by using the step function H(x). In order to express the singularity of temperature gradient in isotropic materials, Duflot  proposed the employment of only the first branch function in Eq. (31) based on the leading term of asymptotic expansion of temperature, which is written as follows r KT 2r θ T =− sin . (38) k π 2 In Eq. (38), KT is the heat flux intensity factor and (r, θ) is the local polar coordinates defined at crack tip, as already mentioned above. However, that is different for orthotropic composite media where the material orientation plays an important role and that has to be considered. Surprisingly, this issue was not mentioned in some previous works for thermo-elastic crack in orthotropic media [32, 31], where the same enriched formulation as Duflot’s proposal was used. In the current work we suggest that not only the first branch function, but also the third branch functions, i.e., Eq. (32) and Eq. (34), should also be employed. Finally, the enriched formulation that is used to approximate the temperature field can be written as follows: X X T h (x) = Ri (x)Ti + Rj (x) (H(x) − H(xj )) cj i∈I s
X k∈K tip
(F α (x) − F α (xk )) dαk ,
One may be noticed that the above improvement generally is to ensure not to lose any physical behavior when dealing with cracks in composite orthotropic media. In fact, the improved scheme of enriched 10
approximation here would be suitable for another relevant problem, for instance, heat flux intensity factor analysis . The authors also notice that the symplectic eigen expansion has been applied by Hu et al. to derive the solution of temperature and heat fluxes containing singularities in multi-material composites [55, 56] and anisotropic materials . Those symplectic analytical solutions are fundamental to study heat transfer across discontinuities and facilitates highly accurate calculation of heat flux intensity factor. This heat flux intensity factor is interesting but is not considered in this study. 3.4.2. Isothermal crack A crack is said to be isothermal when the crack surface is kept at a constant value. In such a circumstance, the temperature is continuous, but its gradient is discontinuous. Therefore, the step function H(x) is no longer sufficient. Instead, the enriched function for the nodes belong to set J split has to be continuous but its derivative is discontinuous. Such a function introduced by Mo¨es et al.  for modeling the material interfaces is used in this study: X G(x) = Ri φi − |φ(x)|, (40) i
in which φ is also the signed distance function as mentioned above. For isotropic materials, the leading term of asymptotic temperature near an isothermal crack can be written as follows  r KT 2r θ T =− cos . (41) k π 2 √ Thus, the branch function r cos 2θ can be used as near-tip enriched function in case of isothermal crack front . Following the Duflot’s suggestion and taking the material orthotropy into account, the second and the fourth branch functions (Eq. (33) and Eq. (35)) are employed to capture the singularities of thermal gradient at crack tip. It should be noticed that the points on crack surface in general are not identical to nodes. Hence, directly applying the prescribed temperature on crack surface is inconsistent under this situation. This issue is the same with that encountered during imposition of Dirichlet boundary condition in the meshfree method EFG . Remedies could be either penalty term or Lagrangian multipliers. In this paper, the penalty term is used on the crack surface, because this kind of treatment does not introduce additional DOFs like the Lagragian multipliers. Ones must be noticed that the crack tip enrichments have no contribution to the penalty term and to avoid possible locking, it is suggested in  that the function G(x) in Eq. (40) should be used to enrich all the nodes belong to J 0 = J split ∪ K tip . Finally, the temperature under condition of isothermal crack is approximated by enriched scheme as follows: X X X X T h (x) = Ri (x)Ti + Rj (x)G(x)ej + Rk (x) (F α (x) − F α (xk )) fkα , (42) i∈I s
Additionally, the weak-form of the governing equations is briefly presented in Appendix. 3.5. Evaluation of SIFs by interaction integral Accurate estimation of the SIFs is a key task in modeling fracture problems as they are important parameters characterizing the strength of stress singularity in the vicinity of crack tips. In 2D problems, 11
the SIFs can be evaluated based on a domain integral (i.e., interaction integral). For orthotropic media, the relation between the interaction integral and SIFs are given by [27, 15] (1) (2) (1) (2) (2) (1) (1) (2) M (1,2) = 2c11 KI KI + c12 KI KII + KI KII + 2c22 KII KII , (43) with c11 c12 c22
S22 z1 + z 2 = − Im 2 z1 z2 S22 1 S11 = − Im + Im (z1 z2 ) 2 z1 z2 2 S11 = Im (z1 + z2 ) 2
(44) (45) (46)
Generally, two states of the cracked body are considered: the present state (state (1)) and an auxiliary state (state (2)) chosen by analysts. Without loss of generality, the auxiliary state can be chosen as the (2) (2) asymptotic fields (see Eqs. (22) - (26)) from either pure mode I (i.e. KI = 1 and KII = 0) or pure (2) (2) mode II (i.e., KI = 0 and KII = 1), and thus the following system of equations is obtained (1)
M (1,pure mode I) = 2c11 KI + c12 KII (1)
M (1,pure mode II) = c12 KI + 2c22 KII
1 are known. By solving these equations, the actual mix-modes SIFs, i.e., KI1 and KII Being derived from the well-known path-independent J-integral, the interaction integral takes the following form in linear thermo-elastic fracture problems [60, 16] ! Z (2) (1) ∂q (2) ∂ui (1) ∂ui (1) (2) (1,2) M = σij + σij − σik εik δ1j dA ∂x1 ∂x1 ∂xj A # (49) Z " (1) (2) ∂T + βij εij qdA, ∂x1 A
with βij = Cijkl αkl , see Eq. (12). Following the definition of J-integral, firstly an arbitrary closed path C surrounding the crack tip and contains no other type of discontinuity within has to be chosen, as illustrated in Figure 3. Because J-integral is path-independent, C can be selected as a circle being centered at the crack tip for simplicity. The area bounded by C and the crack is denoted as A and q is a weight function that serves to convert the line integral J into the domain integral M . The weight function q only takes one of two values: unity inside the curve C and zero outside. 3.6. Numerical integration in element containing cracks In order to numerically evaluate the integration arisen in the governing equations as well as the interaction integral during SIFs computation, Gaussian quadrature is employed. However, if an element is entirely or partially cut by the crack, a direct application of Gaussian scheme would result in loss of accuracy. Instead, in order to enhance the accuracy, further sub-triangulation of those elements is required. The purpose of sub-division is to guarantee that none of the sub-regions is crossed by any 12
Figure 3: Schematic of the domain of interative integral.
discontinuities. Gauss points are then taken within each sub-triangular region, as illustrated in Figure 4. A high number of integration points is selected using Dunavant’s rule  in each sub-triangular region. In particular, 7 integration points are taken in each sub-triangle if the element is completely cut by the crack and 13 points are chosen if the element is partially cut by the crack, i.e., crack tip element.
Figure 4: Sub-division scheme to select integration points for elements that are cut by the crack, either completely (left) or partially (right). Integration points in each sub-triangular region are chosen based on Dunavant’s rule 
3.7. Crack growth modeling Exact prediction of direction where a crack propagates is one important objective of crack growth modeling. Within the framework of LEFM in isotropic materials, various criteria to determine crack growth direction under mixed-mode loading have been introduced [62, 63, 64], such as the maximum circumferential stress criterion, the maximum energy release rate criterion. the minimum strain energy density criterion and the material force criterion. A comparison between various crack growth criteria is reported in . Due to simplicity and efficiency, the maximum circumferential stress criterion  has been commonly used by many authors [41, 67, 68, 10]. The criterion is still valid in case of linear thermo-elastic fracture mechanics as explored in [13, 15, 16]. It states that the crack will propagate in the direction θc (local Cartesian coordinates defined at the crack tip) where the circumferential stress becomes maximum. The maximum circumferential stress criterion has also been re-formulated to apply to orthotropic materials by , in which the angular variation of fracture toughness in the material is taken into account. According to , the fracture toughness along an arbitrary direction θ is calculated from the 13
fracture toughness along the 1- and 2- directions (the direction of material orthotropy as mentioned in Figure 2) θ 1 2 KIC = KIC cos2 θ + KIC sin2 θ. (50) By defining the normalized circumferential stress as ∗ σθθ =
σθθ , θ KIC
the ”maximum circumferential stress criterion” version for orthotropic materials is finding θ that makes ∗ maximum. Because both K 1 and K 2 are positive real numthe normalized circumferential stress σθθ IC IC bers, it is possible to divide both numerator and denominator of the normalized circumferential stress by 1 without changing the variation with respect to θ. Furthermore, study reported in  suggested that KIC the ratio of fracture toughness in both direction is equal to the ratio of elastic moduli 2 KIC E1 = . 1 E2 KIC
Hence, the normalized circumferential stress is finally defined as ∗ σθθ =
σθθ . 2 1 +E E2 sin θ
This criterion has been implemented in numerical investigation of cracked solids subject to mechanical loading [25, 29]. However, as pointed out by Cahill et al. , the obtained results are far from agreement with experiments. Noting that the relative angle of the crack to the material orientation was not considered in ,  proposed to shift the denominator of Eq. (53), in which the normalized circumferential stress is rewritten by ∗ σθθ =
σθθ , 2 1 +E E2 sin θP
where angle θP is defined by θP = θ − β + θcurr ,
in which β is the material orientation and θcurr is the current crack inclination. 4. Numerical results and discussion The proposed formulation is implemented in the Matlab environment and its accuracy is verified through numerical experiments. In this section, numerical results of thermal-mechanical crack propagation problems in orthotropic composite media are presented and discussed to show the accuracy and performance of the proposed approach. In particular, the first four numerical examples are considered with the purpose of verifying the accuracy of the static SIFs of cracked orthotropic materials, and the last three numerical examples are for crack propagation analysis. Different loading conditions and configurations are considered. We shall explore the effects of the thermo-mechanical loads, crack inclination angles, material orientation, mesh size, etc., on the static SIFs, and crack propagation path. 14
4.1. Single edge notched specimen under mechanical tensile load Because of the availability of the experimental data as reported by Jernkvist , the mixed-mode problem of a rectangular plate made of Norway spruce (Picea abies) with an inclined edge notch subject to uniform tensile loading, as depicted in Figure 5, is considered. The initial notch is assumed to be placed along one material principal axis and crack tip is always at the center of the plate. Material properties used for this example are: E1 = 11.84 GPa, E2 = 0.81 GPa, E3 = 0.64 GPa, G21 = 0.63 GPa, ν12 = 0.38, ν13 = 0.56 and ν23 = 0.43. Dimensions of the plate are: length L = 60 mm, width W = 30 mm and thickness t = 20 mm. Under plane strain condition, the normalized SIFs are provided as functions of the crack inclination φ ∈ [0o , 45o ] as follows : ˜ I (φ) = K ˜ II (φ) = K
K √ I = 3.028 − 3.22 × 10−3 φ + 3.73 × 10−4 φ2 − 9.14 × 10−6 φ3 , πa KII √ = sin(2φ)(0.644 + 4.89 × 10−3 φ) πa
W σ Figure 5: Example 4.1: Geometry of a single edge notched specimen under mechanical tensile load
The proposed XCQ4 element is applied to solve this mixed-mode problem, and the vertical displacement, uy , in deformed shape of the plate is then represented in Figure 6, showing the jump across crack surface. The numerical results of the normalized SIFs computed by the developed XCQ4 are compared with both the experimental data  and those derived from the traditional XQ4 using the same mesh of 25 × 49 quadrilateral elements. The comparison is reported in Table 1, showing a good agreement among three solutions. From the table, it is obviously that the XCQ4 offers better accuracy of the normalized SIFs in comparison with the traditional XQ4. This conclusion is consistent with our previous studies. The underlying reason may be due to the better representation of stresses, as evidenced by σyy component for both the XCQ4 and XQ4 in Figure 7. Except the region in the vicinity of crack, which is naturally discontinuous, the stress field provided by XCQ4 element is physically smooth in the entire body. 15
Figure 6: Example 4.1: The vertical displacement uy of a single edge notched specimen under mechanical tensile load obtained by the developed XCQ4, showing the jump across crack surface.
˜ I and K ˜ II with experiments . Table 1: Example 4.1: Comparison of the normalized SIFs K
Methods Experiment  Present XCQ4 XQ4
˜I K ˜I K ˜I K ˜ II K ˜I K ˜ II K
φ = 0o
3.028 0.0 2.948 0.0 2.913 0.0
3.033 0.359 2.969 0.359 2.921 0.356
3.02 0.685 2.956 0.682 2.880 0.675
2.806 0.864 2.704 0.874 2.606 0.862
(b) XCQ4 Figure 7: Example 4.1: Visualization of stress component field (σyy ) obtained by traditional XQ4 and developed XCQ4 elements. The XCQ4 offers smoother stress compared with the XQ4.
4.2. Edge crack in a rectangular glass/epoxy plate under constant flux Next example deals with a strip plate with dimensions W × L (L = 4W) made of glass/epoxy containing a horizontal adiabatic edge crack of length a. A constant heat flux parallel to the crack surface is applied to the strip, by maintaining bottom side at reference temperature, i.e., ∆T = T −Tref = 0, while cooling down the top side by ∆T = −T0 , see Figure 8. The effective material properties of the plate are: E1 = 55 GPa, E2 = 21 GPa, G12 = 9.7 GPa, ν12 = 0.25, α11 = 6.3 × 10−6 K −1 , α22 = 2 × 10−5 K −1 . In this example, the material orientation is aligned with the global axes (i.e., angle β = 0o ). Under the above described boundary conditions, the strip is caused to bend and the crack is open under mode-I.
T-Tref =T0 2 1 L Crack
T-Tref =0 a W Figure 8: Example 4.2: Schematic geometry of a rectangular glass/epoxy plate under constant flux with an edge crack.
In order to verify the accuracy of the proposed approach, four finite element meshes of XCQ4 elements are used including 13 × 25 , 25 × 49, 49 × 99 elements and 99 × 199 elements. In this example, the 22 √ . To study the convergence of numerical SIFs between SIFs are normalized by a factor K0 = α E 22 T0 πa XCQ4 and XQ4 elements with respect to mesh density, data calculated at a/W = 0.2 by both approaches using the above four levels of mesh are presented in Table 2. It is observed that the XCQ4 results are close to reference BEM value taken from , even at a relatively coarse mesh of 25 × 49 elements. The graph of normalized SIFs with respect to number of nodes exhibited in Figure 9 further visualizes the observation in Table 2. Also, the superior of XCQ4 over XQ4 has been demonstrated in our previous works for linear elastic fracture mechanics  and linear thermo-elastic fracture mechanics on isotropic material  In general, due to extra effort in constructing the CIP shape function, XCQ4 requires longer elapsed 18
˜ I calculated by XCQ4 and XQ4 elements for the case a/W = 0.3 by using Table 2: Example 4.2: The normalized SIFs K various finite element meshes. Reference value obtained by BEM is taken from .
13 × 25
25 × 49
49 × 99
99 × 199
Present XCQ4 XQ4
time than XQ4, given the same mesh. However, one must pay attention to the higher accuracy and the need for not using post-processing operation to smoothen gradient fields. Particularly in this example, for the same mesh of 25 × 49 elements, the estimated computational time by XQ4 elements is approximately 103 seconds while that by XCQ4 is more than 621 seconds. However, due to higher accuracy of XCQ4, the SIF value obtained with 25×49 elements is already quite close to reference result. On the other hand, 99 × 199 XQ4 elements are needed to get such accurate value, for which the corresponding elapsed time is nearly 1367 seconds. In short, XCQ4 is more time efficient to achieve the same level of accuracy. 0.64
Normalized mode-I SIFs
0.61 XQ4 XCQ4 BEM
Number of nodes
Figure 9: Example 4.2: Convergence of normalized mode-I SIFs obtained by XCQ4 and XQ4 elements with respect to number of nodes.
Theoretically, the J-integral is path-independent. Without loss of generality, the J-integral contour is simply taken as a circle whose center is located at the crack tip. The radius r of the circle is adjustable p by defining r = f actor × Atip , in which Atip is the area of the element containing the crack tip. f actor is some real positive number chosen by user to control how large the contour is. The region being enclosed by the contour is the domain of J-integral. Here, four J-integral domains for the mesh of 25 × 49 elements are obtained by varying the value of f actor, as shown in Figure 10. In Figure 10, the elements that are crossed by the circular path are marked by red color, while green color is used to mark the area inside the circular path. Table 3 presents the normalized mode-I SIFs numerically evaluated a using the domains illustrated in Figure 10 for the case W = 0.5. As expected, the values of normalized SIFs are almost identical, regardless to the integral contour, suggesting the path-independence of the
˜ I by the present Table 3: Example 4.2: Effect of different J-integral domains represented in Figure 10 on the normalized SIF K a XCQ4, for the case W = 0.5.
Domain 1 0.640
Domain 2 0.632
Domain 3 0.633
Domain 4 0.633
Figure 10: Example 4.2: Different paths selected to form the J-integral domains: (a) domain 1, (b) domain 2, (c) domain 3, and (d) domain 4. A mesh size of 25 × 49 elements is used for both the XQ4 and XCQ4 methods.
The obtained results with respect to various ratios of a/W = 0.2, 0.3, 0.4 and 0.5 are listed in Table 4. For all a/W , corresponding to finer mesh density, the evaluated SIFs tend to increase to a converged value. Reference solution for this example is based on the BEM results reported in . Obviously, a close agreement among the reference BEM results and the XCQ4 with different mesh sizes for four values of a/W is obtained. One sees that even at the coarsest mesh density of 13 × 25 elements, the largest numerical error is only 2.35%. The results gained by the mesh of 99 × 199 XCQ4 elements are identical to those by the mesh 49 × 99 elements and hence, are not listed for brevity. 4.3. A rectangular plate with a slant center crack under constant flux Consider the glass/epoxy plate as described in Example 4.2, but contains an inclined crack as depicted in Fig. 11. Boundary conditions for mechanical and thermal problems are the same as in Example 4.2. This is again an adiabatic crack analysis. The normalized SIFs evaluated at left tip (tip P) and right tip (tip Q) with respect to crack inclined angle θ and a/W are reported in Table 5. Only small differences between the present solution and reference results using BEM  are recorded. Because material 20
˜ I with respect to different values of a/W obtained by the present XCQ4 with Table 4: Example 4.2: The normalized SIFs K different mesh sizes compared with the reference BEM results . a W
Methods BEM  Present XCQ4 Present XCQ4 Present XCQ4
(49 × 99) elements (25 × 49) elements (13 × 25) elements
0.601 0.602 0.599 0.594
0.631 0.631 0.630 0.619
0.645 0.644 0.641 0.630
0.637 0.635 0.633 0.622
orientation is aligned with the global coordinate system, the mode-mixity in this problem is mainly induced by the inclination of crack, θ. As θ increases from 0o to 45o , KII increases while KI decreases. The variation of both normalized mode-I and mode-II SIFs versus inclined crack angle θ, for θ from 0o to 90o are plotted in Figure 12.
P 2 1 T-Tref =0
W Figure 11: Example 4.3: Schematic geometry of a rectangular glass/epoxy plate under constant flux with a slant center crack.
4.4. An anisotropic square plate with two parallel cracks under isothermal condition In this example, the problem of a square plate made of glass/epoxy containing two parallel cracks as shown in Fig. 13 is investigated. The plate boundary is heated by ∆T = T0 , while the crack faces are kept at the reference temperature, i.e., ∆T = 0, leading to isothermal condition. The temperature field calculated by the developed XCQ4 is continuous as clearly depicted in Fig. 14, whereas the heat flux (i.e., temperature gradient) changes sign (jump) across the two crack faces, as visualized in Fig. 15.
˜ I and K ˜ II for different values of crack inclined angle θ and of Table 5: Example 4.3: Comparison of the normalized SIFs K ratio a/W between the BEM  and the present XCQ4.
˜ I , tip P K 
˜ II , tip P K 
θ= 0.3 0.4 0.5
0.361 0.367 0.376
0.361 0.366 0.376
θ = 15o 0.3 0.4 0.5
0.34 0.348 0.359
θ = 30o 0.3 0.4 0.5 θ = 45o 0.3 0.4 0.5
˜ I , tip Q K 
˜ II , tip Q K 
0.001 0.001 0.002
0.001 0.001 0.002
0.361 0.367 0.376
0.361 0.366 0.376
-0.001 -0.001 -0.002
-0.001 -0.001 -0.002
0.338 0.346 0.357
0.086 0.084 0.081
0.087 0.083 0.082
0.339 0.347 0.358
0.337 0.345 0.356
0.084 0.081 0.077
0.086 0.081 0.079
0.280 0.290 0.303
0.278 0.289 0.301
0.152 0.151 0.149
0.151 0.149 0.148
0.277 0.289 0.301
0.276 0.287 0.299
0.149 0.147 0.144
0.153 0.149 0.147
0.192 0.202 0.214
0.192 0.202 0.214
0.181 0.183 0.186
0.179 0.181 0.182
0.191 0.201 0.213
0.189 0.199 0.211
0.178 0.179 0.180
0.180 0.181 0.182
0.4 Mode-I SIF Mode-II SIF
Normalized SIFs [-]
Inclined crack angle [Degree]
Figure 12: Example 4.3: Variation of normalized mode-I and mode-II SIFs with respect to inclined crack angle θ
Analogously to the stress fields presented in Example 4.1, the temperature gradient evaluated by XCQ4 elements is physically smooth.
T-Tref =T0 Y
T-Tref =0 2W
T-Tref =0 2a
T-Tref =T0 2W Figure 13: Example 4.4: Schematic geometry of an anisotropic square plate with two parallel cracks under isothermal condition.
Figure 14: Example 4.4: Continuous temperature distribution of an anisotropic square plate with two parallel cracks under isothermal condition obtained by the developed XCQ4.
Table 6 exhibits normalized SIF values calculated at the right tip of the lower crack, using a uniform mesh of 49 × 49 XCQ4 elements with respect to various cases pf crack lengths and distances between 22 √ . The normalized SIFs values vary the two parallel cracks. The normalization factor is K0 = α E 22 T0 πa according to crack length and the distance between two cracks as well. Both KI and KII are non-zero, but magnitude of KI is about ten times that of KII . Hence, this problem is mode I-dominated. All the normalized SIF values evaluated by developed XCQ4 are higher than those by BEM  but the discrepancies are small. The same tendency of variation of SIFs is observed between the two methods. ˜ I . Variation of K ˜ I with respect to Given the same distance d, longer cracks lead to smaller values of K distance d tends to be smaller when the crack length increases. In case of longest crack, i.e., a/W = 0.7, 23
(a) The traditional XQ4 element
(b) The developed XCQ4 element Figure 15: Example 4.4: Comparison of heat flux field in y-component, qy , between the conventional XQ4 and the XCQ4 elements. The heat flux calculated by the developed XCQ4 is smoother than that by the conventional XQ4.
˜ I and K ˜ II for different values of a/W and of d/W between the Table 6: Example 4.4: Comparison of the normalized SIFs K proposed XCQ4 and reference BEM solution .
˜ I ,  K
˜ I , XCQ4 K
˜ II ,  K
˜ II , XCQ4 K
0.3 0.4 0.5 0.6 0.7
0.142 0.135 0.130 0.124 0.116
0.148 0.140 0.134 0.128 0.120
0.007 0.007 0.005 0.002 -0.003
0.009 0.008 0.007 0.004 -0.001
0.3 0.4 0.5 0.6 0.7
0.153 0.145 0.137 0.128 0.117
0.16 0.151 0.143 0.133 0.121
-0.005 -0.006 -0.007 -0.009 -0.01
-0.0056 -0.0062 -0.0075 -0.0091 -0.0105
0.3 0.4 0.5 0.6 0.7
0.147 0.139 0.133 0.125 0.116
0.153 0.145 0.138 0.129 0.120
-0.009 -0.01 -0.009 -0.007 -0.004
-0.0101 -0.0104 -0.0097 -0.0079 -0.0049
d/W = 0.1
d/W = 0.2
d/W = 0.3
˜ I values are nearly identical for all cases of d/W , indicating little effect of the distances d, which can K be explained by stronger boundary effect near the domain boundary. In Table 6, all the normalized modeI SIFs are close to the reference BEM results, with numerical error less than 5%. Though some of the normalized mode-II SIFs may have disagreement with reference data up to 20%, it should be emphasized that this example is mode-I dominated. Values of mode-I SIF are ten-times greater than mode-II SIFs. Hence, mode-II SIFs are not signifcant. 4.5. Quasi-static crack growth of an anisotropic cracked Brazilian disc specimens under mechanical load Examples for thermal-mechanical crack propagation problems are now studied. The experiments of transversely isotropic cracked Brazilian disc specimens conducted by Ke et al. [24, 70] are replicated by numerical simulation using the proposed XCQ4 element. The disc specimens take a shape of cylinder with diameter 2R = 74 mm and thickness t = 10 mm, and are compressed by point load P under plane stress condition, see Fig. 16. An initial crack length is specified as 2a = 22 mm. Material properties used for this analysis are: E1 = 67.681 GPa, E2 = 78.302 GPa, ν12 = 0.185, ν23 = 0.267, G12 = 25.336 GPa, G23 = 25.336 GPa. This example serves to study the applicability of Cahill’s 25
1 criterion (Eq. (54)) on modeling the evolution of crack in anisotropic materials where ratio E E2 is nearly equal to 1, which has not been discussed yet in . For this purpose, a mesh of 6197 XCQ4 elements is used to discretize the disc domain, see Figure 16. Crack growth is modeled by an incremental scheme, in which the crack is extended from the crack tip by a straight segment of small length ∆a = a/6 after each load step. The direction of new crack segment is determined by Cahill’s criterion and the incremental length is chosen as a small number. According to , ∆a = a/6 is sufficient to provide acceptable numerical representation of crack path in this particular problem. A good agreement between numerical and experimental data indicates the validity of Cahill’s criterion. Despite the material orientation, the crack firstly tends to turn nearly perpendicular to the initial crack tip and then approaches toward the loading point, as depicted in Fig. 17 (a), (b), and (c), respectively for β = 30o , 45o and 60o . This observation does not contradict those by  and . Instead, results suggest that the ratio of elastic 1 modulus E E2 , or equivalently the ratio of fracture toughness (see Eq. (52)) has certain influence on how cracks propagate. When there is strong domination of material stiffness in one direction, cracks would grow along that direction. On the other hand, when material is weakly anisotropic, the mode-mixity would be the main factor that determine the crack paths. The horizontal displacement field of the disc for the case: initial crack inclination θ = 45o and material orientation β = 30o is depicted in Figure 18.
P Figure 16: Example 4.5: (left) Schematic geometry of an anisotropic cracked Brazilian disc specimen under mechanical compression load and (right) The mesh of 6197 quadrilateral elements
4.6. Quasi-static crack growth of an edge crack in a rectangular plate under constant heat flux The next numerical example deals with an edge crack in a rectangular plate under constant flux. This example is same as Example 4.2 and is taken here to study Cahill’s criterion (Eq. (54)) applicability on orthotropic composite materials. To accomplish that purpose, six sets of material properties are set up and they are defined as follows: • Glass-epoxy (Mat0): same data as in Example 4.2
Numerical data (Present) Experimental data (Ke et al.)
(a) Material orientation with β = 30o
Numerical data (Present) Experimental data (Ke et al.)
(b) Material orientation with β = 45o
Numerical data (Present) Experimental data (Ke et al.)
(c) Material orientation with β = 60o Figure 17: Example 4.5: Crack paths of an initial crack inclination θ = 45o for various material orientations computed by the developed XCQ4.
Figure 18: Example 4.5: Distribution of horizontal displacement for the case: initial crack inclination θ = 45o and material orientation β = 30o
• Artificial orthotropic material 1 (Mat1): E2 = 21 GPa, E1 = E2 • Artificial orthotropic material 2 (Mat2): E2 = 21 GPa, E1 = 1.5E2 • Artificial orthotropic material 3 (Mat3): E2 = 21 GPa, E1 = 2E2 • Artificial orthotropic material 4 (Mat4): E2 = 21 GPa, E1 = 10E2 • Artificial isotropic material (MatIso): E = 21 GPa, ν = 0.25, α = 6.3 × 10−6 K−1 For all the orthotropic artificial materials, only E1 is modified, while the rest of material parameters are taken the same as glass-epoxy (denoted by Mat0). In Mat0, the ratio E1 /E2 ≈ 2.62. The same mesh of 49 × 99 XCQ4 elements, which is found to provide ”converged” SIF values in Example 4.2, is used to evaluate the angle where crack propagates. The initial crack angle is placed along the horizontal axis (i.e., angle θ = 0o ) with length a = 0.3 ∗ W . Table 7 presents the computed SIF values KI , KII obtained by the above six sets of material data and the corresponding critical angle θc predicted by Cahill’s criterion with respect to various material orientation β (see Fig. 2). It is noticed that θc is defined in the local system of coordinate originated at crack tip. 1 It can be observed that the critical angle θc varies corresponding to ratio E E2 . When E1 dominates, i.e., E1 is about ten times larger than E2 , θc ≈ β, crack grows along the 1-direction. This observation is identical to results reported by Cahill et al.  where E1 = 114.8 GPa and E2 = 11.7GPa. In 1 the experiment of Jernkvist , where the crack path is found to follow the wood fibers, the ratio E E2 1 is also large (more than ten times). As ratio E E2 gets smaller, the growth direction parts away from 1 the 1-direction. Specially, when E E2 ≈ 1 (Mat1), the behavior is most closest to isotropic media (case MatIso), yet material Mat1 is not isotropic, in which mode-I opening occurs in this spectacular boundary conditions. Fig. 19 represents the crack paths for the case β = 60o after 7 load steps, in which the incremental length of crack at the end of each load step is prescribed to be ∆a = 0.3a, further supports the observation in Table 7.
Table 7: Example 4.6: Numerical results of the SIFs and critical angle θc for different material angles β and six sets of material properties obtained by the developed XCQ4.
1.013 × 10−4
β = 0o Mat1 (E1 /E2 Mat2 (E1 /E2 Mat3 (E1 /E2 Mat0 (E1 /E2 Mat4 (E1 /E2
= 1) = 1.5) = 2) ≈ 2.62) = 10)
26.995 27.040 27.090 27.141 27.368
0.012 0.016 0.018 0.020 0.033
−0.09o −0.09o −0.09o −0.09o −0.09o
Mat1 (E1 /E2 Mat2 (E1 /E2 Mat3 (E1 /E2 Mat0 (E1 /E2 Mat4 (E1 /E2
= 1) = 1.5) = 2) ≈ 2.62) = 10)
28.412 28.409 28.559 28.786 30.696
0.017 0.957 1.695 2.441 6.830
−0.09o 15.77o 20.99o 23.84o 28.74o
Mat1 (E1 /E2 Mat2 (E1 /E2 Mat3 (E1 /E2 Mat0 (E1 /E2 Mat4 (E1 /E2
= 1) = 1.5) = 2) ≈ 2.62) = 10)
28.917 30.442 31.550 32.603 38.150
0.013 1.131 2.012 2.900 8.047
−0.09o 21.82o 30.18o 34.86o 42.97o
Mat1 (E1 /E2 Mat2 (E1 /E2 Mat3 (E1 /E2 Mat0 (E1 /E2 Mat4 (E1 /E2
= 1) = 1.5) = 2) ≈ 2.62) = 10)
28.407 33.422 37.093 40.545 57.086
0.009 1.031 1.913 2.858 9.132
−0.09o 25.32o 37.57o 44.41o 56.67o
β = 30o
β = 45o
β = 60o
MatIso Mat0 Mat1 Mat2 Mat3 Mat4
Figure 19: Example 4.6: The crack paths computed by the XCQ4 for different materials with different ratios the same material orientation β = 60o
4.7. Quasi-static crack growth in a perforated panel with a circular hole under constant heat flux In the last numerical example, we explore crack propagation behavior for a single edge notched panel with a circular hole under thermal load. The geometry dimensions and initial length of the preexisting notch are depicted in Figure 20. The left and right edges of the plate are imposed by equal yet opposite temperature values, resulting in a constant heat flux parallel to the initial notch. The crack surface is adiabatic. Mechanical loading is absent in this example. Plane strain condition is assumed. For numerical simulation, the panel is discretized by 2200 XCQ4 elements. Simulations of crack propagation are conducted for two cases of materials: • Isotropic material: Young’s modulus 30 GPa, Poisson’s ratio ν = 0.3, thermal conductivity k = 5 W/(mK) and thermal expansion α = 15 × 10−6 K −1 . • Composite orthotropic glass/epoxy material: the same data as taken for Example 4.2.
4.7.1. Isotropic material The predicted crack path of the specimen is calculated and represented in Figure 21. It shows that, at first, the crack propagates almost straightforward toward the right edge, and then turns toward the hole. This computed result is consistent with the case reported in Refs. [10, 68], in which the perforated panel is loaded by uniform tensile on top edge. 4.7.2. Orthotropic glass/epoxy material For the glass/epoxy panel, three cases of material orientation are considered: β = 0o , β = 60o and β = −60o . Trajectory of crack growth corresponding to the three cases are hence depicted in Figure 22. It is interesting to see that in the case β = 0o , the crack initially grows straightforward, which is similar to the observation with isotropic material, but then it does not turn toward the hole. Instead, as the crack gets close to the right edge, it tends to turn slightly upward, possibly due to the boundary effect. The crack path in the case β = 60o is as expected. The crack propagates upward following a straight path 30
Figure 20: Example 4.7: Geometry and boundary conditions a perforated panel with a circular hole under constant heat flux.
Figure 21: Example 4.7: Crack path obtained by the present approach for isotropic material.
due to the domination of E1 over E2 . The inclined angle between crack path and horizontal direction is approximately 47o , which is consistent with the observation in Example 4.6. If strong domination of the 1−direction exists, i.e., E1 /E2 ≈ 10, the crack trajectory would follow the 1−direction. On the other hand, when material angle β = −60o , the crack propagates downward. In addition, Figure 23 visualizes the distribution of shear stress component σxy in the perforated panel for the case material orientation β = 0o . 5. Conclusion and outlook In this paper, we have outlined the successful extension of XCQ4 element for 2D fracture problems in orthotropic composite media under thermal and mechanical loadings. We have suggested an improved way for the enriched approximation of the temperature field in both isothermal and adiabatic conditions, and all the desirable features of the developed approach are demonstrated through the numerical examples with different loading conditions. The accuracy and performance of the proposed method have been discussed, and comparisons with experimental data and other numerical methods available in the literature have been made for the static SIFs and quasi-static crack propagation. As expected, higher accuracy can be obtained by the proposed formulation, in comparison with the well-known extended finite element XQ4. The better performance of the developed XCQ4 is originated from the more physically accurate and smooth gradient fields, which in turn are resulted from the enhanced interpolation by the CIP characteristics. The current work also verifies the applicability of Cahill’s criterion of crack growth direction on 1 orthotropic composite media  with different ratios of elastic modulus E E2 . Results are very interesting and they are in good agreement with experimental observations. Both the loading mode-mixity and the elastic modulus ratio have influence on crack propagation. When there is strong domination of material stiffness on one particular direction, crack will open and extend along that direction, irrespective to the mode-mixity. In contrast, when weak anisotropy occurs, loading mode-mixity is the key factor to determine crack path. Recently, Sajjadi et al.  has proposed a modified version of the maximum circumferential stress criterion for isotropic media, which takes into account also the shear stress, allowing better prediction of crack paths. A similar approach is possible for orthotropic composite media, such that the circumferential stress in Eq. (54) is replaced by the effective stress proposed in . Another interesting scientific question that is worth further investigation is the extension of the proposed approach to 3D problems, which brings the method closer to engineering applications. Acknowledgement This research is funded by Ho Chi Minh city University of Technology - VNU-HCM, under grant number C2017-20-06. The authors gratefully thank all the anonymous reviewers for their valuable suggestions/comments to improve the quality of the manuscript.
(a) Material orientation β = 0o
(b) Material orientation β = 60o
(c) Material orientation β = −60o Figure 22: Example 4.7: Crack paths obtained for orthotropic glass/epoxy material with various angle β
Figure 23: Example 4.7: Distribution of shear stress component σxy obtained for glass/epoxy material with angle β = 0o
Appendix After some mathematical transformation, the weak formulation of governing equations are obtained by Z
Z δε : C : εdΩ −
Z δu · bdΩ −
¯ dΓ = 0. δT · q
Z δq : k : qdΩ −
δu · ¯tdΓ = 0,
δT · QdΩ − Ω
In order to prescribe T = T¯ on crack surface, a penalty term is added into the weak formulation of heat transfer equation, i.e. Eq.(59), as follows Z Z Z Z ¯ dΓ + λ δT · T − T¯ dΓ = 0, δq : k : qdΩ − δT · QdΩ − δT · q (60) Ω
where λ is the penalty factor, which is chosen to be a large number with respect to k/L, where L is a representative length.
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• • • • •
The recently developed XCQ4 is further extended to study thermal-mechanical crack propagations in composite media. We introduce an improved way for enriched approximation of discontinuous temperature field taking into account material orientation. Both isothermal and adiabatic loading conditions are considered. Numerical simulations for isotropic and composite orthotropic media with complex geometries are analyzed. Numerical results are validated against the experimental and other numerical methods solutions, showing a good agreement.