The effects of interfacial strength on fractured microcapsule

Abstract

The effects of interfacial strength on fractured microcapsule are investigated numerically. The interaction between crack and microcapsule embedded in mortar matrix is modeled based on cohesive approach. The microcapsules are modelled with variation of core-shell thickness ratio and potential cracks are represented by pre-inserted cohesive elements along the element boundaries of the mortar matrix, microcapsules core, microcapsule shell, and at the interfaces between these phases. Special attention is given to the effects of cohesive fracture on the microcapsule interface, namely fracture strength, on the load carrying capacity and fracture probability of the microcapsule. The effect of fracture properties on microcapsule is found to be significant factor on the load carrying capacity and crack propagation characteristics. Regardless of core-shell thickness ratio of microcapsule, the load carrying capacity of self-healing material under tension increases as interfacial strength of microcapsule shell increases. In addition, given the fixed fracture strength of the interface of microcapsule shell, the higher the ratio core-shell thickness, the higher the probability of microcapsules being fractured.

Keywords

interfacial strength / cohesive elements / microcapsule / core-shell thickness ratio / fracture properties

Introduction

Self-healing systems with microencapsulated healing agents have received a lot of attention nowadays due to its inherent ability of automatic crack detection. This crack endanger the overall durability of structures as aggressive substances can easily flow inside the matrix through fluids or gasses and can lead to structural failure. This new self-healing material allows the early and timely repairing of deep cracks which are not accessible so it can prolong the service life and reduce the cost of maintenance. In general, existing healing methods can be divided into two categories: firstly autogeneous healing where a healing is accomplished by the cement composition itself, and secondly autonomous healing that requires additives to react with cement. The microscapsule based self-healing method, which is claimed among the most promising techniques, belongs to the latter. The size and material of microcapsules need to be designed in a way that they should be able to break when hitted by the crack, so the internal fluid can be released to seal the crack.

Much research work on fracture have been started over the past decade. Different techniques of computational methods of fracture in brittle and quasi-brittle have been developed [¹]. For instance, Rabczuk and Belytschko [²,³] proposed meshfree technique to model discrete cracks. With this technique, the orientation angle of the cracks can be chosed randomly, hence the crack’s arrangement is not practically needed. They applied this technique to some 2D and 3D models, and then validated with experimental results. Rabczuk and Zi [⁴] introduced a meshfree method to study the crack propagation. The state of material controlled the propagation of cracks. This method is then applied to some examples both of static and dynamic problems. The application of cohesive laws in bending problem was introduced by Areias and Rabczuk [⁵]. The crack direction algorithm is controlled by traction-separation law while using thin (Kirchoff-Love) shell models and the extended finite element method.

The new development in XFEM-based to study the propagation of crack are getting more attention [⁶–⁸]. Vu-Bac et al. [⁹] developed a novel numerical method (NS-XFEM) for analyzing 2D fracture problem. They incorporated node-based smoothed finite element into extended finite element method. With this new method, the stress singularity integration at the crack tip can be avoided. Amiri et al. [¹⁰] presented the phase-field model for fracture using thin (Kirchoff-Love) shells with the local maximum-entropy (LME) meshfree method, while the extended local maximum entropy (XLME) combined with enrichment function in partition of unity has also been developed [¹¹]. They showed numerically that convergence of energy norm and stress intensity factors can be optimally achieved. The computational approaches based on cohesive zone modeling are recently developed. The use of cohesive models to investigate fracture process were proposed, for instance, Gui et al. [¹²] simulated fracture process using cohesive model mixed with continuum-discrete element method in rock dynamic tests. Both elastic and inelastic displacements are considered in the cohesive model to control the fracture behavior of the rocks. The hydraulic fracture process in a permeable porous medium was investigated by Nguyen et al. [¹³]. They implemented a unified formulation both of continous and discontinous pressure across the fracture. A new FEM-based simple algorithm for computational fracture in brittle and ductile materials was developed by Areias et al. [¹⁴–¹⁶]. They proposed new algorithm based on the edge rotations with rotating axes at the crack front edges [¹⁴,¹⁵] and also with the injection of continuum softening elements in the process region [¹⁶].

Areias et al. [¹⁷] proposed local remeshing technique based on the phase-field to investigate fracture on plates and shells. An alternative crack propagation algorithm was developed by using a modified screened Poisson model which is exempt of crack path calculations [¹⁸], and 3D edge-based division using tetrahedral meshes refinement and cutting [¹⁹]. A new formulation of dual-horizon peridynamics was introduced by Ren et al. [²⁰,²¹]. This new concept is less sensitive than the conservative one. To reduce the computional cost, a simple adaptive refinement procedure was adopted. An artificial neural network used in predicting the fracture energy of polymer nanocomposites was conducted by Hamdia et al. [²²,²³]. They applied this method along with bayesian method based on different measurements. The investigation in concurrent multiscale methods for fracture was presented by Talebi et al. [²⁴,²⁵]. The 3D cracks at atomistic level are investigated using the coupling scheme between molecular dynamics and extended finite element via bridging technique. Moreover, the fracture simulation in multiscale modeling in solids were also carried out using an open-source software framework. Budarapu et al. [²⁶] modeled the crack growth using the phantom node in adaptive multiscale method. The transition from coarse scale into fine scale was tackled by ghost atom which positions are interpolated from coarse scale results and served as boundary conditions on fine scale. Yang et al. [²⁷] extended the adaptive meshless multiscale method into partition of unity in the coarse scale instead of phantom node method. Moreover, an equivalent continuum model was developed by Budarapu et al. [²⁸]. The aim of this new technique is to make an equivalent coarse grained model from atomic level. The separation of atoms on the surface of the crack were conducted by employing particular parameter. Arash et al. [²⁹,³⁰] developed a model to predict the mechanical properties of carbon nanotube (CNT) based on a coarse-grained (CG) technique. They used a coarse-grained (CG) model to study the interaction between polymer chains and nanotubes for estimating the elastic properties of CNT through molecular simulations.

The research on self-healing materials and the application of microcapsules into cementitious materials was started recently [³¹–³⁴]. The intense research has been focused on studying healing location and capabilities using an autonomous type of self-healing method [³⁵]. Gilabert et al. [³⁶] studied the role of interfaces between capsule and the homogeneous mortar matrix. They analyzed debonding probability when microcapsule with different thickness is used. Experimental research has revealed that the interface between the shell, healing agent and the cement is not perfectly bonded [³⁷]. Voids and cohesion along the interface of the constituents result in a non-uniform healing efficiency and the capsule is forced to break due to its strong adhesive interaction with the mortar matrix [³⁸]. However, all those aforementioned methods developed mostly in finite element procedures but did not address the actual application in encapsulation-based self-healing material. According to the best knowledge of authors, it is difficult to find numerical simulation study in the literature discussed about the effects of interfacial strength in encapsulation-based self-healing system on breakage of microcapsule. One of the key novelty from this study is to investigate the effects of mismatch in interfacial fracture properties on the crack propagation. We investigated the fracture behavior on how an incoming crack interact with the individual constituent at the microstructure level. The key aspect in this interaction is the change of crack trajectory due to the presence of a particle, which in turn in our study depends on the mismatch of interfacial fracture properties of the particle and the mortar matrix. As the consequences, two types of fracture mechanism could be identified such as: an interfacial crack, and fracture of particle. The efficiency of encapsulation-based self-healing material strongly depends on the leakage of the healing fluid, and this can only be achieved with the breakage of microcapsule. In this study, the effects of interfacial strength of microcapsule shell to the fractured of microcapsule are investigated numerically. A rectangular plate under uniaxial tension modeled with single circular microcapsule inclusion with different core-shell thickness ratio and an edge crack is chosen for this study. The microcapsule is considered to be polymeric (PMMA) microcapsule and embedded in the center of mortar matrix. PMMA microcapsule is known as brittle material and even can easily break with small deformation [³⁹]. In order to model the complicated fracture processes in this multi-phase specimen, cohesive elements were inserted between the mortar-, microcapsule cores-, microcapsule shells-elements and at their phase interfaces. The scope of this study is, in particular, (i) to understand the effects of interfacial strength of microcapsule shell on load carrying capacity with different core-shell thickness ratio and (ii) to understand the effects of interfacial strength of microcapsule shell on the probability of microcapsule fracture.

Numerical simulation

Description of the model

Numerical simulations of 2D rectangular plate with single circular microcapsule embedded in a mortar matrix are conducted. The specimen is loaded under uniaxial tension. The dimension of this plate are 50 mm × 25 mm and the diameter of microcapsule is 2 mm with the variation of core-shell thickness ratio, namely, Ratio 1:1, 5:1, 10:1, and 15:1. The microcapsule with smallest core-shell ratio (1:1) have the largest thickness of the shell but smallest volume of the healing agent and vice versa. The initial edge-crack length was fixed to 4 mm. The schematic of this plate complete with the boundary conditions is shown in Fig. 1. Uniform displacements were applied on the top surface of the specimen. The simulation was done in Abaqus/Explicit with displacement-controlled method and loading time t = 0.005 s. All simulations were ended at displacement d = 0.1 mm.

Cohesive zone model

Figure 2 illustrated the constitutive behavior in the cohesive zone using traction-separation law. The cohesive law corresponds to a bilinear relation between traction and separation is used in this study. The constitutive relation describes the relation between the traction on the crack surface and the separation displacement. The total work per unit area needed to complete a fully separated crack (i.e., the total area under constitutive curve) which corresponds to the fracture energy of the material.

(1)G\mathrm{}{\displaystyle {\int }_{\mathrm{0}}^{{\delta }_f}}t(\delta )d\delta \mathrm{}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{t}_o{\delta }_f,

where G is the cohesive fracture energy, t_o is the cohesive strength, and {\delta }_f is the separation displacement. The damage evolution is defined by either the failure displacement, {\delta }_f, the initiation of damage displacement, {\delta }_o, or the fracture energy due to failure, G_c.

When damage evolution criterion is not specified, Abaqus will automatically calculate the response from damage initiation without effect of the cohesive element (i.e., no damage will occur). Unloading and reloading subsequent is assumed to behave linearly until it reaches the softening envelope. When it reaches the softening envelope, the next reloading will follow the path as marked by the arrow as shown in Fig. 2. The damage in the cohesive elements is assumed to initiate when a quadratic nominal stress ratio reaches a value of one is used in this study.

(2){\left\{{\displaystyle \frac{\langle t_n\rangle }{t_{n\mathrm{0}}}}\right\}}^{\mathrm{2}}+{\left\{{\displaystyle \frac{t_s}{t_{s\mathrm{0}}}}\right\}}^{\mathrm{2}}1,

where t_n and t_s are the traction components in normal and tangential direction; the detailed procedures of relation between the traction and separation in the cohesive law can be found in Ref. [⁴⁰].

Weak form and discretization

The weak form of the equilibrium equation can be formulated as follows:

(3){\displaystyle {\int }_V}{(\nabla w)}^c\mathrm{}S dV\mathrm{}{\displaystyle {\int }_V}\nabla w\cdot q\hspace{0.17em}dV+{\displaystyle {\int }_A}w\cdot t\hspace{0.17em}dA\hspace{0.17em}+{\displaystyle {\int }_A}\mathrm{}w\mathrm{}\cdot t_c\hspace{0.17em}dA,

with gradient operator \nabla, stress tensor S, body force vector q, boundary traction t, cohesive tractions t_c and an arbitrary continuous weighing function w; \left[[\cdot ]\right] denotes the jump operator. The last term on the RHS refers to the cohesive energy. In contrast to partition-of-unity enriched approaches as proposed in [⁴¹–⁴⁹], the jump term is computed by pairwise nodes in cohesive elements. Therefore, a simple node splitting techniques is employed after a certain fracture criterion is reached. Substituting the finite element discretization of the trial and test functions

u^h(X\mathrm{}N(X)\hspace{0.17em}d,

(4)w^h(X\mathrm{}N(X)\hspace{0.17em}w,

into the weak form leads finally after linearization the following well-known system of equations:

KD\mathrm{}{F}_{\mathrm{ext}}-F_{\mathrm{int}},

K\mathrm{}{\displaystyle {\int }_V}{B}^T{C}^{\tan }B\hspace{0.17em}dV,

F_{\mathrm{int}}\mathrm{}{\displaystyle {\int }_V}{B}^{T}S\hspace{0.17em}dV,

(5)F_{\mathrm{ext}}\mathrm{}{\displaystyle {\int }_A}{N}^{T}q\hspace{0.17em}dA+{\displaystyle {\int }_A}{N}^{T}t\hspace{0.17em}dA+{\displaystyle {\int }_A}{N}^T{t}_c\hspace{0.17em}dA,

where N is the vector contain the finite element shape functions and the vectors d and w contain the nodal parameters of the displacement field and virtual displacement field which are identical to the physical values in FEM in contrast to other approaches such as meshless methods [⁵⁰,⁵¹] or isogeometric analysis [⁵²–⁵⁸]. The matrix B contains the spatial derivatives of the shape functions. The non-linear system of equations is solved with Newton method.

Cohesive elements

The samples are meshed with T3 elements assuming plane stress conditions. All the samples are meshed using two length element scale consisting of a fine mesh (the approximate size of element length is 0.2 mm) in the inner region containing of pre-existing crack and microcapsule and a coarse mesh (the approximate size of element length is 0.5 mm) in the outer region as shown in Fig. 3(a). 4-noded cohesive interface elements are inserted into the generated initial mesh along all continuum elements, namely, cohesive elements inside the mortar (COH-MM), inside the microcapsule cores (COH-CC), inside the microcapsule shells (COH-SS), between the microcapsule cores and the shells (COH-CS), and between the microcapsule shells and the mortar (COH-SM). With this method, the initiation and propagation of the crack is allowed at any points with an arbitrary direction. So, theoretically, modeling a pre-existing crack with this type of analysis is not required. It is done for solely comparison purpose when a crack start to initiate from the same edge. A typical discretization around the constituents of the specimen after the insertion of cohesive elements is shown in Fig. 3(b).

All material properties based on [³⁹,⁵⁹,⁶⁰] are listed in Table 1. In this study, the cohesive strength and fracture energy of microcapsule core and its interface are assumed less than microcapsule shell, and the shear fracture properties and the normal ones were assumed equal.

Study of mesh size

In order to investigate the mesh dependence of simulation results, three meshes with different density were modelled as shown in Fig. 4. The coarse mesh has 5106 elements, the medium mesh 15,189 elements, and the fine mesh 31,876 elements. Figure 5 shows the fracture energy curves for each mesh density are compared. The medium mesh and the fine mesh predicted very close in term of fracture energy curves. Therefore, we subsequently present results only for simulations based on medium mesh density.

Parametric studies

In encapsulated-based self-healing system, the breakage of microcapsule is important. The normal and shear cohesive strengths t_n and t_s (t_s = t_n) and its corresponding failure separations are the most significant components governing the cohesive model. When traction-separation law is used along with linear softening, the failure separations can be directly calculated from the fracture energy G_f. In order to investigate the effects of interfacial strength of microcapsule shell, parametric studies of four different core-shell thickness ratio with different material inputs for t_n and G_f were carried out as described in Fig. 6.

The default values of material parameters in Table 1 are assigned to cohesive elements between microcapsule shell and mortar (i.e., the interfacial transition zone, itz). Only two parameters, namely, t_{n-SM} and G_{f-SM} for microcapsule shell and mortar interfaces (itz) were varied relative to the the properties of the mortar for each simulation while the other parameters were fixed. It is assumed that the relative percentage of fracture strength and fracture energy of the interface (t_{n-SM} and G_{f-SM}) to the mortar matrix are kept constant.

Results and discussion

Effects of fracture properties on the load carrying capacity

Figure 7 shows the effects of variation in fracture properties of the interface (itz) on the load carrying capacity from four samples with different core-shell thickness ratio. It is obvious that the specimen strength is highly influenced by interfacial cohesive strength.

Figures 7(a)‒7(d) show the strength of the interfacial zone (itz) ranging from 0.6 MPa (i.e., 10% of mortar strength) to 6.0 MPa (same as mortar matrix) from microcapsule core-shell thickness ratio 1:1, 5:1, 10:1, and 15:1, respectively. Figure 7(a) shows the effects of itz strength on specimen strength from microcapsule with ratio 1:1. It is obvious that the strength of itz is the dominant factor governing the specimen strength. The specimen strength jumps from 103.8 N for itz= 10% to 113 N for itz= 100%. The same phenomenon also can be found from Figs. 7(b), 7(c), and 7(d) for microcapsule with ratio 5:1, 10:1, and 15:1, respectively. It is clear that when the cohesive strength on the interface of microcapsule and the mortar matrix are the same, the specimen strength will reach the higher value. It can be seen from the curves that despite of core-shell thickness ratio, the higher the ITZ strength, the larger the load carrying capacity is and vice versa. The relationship between peak load and percentage of itz strength with respect to the mortar matrix strength from different core-shell thickness ratio can be observed in Fig. 8. It is shown that there is a clear trend that the higher percentage of itz strength lead to the higher peak load. Moreover, the increasing of load carrying capacity remain linear when the itz strength is ranging from 0%‒50% of the strength of mortar matrix. Furthermore, from Fig. 8 it can also be found that the core-shell thickness ratio of microcapsule has no significant influence the load carrying capacity of specimen.

Effects of fracture properties on the crack pattern

Figures 9, 10, 11, and 12 show the effects of variation in fracture properties of the interface (itz) on the crack pattern for specimen with core-shell thickness ratio 1:1, 5:1, 10:1, and 15:1 respectively. The fracture properties of the interface (itz) is calculated as percentage to the properties of mortar matrix. The samples with core-shell thickness ratio 1:1 and 5:1 produced the similar crack paths regardless of whether the strength of interface is increased or reduced, as shown in Figs. 9 and 10. The main approaching crack could not penetrate through microcapsule shell even when the strength of interface and mortar matrix are the same. As can be observed from Figs. 9(a) and 9(b), when the percentage of itz with respect to the strength of the mortar matrix ranging from 0%‒50%, an interfacial crack occurs. When the percentage of itz larger than 50% of the mortar matrix, the incoming crack that reaches microcapsule shell initially become an interfacial crack and suddenly deflects away from the microcapsule continuing its path, as illustrated in Figs. 9(c) and 9(d). The same fracture behavior with respect to the crack trajectory also could be found from specimen with microcapsule core-shell ratio 5:1 as can be observed in Figs. 10(a)‒10(d).

For the sample with core-shell thickness ratio 10:1 and 15:1, it is clear that when the fracture properties of the interface (itz) less than on the mortar matrix 0\le \mathrm{itz}\le 75, the crack paths are found similar, which crack paths have tendency to divert its direction to the interface of microcapsule shell as shown in Figs. 11(a)‒11(c) and 12(a)‒12(c). As can be observed from Figs. 11(a) and (b), when the percentage of itz with respect to the strength of the mortar matrix ranging from 0%‒50%, an interfacial crack occurs. This phenomenon also occured to the microcapsule with core-shell thickness ratio 15:1 (Figs. 12(a) and (b)). When the percentage of itz is equal to 75% of the mortar matrix, the incoming crack that reaches microcapsule shell initially become an interfacial crack and suddenly deflects away from the microcapsule continuing its path and then leave the microcapsule intact without successfully healed, as illustrated in Figs. 11(c) and 12(c). In other words, the main approaching crack could not penetrate through microcapsule shell and break it, only just remain going around the interface and then deflect away from the microcapsule.

On the contrary, when the fracture properties of the interface (itz) and on the mortar matrix are the same (itz= 100%), the main approaching crack could penetrates into the microcapsule as shown in Figs. 11(d) and 12(d). As can be seen from Fig. 11(d), when the percentage of itz with respect to the strength of the mortar matrix is equal to 100%, fracture in microcapsule core-shell thickness ratio 10:1 is observed. This phenomenon also occured to the microcapsule with core-shell thickness ratio 15:1 (Fig. 12(d)). Both of crack paths shown in Fig. 11(d) and Fig. 12(d) are similar which advance from left to right. In this case, the main crack is attracted to penetrate the microcapsule shell because of good fracture properties agreement between microcapsule shell interface and the mortar matrix. It means that the incoming crack successfully reaches the microcapsule shell and able to break it. This process would lead to the leakage of healing agents to seal the crack and cures material internally.

Conclusions

Numerical simulations have been carried out to investigate the effects of interfacial strength of microcapsule shell to the fractured microcapsule. A specimen is discretized as four-phase composite composed of mortar, microcapsules core, microcapsules shell, and interface between them. To represent the interaction between these components and to predict more realistic crack paths, pre-inserted cohesive elements are used. The effects of ratio core-shell thickness of microcapsules on the specimen strength and breakage of microcapsules shell are also investigated. Some concluding remarks are made as follow:

1) The mismatch in strength between the interface of microcapsule shell and the mortar matrix has significant influence on the load carrying capacity of the self-healing material.

2) Despite of core-shell thickness ratio of microcapsule, the load carrying capacity of self-healing material under tension increases as interfacial strength (itz) of microcapsule shell increases.

3) At fixed value of interfacial strength, the variation core-shell thickness ratio of microcapsule has no significant effect on the load carrying capacity of self-healing material.

4) The mismatch in strength between the interface of microcapsule shell and the mortar matrix has adverse effects on the fracture behavior of microcapsule.

5) The crack path is significantly determined by the fracture properties of the interface of microcapsule shell. Further, having the fracture properties of microcapsule shell interface lower than the mortar matrix, highly favors debonding of the microcapsule.

6) The variation of core-shell thickness ratio of microcapsule has significant influence on the fracture probability of microcapsule. The higher the ratio core-shell thickness, the higher the probability of microcapsules being fractured.

The present work focused on the fracture behavior based on the interaction of an incoming crack and the individual constituent at the microstructure level. The effects of interfacial strengths variation around microcapsule shell with respect to the mortar matrix were investigated. For design purposes, in particular for encapsulation-based self-healing material, the present study found that a very good bonding between interface of microcapsule shell and the mortar matrix is required. The fracture strength of microcapsule shell interface should be higher or at least equal (with particular core-shell thickness ratio) than the fracture strength of the mortar matrix. With this combination, self-healing triggered by the breakage of microcapsule shell would be successful. Furthermore, considering the lack of references for microcapsule input paramaters, in order to study how sensitive are the numerical results with respect to the small changes in the input parameters, the sensitivity analysis of different properties in input parameters of microcapsule should be studied in the future. With this analysis, uncertain microcapsule key input parameters of an output of interest which affects the macroscopic behavior of fracture could be determined [⁶¹,⁶²]. Also the future works will focus on investigating the effects of mismatch in fracture properties from microcapsule shell, microcapsule core, and the surrounding mortar matrix on the fracture behavior of specimen.

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[42] Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71

[43] Rabczuk T, Song J H, Belytschko T. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics, 2009, 76(6): 730–741

[44] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[45] Rabczuk T, Zi G, Gerstenberger A, Wall W. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599

[46] Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6‒8): 641–658

[47] Rabczuk T, Areias P, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

[48] Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495

[49] Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23‒24): 1391–1411

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[54] Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based xiga for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[55] Nguyen-Thanh N, Valizadeh N, Nguyen M, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on kirchhoff-love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

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[57] Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger K, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using pht-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47‒48): 3410–3424

[58] Ghasemi H, Park H, Rabczuk T. A level-set based iga formulation for topology optimization of exoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258

[59] Quayum M S, Zhuang X, Rabczuk T. Computational model generation and RVE design of self-healing concrete. Frontiers of Structural and Civil Engineering, 2015, 9(4): 383–396

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[62] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

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