A comparative study of cohesive zone models for predicting delamination fracture behaviors of arterial wall

2.1 Bulk material model: HGO model

Free energy, Ψ , the potential energy per unit volume of arterial material defined in decoupled form, is expressed as follows:

The volumetric part, Ψ vol , is given, according to Gasser et al. [16], as follows:

where 1 D approximates to the bulk modulus of the material, and J = det ( F ) , where F is the deformation gradient.

The isochoric potential is given as follows:

which is composed of two parts. One is used to describe the mechanical response of the matrix:

Another is implemented to represent the mechanical behavior of the two families of collagen fibers:

where I ¯ 1 = tr ( C ¯ ) denotes the first invariant of C ¯ and μ is the neo-Hookean parameter, which characterizes the shear modulus of the material without fibers; I ¯ 41 = a 01 ⋅ C ¯ a 01 and I ¯ 42 = a 02 ⋅ C ¯ a 02 are the tensor invariants, which are equal to the square of the stretch in the directions of [ a 01 ] = [ cos γ , sin γ , 0 ] and [ a 02 ] = [ cos γ , − sin γ , 0 ] , respectively. Note that the constitutive parameter k 1 is related to the relative stiffness of the fibers, which is determined from the mechanical tests of tissue, and k 2 is the dimensionless stiffness. As described in reference [17], the parameter κ is the dispersion parameter, which characterizes the dispersion of the two families of fibers along the two average distributed directions, and 0 ≤ κ ≤ 1 / 3 , when κ = 0 , each family of collagen fibers is parallel to each other, and when κ = 1 / 3 , the random distribution of the fibers is equivalent to an isotropic material. γ represents the angle between the mean fiber orientation of one family of fibers and the circumferential direction of the aorta [18].

2.2 Interfacial CZM

Four types of CZMs are implemented in this study through a user-defined subroutine UEL in ABAQUS [19]. With the CZM parameter, which characterizes the relationship between the cohesive force of the interface and the separation force of the interface, the “effective” opening displacement δ is given by:

where λ is a scalar parameter being introduced to assign different weights to the opening displacement δ n and the sliding displacement δ s . The parameter δ s is denoted by:

where δ s 1 and δ s 2 represent the components of the two directions of the sliding displacement δ s passing through the cohesive surfaces.

According to the theory of CZM, the delamination fracture failure of arterial layers contains three steps. The first step is damage initiation. When the “effective” opening displacement reaches δ 0 (for triangular, linear–exponential, and exponential–linear CZMs; Figure 1) or δ 1 (for trapezoidal CZM; Figure 1), the cohesive elements start to degrade. Meanwhile, the effective traction t (Cauchy stress) reaches the cohesive strength σ c of the interface. The expressions of effective traction t and the first-order partial derivative of the effective traction t to the effective opening displacement − ∂ t ∂ δ are given in chapter 3 of reference [17] (Appendix A). The second step is damage evolution. In this step, the accumulation of damage occurs in the cohesive element. The damage variable d is introduced to quantitatively evaluate the damage. Evidently, d ranges from 0 to 1, representing the cohesive surfaces without damage and the complete separation of the cohesive surfaces, respectively. The loading and unloading processes of the cohesive traction–separation criterion are illustrated in Figure 1. When the effective traction t exceeds σ c , the damage accumulates, as the point A reflects in Figure 1. When the curves unloaded at points A and B, respectively, permanent damage occurs at the cohesive surfaces at points A and B. Therefore, the traction–separation relationship curve will travel along the lines AO and BO. Conversely, when no damage occurs on the cohesive surface, the traction–separation curve will follow the original curve. The third step is that the cohesive elements completely failed after the effective displacement reaches the critical effective separation value δ c (as shown in Figure 1). Meanwhile, the critical energy release rate G c is attained. All the expressions for the four types of CZMs can be found in chapter 3 of reference [17] (Appendix A).

Figure 1

Four types of irreversible cohesive model denoted with effective traction and displacement: (a) triangular CZM, (b) trapezoidal CZM, (c) linear–exponential CZM, and (d) exponential–linear CZM.

2.3 Arterial tissue delamination

Three different arterial tissue delamination experiments were performed in different research studies to find the mechanism of arterial tissue failure. With regard to the media delamination processes, the geometrical model is reconstructed based on the experiments of the peeling test of human aortic media [8] (Figure 2a–c). The geometrical dimensions of the arterial wall are 4 mm of length, 1.2 mm of width, and 0.05 mm of thickness. For the mouse plaque delamination test (Figure 2d), an exposed aorta was fastened to a small plate connected to the load cell for load data recording and sequential loading–delamination–unloading cycles were applied to cause the plaque delaminating from the underlying arterial tissue [14]. For the human fibrous cap delamination test, as shown in Figure 2e, the delamination of human fibrous cap presents as a pair of micro–clamps. The sequential loading–delamination–unloading cycles are applied to make the fibrous cap separate from the underlying plaque tissue [15]. Although there are three types of delamination experiments, the HGO models and CZM models used for the simulation are the same, except for the human aortic media delamination test (including viscoelastic material model parameters for human arterial tissue) and the material parameter values for the three cases.

Figure 2

A schematic of the arterial layer peeling test setup. (a) A part of aorta; (b) a flat of arterial layer after cutting from (a); (c) an arterial strip, which was peeled at one end, and the geometric model of the peeling test for the simulation (in the red line box); (d) the simulation predicted load–load point displacement curves of the mouse atherosclerotic plaque delamination; and (e) the simulation predicted load–load point displacement curves of the human fibrous cap delamination under loading–delamination–unloading cycles.

3.1 Arterial wall delamination modeling using CZMs

In this section, to predict the delamination fracture behaviors of the arterial tissue, a part of the arterial wall is utilized to perform experiment and simulation. The sampling method for the experiment is shown in Figure 2. The geometrical model is reconstructed based on the experiments of the peeling test of human aortic media [8], as shown in the red box of Figure 2c. The geometrical dimensions of the arterial wall are 4 mm of length, 1.2 mm of width, and 0.05 mm of thickness. The front and back surfaces of the strip (Figure 2c, in the red line box) are fixed along the z direction for the sake of computational efficiency. The delamination behavior is modeled as plane strain deformation.

To evaluate the performance of the arterial wall dissection, the finite element (FE) simulations are implemented using ABAQUS [19,20,21], associated with a user-defined subroutine modeling the delamination fracture of the cohesive interface in the arterial layer. The arterial layer is meshed with 1920 eight-node hexahedral elements, C3D8H, where the cohesive interface is meshed with 72 zero-thick eight-node 3D user-defined elements. A layer of cohesive elements is placed along the mode I delamination (crack growth) path starting from the initial peeling front to the end of arterial strip (Figure 2c). The size of the elements for the FE model is 0.05 mm, based on the parametric analysis. Presetting the peeling path can increase the computational efficiency of the CZM model in the practice of simulating interface material separation events. The peeling path is obtained from the experimental observations of this study. Meanwhile, the experimental setup and simulation modeling details of the mouse plaque delamination and human fibrous cap delamination processes are presented in references [14,15].

3.2 Parameter selection in modeling

The parameter values for the HGO and CZM models were taken from references [4,10,22]. The value of the interfacial stiffness K was assigned as the same one from reference [14]. With regard to the simulations of the mouse plaque delamination processes, the material parameter values for the HGO and CZM models were adopted from reference [14] and those values of the human fibrous cap delamination were obtained from reference [15] (including viscoelastic material model parameters for human arterial tissue), as shown in Table 1.

3.3 Results of simulations and experiments

The parameters used for the simulations of the arterial layer peeling test (Figure 2) are set with the same values as shown in Table 1. The pulling forces divided by the width of the specimen vs the load point displacement are shown in Figure 3. The pulling forces per unit of width are between 25.5 and 28.5 mN/mm, which are in the domain of the experimental results (23−35 mN/mm) [8].

Figure 3

Comparison of the load/width–load point displacement curves between the simulation and average experimental results.

The deformed configurations of the aortic media delamination are shown in Figure 4a. The contour levels shown in the figure represent the normal component of the Cauchy stress to the delamination plane. U is defined as the load point displacement (five stages). The traction–separation (displacement) relationships in terms of a cohesive element in the middle of the crack path are presented in Figure 4b.

Figure 4

(a) Tensile stress contour of the arterial wall at five levels of the aortic media peeling. (b) The traction–separation (displacement) relationship of a cohesive element in the middle of the crack path.

The load–load point displacement curves are shown in Figures 5a and 6a. The traction separation (displacement) relationships of a cohesive element in the middle of the crack growth path for the plaque and the fibrous cap are shown in Figures 5b and 6b.

Figure 5

(a) The simulation predicted load–load point displacement curves of the mouse atherosclerotic plaque delamination are compared with the experimental measured curves. (b) The traction–separation curves of a cohesive element in the middle of the crack path.

Figure 6

(a) The simulation predicted load–load point displacement curves of the human fibrous cap delamination under loading–delamination–unloading cycles are compared with the experimental measured curves. (b) The traction–separation curves of a cohesive element in the middle of the crack path.

4.1 Aortic media delamination

The comparison of the four shapes of CZM laws for modeling the aortic media delamination was described in terms of the load/width–load point displacement curves, tensile stress contour, and traction–displacement curves. This study simulated the aortic media delamination experiments in Sommer et al. [8] by four types of CZMs and further compares the results with the simulations in Gasser and Holzapfel [10] and Ferrara and Pandolfi [4]. Figure 3 shows that some research records in this study are within the range of the known results of the previous studies. The pulling forces per unit of width (load/width) are between 25.5 and 28.5 mN/mm, which are in good agreement with the numerical results by Gasser and Holzapfel (23 mN/mm) [10] and Ferrara and Pandolfi (28.8 mN/mm) [4]. Also, the CZM simulation of peeling load/width ratio is in the range of 22.9–34.8 mN/mm, which is based on the peeling tests of the rectangular-shaped medial strips along the circumferential and axial directions by Sommer et al. [8]. For the traction–separation relationships, it is observed that the shape of the softening stage (after the initial damage) is almost the same as the shape of the corresponding CZM, as shown in Figure 4b.

4.2 Mouse plaque delamination and human fibrous cap delamination

Besides, the comparison of the results of the experiment of the aortic media delamination and the simulation of mouse plaque delamination [14] and human fibrous cap delamination [15] was carried out to compare with the simulation results obtained using four types of CZMs. A mesh dependency study of the mouse plaque and human fibrous cap delamination was conducted in the previous works [14,15,23], in which the selected mesh refinement is enough to obtain the convergence solutions. For these two delamination simulations, the magnitude of the maximal force during delamination predicted by the linear–exponential CZM is lower by nearly 10%, while the magnitude predicted by the trapezoidal CZM is higher by nearly 4% than that reported in the previous studies [14,15,24] (Figures 5a and 6a). The load–load point displacement curves of other two types of CZMs (triangular and exponential–linear) match well with the experiments and simulations of the previous studies [14,15]. The minimum values of the traction (linear–exponential CZM) compared with the other shapes of the CZMs reveal that the reaction forces (Figures 5a and 6a) during the plaque and fibrous cap delamination are less than the other types of the CZMs. Figures 5b and 6c show the traction–separation relationships in four types of CZMs; the traction of the elastic stage of the CZM for the plaque delamination rapidly increases to the maximum value (Figure 5b). For the fibrous cap delamination, the tractions of the elastic stage of the CZMs are similar to the shapes of the CZMs (Figure 6b). For both delamination, it shows that the shapes of the softening stage are almost the same as the shapes of the corresponding CZMs. The unloading and loading processes of the element under unloading and loading cycles are shown in Figure 6b, indicating that the softening of the element experiences a long loading displacement.

4.3 Effects and features of four shapes of CZMs in simulating the delamination behaviors

The shapes of the traction–separation curves of an element at the front of the crack are in good agreement with the shape of CZM laws used to simulate the aortic delamination process (Figure 4b), showing that the extent of damage evolution is the largest for the linear–exponential CZM, followed by the triangular, exponential–linear, and trapezoidal CZM laws. However, this respective order does not represent the order of the predicted load/width values (Figure 3). The trapezoidal law shows higher values of transmission of the load/width at the initial stage of damage, because the maximum traction kept constant under the longest displacement load during the damage evolution process compared with the other shapes of CZM laws. The triangular CZM can obtain the predicted load/width or load with an acceptable accuracy, and it is the easiest CZM law to use in terms of simulation of the arterial tissue fracture failure. At the stage when a cohesive element is totally failed, the release area of the bulk material will induce a contraction, which indicates that the oscillation for the predicted load using the trapezoidal CZM is more obvious than that by other CZM laws.

The iterative process and the time for acquiring the solutions are influenced by the shape of CZM laws. The degree of difficulty to get convergence solutions in the iterative process is the highest for the trapezoidal, followed by the linear–exponential, exponential–linear, and triangular CZMs.

The elastic deformation of the cohesive element is governed by the mechanical properties of its structural constituents and the interfacial strength of the material (e.g., triangular and trapezoidal CZM). The nonlinearity of the traction–separation relationship of the cohesive element is largely attributed to the gradual recruitment of the load-bearing collagen fibers with increasing pressures [25,26,27]. The softening of the traction–separation relationship is attributed to the softening behavior of the medial interfacial strength, plaque, or fibrous cap [28,29,30]. Hence, these CZMs are able to characterize the mechanical response of the cohesive element during the crack propagation.