Experimental and FE Modeling of Mixed-Mode Crack Initiation Angle in High Density Polyethylene

Period. Polytech. Mech. Eng. A. Zerrouki, A. Boulenouar, M. Mazari, M. Benguediab Abstract In this paper, an experimental and a numerical analysis were carried out using High density polyethylene (HDPE). Sheets with an initial central crack (CCT specimens) inclined with a given angle are investigated and compared to the loading direction. The kinking angle is experimentally predicted and numerically evaluated under mixed mode (I+II), as a function of the strain energy density (SED) around the crack-tip, using the Ansys Parametric Design Language (APDL).According to the experimental observations and numerical analysis, the plan of crack propagation is perpendicular to the loading direction. Moreover, as suggested by Sih in the framework of linear elastic fracture mechanics (LEFM), the minimum values Sminof the factor S are reached at the points corresponding to the crack propagation direction. These results suggest that the concept of the strain energydensity factor can be used as an indicator of the crack propagation direction.

et al. [26], Berto et al. [27] and Campagnolo et al. [28] to assess the fracture strength of different materials, characterized by different control volumes and subjected to wide combinations of static loading [29][30][31][32]. Moreover, it has been successfully employed for the fatigue strength of welded joints [33][34] and notched components [35,36]. As shown by Lazzarin et al. [37], the SED can be easily evaluated numerically through finite element analysis by using coarse meshes, and it allows automatically to take into account higher order terms and three-dimensional effects [38,39]. These considerations are among the main advantages of the SED approach.
The objective of this paper is to present an experimental and numerical modeling of crack initiation angle for ductile fracture of HDPE, under mixed mode (I+II). Using the Ansys Parametric Design Language (APDL), the crack direction is evaluated as a function of the minimum strain energy density (dW/dV) min around the crack tip. The results obtained by numerical approach are compared with those obtained in experiments.

Strain energy density theory
The SED fracture criterion locally focuses on the continuum element ahead of the crack,it is based on the notion of weakness or severity experienced by the local material. Failure occurs when a critical amount of strain energy dW is accumulated within the element volume dV and the crack is then advanced incrementally in the corresponding direction [3,40]. The strain energy density function (dW/dV) is assumed to have the following form dW dV S r = Where S is the strain energy density factor and r is the distance from the crack tip. The minimum of the strain energy density factor S min around the crack tip determines the likely direction of crack propagation.
The strain energy density can be determined directly from the relationship.
Where σ ij and ε ij are the stress and strain components respectively [41]. The strain energy per unit volume dW/dV can be further decomposed into two parts: (dW/dV) d represents the distortional strain energy per unit volume, corresponding to the deviatoric stress tensor that is associated with distortion (shape change) of an element and is responsible for macroyielding and the creation of slip planes or microcracks.
(dW/dV) v represents the part of the strain energy per unit volume associated with dilatation (volume change) and is responsible for macrofracture (crack growth) and the creation of the cleavage planes which are perpendicular to the direction of tension.
The computed discrete values for S are fitted with approximation function which enables simple determination of the local minimum. The strain energy density function has several local minimums around the crack tip, where the global minimum is not necessarily the true solution, as it can be observed from Fig. 1. The minimum of strain density function S min can be found numerically by incrementalsearch for a local minimum in possible crack extension directions θ i in the range ±π around the crack tip [42].
Fajdiga et al. [42] used the position of integration points to define the corresponding angle of calculated strain energy density and strain energy density factor S around the crack tip. In this study, the minimum strain energy density (dW/dV) min is computed by introducing a ring of elements around the crack tip. At each crack increment, the crack direction is evaluated as a function of the angle between the centre of the element and the crack axis (Fig. 2).

Materials, specimens, tests
The experiments were carried out in CHIALI Group using HDPE. Fracture tests were performed using specimens containing an inclined central crack introduced by a razor blade.
The dimensions of these specimens are: length h=120mm, width w = 75mm and thickness B = 4mm (Fig. 3). The considered crack length is a = 20mm with four orientations defined by the angle α = {0°, 30°, 45°, 60°}. Specimens were loaded in a uniaxial direction under controlled load F=10KN with a strain rate of v = 50mm/min, but a mixed mode (I+II) could be induced around the crack tip because of the crack inclination. Fig. 3 shows the geometry of the center cracked specimens used in fracture tests under mode-I and mixed mode loadings.

Crack initiation angle
The crack initiation angle θ 0 is the direction in which the crack propagates from the original crack ( Fig. 4(a)). For different α, the value of θ 0 is predicted by each fracture test. Fig. 4(b) shows the crack initiation angle evaluated in this study (Table1).

.1 Model and meshing
The geometry of the centre cracked plate (120 mm x 75mm) with an initial crack (a = 20mm) is considered for 2-Dimensional finite element analysis. The pre-existing central crack is inclined to the horizontal axis with angle α = {0°, 30°, 45°, 60°}.The FE calculation was achieved by gradually increasing of the displacements (d) applied to the nodes located at the top of the plate.
All the above mentioned fracture tests were numerically simulated using the ANSYS® finite element code [43].
For the mesh generation of the cracked plate, the element type 'PLANE183' is used, as shown in Fig. 5(a).
It is a higher order two dimensional, 8-node element having two degrees of freedom at each node (translations in the nodal x and y directions), quadratic displacement behavior and the capability of forming a triangular-shaped element, which is required at the crack tip areas.
Due to the singular nature of the stress field in the vicinity of the crack, the singular elements proposed by Barsoum [44] are considered at each crack tip area. Fig. 5(b) shows the typical finite element mesh used for numerical analysis. The material properties used in this study were those corresponding to the stress-strain curve obtained from experimental tensile tests obtained by Blaise [45].
The stress-strain curve is modelled by elastic-plastic behavior with multi-linear isotropic hardening according to the experimental plastic curve [46,47].
The rings of the elements surrounding the tip of the crack were employed. This mesh will be used in order to determine the strain energy density in these elements thus to determine the kinking angle θ in the direction for which this energy density is minimal (dW/dV) min . Fig. 6 The stress-strain curve obtained from experimental tensile tests [45] Fig. 7 illustrates the description of the parameters: r, α and θ. Where: r is the distance from the crack-tip, α is the crack inclination angle and θ is the initial crack propagation.
The specifications of the crack tip mesh and a close up view for crack inclination angle α = 45° are shown in Fig. 8. In this figure, 'r i ' represents the distance between the crack tip and the center of the element 'c'.

Results and discussion 5.1 Strain energy density
The obtained results are traced for several values of the radius r i and for four orientations of initial crack α = 0°, 30°, 45° and 60° (Fig. 9).
These figures highlight that, out of the core region surrounding the crack tip (in our caser ≥ r 1 ), the minimum of (dW/dV) is reached for a constant value independently of distance r. The angle θ 0 corresponds to the horizontal direction perpendicular to the loading.
In general, a material element is subjected to both distortion and dilatation. Distortion and dilatation vary in proportion, depending on the load history, location, and nonuniformity in stress or energy fields. For the macrocrack under tension, the macrofracture coincides with the direction in which (dW/dV) v > (dW/dV) d and macroyielding with the direction in which (dW/dV) d > (dW/dV) v . In the case of 2-D analysis, the continuum mechanics solution of the stress problem shows that the Fig. 9 Evolution of dW/dV as a function of θ for several values of the radius r (with α = 0°, 30°, 45° and 60°) Fig. 7 Description of the parameters r, α and θ Fig. 8 Ring of elements around the crack tip principal stresses σ 1 and σ 2 are equal (σ 1 =σ 2 ) in the element ahead crack, under this condition the greatest change in volume occurs ((dW/dV) v > (dW/dV) d ).
That means the domination of the macrodilatation, which is responsible to the cleavage process in the plane perpendicular to the loading direction [41].
The relative local minimum of dW/dV corresponds to large change in volume and is identified with the region dominated by macrodilatation leading to fracture i.e. the appearance of cleavage planes perpendicular to the direction of tension.
The angle values obtained numerically are compared with those predicted experimentally. The obtained result shows a good correlation between the two approaches. Table 2 summarizes the crack initiation angle θ 0 of fracture tests and FE analysis for different crack inclination angles α.

Effect of elements number surrounding the crack tip
For the strain energy density criterion, the precision is strongly related to the number of elements surrounding crack tip zone [35]. For this purpose, we examined the influence of the mesh size (or number of the elements) on the strain energy density variation. For that, the density (dW/dV) is evaluated for different number of elements surrounding the crack tip. Inthis study, the numbers of elements considered are 20, 36, 48 and 60 (Fig. 10). For different number of elements surrounding crack tip zone and for several values of the radius r, Fig. 11 illustrates the evolution of the strain energy density (dW/dV) as function of the initial angle of propagation θ. The results obtained show that the precision is related to the elements sizes around the crack tip. To better show the influence of the elements sizes around the crack tip on the determination of initial angle of crack propagation θ 0 , Fig. 12 illustrates the variation of angle θ 0 as a function of number of elements of elements surrounding the crack tip for various distances.
The curves obtained show that more there will be elements around the crack tip; more the crack direction θ 0 at each crack increment length and the final path of crack propagation will be precise.

Conclusion
The study has been conducted to analyze and simulate the initial crack propagation angle in HDPE, under mixed mode (I+II). Using the Ansys Parametric Design Language (APDL), the strain energy density approach is investigated. For various crack inclined angle, the kinking angle is evaluated as a function of the Minimum Strain Energy Density (MSED) around the crack tip. The obtained results allow us to deduce the following conclusions: • The quarter-point singular elements proposed by Barsoumare used to consider the singularity of stress and deformations fields at crack tip. • Out of a core region surrounding the crack tip, the minimum of SED is reached for a constant value independently of distance r. The obtained results show a convenient agreement between experimental and numerical approach. • The minimal value of SED, corresponding to the direction of crack propagation, is always reached in the plan perpendicular to the loading axis, independently of the initial crack orientation. • The value of the angle of initial crack direction is related to the number of the elements surrounding the crack tip. • Consequently, the SED approach, developed in the linear elastic fracture problems, could be extended to highly non-linear deformable materials as an indicator of the crack propagation direction.