- Research
- Open Access
3D Progressive Damage Based Macro-Mechanical FE Simulation of Machining Unidirectional FRP Composite
- Yan-Li He^{1}Email author,
- Joao-Paulo Davim^{2} and
- Hong-Qian Xue^{1}
https://doi.org/10.1186/s10033-018-0250-5
© The Author(s) 2018
- Received: 21 April 2016
- Accepted: 9 June 2018
- Published: 26 June 2018
Abstract
Finite element (FE) simulation is a powerful tool for investigating the mechanism of machining fiber-reinforced polymer composite (FRP). However in existing FE machining simulation works, the two-dimensional (2D) progressive damage models only describe material behavior in plane stress, while the three-dimensional (3D) damage models always assume an instantaneous stiffness reduction pattern. So the chip formation mechanism of FRP under machining is not fully analyzed in general stress state. A 3D macro-mechanical based FE simulation model was developed for the machining of unidirectional glass fiber reinforced plastic. An energy based 3D progressive damage model was proposed for damage evolution and continuous stiffness degradation. The damage model was implemented for the Hashin-type criterion and Maximum stress criterion. The influences of the failure criterion and fracture energy dissipation on the simulation results were studied. The simulated chip shapes, cutting forces and sub-surface damages were verified by those obtained in the reference experiment. The simulation results also show consistency with previous 2D FE models in the reference. The proposed research provides a model for simulating FRP material behavior and the machining process in 3D stress state.
Keywords
- FRP
- Macro mechanical
- Machining
- Finite element
- Progressive damage
1 Introduction
Fiber reinforced polymer composite (FRP) material is finding increasing applications in modern aerospace, automobile and sports industries. A significant amount of machining work is necessary as material removal is required for the FRP component to meet the dimensional requirements before assembly. To understand the machining process and to avoid machining damages such as fiber breakage, debonding, or delamination, many experiments have been performed in past years [1–4].
On the other hand, finite element (FE) simulation has been widely used for investigating the machining or forming process of homogeneous and isotropic materials [5, 6]. This technology is also extended to inhomogeneous and anisotropic FRP composite when material behavior is defined appropriately [7]. Most existing work in this area focused on orthogonal cutting of unidirectional FRP. Generally, these FE models are based on either macro-mechanical or micro-mechanical approaches. In macro-mechanical FE machining models, the FRP material is modeled as an equivalent homogeneous anisotropic material (EHM). In micro-mechanical models, multiphase and different constituents (matrix and fiber) in FRP are modeled separately [8–10]. There are also studies which combine these two approaches in one model to leverage the advantages of both methods [11, 12].
Many macro-mechanical FE models are built as two-dimensional, using various material damage models and failure criteria. The Tsai-hill criterion was used in the 2D FE machining models developed by Arola et al. [13, 14], Mahdi et al. [15], and Mkaddem et al. [16]. In the 2D FE model developed by Lasri et al. [17], damage analysis was conducted using the Hashin, Maximum stress and Hoffman failure criteria together with the instantaneous stiffness degradation strategy. Other 2D plane stress FE models [18–20] apply the Hashin criterion together with the progressive damage model built in Abaqus software. In these models, the predicted principle cutting force generally agreed with experimental records, but a significant difference was observed between the predicted and experimental thrust forces [14, 17, 18].
3D FE models for FRP machining simulation are not frequently found in the literature. Mahdi and Zhang [21] presented a 3D FE model in which the maximum shear stress was used as the material separation criterion. Rao et al. [22] developed a 3D FE model with the Tsai-hill failure criterion to simulate the cutting force and chip formation for orthogonal machining of UD-CFRP composite. Santiuste et al. [23] developed a 3D laminate model based on the three-dimensional Hoffman criterion. In these limited research models, the FRP is modeled as an elastic material with instantaneous failure behavior, and neither the chip shape nor the sub-surface damage was studied in a quantitative manner.
Recently, finite element modeling was also used in simulation of drilling or milling of FRP composite. In the meso-scale CFRP drilling FE models developed by Isbilir et al. [24] and Feito et al. [25], sudden stiffness degradation was assumed in intra-lamina failure analysis with the 3D Hashin and Hou criteria, respectively. Phadnis et al. [26] developed a meso-scale FE model for CFRP drilling simulation. The 3D Hashin criterion was used in fiber failure prediction, and the Puck criterion was used to model matrix failure. Still, a sudden degradation model was applied in intra-ply damage analysis such that the material point offers no resistance to deformation if any damage initiation condition was met.
From the existing research studies, there lacks a FE machining simulation work that analyses the 3D progressive damage model of FRP. As such, the FRP material behavior during cutting is not fully investigated in a general stress state, because the 2D damage model is only applicable with plane stress assumption. Additionally, different failure criteria were used in different FE models, but the influence of failure criterion on the simulation results was not discussed. To reveal the 3D material behavior that explains the phenomena in FRP cutting, a 3D macro-mechanical FE model was developed according to the orthogonal cutting experiment by Nayak et al. [3]. An energy based progressive damage model was proposed with the Hashin-type and Maximum stress criteria for continuous stiffness degradation of UD-GFRP. The material damage model was implemented in a user subroutine (VUMAT) and incorporated into the FE models for machining simulation.
The proposed 3D macro-mechanical FE model was verified by the reference experiment [3] in terms of chip formation, cutting forces, and sub-surface damages. In addition, the simulation observations were also consistent with the results from existing FE models in the literature. The influences of failure criterion and fracture energy dissipation on the simulation results were also studied. The proposed damage model can also be used in FRP structure analysis (e.g., under low-velocity impact load).
2 Finite Element Modeling
2.1 Assumptions, Control Volume and Boundary Conditions
The machining process is simulated as a quasi-static explicit analysis. The cutting tool is modeled as a rigid body with a predefined velocity v in the negative x-direction. A reference point at the top right corner is defined to control the movement of the tool and offer output of the reaction force. The UD-GFRP material is modeled as an equivalent homogeneous and orthotropic material with elastic-failure behavior, and is meshed with eight-node brick elements (C3D8R). The mesh contains 20 layers of elements through the thickness direction, and was refined in the zone surrounding tool tip with a size of approximately 5 microns. The total element number is 160000. The mesh will influence the accuracy and step increment of the simulation, but the sensitivity analysis of the numerical parameter is outside the scope of this study.
The interaction between the cutting tool and workpiece is modeled as surface-node surface contact available in Abaqus/Explicit. The contact is assumed to follow the Coulomb friction law. The experiment shows that friction increases with the fiber orientation [9], so six different friction coefficients, i.e., 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9, were set for fiber orientations of 15°, 30°, 45°, 60°, 75°, and 90°, respectively in the simulation. It is noted that these coefficients are obtained from a pin-on-disc experimental test with a normal load of 120 N, and thus they are used in machining simulation as an approximate estimation that reflects the influence of fiber orientation on the friction. The thermal effect is neglected since the cutting process is conducted at a rather low speed.
2.2 Material Separation for Chip Formation
In the literature, the chip can be simulated by means of material separation criterion (node/element splitting), element deletion, or a pure deformation process based on adaptive remeshing [27–29]. For remeshing, the state variables have to be frequently interpolated from the old mesh to the new mesh, leading to error accumulation and deterioration of results. To avoid this problem, chip formation is achieved based on the material failure criterion and element deletion approaches, which were also applied in previous machining simulation studies [17–19]. When all material points in an element fail, the element is deleted and removed from the mesh. No separation line along the tool tip path is predefined as required in the node splitting technology.
It should be noted that element deletion induces loss of volume, which violates the law of continuity, and thus the mesh design is notably fine to minimize this volume loss. The issue of convergence is important for quasi-static simulation, as indicated by the energy approximation balance in cutting simulation. The energy balance is checked for each simulation run to ensure that the ratio of kinetic energy to internal energy meets the requirement.
3 Progressive Degradation Model
3.1 Material Constitutive Law and Properties
Material properties of the GFRP workpiece
Mechanical properties | Value |
---|---|
Longitudinal modulus E_{1} (GPa) | 48 |
Transverse modulus E_{2} (GPa) | 12 |
Transverse modulus E_{3} (GPa) | 12 |
In-plane shear modulus G_{12} (GPa) | 6 |
In-plane shear modulus G_{13} (GPa) | 6 |
Transverse shear modulus G_{23} (GPa) | 5 |
Poisson’s ratio v_{12} | 0.19 |
Poisson’s ratio v_{13} | 0.19 |
Poisson’s ratio v_{23} | 0.26 |
Longitudinal tensile strength X_{T} (MPa) | 1200 |
Longitudinal compressive strength X_{C} (MPa) | 800 |
Transverse tensile strength Y_{T} (MPa) | 59 |
Transverse compressive strength Y_{C} (MPa) | 128 |
Lamina tensile strength Z_{T} (MPa) | 59 |
Lamina compressive strength Z_{C} (MPa) | 128 |
Shear strength in 1–2 plane S_{12} (MPa) | 25 |
Shear strength in 1–3 plane S_{13} (MPa) | 25 |
Shear strength in 2–3 plane S_{23} (MPa) | 20 |
Density ρ (kg/m^{3}) | 1800 |
3.2 Failure Criterion
3D Hashin-type and Maximum stress failure criteria
Failure mode | 3D Hashin-type | 3D maximum stress |
---|---|---|
Fiber breakage \((\sigma_{11} \ge 0)\) | \(f_{{ 1 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{11} }}{{X_{\text{T}} }}} \right)^{2}\) | \(f_{{ 1 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{11} }}{{X_{\text{T}} }}} \right)^{2}\) |
Fiber crushing \((\sigma_{11} \, < \,0)\) | \(f_{{1{\text{C}}}} = \left( {\frac{{\sigma_{11} }}{{X_{\text{C}} }}} \right)^{2}\) | \(f_{{1{\text{C}}}} = \left( {\frac{{\sigma_{11} }}{{X_{\text{C}} }}} \right)^{2}\) |
Matrix cracking in Dir. 2 \((\sigma_{22} \, \ge \,0)\) | \(f_{{2{\text{T}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{T}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 2 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{T}} }}} \right)^{2}\) |
Matrix crushing in Dir. 2 \((\sigma_{22} \, < \,0)\) | \(f_{{ 2 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{C}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 2 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{C}} }}} \right)^{2}\) |
Matrix cracking in Dir. 3 \((\sigma_{33} \, \ge \,0)\) | \(f_{{ 3 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{T}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 3 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{T}} }}} \right)^{2}\) |
Matrix crushing in Dir. 3 \((\sigma_{33} \, < \,0)\) | \(f_{{ 3 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{C}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 3 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{C}} }}} \right)^{2}\) |
Shear failure in 1–2 plane | N/A | \(f_{12} \, = \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2}\) |
Shear failure in 1–3 plane | N/A | \(f_{13} \, = \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2}\) |
Shear failure in 2–3 plane | N/A | \(f_{23} \, = \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) |
The fiber failure mode is associated with the longitudinal direction (Dir. 1), with f_{1T} and f_{1C} denoting tension and compression failure indexes respectively [17, 31]. The matrix failure is associated with the transverse (Dir. 2) and thickness (Dir. 3) directions, with failure indexes represented as f_{2T}, f_{2C} and f_{3T}, f_{3C}. The mechanism of delamination damage is ignored. This assumption is reasonable since the out-of-plane stress is not significant and plane stress was used in most orthogonal cutting FE models [14–20].
When the failure criterion is met (i.e., the failure index in Table 2 exceeds 1.0), the material point reaches the onset of damage and stiffness degradation is initiated. Two types of material degradation strategies, i.e. instantaneous failure and progressive failure, have been widely used [32].
Property degradation rules for instantaneous degradation
Failure mode | Rule 1 | Rule 2 |
---|---|---|
Fiber failure | E_{11} = E_{22} = E_{33} = G_{12} = G_{13} = G_{23} = v_{12} = v_{13} = v_{23} = 0 | E_{11} = G_{12} = G_{13} = v_{12} = v_{13} = 0 |
Matrix failure Dir. 2 | E_{22} = G_{12} = G_{23} = v_{12} = v_{23} = 0 | E_{22} = G_{12} = G_{23} = v_{12} = v_{23} = 0 |
Matrix failure Dir. 3 | E_{33} = G_{13} = G_{23} = v_{13} = v_{23} = 0 | E_{33} = G_{13} = G_{23} = v_{13} = v_{23}=0 |
For the progressive damage model, the material property is gradually degraded. Various approaches address progressive degradation using continuum damage mechanics (CDM) theory [34–36]. The CDM replaces the mechanical properties of the damaged materials with those of the homogenous materials by associating the damage mechanisms with their effects on the elastic constants of the materials. Several energy based stiffness degradation models are presented by Fang et al. [37], and Lapczyk et al. [38], while this study extends previous studies [38, 39]. For each failure mode, the damage evolution in the post-damage initiation phase is indicated by a damage variable.
3.3 Damaged Material Response
3.4 Damage Evolution
The evolution of the damage variable is based on the fracture energy dissipated during the damage process. The crack band model is adopted to alleviate the mesh dependence of the energy dissipated in the finite element result [37, 38]. The fracture energy is preserved and is the function of stress, the characteristic element length and the strain at strength limit.
The equivalent displacement and equivalent stress are calculated by the following expressions, where L^{c} represents the characteristic element length, and operator \(\left\langle x \right\rangle = (x + \left| x \right|)/2.\)
(1) Hashin-type criterion
(2) Maximum stress criterion
The exact fracture energy for progressive damage is difficult to obtain. Due to the absence of experimental data, the energies for various modes were assumed for the numerical computations. The energies for fiber tension, fiber compression, matrix tension and matrix compression modes were specified as 8 N/mm, 8 N/mm, 1 N/mm, and 2 N/mm, respectively. For the Maximum stress criterion, an additional value of 0.5 N/mm was assumed for each shear damage mode. Moreover, other energies were also specified to investigate the influence of the fracture energy level on the simulation results. In future, more accurate degradation parameters could be obtained through experiments.
3.5 Subroutine Implementation
All damage initiation indexes and damage variables are stored as solution dependent state variables (SDSV) in VUMAT. To avoid numerical difficulties, the damage variable was limited to a maximum value of 0.999 for each failure mode. In explicit analysis, a material point fails when it reaches either the fiber tensile or fiber compression mode. As such, the material point is flagged as failed and is removed from the mesh when d_{1} reaches the critical value of 0.999.
4 Results and Discussion
4.1 Chip Formation Process
The chip formation simulation using the Maximum stress criterion is demonstrated in Figure 5. For the same failure mode, the damage generally initiates later because the material is less likely to reach the damaged state when evaluated with the Maximum stress criterion. Among all failure modes, the fiber damage in Dir. 1, the matrix damage in Dir. 3, and shear damage in the 1–3 plane are rarely observed. The transversal matrix damages in Dir. 2 are more obvious, but are limited to the vicinity of the cutting tool. Matrix cracking damage exists below the tool flank and extends vertically. The matrix crushing damage initiates near the tool tip-workpiece contact area and propagates in the vicinity of the tool edge. Both matrix damage patterns are different from those in the Hashin-type model in which the damage progresses along fiber direction, because the shear stress does not contribute to transversal damage initiation in the Maximum stress criterion. The most obvious damage is shear damage in the 1–2 plane which propagates along fiber direction to the top surface of the workpiece due to the nature of the shear damage initiation criterion and its evolution mechanism. The damage extension pattern implies that the shear damage in the 2–3 plane is primarily induced by transversal matrix cracking and crushing.
4.2 Cutting Forces
Oscillation in the value of the cutting forces occurs in each model due to material stiffness reduction upon failure. Additionally, it is observed that the cutting force simulation reaches steady state earlier in the Hashin-type model than that in the maximum stress model.
The thrust forces predicted by the two criteria are also different. Both thrust forces remain negative at the early stage but increase gradually and turn into positive, reaching nearly 0.5 N and 3.5 N, respectively, at a later stage. In the beginning, the thrust force is negative because at the initial stage of penetration of the tool edge into workpiece, the workpiece contacts the tool edge and the rake area. Besides the reaction forces ahead in the horizontal direction, the tool also encounters a reaction force from upper left perpendicular to the tool rake. The resultant force has a negative (downwards) thrust force component. However, with additional advances of the tool, when the contacts between the workpiece and the bottom of tool edge and the tool flank are established, the reaction force from the tool bottom becomes dominant, which results in a positive thrust force.
The experimental thrust force is approximately 32 N per unit width for this condition. Both models predict far smaller thrust forces than the experiment, as is consistent with the 2D FE models in Refs. [9, 17, 18] in which significant differences are observed between the predicted and experimental thrust force results. According to Wang and Zhang [2], this difference can be explained by the “bouncing back” mechanism of the material under the tool flank. During cutting, the material under the tool tip is pushed down and springs back partially elastically after the tool has passed. The real depth of cut is different from the nominal one, and thus the experimental thrust force is strongly influenced.
As shown in Figure 12(b), the thrust forces predicted by both models are smaller than experiment for all fiber orientations, due to the bouncing mechanism explained previously. The same phenomenon was observed in the 2D FE model in Refs. [9, 17, 18]. However, the trends in both models agree with the experimental observations.
4.3 Sub-surface Damages
In VUMAT, the damage variables for different failure modes are stored in different SDSVs and updated in each increment. The progression of various sub-surface damage modes can also be observed in Figure 4 and Figure 5. Damage occurs where the corresponding damage variable is larger than 0. For 45° fiber orientations, the sub-surface damage in the Hashin-type criterion model is characterized by transversal matrix damage in Dir. 2 and subsequent shear damages in 1–2 and 2–3 planes. In the Maximum stress model, the sub-surface damage is attributed to shear damage in the 1–2 plane, the transversal matrix damage and subsequent shear damage in 2–3 plane.
In both models, the simulated sub-surface damages increase with increasing fiber orientation, and the values and trends are close to the experiment for fiber orientations below 75°. However, the predicted damage drops when the fiber orientation increases further to 90°, leading to a large discrepancy between the simulated and experimental sub-surface damage at larger fiber orientations. This result occurs because the simulated sub-surface damage is only related to the material failure (fiber/matrix cracking or crushing). However during the cutting experiment, fiber-matrix interface debonding also contributes to sub-surface damages. This mechanism plays a minor role for small fiber orientations but a significant role at 90° orientation, where the interface debonding can extend severely below the machined surface [1–3]. The fiber-matrix interface and its damage process are not modeled and simulated appropriately in the macro-mechanical FE models, and a multiphase micro-mechanical model is thus required.
5 Conclusions
- (1)
The failure criterion has a strong influence on the simulation results, especially on the cutting force. The Maximum stress criterion always predicts thicker chip shapes, higher cutting forces, and larger sub-surface damages due to the difference in expression for failure initiation and damage evolution.
- (2)
For both criteria, an increase in the fracture energy level results in increases in chip thickness, cutting and thrust forces, as well as sub-surface damage predictions. However, when the energy reaches a critical value, the simulation results tend to remain stable, and no obvious change is observed if the energy is increased further.
- (3)
For the assumed fracture energy value, the chip shapes simulated by both FE models are similar to the mechanisms. The simulated chip length decreases with increasing fiber orientation. The trends of cutting force and thrust forces with respect to fiber orientations agree reasonably with the experiment, with the cutting force increasing with fiber orientation, and the thrust force increasing initially and subsequently decreasing. Consistent with other 2D macro-mechanical FE models in the literature, the simulated thrust force has an order of magnitude smaller than the experiment.
- (4)
The sub-surface damages predicted by both models show an increasing tendency with fiber orientation and agree with the experimental results for fiber orientations up to 75°. A significant difference exists between the predicted and experimental results at higher fiber orientation (above 75°) since the macro-mechanical model does not simulate the fiber/matrix interface debonding behavior, revealing one of the limitations of the macro-mechanical FE model.
Declarations
Authors’ Contributions
Y-LH was in charge of the whole modeling work and the manuscript; JPD helped the manuscript composition; H-QX assisted with data analyses. All authors read and approved the final manuscript.
Authors’ Information
Yan-Li He, born in 1978, is currently an associate professor at Key Laboratory of Contemporary Design and Integrated Manufacturing Technology of Ministry of Education, School of Mechanical Engineering, Northwestern Polytechnical University, China. He received his PhD degree from Northwestern Polytechnical University, China, in 2007. His research interests include manufacturing of composite, intelligent machining process and manufacturing systems. Tel: +86-29-88494945; E-mail: heyl@nwpu.edu.cn.
Joao-Paulo Davim, is currently a professor at Department of Mechanical Engineering, the University of Aveiro, Portugal. He received his PhD from the University of Porto, Portugal, in 1997. His research interests include machining process, tribology, sustainable manufacturing, etc. E-mail: pdavim@ua.pt.
Hong-Qian Xue, born in 1967, is currently a professor at School of Mechanical Engineering, Northwestern Polytechnical University, China. He received his PhD degree from Conservatoire National Des Arts et Métiers, France, in 2005. His research interests include fatigue and fracture of structure material, mechanical assembly. Tel: +86-29-88492843; E-mail: xuedang@nwpu.edu.cn.
Competing Interests
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Not applicable.
Funding
Supported by Science Foundation of NPU (Grant No. 3102015JCS05009) and Chinese Foreign Talents Introduction and Academic Exchange Program (Grant No. B13044).
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