Micro-macro-mechanical model and material removal mechanism of machining carbon fiber reinforced polymer

Author’s Accepted Manuscript Micro-Macro-Mechanical Model and Material Removal Mechanism of Machining Carbon Fiber Reinforced Polymer Bin Niu, Youliang Su, Rui Yang, Zhenyuan Jia www.elsevier.com/locate/ijmactool

PII: DOI: Reference:

S0890-6955(16)30262-0 http://dx.doi.org/10.1016/j.ijmachtools.2016.09.005 MTM3195

To appear in: International Journal of Machine Tools and Manufacture Received date: 15 June 2016 Revised date: 12 September 2016 Accepted date: 13 September 2016 Cite this article as: Bin Niu, Youliang Su, Rui Yang and Zhenyuan Jia, MicroMacro-Mechanical Model and Material Removal Mechanism of Machining Carbon Fiber Reinforced Polymer, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2016.09.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Micro-Macro-Mechanical Model and Material Removal Mechanism of Machining Carbon Fiber Reinforced Polymer Bin Niu, Youliang Su, Rui Yang*, Zhenyuan Jia Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, School of Mechanical Engineering, Dalian University of Technology, 116024, Dalian, People’s Republic of China [email protected] [email protected] [email protected] [email protected] *Corresponding author. Tel./Fax: 0086 411 84707862 Keywords Machining, Carbon fiber, Composite, Elastic foundation beam, Micro-Macro-mechanical model

ABSTRACT The present paper studies the material removal mechanism of machining carbon fiber reinforced polymer (CFRP) by a micro-mechanical model, and proposes prediction models of cutting forces from the microscale to the macroscale. At the microscale, the micro-mechanical model for cutting a fiber in orthogonal cutting CFRP is established via the elastic foundation beam theory with explicit description of the carbon fiber and the matrix. The deflection and failure of the fiber constrained by the surrounding composite are analysed under the cutting effects by the tool edge. In addition, the fiber failure under the pressing of the flank face is analysed based on the undulating fiber theory. Analytical expressions are established at the microscale for evaluating the force for cutting a single fiber and the compression force for a single fiber from the flank face. At the macroscale, the chip length is determined by analyzing the characteristics of the cutting force signals

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of orthogonal cutting experiments. The characteristic chip length is used for establishing the trans-scale prediction model of cutting forces from the microscale to the macroscale. The total cutting and thrust forces at the macroscale during the formation of a chip are predicted based on the micro-mechanical results and the characteristic chip length, which agree well with the experimental results for orthogonal cutting of CFRP. Furthermore, the fiber failure modes and the debonding between the fiber and the matrix under different supporting conditions are discussed by the micro-mechanical model, by which subsurface damages are recognized.

1. Introduction Carbon fiber reinforced polymers (CFRPs) exhibit excellent mechanical properties, such high specific stiffness and strength, which result in increasing application in aerospace structures. However, the material removal mechanism of this kind of composites by machining is quite different from that of metals due to the heterogeneous and trans-scale characteristics of the composites [1]. The direct derivation of the mechanism of metal machining can not reveal the complex interaction effect between the fiber and the matrix during machining, and can not illustrate the different material failure and removal behaviour of the fiber, the matrix and the interface. The particular material removal mechanism of machining the composites has been attracted increasing attention in academia and industry. Extensive studies on material removal mechanism of machining CFRPs have been conducted in literature by theoretical, numerical and experimental approaches, see recent reviews and books [2-6]. Due to the convenience in observation and measurement, the simple two-dimensional orthogonal cutting is widely used to investigate the general material removal mechanism and mechanics in the common threedimensional and complex cutting operations. Orthogonal cutting experiments were carried out, e.g. in Refs. [7-9], to measure cutting forces and calculate the specific cutting energy. Cutting forces may be predicted from the specific cutting energy functions and chip geometry [5], where the specific cutting energy functions

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can be expressed as an exponential form or polynomial form in terms of the depth of cut and the fiber orientation. Empirical models of cutting forces have been obtained by fitting the experimental data of cutting tests [10, 11], which can be categorized as mechanistic modelling methods for predicting cutting forces. Mechanistic modelling methods can be used to predict the cutting forces in terms of a number of relevant machining process parameters including the chip dimension effects, process conditions, tool and workpiece geometries, tool-workpiece interaction effects, and other process noise effects [5, 6, 12, 13]. The mechanistic modelling method has also been used in other cutting process such as drilling, milling, etc. to predict cutting and thrust forces, or torque, see detailed references cited in the review [6]. The model based on the mechanistic methods can not reveal the physical essence of the material failure and removal. In comparison with the mechanistic model, the analytical method can consider directly the macroscopic homogenized properties of the work piece material, and even the microscopic mechanical properties of the constituent materials. Therefore, analytical studies of cutting forces, failure modes of constituent materials and the deformation regions receive particular attention. In earlier works of analytical studies on modelling, the classical theory of metal cutting is directly extended to study the machining of CFRPs. Everstine and Rogers [14] presented an analytical expression of the cutting force for the 0° fiber orientation based on the thick shear plane theory in metal cutting, where the composite was treated as a homogenized incompressible and inextensible continuum. Takeyama and Iijima [15], Bhatnagar et al. [16] derived the cutting forces for unidirectional laminates with fiber orientations in the range 0° to 90° using the classical Merchant’s thin shear plane theory in metal cutting, but different methods were used in their works to compute shear plane angle. Zhang et al. [17] made further extension for the prediction of cutting forces by considering three distinctive cutting regions, i.e., chipping, pressing and bouncing regions. In Zhang’s model [17], the total force was calculated by summing the contributions from these regions, and fiber orientations were limited to the range from 0° to 90°. Using a generalized plane strain anisotropic elasticity formulation, Gururaja and Ramulu [18] studied analytically the subsurface stress during orthogonal cutting of unidirectional laminate, where the unidirectional laminate was modelled as a continuous homogeneous

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elastic half-space, the cutting forces were represented by an inclined line load, but the chip formation was not considered. These studies [14-18] can be categorized as the macro-mechanical model, which directly uses the classical theory of metal cutting, continuum mechanics at the macroscale or conducts some extensions of the classical theory. However, the formation of a chip during machining CFRPs includes the fracture of fibers and the matrix, and the debonding between the fibers and matrix. The material failure and removal mechanism during machining CFRPs are more complicated than the classical metal cutting. These macromechanical models do not consider the microscopic characteristics of the composites, which implies that the detailed behaviour of the constituents of the composite including individual fibers, the surrounding matrix and interface is ignored, therefore they are not able to explain the material removal at the microscale. The fiber deformation and fracture at the microscale need to be taken into account for reflecting the trans-scale characteristic effects, and explaining the microscopic material removal and induced damage, which have been observed in experiments [9, 19]. Pwu and Hocheng [20] proposed prediction formulae of cutting forces by modelling a chip as a cantilever beam for the case while cutting perpendicular to the fiber axis. They discussed the bending failure of fibers during cutting, but detailed quantitative analysis of individual fibers and interface at the microscale was not performed. From the microscopic point of view, Jahromi and Bahr [21] derived the force expressions for cutting each individual fiber via the classical beam theory by modeling the fiber as a cantilever beam. The effects of fiber orientations on the cutting forces were studied, but the supporting effect of the surrounding composite on the failure modes of the fiber to be cut was not considered. In Jahromi and Bahr [21], the classical Merchant’s shear plane theory was used to determine the shear angle of the chip formation, by which the length of chip and the number of fibers in a chip were determined and used to compute the total cutting forces together with the cutting force for a single fiber. The constraint effect of the surrounding composite on the fiber has significant effect for the deformation and failure modes of the fiber, which is considered to be quite important for revealing the material removal mechanism, but a limited number of studies [22, 23] have been carried out about this point. Xu and Zhang [22] studied vibration-assisted cutting of the unidirectional laminate only with the 90° fiber orientation, and

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presented the cutting force predictions for cutting individual fiber with consideration of the supporting effect from the surrounding composite around the fiber. Their research [22] is quite important since the constraint effect is explicitly introduced first in calculating the cutting forces, but the transition from cutting off the individual fiber at the microscale to the chip formation at the macroscale was not studied in detail, which implies that the model needs to improve for deepening the understanding of the trans-scale characteristics in cutting force prediction for machining CFRPs. Qi et al. [23] extended the model by Jahromi and Bahr [21] by considering the supporting effect during cutting off a single fiber, and used the same method as in Ref. [17] to compute the shear angle and the contact force between the flank face and the workpiece. The cutting force prediction was studied in Qi et al. [23] for the fiber orientation in the range 0° to 90°. Actually, the trans-scale transition from cutting off a single fiber to a chip formation was not investigated in [23], and the debonding between the fiber and matrix can not be considered in this model, therefore the bonding effect for cutting a fiber was ignored. Moreover, the following issues on the microscale model have not been studied in depth in literature, including difference in failure modes of fibers under different supporting conditions, and theoretical analysis of the fiber fracture positions at different cutting conditions. In the present paper, the material failure modes and removal mechanism at the microscale are studied for varying depths of cut, different fiber orientations, and supporting effects. Prediction models of cutting forces from cutting a single fiber at the microscale to a chip formation at the macroscale are proposed. To the authors’ best knowledge, few work in literature has been done to establish the relation from the microscale to the macroscale during cutting CFRPs. As a view to understand the cutting process and derive the cutting forces from the cutting physical essence at the microscale, the micro-mechanical model and the transition from the microscale to the macroscale are studied. At the microscale, the micro-mechanical model of a representative volume element (RVE) in machining a unidirectional laminate is established with explicit description of the carbon fiber and the matrix. The fiber in the RVE is subjected to cutting forces, and is supported by the rest of the composite. Thus the deflection of the fiber is constrained by the composite surrounding the fiber, which can be modelled as a beam on an elastic foundation. The reaction from the

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supporting composite is applied on the fiber together with the cutting forces. Analytical expressions are established for evaluating the cutting forces and the debonding between the fiber and the matrix using the micro-mechanical model. Different failure modes of fibers with the constraint effects of the surrounding composite can be recognized from the analytical solutions. By revealing the fiber deformation, fiber fracture and debonding between the fiber and the matrix, the particular material removal mechanism is studied for machining carbon fiber reinforced polymer. The compressive force for a single fiber below the flank face is solved from the undulating fiber model [24] under compression. Furthermore, the cutting force and the compressive force at the microscale are used for the prediction of the total cutting and thrust forces at the macroscale by relating the single fiber fracture at the microscale and the chip formation at the macroscale. The chip formation is observed in experiments to be apparent intermittent and approximately periodic. By performing the characteristic analysis of the chips, the relation is established for the chip length, the depth of cut as well as the fiber orientation with respect to the cutting direction. The total cutting and thrust forces at the macroscopic scale during the formation of a chip are predicted by the summation of the forces of cutting a single fiber over all fibers in the chip. The cutting forces from the theoretical predictions and the experimental results are compared for several examples of orthogonal cutting. The effects of variable depths of cut and different fiber orientations on the material removal are discussed in the examples. The organization of the rest of this paper is as follows. Section 2 presents a description of the micromechanical modelling of cutting CFRPs, the detailed formulations of the fiber deformation based on the elastic foundation beam model, the solving procedures to obtain the analytical cutting forces and the debonding conditions, and the compressive force for a single fiber using the undulating fiber model. The total cutting and thrust forces are obtained in Section 3 by summing the analytical cutting and compressive forces of a single fiber over all fibers during a chip formation. The theoretical and experimental results of orthogonal cutting are compared in Section 4. The theoretical cutting force for a single fiber derived is analyzed in Section 5 for different fiber supporting conditions and different depths of cut. Finally, a section with discussion and conclusions closes the paper.

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2. A Micro-mechanical Model of Machining Fiber Reinforced Polymer 2.1 Elastic foundation modeling of cutting a single fiber Firstly, a two-dimensional micro-mechanical model of a representative volume element (RVE) in machining a unidirectional laminate of CFRPs is established with explicit description of the fiber and the matrix [1, 22]. The fiber in the RVE is subjected to cutting forces, and is supported by the rest of the composite. As indicated in Fig. 1, the cutting force component f Ay is perpendicular to the fiber, and the component f Ax is along the fiber. The deflection of the fiber is modeled as a beam on an elastic foundation, which reflects the constraint effect of the composite surrounding the fiber. The reaction force pm from the supporting composite is applied on the fiber together with the cutting force. At the same time, the interface between the fiber and the matrix may debond due to the tension effect. The possible debonding depth is indicated by h in Fig. 1. The intensity of the bonding force between the fiber and the matrix is indicated by

qb . The deflection of the fiber caused by applied cutting forces is constrained from both sides, i.e. the forces from the supporting composite and the bonding forces. In Fig. 1, ac is the nominal depth of cut,  is the bouncing back of the work piece in the finished surface, and re is the radius of the cutting tool edge.

Actions of the supporting composite and the bonding layer on the fiber from the both sides can be obtained from Ref. [25]. The reaction force pm from the supporting composite per unit length is expressed as

pm  km w  km1

d 2 w x  dx 2

(1)

where the coordinate x is along the fiber direction, w is the deflection of the fiber in the y direction. The local coordinate x  y of the fiber is indicated in Fig. 1. The symbol km is the modulus of the supporting composite, which is called the Winkler foundation modulus reflecting the normal effect from the supporting

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composite. The symbol km1 is the second parameter of the supporting composite, which can be attributed to the shear effect. The fiber to be cut is modeled as a beam on a two-parameter elastic foundation. The supporting composite can be treated as an Equivalent Homogeneous Material (EHM) [22]. The modulus of the EHM may be calculated from the homogenization method [26] or the RVE (Representative Volume Element) method [27]. If km1  0 , the model reduces to the classical Winker foundation model with the only one parameter km .

Fig. 1 A schematic illustration of the micro-mechanical model of orthogonal cutting CFRP

Similarly the intensity of the bonding force qb between the fiber and the matrix is expressed as qb  kb w

(2)

The symbol kb is the equivalent modulus of the fiber-matrix bonding layer. At the onset point of the debonding between the fiber and matrix, the intensity of the bonding force is equal to the bonding strength between the fiber and the matrix.

Considering the equilibrium of an infinitesimal element of a length dx of the fiber, we can obtain the governing equation [25]

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Ef I f

d 4w  x  d 2w  x   k   k m  kb  w  x   0 m 1 dx 4 dx 2

(3)

where E f and I f are the Young's modulus and the moment of inertia of the fiber, respectively. A generalized model of the beam on an elastic foundation is derived here through including the second parameter which can be used for considering the shear effect of the supporting composite.

2.2 Solution of the deflection of a single fiber The deflection of the fiber and the stress in the fiber caused by applying cutting forces are solved analytically in this sub-section with consideration of the constraint effect of the supporting composite and the bonding effect. The solution of Eq. (3) can be written as in Eq. (4), see Ref. [28], while km1   4kE f I f



1 2

,

where the ratio reflects the relation of the shear effect and the normal effect of the foundation. w x   C1 cos  x cosh  x  C2 cos  x sinh  x  C3 sin  x cosh  x  C4 sin  x sinh  x

(4)

where C1 , C2 , C3 , and C4 are constants of integration, and the symbols  and  are computed as

  2   ,   2  

(5)

The symbols  and  are expressed as

4

km  kb k ,   m1 4E f I f 4E f I f

(6)

From the interaction between the fiber and the cutting tool, the fiber in question has varying supporting conditions along its axis, thus the fiber is divided into three segments, OA, AE, and EC as illustrated in Fig. 1. For the first segment OA of the fiber above the tool-fiber contact point A ( x  ac  re ), and the second

9

segment AE of the fiber between the points A and E ( ac  re  x  ac    h ), the governing equation for these two segments reduces to Eq. (7) since there is no the bonding effect.

Ef I f

d 4 w x  dx 4

 km1

d 2w x  dx 2

 km w  x   0

The solution of Eq. (7) is expressed similarly as in Eq. (4), but   4

(7)

km k and   m1 should be 4E f I f 4E f I f

used instead. For the third segment EC of the fiber below the point E ( x  ac    h ), the governing equation is the same as in Eq. (3), and the solution is expressed in Eq. (4). By use of the expressions of hyperbolic functions, Eq. (4) can be rewritten in Eq. (8). w  x   e x  B1 cos  x  B2 sin  x   e x  B3 cos  x  B4 sin  x 

(8)

where the constants B1 , B2 , B3 , and B4 can be expressed by C1 , C2 , C3 , and C4 in Eq. (4). At the bottom of the fiber ( x   ),

w

x 

0

(9)

Using this boundary condition, we can obtain B1  B2  0 in Eq. (8).

In order to solve the deflection of the fiber, the length h of damage should be determined a priori. At the point E, i.e. the onset point of the debonding, the intensity of the bonding force is calculated from Eq. (10), which is equal to the bonding strength

 b  kb wE

10

(10)

where wE is the deflection of the fiber at the debonding point E. Eq. (10) provides a condition to calculate the critical damage length h. With the damage length, the deflection w  x  along the fiber can be derived analytical from the global displacement equation in Eq. (11) for the three segments OA, AE, and the bottom segment starting from E. The global displacement equation is established by assembling the element formulations. The element formulations for the top, the middle and the bottom segments defined above can be derived analytically from the exact displacement formulations considering the elastic foundation effect, which accounts for the stiffness of the beam element and the stiffness of the foundation directly, i.e. the contribution of the supporting composite and the bonding layer is included.

T

dwO dwA dwE   1 u =  wO wA wE  K R dx dx dx  

(11)

where the global stiffness matrix is denoted by K , and the load vector is R . Then, the maximum stress in each section of the beam is calculated from

  Ef r

where r is the radius of the fiber. The curvature

d 2w dx 2

(12)

d 2w should be evaluated piecewise for the beam segments dx 2

OA and AE, and for the bottom beam segment starting from E.

2.3 Iterative algorithm to evaluate the damage length h and the cutting force Since the damage length h is unknown, the computation of Eqs. (11) and (10) should be conducted iteratively. The cutting force f Ay will be determined from the fracture conditions of the fiber, which requires the computation of the maximum stress in the beam. The solving procedures are summarized below.

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[1]. Assume an undamaged fiber with h = 0, there is no debonding between the fiber and the matrix so that only two beam segments need to be calculated, i.e. segment OA, and the infinite segment starting from A;

[2]. Gradually increasing the cutting force f Ay until the maximum stress in the fiber reaching the tensile strength; [3]. Evaluate the displacement and stress at the point A to check if the debonding happens;

[4]. From Step [3], if no debonding, we get the force f Aydf from Step [2], which is the minimum force required to cut off the fiber; Complete the iteration, go to Step[8].

[5]. From Step [3], if debonding happens, we should reduce the applying force f Ay to a value f Aydebon which is going to create the debonding;

[6]. From Step [5], we know the critical force f Aydebon resulting into debonding. It implies that the three beam segments are needed to compute simultaneously. The damage length h and the applying force should be increased iteratively to find the final force f Aydf (with debonding and fiber fracture) satisfying

 d 2 w  h, f Ay      tensile with the constraint kb wE  h, f Ay    b ; max  E f r 2   dx  

[7]. The damage length h and the cutting force f Ay are solved from the following optimization problem by an optimality iterative algorithm. 2     d 2 w  h, f Ay        tensile   min  max  E f r 2 h , f Ay      dx      

subject to :

kb w3  h, f Ay    b

12

(13)

The initial value of the cutting force f Ay for the iterative process is f Aydebon obtained in Step [6]. Finally, we solve the critical cutting force f Aydf , and the final damage length h final . The position of the maximum stress in the fiber, i.e. the position where fracture happens, is solved at the same time. [8]. Stop.

2.4 Compressive force for a single fiber

Fig. 2 A schematic illustration of the compression zone of the work piece below the cutting edge

As shown in Fig. 2, the fiber to be cut deforms during the cutting process. When the fiber is cut off, the lower part of the fiber under the fracture position already deforms before the further compression by the nose and the flank face of the cutting tool. The undulating fiber model in Reifsnider and Case [24] is used to predict the compressive force. In this model, the initial bending of the fiber due to the effect of the rake face is considered. The fibers below the cutting edge may crush due to the compression of the tool.

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Two critical cases should be considered in this model, i.e., failure controlled by local shear stress of the composite and failure controlled by the bending stress in the fiber. Therefore, two stresses  cshear and

 cbending corresponding to the two critical cases are expressed in Eqs. (14) and (15)

 cshear  GLT

 cbending  GLT

1  f G 1    0  LT  l  t

1  f d E 1   2  02  f f  2l   c

(14)

(15)

where GLT is the shear stiffness for this failure mode, f 0 is the maximum deflection of the initial deformed fiber,  t is the shear strength of the fiber,  c is the compressive strength of the fiber, d f is the diameter of the fiber, l is the length of the deformed fiber below the cutting edge, and E f is the Young's modulus of the fiber in the longitudinal direction.

The shear stiffness GLT can be expressed in Eq. (16) from Reifsnider and Case [24]

GLT

2 2 E f Vf  d f    d f  E f 1      1    2  l    l  Gf 4   

(16)

where V f is volume ratio of the fiber in the composite, and G f is the shear modulus of the fiber.

The larger one of the two critical stresses in Eqs. (14) and (15) multiplied by the area occupied by a single fiber is chosen as the compressive force for a single fiber along the fiber direction, i.e.

f  max  t x

shear c

14

,

bending c



 d 2f 4

(17)

3. Total cutting and thrust forces The cutting force and the compressive force for a single fiber are established in the previous section. The total cutting and thrust forces should be calculated for forming a chip macroscopically. The total cutting force can be calculated by multiplying the force for breaking a single fiber with the numbers of fibers in a chip. The total thrust force can be obtained by multiplying the force for compressing a single fiber in the thrust direction with the numbers of fibers below the cutting edge.

3.1 Average length of chips Formation of a chip includes the process of breaking all fibers in the chip, therefore the total cutting force for such a chip can be considered as a sum of the forces of breaking each single fiber in the chip. The characteristic length of the chip should be determined in order to calculate the numbers of fibers in a chip. The depth of cut ac , and the chip length lchip are defined and illustrated in Fig. 3. The width of cut is defined as the same as the width of the work piece.

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Fig. 3 Illustration of the definitions of width and depth of cut, and length of a chip

In the present study, the statistical analysis of the chips in the experiment of orthogonal cutting uni-directional CFRP is performed to find the relation between the chip length, the depth of cut, and the fiber orientation with respect to the cutting direction, which is defined as the cutting angle. By observing the experiments of orthogonal cutting, the formation of chips presents apparent intermittent and approximately periodic variation, which are reflected in the time history of cutting forces, see Fig. 4. In the inset of Fig. 4, the enlarged local signal of the cutting force is shown. The time history of cutting forces during the steady machining process is analyzed to determine the average time of chip formation. The average length of a chip is chosen as the average of the chip's lengths in five intermittent periods. For the given rake angle (25°) and the clearance angle (5°) of the cutting tool, and the given cutting speed (0.5 m/min), the average lengths lchip of chips for different depths of cut and different cutting angles 45°, 60°, 90°, 95° and 110° are obtained from the experiments, see Fig. 5. An artificial neural network (ANN) model of the average chip length lchip is established based on the experimental data. From the ANN model, the chip length can be predicted at different depths of cut as shown in Fig. 6 for the cutting angle in the range of from 45° to 110°.

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(a) 45°

(b) 90°

Fig. 4 Time history of cutting forces with a depth of cut 50 m, and for different cutting angles: (a) 45°, (b) 90°

Fig. 5 The chip lengths lchip for different depths of cut ac , and cutting angle 

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Fig. 6 Predicted chip length by the ANN model

3.2 The number of fibers in a chip and the number of fibers below the cutting edge

From the average chip length, the number of fibers n f in the chip can be calculated approximately as

nf 

tlchipV f sin 

 rf2

(18)

where the symbol t is the width of the work piece, rf is the radius of the fiber,  rf2 is the cross-section area of a single fiber,  indicates the cutting angle, and V f is the volume ratio of fibers in the composite. The distribution of the fibers in CFRP is observed under Scanning Electrical Microscope, as shown in Fig. 7.

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Fig. 7 Unidirectional carbon fiber composite observed under Scanning Electrical Microscope

The numbers of fibers below the cutting edge can be calculated from the contact area sc in Eq. (19) between the cutting edge and the work piece on the machined surface. As shown in Fig. 3, lr  re is the contact length below the cutting edge radius with the work piece, re is the edge radius of the cutting tool,

lh 

dp tan 

is the contact length below the flank face of the tool with the work piece,  is the clearance

angle of the tool, and d p is the bouncing back value and assumed to equal to the cutting edge radius. The contact area sc is computed as

sc  t (lr  lh )

(19)

The number of fibers below the cutting tool is obtained as

nu 

t (lr  lh )V f sin 

 rf2

19

(20)

3.3 Total cutting and thrust forces in the global coordinate When the cutting angles are other than 90o, the derivation of cutting forces can be extended from the results in Section 2.2. The cutting angle  is defined with respect to the cutting direction, which is the same as the positive direction of the X coordinate, as shown in Fig. 1. The local coordinate x-y of the fiber is denoted in the Fig. 1. The cutting force f Aydf applying on the fiber is expressed in the y direction of the local coordinate x-y, i.e. in the transverse direction with respect to the fiber. The components in the global coordinate X-Y corresponding to f Aydf are expressed as

f X  f Aydf sin 

(21)

fY  f Aydf cos 

(22)

By multiplying the number n f of fibers in a chip calculated in Eq. (18), the cutting forces on the cutting tool edge are expressed as

FXedge  f Aydf n f sin 

(23)

FYedge  f Aydf n f cos

(24)

Similarly, using the compressive force f xt in Eq. (17) for a single fiber, the compressive and friction forces below the cutting edge are expressed in the global coordinate as

FXclearance    f xt cos    f xt sin   nu

(25)

FYclearance   f xt sin    f xt cos   nu

(26)

The friction forces on the flank face are considered by assuming a friction coefficient  between the cutting tool and the work piece. It is emphasized that a coordinate transformation is required to obtain the force components in the global coordinate, because the compressive force f xt and the corresponding friction

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force  f xt are expressed in the local coordinate as obtained in Section 2.4, i.e. f xt is along the fiber direction and the friction force is transverse to the fiber. The deformation of the macro chip before releasing from the work piece should be considered in the calculation of the total cutting force FXc and the thrust force FYt . A relatively simplified model is applied for evaluating the chip deformation, where uniform shear strain is assumed as shown in Fig. 8.

Fig. 8 A simplified illustrative model for evaluating the macro chip deformation

The energy stored by the deformation of a macro-chip before releasing is equal to the work applied by the rake face. An equivalent load F rake of the rake surface is expressed in Eq. (27), which is required for causing the macro-chip deformation before releasing from the work piece

F

rake

1 Gs aclchip 2 G a  2 2  s c 1 tan  lchip tan  2

(27)

where  is the shear strain, Gs is the shear modulus, and  is the angle of the chip shear deformation. The components in the local coordinate x-y resulted from the equivalent load F rake can be written as

21

Fxrake   F rake cos 

(28)

Fyrake  F rake sin 

(29)

The equivalent load F rake can be expressed in the global coordinate X-Y as

FXrake  F rake cos  cos  F rake sin  sin   F rake cos    

(30)

FYrake   F rake cos  sin   F rake sin  cos  F rake sin    

(31)

By combining the contributions of the cutting force in Eqs. (23) and (24), the thrust force in Eqs. (25) and (26), and the equivalent load on the rake face in Eqs. (30) and (31), the total cutting force FXc and the total thrust force FYt in the global coordinate X-Y can be evaluated as in Eqs. (32) and (33)

FXc  FXedge  FXclearance  FXrake

(32)

FYt  FYedge  FYclearance  FYrake

(33)

4. Experimental verification As a view to validate the theoretical model derived in the previous sections, the total cutting and thrust forces are measured in orthogonal cutting experiments of unidirectional carbon fiber reinforced polymers, and compared with the theoretical predictions from Eqs. (32) and (33). Machining experiments are carried out on an experimental system of orthogonal cutting. A series of composite coupons to be cut have the same dimension as a length 100 mm, a width 40 mm, and a thickness 3 mm. The composite coupon is installed on a particular designed fixture, which is mounted on the bottom with a KISTLER 9257B dynamometer to measure the forces together with a charge amplifier and a data acquisition device. The cutting process is visually recorded using a high speed video camera Photron FASTCAM SA5. The entire experimental setup is illustrated in Fig. 9.

22

(a)

(b)

Fig. 9 Experimental setup of machining unidirectional carbon fiber reinforced polymer: (a) the entire experimental setup, (b) the close-up of the clamped workpiece and the cutting insert mounted in the tool holder

The material properties of the composite samples are listed in Table 1. The samples are manufactured using the autoclave curing process. The first parameter km of the supporting composite around a fiber is computed using Biot's formula [29]

  Em* d 4  km  1.23  2  C 1  v  E f I 

0.11

Em* C 1  v 2 

(34)

where C is a coefficient varying from 1 for uniform pressure distribution to 1.13 for uniform deflection. Here C=1.06 is simply adopted. In the analytical studies, the second parameter of composite is ignored by assuming km1 =0, which implies that a classical Winkler foundation is adopted. Several YG8 carbide tools with a rake angle 25o and a clearance angles 5o , an edge radius 10 m and an edge width 5 mm, are used for the machining experiments. The cutting speed is kept at 500 mm/min. The cutting conditions are summarized in Table 2.

Table 1 Material properties of workpiece made of carbon fiber reinforced polymer

23

Items

Symbol

Value

Young's modulus of fiber

Ef

295 GPa

Shear modulus of fiber

Gf

103 GPa

Fiber diameter

d  2r

6.5  m

Poisson's ratio of fiber

vf

0.3

Moment of inertia of fiber

I

d4 64

First parameter of supporting composite

km

in Eq. (34)

Second parameter of supporting composite

km1

0

Tensile strength of fiber

t

4.0 GPa

Compressive strength of fiber

c

4.0 GPa

Shear strength of fiber

s

0.38 GPa

Young's modulus of matrix

Em

3.45 GPa

Poisson's ratio of matrix

vm

0.3

Equivalent modulus of the interface

kb

115 GPa

Bonding strength

b

50 MPa·m

24

Items

Symbol

Value

Transverse effective elastic modulus of EHM

Em*

9.65 GPa

Shear modulus of EHM

Gs

6.21 GPa

Poisson's ratio of EHM

v

0.3

Glass transition temperature of the matrix

Tg

165℃

Table 2 Cutting parameters

Tool & workpiece

Value

Tool material

Cemented carbide YG8

Work piece material

Uni-CFRP (T800H fiber, matrix 3900-2)

Cutting tool edge radius

10 m

Tool rake angle

25°

Tool clearance angle

5°

Cutting angle

45°~110°

Fiber volume fraction

65%

Nominal depth of cut

10 m, 30 m, 50 m

25

Tool & workpiece

Value

Friction coefficient between the tool flank face and the finished workpiece surface

0.1

Cutting speed

500 mm/min

Fig. 10 shows samples of the measured cutting and thrust forces in the steady stage of cutting CFRP with the cutting angles 90° and 45°. The average total cutting and thrust forces over the measured time can be calculated for the comparison with the theoretical predictions.

26

Fig. 10 Measured total cutting and thrust forces for different cutting angles using the cutting tool with the rake angle 25° and the clearance angle 5°, (a) for the cutting angle   90o and depth of cut 50 m, (b) for the cutting angle   45o and depth of cut 50 m

Fig. 11 shows the comparison of the total cutting and thrust forces using the theoretical method derived in Sections 2 and 3, and the experimental results of the average total cutting and thrust forces for different cutting angles and different depths of cut. For each cutting angle and depth of cut, three experiments are repeated as shown in Fig. 11. It is found that the theoretical predictions can compare reasonably well with the experimental results at several different cutting angles 45°, 60°, 90°, 95°, and 110°. From the predicted results in Fig. 11, it is seen that the amplitude of the cutting force generally increases with increasing the cutting angle, but the amplitude of the thrust force increases while the cutting angle increasing from 45° to 60°, and then decreases generally with the rising of the cutting angle. Observed from Fig. 11(c) with a depth of cut 50 m, the cutting force is generally small for the cutting angle less than 90° though the cutting force is rising with the increase of the cutting angle. There is an obvious rise of cutting force for a larger cutting angle, which is also reflected from the larger length of the chip as shown in Fig. 5. The thrust force in Fig. 11(c) changes from a relatively large positive value to smaller, even a negative value. The negative thrust force for the cutting angle larger than 90° may be due to the combined effect from the rake face and the flank

27

face. Because the rake face of the tool has a strong squeezing effect to lift the chip away from the work piece, the combined thrust force may become negative. It is also found from the experiments that it is difficult to conduct orthogonal cutting of unidirectional CFRPs with the cutting angle around 135° for a relatively larger depth of cut, e.g. larger than 50 m, because the tool edge may break due to very large load. Correspondingly, the subsurface damage is quite severe for this case, and even the sample may fracture into two pieces completely due to quite large cutting force, therefore it is difficult to perform a continuous and complete cutting experiment for this case.

200

200 Experimental cutting force Experimental thrust force

180 160

Force (N)

100 80

120 100 80

60

60

40

40

20

20 60

70

80

90

100

110

0 40

120

50

Cutting angle (/°)

60

70

90

100

(b)30 m

1000 Experimental cutting force Experimental thrust force

800

Theoretical cutting force Theoretical thrust force

600 400 200 0 -200 40

80

Cutting angle (/°)

(a)10 m

Force (N)

50

Theoretical cutting force Theoretical thrust force

140

120

0 40

Experimental cutting force Experimental thrust force

160

Theoretical cutting force Theoretical thrust force

140

Force (N)

180

50

60

70

80

90

Cutting angle (/°)

(c)50 m

28

100

110

120

110

120

Fig. 11 The comparison of the theoretically predicted and experimental total cutting and thrust forces: (a) 10 m, (b) 30 m, and (c) 50 m

5. Discussion on the cutting force for a single fiber with different supporting conditions The theoretical cutting force for a single fiber derived in Section 2 is analyzed in this section for different supporting conditions and different depths of cut. The cutting model of a single fiber with the cutting angle 90° is taken as an example, as presented in Fig. 1. The two nominal depths of cut 50 m and 10 m are selected for the analysis. Material properties of the fiber and the matrix materials are the same as in Table 1. The variation of the deflection in the local x-y coordinate and the maximum stress of the fiber along the fiber are shown in Fig. 12 (a, b) for the depth of cut 50 m, which reflects a relatively strong supporting condition for the fiber. It is observed that the deflection of the fiber reaches the maximum at the cutting position A, and the maximum stress in the fiber also appears at the cross-section of the fiber at the position A. The fiber will be cut off at the position A since the maximum stress at the position reaches the tensile strength of the fiber.

O A

Position along the fiber

Position along the fiber

O A

C 4

3

2 1 Deflection (mm)

0

C 0

-1 -4

1000

2000

3000

Stress (MPa)

x 10

(a)

(b) 29

4000

5000

Fig. 12 Variation of the deflection and the stress along the fiber for a strong constraint effect of the surrounding composite and a nominal depth of cut 50 m, (a) The deflection of the fiber, (b) The variation of maximum stress of the fiber along the fiber direction

As a comparison with the deformation of fiber with the relatively strong constraint effect in Fig. 12, the deformation of the fiber with a low supporting stiffness of the surrounding composite is studied. When the modulus Em* of the surrounding composite is reduced to 50 MPa, the variation of the deflection in the local x-y coordinate and the maximum stress of the fiber along the fiber are shown in Fig. 13 (a, b). By comparing the deflection of the fiber in Fig. 12 (a) and Fig. 13 (a), it can be concluded that when the supporting effect is strong, the fiber deflects significantly around the cutting position, which reflects the strong constraint of the surrounding composite. However, when the constraint effect is weak as in Fig. 13 (a), the deflection curve of the fiber is not confined at the local area of the cutting position. The stress distribution in Fig. 12 (b) and Fig. 13 (b) for the strong and weak supports is different. The distribution of the stress in Fig. 12 (b) is approximately symmetrical with respect to the cutting position where the depth of cut 50m is adopted. In Fig. 13 (b), the stress reaches the maximum at the cutting position, and the second largest stress (the second highest peak) occurs below the machining surface, which may induce some damage. The same modulus Em* = 50 MPa is assumed, but a smaller depth of cut 10m is adopted, the variation of the deflection in the local x-y coordinate and the maximum stress of the fiber along the fiber are shown in Fig. 14 (a, b). In Fig. 14 (a), the deflection curve of the fiber is very similar to that of a cantilever beam, where the maximum displacement occurs at the tip point O of the fiber. The stress distribution in Fig. 13 (b) and Fig. 14 (b) for larger and smaller cutting depths is quite different. In Fig. 14 (b), the stress reaches the maximum below the cutting position, which implies that the fiber may fracture below the machined surface.

30

O

A

A

Position along the fiber

Position along the fiber

O

C 6

C 4

2 Deflection (mm)

0

-2

0

1000

-3

2000

3000

4000

5000

Stress (MPa)

x 10

(a)

(b)

Fig. 13 Variation of the deflection and the stress along the fiber for a weak constraint effect of the surrounding composite and a nominal cutting depth 50 m, (a) The deflection of the fiber, (b) The variation of maximum stress of the fiber along the fiber direction

O A

Position along the fiber

Position along the fiber

O A

C 15

10

5 Deflection (mm)

0

-5 -3

C 0

1000

2000

3000

4000

Stress (MPa)

x 10

(a)

(b)

Fig. 14 Variation of the deflection and the stress along the fiber for a weak constraint effect of the surrounding composite and a nominal cutting depth 10 m, (a) The deflection of the fiber, (b) The variation of maximum stress of the fiber along the fiber direction

31

The failure modes of fibers with different constraint effects of the surrounding composite are recognized from the theoretical analysis. When there is no damage or thermal degradation in the surrounding composite, the supporting effect for the fibers can be considered as relatively strong, at these conditions the fiber fractures at the cutting position. When the supporting effect becomes weaker due to some damage or thermal degradation, the fiber may fracture below the machined surface for a relatively small depth of cut, which implies that the fiber seems to pull out, and it may induce some subsurface damage. For a weaker support and a large cutting depth, debonding may happen between the fiber and the surrounding composite. By revealing the fiber deformation, the particular material removal mechanism at the microscopic scale is found for machining fiber reinforced polymer composite. Actually, machining CFRPs may result into high and localized temperature due to the friction between the tool and the workpiece. The material properties in the zone exposed to considerable heat may degrade so that the supporting effect of the surrounding composite in the cutting zone may become weak. Particularly, when the glass transition temperature is reached in the heat affected zone, thermal damage may be quite severe and the supporting effect on the fiber is considerably weaker.

6. Conclusions An elastic foundation beam model is proposed to study the material removal of machining CFRP at the microscopic scale. By explicit description of the carbon fiber and the matrix, the micro-mechanical model of machining a unidirectional CFRP laminate is established by considering the supporting and the bonding forces from the supporting composite together with the cutting force on the fiber. Analytical expressions are derived for evaluating the cutting forces and the debonding between the fiber and the matrix using the micromechanical model.

32

The cutting force for a single fiber is solved analytically by the present micro-mechanical model. The compressive force for a single fiber below the flank face is solved from the undulating fiber model under compression. At the macroscale, the total cutting and thrust forces are further evaluated by summation over all fibers in a chip. The characteristic chip length is obtained from the statistical analysis of machining experiments. By the present theoretical method, the total cutting and thrust forces are predicted from the material properties and the failure mechanism at the microscopic scale. The theoretically predicted total cutting and thrust forces compare reasonably well with the experimental results for variable depths of cut and different cutting angles. At the same time, the material removal during machining is analyzed for different cutting angles and different depths of cut. The present study establishes a prediction model of the cutting and the thrust forces from the microscopic scale to the macroscopic scale, which provides a trans-scale way to understand the cutting process from the fiber fracture to the formation of the macroscopic chip. Different failure modes of fibers with the constraint effect of the surrounding composite are recognized from the analytical solutions. When the supporting effect is strong and a relatively large depth of cut is adopted, the fiber fractures at the cutting position. When the supporting effect is relatively weak and a small depth of cut is adopted, the fiber may fracture below the machined surface, which implies that the fiber appears to pull out. For a weaker support and a large depth of cut, debonding may happen between the fiber and the surrounding composite. Subsurface damage may be induced due to fiber fracture below the machined surface or debonding. By revealing the fiber deformation, the material removal mechanism is built at the microscale for machining fiber reinforced polymer composite. In the further studies, the thermal damage may be included in the micro-macro-mechanical model by considering the temperature dependent material properties. The effect of cutting speed on the chip formation and the variation of cutting forces is also worth studying in the future by considering the strain-rate effect of the matrix.

33

Acknowledgements This work was partially supported by the National Natural Science Foundation of China (no. 51505064), 973 Project of China (2014CB046503), and the National Natural Science Foundation of China (no. 51321004). These supports are gratefully appreciated. The authors also gratefully extend their acknowledgements to AVIC Shenyang Aircraft Corporation of China for supplying the unidirectional CFRP.

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Bin Niu received his Ph.D. degree in engineering mechanics at Dalian University of Technology in 2010, and worked as a postdoctoral researcher at Aalborg University, Denmark from 2010 to 2013. Starting from September 2013, he has been working as an associate professor at the School of Mechanical Engineering, Dalian University of Technology. His research interests include mechanics of composite materials, machining mechanism of CFRP and high quality and high efficient machining of CFRP.

Youliang Su is a Ph.D. student at the School of Mechanical Engineering, Dalian University of Technology. His research interests focus on the theoretical and experimental studies on machining mechanism of CFRP.

Rui Yang received his Ph.D. degree in mechanical engineering at Dalian University of Technology in 2004. He is working as an associate professor at the School of Mechanical Engineering, Dalian University of Technology. His research interests include mechanics of composite materials and structures, machining mechanism of CFRP and manufacturing of composite structures.

Zhenyuan Jia received his PhD degree in mechanical engineering from Dalian University of Technology in 1990. He is currently working as a professor at the School of Mechanical Engineering, Dalian University of Technology. His research interests include theory and technology for high quality and high efficient machining of CFRP, precision measurement and control of manufacturing process, numerical control technique, and applications of smart material in sensor and actuator.

Highlights

36



A micro-mechanical model based on the elastic foundation beam theory is established to study cutting a single fiber with the supporting effect by surrounding composite.



Trans-scale prediction models of cutting forces from the microscale to the macroscale are proposed for machining carbon fiber reinforced polymer.



The relation from cutting a single fiber at the microscale to the chip formation at the macroscale is established based on the micro-mechanical model and the characteristic analysis of the chip formation as a view to evaluate the total cutting forces at the macroscale.



Different failure modes of fibers and debonding under different supporting conditions are analyzed, and subsurface damage are recognized from the micro-mechanical model.

37