Investigation on influences of initial residual stress on thin-walled part machining deformation based on a semi-analytical model

Accepted Manuscript Title: Investigation on Influences of Initial Residual Stress on Thin-walled Part Machining Deformation Based on a Semi-Analytical Model Authors: Hanjun Gao, Yidu Zhang, Qiong Wu, Bianhong Li PII: DOI: Reference:

S0924-0136(18)30144-4 https://doi.org/10.1016/j.jmatprotec.2018.04.009 PROTEC 15709

To appear in:

Journal of Materials Processing Technology

Received date: Revised date: Accepted date:

25-1-2018 28-3-2018 5-4-2018

Please cite this article as: Gao H, Zhang Y, Wu Q, Li B, Investigation on Influences of Initial Residual Stress on Thin-walled Part Machining Deformation Based on a Semi-Analytical Model, Journal of Materials Processing Tech. (2010), https://doi.org/10.1016/j.jmatprotec.2018.04.009 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 proof before it is published in its final 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.

Investigation on Influences of Initial Residual Stress on Thinwalled Part Machining Deformation Based on a Semi-

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Analytical Model

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Hanjun Gao1, Yidu Zhang1, Qiong Wu1*, Bianhong Li2

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1. School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191, PR China; 2. School of Mechatronical Engineering, Beijing Institute of Technology, Beijing City, 100081, P R China;

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Abstract

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Graphical Abstract

The manufacturing accuracy of the thin-walled parts is significantly affected by the residual

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stress in the blank and the resulting machining deformation. In recent years, the prediction and control of the machining deformation have received extensive attentions. A semi-analytical machining deformation prediction model is proposed for the thin-walled parts in terms of the equivalent bending stiffness calculated by finite element method (FEM) simulation and theory 1

of plates and shells. Machining deformations of seven typical study cases are predicted by the semi-analytical model. Corresponding experiments and FEM simulations are conducted to validate the proposed model. Then, the influences of the initial residual stress and equivalent

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bending stiffness on the machining deformation are investigated based on the quantitative results of the proposed model, which is difficult to be obtained by the previous FE models. The

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results suggests that 1) improving the bending stiffness of the part can effectively reduce the

machining deformation; 2) decreasing the stresses in the top part of the blank and in the length

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direction is more advantageous to reduce the deformation than other parts and directions; 3) the

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final machining deformation is basically determined by the residual stress within a certain

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mid-plane in thickness direction.

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thickness under the blank surface when the residual stress symmetrically distributes along the

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Keywords: machining deformation; thin-walled parts; semi-analytical model; initial residual

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1. Introduction

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stresses; equivalent bending stiffness;

The manufacturing accuracy of the thin-walled parts is significantly affected by the residual

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stress in the blank and the resulting machining deformation because of the low rigidity and high removal rate. Residual stresses are inevitably induced into the blank due to the mechanical and thermal operation during forming process. The original equilibrium of residual stress is broken during the machining process because of the removal of the material, and the internal stress 2

redistribution as well as distortion occurs to the remaining workpiece. In recent years, the prediction and control of the machining deformation have received extensive attentions of scholars.

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Finite element method (FEM) simulation is a common numerical method for machining deformation and residual stress prediction. Outeiro et al. (2006) analyzed the influences of tool

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and processing parameters on the generation of the induced residual stress of AISI 316L steel

part via FEM simulations and experiments. Wei and Wang (2007) built a FE model to study the

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machining distortion and residual stress redistribution of aviation thin-walled parts. Dong and

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Ke (2006) simulated the quenching, stretching and machining process to predict the machining

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deformation by establishing a FE model. Liu et al. (2015) investigated the distortion of an

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aluminum alloy 7085-T7452 windshield frame part caused by forging residual stresses via FEM

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simulations and machining experiments. Alvise et al. (2015) developed a FE machining distortion prediction model based on the conforming meshes and faces technology. Cheng et al.

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(2015) proposed a theoretical model of the milling tools based on their analysis of the

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deformation performance and cutting force to investigate the distortion of titanium alloy parts. Izamshah et al. (2011) presented a high-efficient methodology based on a combination of the

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FEM simulation results and statistical experimental data to obtain the machining deflection of the side wall of Ti-6Al-4V alloy part. Jiang et al. (2014) presented an integrated method to calculate the deformation given rise to the induced residual stresses. Yang and Liu (2015) presented a unique and generic technique based on the depth of plastic strain in the surface to 3

analyze the machining distortions of the workpiece. Guo et al. (2009) introduced a “housebuilding frame modeling” method based on FE model to study the milling distortion of monolithic thin-walled components. Diez et al. (2015) developed an on-line system in terms of

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the predicted machining deformation and corrected the relative location between the cutting tool and workpiece for compensating machining errors. Cerutti·and Mocellin (2015) introduced

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a parallel FE tool to obtain the workpiece non-linear performance caused by the clamping deformation.

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Tang et al. (2013) developed a FEM prediction model for the thin-walled part machining

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deformation that considers the coupling effects of four factors, namely the initial residual stress,

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cutting load, induced residual stress, and clamping force. Rai and Xirouchakis (2008) presented

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deflections of the thin-walled part.

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a FE model considering cutting regime, clamping force, and processing path to calculate the

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Moreover, some scholars carried out the research on controlling and reducing the machining deformation. Li et al. (2015) found that the magnitudes of the distortion are significantly

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affected by the cutting depth during roughing and finishing. Li and Wang (2017) concluded that decreasing the residual stresses magnitude can effectively reduce the machining deformation of

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aeronautical aluminum alloy parts. Cerutti et al. (2017) found that adjusting the process plans according to the initial residual stress distribution is an effective method to decrease the machining deformation. Zhang et al. (2014) also found that the distortion can be decreased by optimizing the machining sequence in terms of the stress map of the part section. Yang et al. 4

(2014) summarized that the residual stress and cutting load are the main causes of the machining deformation of aluminum and titanium alloy thin-walled parts, respectively. Masoudi et al. (2014) found that the distortion increases with the imbalance of the residual

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stress, and decreases with the part thickness. Husson (2014) concluded that shaft bending is reduced by conducting a 600 °C thermal stress relief for 4 hours. Khan et al. (2017) found that

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a 60% reduction in the 6061 aluminium alloy machining deformation occurs after treated by thermal stress relieving with 290 °C re-heating temperature. Wu et al. (2016) used a quasi-

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symmetric machining method to improve machining precision, in which the maximum

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machining deformation is decreased from 0.3246 mm to 0.0589 mm.

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In summary, most of the machining deformation prediction models are developed based on

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FEM. As a classical numerical simulation method, FEM can effectively simulate the machining

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process and calculate the deformation and stress during manufacturing. However, the

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quantitative relation between the residual stress and deformation is difficult to be obtained via FE models. Nevertheless, the analytical model is suitable for quantitative research. Shin (1995)

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proposed an analytical machining deformation prediction model that considers uniaxial residual stress for plate parts. Gao et al. (2017) presented an analytical model considering the initial

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residual stresses in length and width directions for plate parts. Existing analytical models are only applicable to rectangle plate parts. The application scope of the analytical model needs to be extended to the more general thin-walled parts. In the present study, a semi-analytical prediction model is proposed on the basis of the 5

equivalent bending stiffness calculated by FEM and theory of plates and shells. The proposed model that considers the biaxial initial residual stress applies to the thin-walled parts, which has a wider application scope than the previous model that applies to the rectangle plate parts. Then,

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the model is validated by experiments and FEM. The quantitative relationship between the initial residual stress and machining deformation, which is difficult to be obtained by the FE

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models, is established based on the proposed model. Eventually, the influences of the initial residual stress and equivalent bending stiffness on the machining deformation are investigated.

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2. Modelling

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In this section, a semi-analytical prediction model of the machining deformation is presented.

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The model is applicable to the thin-walled parts machined from rectangular plate blanks. This

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model is proposed in terms of the equivalent bending stiffness calculated by FEM and theory

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of plates and shells. Blank size, workpiece shape, material properties, and biaxial initial residual

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stresses are considered in the model, and the following assumptions are made: 1) the residual stress distributes uniformly at the same depth in the blank and 2) the stress in the thickness

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direction is negligible. Furthermore, the effects of machining-induced residual stress, fixture,

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cutting force, and cutting temperature are not considered.

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2.1 Equivalent bending stiffness

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According to Eqs. (A1) - (A3) (Appendix), the bending deformation is determined by the

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bending stiffness D of the rectangle plate with a given load and material, and the bending

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stiffness is only related to the plate thickness, excepted for the elastic modulus E and Poisson’s

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ratio μ. Thus, the equivalent bending stiffness and thickness of the thin-walled part should be obtained via FEM before calculating the deformations.

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The general FEM simulation steps are summarized as follows: 1) importing the geometry

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model of the thin-walled part into an FEM software (ANSYS 17.0 in present study); 2) defining the material properties; 3) generating the FE meshing model; 4) applying the fixed constraint to one end face perpendicular to the length direction and uniform load qx (parallel to thickness

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direction) to one edge on the opposite side; 5) solving the load step and obtaining the maximum deformation disx; and 6) applying the constraint and uniform load qy to the face and edge perpendicular to the width direction and obtaining the corresponding deformation disy (Fig. 1).

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Fig. 1 Finite element (FE) model for calculating the equivalent thickness

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Cantilever beam deflection formula in materials mechanics is used to calculate the equivalent thickness (Gere and Goodno, 1972), because there is no standard analytical solution exists for

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cantilever plate bending in elastic theory (Xv, 2016). The maximum deflection of a cantilever

4 F  len 3 , E  bl  hl 3

(1)

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wl 

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beam with concentrated load F on the cantilever end is expressed as:

where wl is the maximum deflection, and len, bl, and hl are the length, width, and thickness of the beam, respectively.

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Based on the deformation results of the FEM simulations, the equivalent thickness (heqvx and

heqvy) and bending stiffness (Deqvx and Deqvy) of the thin-walled part are presented in Eqs. (2) (5):

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heqvy  (

Deqvx  Deqvy 

1 4 Fx L3 )3 , disx  E W

4 FyW 3

1

)3 ,

(3)

,

(4)

,

(5)

dis y  E  L E  heqvx 3

12(1   2 ) E  heqvy 3 12(1   2 )

(2)

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heqvx  (

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where L is the workpiece length, W is the workpiece width, Fx=qx·W, and Fy=qy·L. 2.2 Deformation curvature

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The initial residual stress in the blank is in a stress equilibrium state before machining. The

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machining process breaks the equilibrium state because of the material removal. Residual stress

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redistributes in the remaining part, and the deflection occurs. The deflection caused by the stress

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release is equivalent to that caused by the uniform moment on two opposite edges (Zhang et al.,

deformation problem.

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2014). Thus, the boundary condition in Appendix can be used to solve the machining

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The rectangle blank is evenly divided into n layers along the thickness direction. The

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thickness of each layer is t= h1/n (h1 is the blank thickness). The biaxial initial residual stresses in the layers 1 - n are denoted by σx01, σx02 …, σx0,n-1, σx0n, σy01, σy02…, σy0,n-1, and σy0n, respectively.

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The relation between residual stress and moment after removal of the material in the first layer is presented in Eqs. (6) - (7) (Shin, 1995):  x 01  t 

h1 dy  M x dy, 2

(6)

 y 01  t 

h1 dx  M y dx, 2

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Eqs. (6) - (7) are substituted into Eqs. (A1) - (A2), respectively, and h is replaced by the equivalent thicknesses of the workpiece after the removal of the first layer, namely heqvx1 and heqvy1. Thus, the variation of bending curvature is expressed as: 6th1 f1 ( x 01   y 01 ) 1 1   , 3 Rx1 Rx 0 E  heqvx 1

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(8)

6th1 f1 ( y 01   x 01 ) 1 1   , 3 R y1 R y 0 E  heqvy 1

(9)

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where Rx0 and Rx1 are the X direction curvature radii before and after the first layer removal, respectively; Ry0 and Ry1 are Y direction curvature radii before and after the first layer removal,

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respectively; and f1 is the volume removal ratio of the first layer.

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2.3 Residual stress redistribution

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New stresses are generated in each layer after the removal of the first layer. The coupling of

equilibrium state.

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the new stress and residual stress forms the new stress distribution, thus achieving a new

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In this study, curvature is regarded as positive and negative when the part bends upward and downward, respectively. The curvature is assumed to be positive after the removal of the first

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layer. In this case, the lengths in the top layers of the remaining workpiece increase, and those in the bottom layers decrease. The remaining workpiece must have a neutral surface with

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unchanged length. As shown in Fig. 2, the length and stress variations in the layers from top to bottom are an arithmetic sequence (Gao et al., 2017). Moreover, Eqs. (10) - (11) are deduced from Eqs. (8) - (9) and generalized Hooke’s law (Gao et al., 2017):

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S y1  

6th1 x 01a x1 , 3 heqvx 1

6th1 y 01a y1 3 heqvy 1

(10)

(11)

.

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S x1  

Thus, Eqs. (12) - (15) are obtained according to the equilibrium conditions of the internal

6h1h2

 0.5h2 ,

3 6th1 x 01 h eqvx1 (  0.5h2 ), 3 h eqvx1 6h1h2

6th1 y 01 3 heqvy 1

(

h 3eqvy1 6h1h2

 0.5h2 ).

(13)

(14)

(15)

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S y1  

(12)

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S x1  

 0.5h2 ,

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h 3eqvy1

6h1h2

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a y1 

h 3eqvx1

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a x1 

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residual stress (Gao et al., 2017) and Eqs. (10) - (11):

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where Sx1 and S’x1 are the X direction residual stresses newly generated in the top and bottom surface of the workpiece, respectively; Sy1 and S’y1 are the Y direction residual stresses newly

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generated in the top and bottom surface, respectively; ax1 is the distance between the neutral surface and top surface in the X direction; bx1 is the distance between the neutral surface and

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bottom surface in the X direction; ay1 is the distance between the neutral surface and top surface in the Y direction; by1 is the distance between the neutral surface and bottom surface in the Y direction; and h2 = h1-t. Thus, the residual stresses in the remaining workpiece can be calculated using Eqs. (16) 11

(19) (in Eqs. (16) - (19), j = 1, 2, …, n-1), S x1, j 

S x1 t [a x1  ( j  1  )], a x1 2

(16)

 x1, j   x0, j 1  S x1, j , S y1 a y1

t [a y1  ( j  1  )], 2

(18)

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S y1, j 

(17)

 y1, j   y 0, j 1  S y1, j ,

(19)

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where Sx1,j is the newly generated stress in the X direction in the jth layer; Sy1,j is the newly generated stress in the Y direction in the jth layer; σx1,j is the X direction residual stress in the jth

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layer, and σy1,j is the Y direction residual stress in the jth layer.

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Fig. 2 Stress redistribution in the remaining workpiece 2.4 Summary of the semi-analytical model The residual stress in the remaining workpiece is regarded as the initial stress before the

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removal of the next layer. When total k layers should be removed, the stress distribution and deformation in the X direction after the removal of ith layer (i=1, 2, 3,…,k) are presented in Eqs. (20) - (24) (the subscript x denotes the X direction in Eqs. (20) - (24)),

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Rx ,i 1

a x ,i 



6thi fi ( x ,i 1,1   y ,i 1,1 ) 1  , 3 Rx ,i Eheqvx ,i

h 3eqvx ,i 6hi hi 1

S xi  

S x ,i , j 

(20)

 0.5hi 1 ,

6thi x ,i 1,1a x ,i h 3eqvx ,i

(21)

(22)

,

S xi t [a xi  ( j  i  )] a xi 2 i

 x ,i , j   x ,i  j   S x ,k ,i  j 1

 j  i, i  1, n  1 , j

(23)

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1

1, 2 , n  i  .

(24)

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k 1

where heqvx,i is the equivalent thickness after the ith layer removal; fi is the volume removal ratio

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of the ith layer; Rx,i is the curvature radius after the ith layer removal; σx,i-1,1 is the residual stress

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in the first layer of the workpiece before the removal of the ith layer; σy,i-1,1 is the Y direction

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residual stress in the first layer; Sxi is the stress newly generated in the top surface; Sx,i,j is the

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residual stress newly generated in the jth layer; σx,i,j is the residual stress in the jth layer; and hi

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=(n+1-i)t.

It is worth noting that Eqs. (21) - (22) are suitable for the model when (heqvx,k – hk+1) is greater

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than 3·t (3 times of the layer thickness). When (heqvx,k – hk+1) is less than 3·t, the equivalent

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bending stiffness of the part is closed to the bending stiffness of a rectangle plate. The heqvx,i in Eqs. (21) - (22) should be replaced by hi+1 to enhance the calculation accuracy.

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Likewise, the curvature and stress in the Y direction can be deduced by iteration. Thus, the

deflection function after the removal of the ith layer is obtained using Eq. (A4),

1 1 2 1 2 L L W W wi ( x, y )  ( x  y ) (   x  ,   y  ). 2 Rx ,i Ry ,i 2 2 2 2

(25)

In summary, the flowchart of the semi-analytical model is shown in Fig. 3. 13

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Fig. 3 The flowchart of the semi-analytical model

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3. Simulations and experiments 3.1 Case study

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The proposed model was employed to calculate the machining deformations of seven typical

thin-walled parts, and corresponding experiments and numerical simulations were conducted to verify the proposed model. The size and shape of the study cases are shown in Fig. 4. The blanks of Cases 1, 2, 3, 5, 6 and 7 were 300×150×30 mm rectangle plates, and that of Case 4 14

blank was a 200×150×30 mm rectangle plate. The plates were taken from a 7075-T6 aluminium alloy rolling plate with 30 mm thickness. The length direction was parallel to the rolling direction. Two 200×150×30 mm specimens from the same rolling plate were employed to detect

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the initial residual stress. The mechanical properties of 7075-T6 aluminium alloy are presented

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in Tab. 1.

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Fig. 4 Size and shape of Cases 1-7

Tab. 1 Material properties of 7075-T6 aluminium alloy Elastic modulus/

Poisson’s ratio

Density/

Yield strength/ 16

(GPa) 71.7

0.33

(kg/m3)

(MPa)

2810

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Prism (Stresstech Group) is a residual stress measurement device on the basis of hole-drilling and electronic speckle pattern interferometry technique. It was applied to measure the residual

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stress from specimen surface to 2 mm depth by drilling a small hole. Layer removal method was used for measuring the internal residual stresses. A XK7132 vertical milling machine

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(Shandong Lunan Machine Tool Co. Ltd.) with a batch of cemented carbide Φ14 mm end mills was used for the material removal. The spindle speed was 3000 r/min; the feed rate was 300

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mm/min; and the cutting depth was 1 mm. Thus, the initial residual stress profile along the

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thickness direction was obtained by computing the mean value of the measurements of two

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specimens, and the residual stress in X and Y directions can be expressed by the depth beneath

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the surface using piecewise polynomial fitting method (Fig. 5).

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The equivalent thickness and stiffness of Cases 1-7 during the machining process were

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calculated using ANSYS Workbench 17.0 and Eqs. (2) - (5). Computing programs of the proposed model were written using MATLAB 2014b. Layer number n was set as 30, and total

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24 layers for each case were removed from the blank (k=24). Thus, the predicted results of

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deformation and stress could be calculated using the proposed model.

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Fig. 5 Initial residual stress profile 3.2 FEM and experiment validation

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FEM simulation and machining experiments were conducted to validate the semi-analytical

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model. FEM simulation steps using ANSYS Parametric Design Language (APDL) 17.0 were

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described as below: 1) choosing the element type as Solid 186, which is a second-order hexahedral solid element with 20 nodes; 2) defining the material properties as shown in Tab. 1;

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3) establishing the blank geometric model that comprises the areas to be and not to be machined;

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4) evenly dividing the geometric model into 30 layers in the thickness direction; 5) assigning the stresses to different layers according to Fig. 5; 6) removing each layer of the workpiece

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from top to bottom utilizing element birth and death technology and solving; and 7) outputting the results.

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Machining experiments were conducted using a BV100 vertical machining center (Beijing

Research Institute of Mechanical and Electrical). A batch of Φ14 three-flute cemented carbide end milling cutters and Φ4 two-flute cemented carbide end milling cutters were used for face milling and fillet machining, respectively. HY-2 type emulsified oil was used for coolant. The 18

spindle speed was 7000 r/min; the feed rate was 2000 mm/min; and cutting depth was 1 mm. Fig. 6 presents the setups of the machining experiments. The material was removed from the blank layer by layer, and the machining tool paths of Cases 1-7 are shown in Fig. 7.

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For Cases 4-7, the fixture was released for deformation measurement after each 4 mm depth was machined. The machining of Cases 1-3 was finished with one-time clamping, and only the

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final deformations were measured. A dial indicator with 0.001 mm accuracy was used for

deformation measurement (Fig. 8). The dial indicator was fixed to the spindle of a 3-axis

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machine tool via a bracket. The workpiece was placed upside down on the working table. Total

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31 measuring points were marked on the workpiece bottom (Fig. 8). The pointer of the dial

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indicator was first located at P27 through the moving of the spindle, and was moved to 30 other

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points with the same position in vertical direction to measure the deflections of the bottom of

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the workpiece.

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Fig. 6 Machining experiments

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Deformation

Machining

strategies

of

Cases

1-7

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Fig.

Fig. 8 Deformation measurement setup and measuring points

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4. Results and discussions 4.1 Validation results The deformations of the vertexes on the bottom surface are compared in Fig. 9. The

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experimental results present the average measured deformations of the four vertexes (P21, P22, P23 and P24). The final deformation surfaces are compared in Fig. 10. The measured

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deformation data in Fig. 10 were processed as follows: the average final deformations of the measuring points symmetrical to the midline parallel to the Y direction were calculated, and the

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corresponding deformation curves in the X direction were obtained by quadratic fitting method.

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These deformation curves constituted a deformation surface.

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Fig. 9 and 10 show that the deformation results of the semi-analytical model after the removal of each layer agree well with FEM and experimental results. Compared with experimental

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results, the relative errors of the semi-analytical model of the vertex final deformation for Cases

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1-7 are -6.31%, 10.66%, -15.53%, -13.30%, 9.48%, -14.06%, and -11.25%, respectively, and the maximum relative errors in the machining process for Cases 4-7 are -35.98% (4th layer),

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25.00% (20th layer), -14.06% (4th layer), and -29.49% (4th layer), respectively. Compared with FEM results, the maximum relative errors (the relative error where corresponding absolute error

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is less than 0.015 mm is not considered) for Cases 1-7 are 15.85% (23th layer), 13.68% (23th layer), -22.51% (24th layer), 20.04% (23th layer), 20.08% (23th layer), -10.00% (2nd layer), and -7.16% (23th layer), respectively. The deformations initially increase and then decrease with the increase in cutting depth for Cases 1-6. Despite of certain errors, the final deformation surface 22

results of the proposed model agree well with those of FEM and experiments (Fig. 10). Measured deformation results are generally larger than predicted results of the semianalytical and FE model. The main reason for the deviations is that high speed, small cutting

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depth and the use of the coolant greatly decrease the influences of the cutting loads and inducedmachining residual stress. However, the influences of these factors which are not considered in

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the semi-analytical and FE models can’t be completely eliminated in machining experiments.

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Furthermore, the influence of the clamping force also results in errors.

(b)

(c)

(d)

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(a)

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(f)

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(e)

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(g)

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Fig. 9 Machining deformations of vertex on bottom surface versus cutting depth in Cases (a)

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1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, and (g) 7.

(a)

(b) 24

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(d)

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(c)

(f)

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(e)

(g)

Fig. 10 Comparison of final deformation surfaces of Cases (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, and (g) 7. 25

4.2 Influence study of equivalent bending stiffness and initial residual stress Fig. 11 shows the equivalent thicknesses of Cases 1-7 during the machining process. Figs. 9 and 11 indicate that machining deformation is negatively correlated with the equivalent

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thickness for the different thin-walled parts with the same residual stress profile. The equivalent thickness and the equivalent bending stiffness both decrease with the increase in the cutting

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depth. For Cases 1, 2, 4, 5, and 6, the equivalent thickness in X direction decreases slowly after the cutting depth reaches 12 mm due to the stiffening ribs parallel to the X direction, whereas

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that of Cases 3 and 7 decrease linearly due to lack of stiffening ribs in the X direction. Similarly,

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the equivalent thickness in Y direction of Case 1 to 6 decreases slowly after the cutting depth

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reaches 12 mm due to the stiffening ribs in Y direction, whereas that of Case 7 decreases linearly.

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Hence, improving the bending stiffness can effectively reduce machining deformation, and

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the stiffening ribs in one direction can evidently enhance the bending stiffness of this direction.

Fig. 11 Equivalent thicknesses in the (a) X direction, and (b) Y direction. Quantitative relation between the initial residual stress and machining deformation is established using the proposed model. In addition to the initial residual stress σx01, σx02 …, σx0,n26

1, σx0n, σy01, σy02…, σy0,n-1, and σy0n, all other parameters, including equivalent thicknesses, material

properties, and size parameters, are regarded as constants. Subsequently, the vertex deformation is expressed as below: k L W ,  )   (cx ,i x ,0,i c y ,i y ,0,i ), 2 2 i 1

(26)

IP T

w( 

SC R

where cx,i and cy,i are the influence coefficients of the X and Y directions residual stresses in the ith layer, respectively, which are determined by equivalent thicknesses, material properties, and

U

size parameters.

N

Fig. 12 shows that the absolute value of the influence coefficient decreases approximately

A

linearly with the increase in the number of layer. The influence coefficient in X direction cx,i is

M

much larger than the corresponding Y direction coefficient cy,i, due to the difference between

ED

the blank length and width. It also indicates that the residual stress in the top part of the blank has greater influence on the final machining deformation than that in the bottom part with the

PT

same stress magnitude, and the residual stress in the length direction has greater influence than

CC E

that in the width direction. The values of cx,1/cx,24 for Cases 1-7 are 11.78, 11.07, 6.13, 11.34, 11.71, 10.37, and 8.64, respectively, and the absolute values of cx,1/cy,1 for Cases 1-7 are 18.36, 21.61, 7.86, 2.07, 27.44, 45.15, and 9.23, respectively. Therefore, decreasing the residual

A

stresses in the top part and in the length direction is more advantageous to reduce the deformation than other parts and directions.

27

IP T

SC R

Fig. 12 Influence coefficients of initial residual stress, (a) cx,i, and (b) cy,i.

Moreover, the residual stress profile of many types of plate blanks, such as rolled, extruded,

U

and quenched plate, can be considered as the symmetrical distribution along the mid-plane in

N

the thickness direction (Yang et al., 2014), and the stress distribution meets the stress

A

equilibrium condition, that is, the algebraic sum of the stresses in one direction is equal to zero.

M

Fig. 5 shows that the residual stress profile is a typical stress distribution of the rolled plate.

ED

Thus, the σx0,16, σx0,17, …, σx0,29, σx,0,30, σy0,16, σy0,17 …, σy0,29, and σy,0,30, in Eqs. (26) can be

PT

substituted by σx0,15, σx0,14, …, σx0,2, σx,0,1, σy0,15, σy0,14 …, σy0,2, and σy,0,1, respectively. Thus, the stress equilibrium equation is expressed as, (27)

 y ,0,1   y ,0,2  ... y ,0,14   y ,0,15  0.

(28)

CC E

 x ,0,1   x ,0,2  ... x ,0,14   x ,0,15  0,

A

Moreover, the coefficient series (cx,1, cx,2, cx,3,… and cx,24, or cy,1, cy,2, cy,3,… and cy,24,) is

approximately regarded as an arithmetic series, because the influence coefficient changes approximately linearly with the layer number. Thus, Eqs. (29) - (30) are obtained as follows: cx ,15  cx ,16  cx ,14  cx ,17  ...  cx ,i  cx ,n1i  ...  cx ,8  cx ,23  cx ,7  cx ,24 ,

(29) 28

c y ,15  c y ,16  c y ,14  c y ,17  ...  c y ,i  c y ,n1i  ...  c y ,8  c y ,23  c y ,7  c y ,24 .

(30)

Thus, Eq. (26) can be changed into Eq. (31) by considering Eqs. (27) to (30), n

n

(31)

n k n k L W w(  ,  )   (cx ,i  cmx ) x ,0,i   (c y ,i  cm y ) y ,0,i 2 2 i 1 i 1

(32)

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n k n k 2 2 L W w(  ,  )   (cx ,i  cmx ) x ,0,i   rex ,i x ,0,i   (c y ,i  cm y ) y ,0,i   rey ,i y ,0,i 2 2 i 1 i n k 1 i 1 i n k 1

SC R

where cmx= cx,15 + cx,16, cmy= cy,15 + cy,16, rex,i = (cx,15 + cx,16) - (cx,i + cx,n+1-i), and rey,i = (cy,15 + cy,16) - (cy,i + cy,n+1-i), (n=30 and k=24 in the study cases).

U

The rex,i and rey,i are relatively small according to Eqs. (29) and (30). Hence, the final

A

N

deformation can be approximately expressed by Eq. (32). The deformation calculated by Eqs.

M

(26) and (32) are compared in Tab. 2. As is shown in Tab. 2, the difference between the results of Eqs. (26) and (32) is relatively small. The maximum relative error of the seven cases is 5.80%,

ED

which indicates that the final machining deformation is basically determined by the initial

PT

stresses in layers 1-6 and 25-30, whereas the stresses in layers 7-24 have only a slight effect on

CC E

final machining deformation.

Tab. 2 Comparison of results calculated by Eqs. (26) and (32)

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

By Eq. (26) By Eq. (32) Absolute error Relative error

0.322 0.341 0.019 5.79%

0.344 0.363 0.019 5.39%

0.477 0.465 -0.013 -2.64%

0.193 0.204 0.011 5.64%

0.325 0.344 0.019 5.80%

0.374 0.391 0.017 4.56%

1.332 1.311 -0.021 -1.55%

A

Deformations (mm)

Therefore, when the residual stress symmetrically distributes along the mid-plane in

29

thickness direction, the final machining deformation is basically determined by the initial stress within a certain thickness under the blank surface. The residual stress in the middle part has a slight effect on final machining deformation. The thickness of the layer is equal to remaining

IP T

thickness of the bottom plate (This thickness in the present study is 6 mm). In these cases, reducing or eliminating the stresses near the blank surface will evidently reduce the machining

SC R

deformation.

In summary, the aforementioned comparison results validate the proposed model. The

U

deformations calculated by the proposed model agree well with those predicted by FEM

N

simulations and measured by experiments. The semi-analytical model is also more efficient

A

than the FE model. Instead of nonlinear calculation and application of the element birth and

M

death technology, only linear static FEM simulations and analytical calculations are required

ED

for the proposed model. The total computing time is approximately 1/5 ~ 1/2 of that of the FE model. Moreover, the quantitative relation between the initial residual stress and machining

CC E

model.

PT

deformation, which are difficult to calculate using FEM, can be obtained via the semi-analytical

The proposed model has certain limitations. The model is based on the basic hypotheses in

A

Section 2 and effective only in the bending deformation prediction of thin-walled parts with high removal rates (larger than 60% each layer) and machined from rectangular plate blanks. 5. Conclusions A semi-analytical machining deformation prediction model for thin-walled parts machined 30

from rectangle plate blanks is proposed on the basis of the equivalent bending stiffness calculated by FEM and theory of plates and shells. Machining deformations of seven typical study cases are predicted by the semi-analytical model, and corresponding experiments and

IP T

FEM simulations are carried out to validate the prediction model. Results show that the maximum and final deformation surfaces predicted by the proposed model agree well with FEM

SC R

simulations and experiments. Several conclusions are drawn via investigating the influences of equivalent bending stiffness and initial residual stress on the machining deformation:

U

(1) Improving bending stiffness can effectively reduce the machining deformation, and the

N

stiffening ribs in one direction can evidently enhance the bending stiffness in this direction.

A

(2) The absolute value of the influence coefficient decreases approximately linearly with the

M

increase in layer number, and the influence coefficient in the X direction (cx,i) is much larger

ED

than the corresponding Y direction coefficient (cy,i). Thus, reducing the stresses in the top part

PT

of the blank and in the length direction is more advantageous to control the deformation than other parts and directions.

CC E

(3) When the residual stress symmetrically distributes along the mid-plane in the thickness

direction, the final machining deformation is basically determined by the initial stresses within

A

a certain thickness under the blank surface. The residual stresses in the middle part have a slight effect on machining deformation. The thickness is equal to that of the uncut thickness of the workpiece. In these cases, reducing stresses near the blank surface will evidently reduce the machining deformation. 31

The limitations of the proposed model are that the model is proposed based on the basic hypotheses in Section 2, and it is effective only in the deformation prediction of thin-walled parts with high removal rate (larger than 60% each layer) that machined from the rectangle plate

IP T

blanks. Acknowledgements

SC R

The present work is financially aided by Beijing Municipal Natural Science Foundation (3172021).

U

Appendix. Bending deformation of rectangle plates

A

N

When a rectangle plate meets the condition of (1/80-1/100) < h/b (thickness/width) < (1/5-

M

1/8), the theory of plates and shells is applied for solving bending deformations. When two sets of opposite edges are subject to uniform moments (Fig. A1), the bending curvature can be

ED

expressed by the second derivatives of the deflection function (the geometry center of the plate

PT

is defined as the coordinate origin) or the moment, bending stiffness D, Poisson’s ratio μ, and elastic modulus E (Timoshenko, 1959): (A1)

1  2 w( x, y ) M y   M x   , Ry y 2 D(1   2 )

(A2)

CC E

1  2 w( x, y ) M x   M y   , Rx x 2 D(1   2 )

Eh 3 , 12(1   2 )

A

D

(A3)

where Mx and My are the uniform moments that cause X and Y directions bending deformation, respectively, and w(x,y) is the deflection function.

32

In present study, w(0,0) is regarded as 0. Thus, the deflection function is expressed as, M x  M y 2 D(1   ) 2

x2 

M y  M x 2 D(1   2 )

y2 

1 2 1 2 x  y . 2 Rx 2 Ry

(A4)

SC R

IP T

w( x, y ) 

A

CC E

PT

ED

M

A

N

U

Fig. A1 Rectangle plate subjected to bending moment on two sets of opposite edges

33

References Cerutti, X., Mocellin, K., 2015. Parallel finite element tool to predict distortion induced by initial residual stresses during machining of aeronautical parts. Int. J. Mater. Form. 8, 255–

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268. doi:10.1007/s12289-014-1164-0 Cerutti, X., Mocellin, K., Hassini, S., Blaysat, B., Duc, E., 2017. Methodology for aluminium

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part machining quality improvement considering mechanical properties and process conditions. CIRP J. Manuf. Sci. Technol. 18, 18–38. doi:10.1016/j.cirpj.2016.07.004

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Cheng, Y., Zuo, D., Wu, M., Feng, X., Zhang, Y., 2015. Study on simulation of machining

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8, 401–410. doi:10.14257/ijca.2015.8.1.38

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deformation and experiments for thin-walled parts of titanium alloy. Int. J. Control Autom.

D’Alvise, L., Chantzis, D., Schoinochoritis, B., Salonitis, K., 2015. Modelling of part distortion

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due to residual stresses relaxation: An aeronautical case study. Procedia CIRP 31, 447–452.

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Diez, E., Perez, H., Marquez, J., Vizan, A., 2015. Feasibility study of in-process compensation of deformations in flexible milling. Int. J. Mach. Tools Manuf. 94, 1–14. doi:10.1016/j.ijmachtools.2015.03.008

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Dong, H., Ke, Y., 2006. Study on Machining Deformation of Aircraft Monolithic Component by FEM and Experiment. Chinese J. Aeronaut. 19, 247–254. doi:10.1016/S10009361(11)60352-X

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Gao, H., Zhang, Y., Wu, Q., Song, J., 2017. An analytical model for predicting the machining deformation of a plate blank considers biaxial initial residual stresses. Int. J. Adv. Manuf. Technol. 1–14. doi:10.1007/s00170-017-0528-2

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aero-multi-frame parts. Mater. Sci. Eng. A 499, 230–233. doi:10.1016/j.msea.2007.11.137

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Husson, R., Baudouin, C., Bigot, R., Sura, E., 2014. Consideration of residual stress and

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geometry during heat treatment to decrease shaft bending. Int. J. Adv. Manuf. Technol. 72, 1455–1463. doi:10.1007/s00170-014-5688-8

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Izamshah, R., Mo, J.P.T., Ding, S., 2011. Hybrid deflection prediction on machining thin-wall

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monolithic aerospace components. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf. 226, 592– 605. doi:10.1177/0954405411425443

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Jiang, Z., Liu, Y., Li, L., Shao, W., 2014. A novel prediction model for thin plate deflections considering milling residual stresses. Int. J. Adv. Manuf. Technol. 74, 37–45.

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Li, J. guang, Wang, S. qi, 2017. Distortion caused by residual stresses in machining aeronautical aluminum alloy parts: recent advances. Int. J. Adv. Manuf. Technol. 89, 997–1012. doi:10.1007/s00170-016-9066-6

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Liu, L., Sun, J., Chen, W., Sun, P., 2015. Study on the machining distortion of aluminum alloy parts induced by forging residual stresses. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf.

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parts: Thermal modeling and mechanical implications. Int. J. Mach. Tools Manuf. s, 118–

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Masoudi, S., Amini, S., Saeidi, E., Eslami-Chalander, H., 2014. Effect of machining-induced residual stress on the distortion of thin-walled parts. Int. J. Adv. Manuf. Technol. 76, 597–

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Outeiro, J.C., Umbrello, D., M’Saoubi, R., 2006. Experimental and numerical modelling of the residual stresses induced in orthogonal cutting of AISI 316L steel. Int. J. Mach. Tools

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distortion for aluminum alloy and titanium alloy aircraft monolithic component. Int. J. Adv. Manuf. Technol. 70, 1803–1811. doi:10.1007/s00170-013-5431-x

ED

Zhang, Z., Li, L., Yang, Y., He, N., Zhao, W., 2014. Machining distortion minimization for the

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manufacturing of aeronautical structure. Int. J. Adv. Manuf. Technol. 73, 1765–1773.

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doi:10.1007/s00170-014-5994-1

38

N

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Figures

A

CC E

PT

ED

M

A

Fig. 1 Finite element (FE) model for calculating equivalent thickness

39

IP T

A

CC E

PT

ED

M

A

N

U

SC R

Fig. 2 Stress redistribution in the remaining workpiece

40

IP T SC R U N A M ED PT

A

CC E

Fig. 3 The flowchart of the semi-analytical model

41

IP T SC R U N A M ED PT CC E A Fig. 4 Size and shape of Cases 1-7

42

IP T SC R

A

CC E

PT

ED

M

A

N

U

Fig. 5 Initial residual stress profile

43

IP T SC R U N A M

A

CC E

PT

ED

Fig. 6 Machining experiments

44

IP T SC R U N A M ED

A

CC E

PT

Fig. 7 Deformation Machining strategies of Cases 1-7

45

IP T

A

CC E

PT

ED

M

A

N

U

SC R

Fig. 8 Deformation measurement setup and measuring points

46

IP T

(b)

(c)

(d)

(e)

(f)

A

CC E

PT

ED

M

A

N

U

SC R

(a)

47

IP T SC R

(g)

Fig. 9 Machining deformations of vertex on bottom surface versus cutting depth in Cases (a)

A

CC E

PT

ED

M

A

N

U

1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, and (g) 7.

48

IP T

(b)

(d)

A

CC E

PT

ED

(c)

M

A

N

U

SC R

(a)

(e)

(f)

49

IP T SC R

(g)

Fig. 10 Comparison of final deformation surfaces of Cases (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f)

A

CC E

PT

ED

M

A

N

U

6, and (g) 7.

50

IP T

A

CC E

PT

ED

M

A

N

U

SC R

Fig. 11 Equivalent thicknesses in the (a) X direction, and (b) Y direction.

51

IP T

A

CC E

PT

ED

M

A

N

U

SC R

Fig. 12 Influence coefficients of initial residual stress, (a) cx,i, and (b) cy,i.

52

Tables Tab. 1 Material properties of 7075-T6 aluminium alloy Elastic modulus/ (GPa)

Poisson’s ratio 0.33

2810

Yield strength/ (MPa) 518

A

CC E

PT

ED

M

A

N

U

SC R

IP T

71.7

Density/ (kg/m3)

53

Tab. 2 Comparison of results calculated by Eqs. (26) and (32)

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

By Eq. (26) By Eq. (32) Absolute error Relative error

0.322 0.341 0.019 5.79%

0.344 0.363 0.019 5.39%

0.477 0.465 -0.013 -2.64%

0.193 0.204 0.011 5.64%

0.325 0.344 0.019 5.80%

0.374 0.391 0.017 4.56%

1.332 1.311 -0.021 -1.55%

A

CC E

PT

ED

M

A

N

U

SC R

IP T

Deformations (mm)

54