A novel method to alleviate flash-line defects in coining process

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<p>A</p><p>J</p><p>C</p><p>a</p><p>A</p><p>R</p><p>R</p><p>A</p><p>A</p><p>K</p><p>C</p><p>F</p><p>C</p><p>R</p><p>F</p><p>1</p><p>a</p><p>e</p><p>d</p><p>i</p><p>t</p><p>O</p><p>s</p><p>r</p><p>c</p><p>t</p><p>t</p><p>t</p><p>s</p><p>o</p><p>w</p><p>T</p><p>p</p><p>A</p><p>h</p><p>h</p><p>e</p><p>t</p><p>0</p><p>h</p><p>Precision Engineering 37 (2013) 389– 398</p><p>Contents lists available at SciVerse ScienceDirect</p><p>Precision Engineering</p><p>jou rna l h om epage: www.elsev ier .com/ locate /prec is ion</p><p>novel method to alleviate flash-line defects in coining process</p><p>iang-ping Xu, Kamran A. Khan, Tamer El Sayed ∗</p><p>omputational Solid Mechanics Laboratory (CSML), Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Saudi Arabia</p><p>r t i c l e i n f o</p><p>rticle history:</p><p>eceived 24 June 2012</p><p>eceived in revised form 6 November 2012</p><p>ccepted 7 November 2012</p><p>vailable online 1 December 2012</p><p>a b s t r a c t</p><p>We employ a finite element framework based on a dynamic explicit algorithm to predict the flash-</p><p>line defects in the coining process. The distribution of the flash-line is obtained by building a radial</p><p>friction work model at the element level. The elasto-plastic behavior of porous materials undergoing</p><p>large deformations is considered where the constitutive level updates are the result of a local variational</p><p>minimization problem. We study the material flow at different strokes of the die across the entire coining</p><p>eywords:</p><p>oining</p><p>lash line</p><p>onstitutive model</p><p>adial friction work</p><p>inite element analysis</p><p>process and observe that the change in the flow direction of the material in the rim region may contribute</p><p>to the flash lines. Our proposed framework shows that a part of the rim region in which the flash-line</p><p>defects appear is consistent with the reported experimental results. We also propose a novel method of</p><p>redesigning the rim geometry of the workpiece to alleviate the flash-line defects which also shows good</p><p>agreement with experiments.</p><p>© 2012 Elsevier Inc. All rights reserved.</p><p>. Introduction</p><p>In the manufacturing of coins, the rejection rate of products is</p><p>pproximately 10–30% and up to 50% for the coins with large diam-</p><p>ters [1]. One factor relevant to this high rejection rate is flash-line</p><p>efects. As shown in Fig. 1.1, flash lines are distributed as scratches</p><p>n the radial direction on a coin surface. Engineers have been unable</p><p>o capture the origin and formation mechanism of these flash lines.</p><p>nce the defect occurs, some treatments, such as trimming the die</p><p>urface, improving the die surface condition by using lubrication or</p><p>edesigning the geometry of the workpiece with the assistance of</p><p>oin designers, are utilized but without the guidance of mechanical</p><p>heory. Sometimes, the defects automatically vanish after a certain</p><p>ime in the production process with the help of one or more of these</p><p>reatments. With these treatments, the defects can be avoided to</p><p>ome extent, but they remain an unresolved issue due to the lack</p><p>f knowledge of their origin. For instance, if the material of the</p><p>orkpiece or the dies is changed, the defects likely occur again.</p><p>his instability in quality control hinders the development of new</p><p>roducts and increases the cost in producing gold and silver coins.</p><p>s described in the literature [2], surface engineering technologies</p><p>ave been used to increase the life of coin dies and experiments</p><p>ave been conducted to validate the technologies. In making coins,</p><p>specially commemorative coins, the weight error is critically con-</p><p>rolled by the mint [3]. Surface treatments, i.e., polishing, burning</p><p>∗ Corresponding author. Tel.: +966 544700060.</p><p>E-mail address: tamer.elsayed@kaust.edu.sa (T. El Sayed).</p><p>141-6359/$ – see front matter © 2012 Elsevier Inc. All rights reserved.</p><p>ttp://dx.doi.org/10.1016/j.precisioneng.2012.11.001</p><p>and coating, are typically not used on coins because they add to or</p><p>reduce the weight of coins.</p><p>With the development of the finite element method (FEM), it is</p><p>possible to simulate the forging and stamping process [4,5]. Only a</p><p>limited number of studies have been performed to simulate coin-</p><p>ing, however, due to the proprietary nature of the work. Several</p><p>independent FE frameworks have been developed by assuming</p><p>two-dimensional (2D) [6], three-dimensional (3D) [7] and axisym-</p><p>metric geometries [8] to simulate the precision coining process.</p><p>However, the rigid-plastic constitutive model used by Marques and</p><p>Martins [6] and Choi et al. [7] cannot describe the material flow</p><p>accurately. Most FE simulations in the literature are conducted by</p><p>using axisymmetric FE models that analyze the metal flow between</p><p>the dies and determine the best geometry of the workpiece [8]. A</p><p>2D dynamic implicit FE model was developed by Fabbri [9,10] to</p><p>analyze the material flow and stress distribution for axisymmetric</p><p>coins. As we know, there are many coins that are not axisymmetric</p><p>due to the complexity of the patterns on the coins’ surfaces. Based</p><p>on the commercial software ABAQUS, a finite element design plat-</p><p>form was developed by Guo [11] to study the material flow during</p><p>the coining process and tools design in the manufacture of coins.</p><p>By using FEM with the variational constitutive model proposed by</p><p>Siddiq et al. [12], the embossing force was predicted and verified</p><p>with experimental results by Xu et al. [13]. Based on the J2 plasticity</p><p>model, a modified friction model was used to describe the friction</p><p>behaviors and a radial friction work (RFW) model was employed to</p><p>study the flash lines by Zhong et al. [14].</p><p>To our best knowledge, there is no model available that is able</p><p>to address flash-line defects during the manufacturing process of</p><p>coins except for Zhong et al. [14] and Xu et al. [15]. Recently, Xu</p><p>dx.doi.org/10.1016/j.precisioneng.2012.11.001</p><p>http://www.sciencedirect.com/science/journal/01416359</p><p>http://www.elsevier.com/locate/precision</p><p>mailto:tamer.elsayed@kaust.edu.sa</p><p>dx.doi.org/10.1016/j.precisioneng.2012.11.001</p><p>390 J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398</p><p>e</p><p>C</p><p>i</p><p>t</p><p>d</p><p>J</p><p>m</p><p>u</p><p>l</p><p>o</p><p>e</p><p>s</p><p>e</p><p>w</p><p>u</p><p>l</p><p>a</p><p>t</p><p>f</p><p>2</p><p>w</p><p>d</p><p>c</p><p>p</p><p>s</p><p>p</p><p>b</p><p>a</p><p>2</p><p>t</p><p>f</p><p>fi</p><p>r</p><p>t</p><p>2</p><p>s</p><p>N</p><p>Fig. 1.1. An example of flash-line defects.</p><p>t al. [15,16] and Xu [17] developed a commercial software called</p><p>oinForm to analyze the material flow during the coining process</p><p>ncluding the prediction of the embossing force and the optimiza-</p><p>ion of the geometry of the workpiece as well as the gaps among</p><p>ies. In previous studies, the homogeneous material following the</p><p>2 plasticity model was employed. In this study, we assume that the</p><p>aterial contains defects, such as voids, and we account for the vol-</p><p>metric expansion and contraction of the voids under mechanical</p><p>oading. As a continuation of previous work [13], this study focuses</p><p>n flash-line defects. In our previous work [13], we considered the</p><p>ffects of the parameters used in this constitutive model on the</p><p>imulation results. It must be stressed that the constitutive model</p><p>mployed in [14,15] was based on the classic J2 plastic model, while</p><p>e employ a variational constitutive model [12] for constitutive</p><p>pdates of porous elasto-plastic materials. Here, we study flash-</p><p>ine defects in the view of material flow as well as the RFW model</p><p>nd propose a novel solution of redesigning the rim geometry of</p><p>he workpiece, which differs from the solution of improving the</p><p>riction condition [14].</p><p>The remainder of this paper is organized as follows. In Section</p><p>, we present the integration of the variational constitutive model</p><p>ith a dynamic explicit finite element algorithm. We present a brief</p><p>escription of the variational model, free energy expressions and</p><p>onstitutive updates. In Section 3, we introduce the RFW model for</p><p>redicting the distribution of the flash lines and then describe a</p><p>tudy of material flow to observe the origin of the defects. We then</p><p>ropose a solution to mitigate these defects and verify the solution</p><p>y experimental results in Section 4. Finally, we offer conclusions</p><p>nd future directions for research in Section 5.</p><p>. Integration of the variational constitutive model with</p><p>he dynamic explicit finite element</p><p>framework</p><p>Dynamic explicit FE algorithms are widely used in sheet metal</p><p>orming [18] and stamping [19] as well as other manufacturing</p><p>elds to avoid convergence problems encountered in implicit algo-</p><p>ithms. In this study, we use explicit time integration to simulate</p><p>he coining process.</p><p>.1. Dynamic equilibrium equation</p><p>In the present FE framework, we use the 8-node solid element</p><p>hown in Fig. 2.1. The shape function matrix is expressed as follows:</p><p>= [N1I, N2I, N3I, N4I, N5I, N6I, N7I, N8I], (2.1)</p><p>Fig. 2.1. The 8-node solid element in a natural coordinate system.</p><p>where I is the unit matrix in three dimensions and</p><p>Ni = (1 + �i�)(1 + �i�)(1 + �i�)/8, i = 1, 2, . . ., 8.</p><p>The global finite element equation can be expressed as [20]:</p><p>MÜ + CU̇ = P − F, (2.2)</p><p>where U̇ and Ü are the velocity and acceleration vectors of the</p><p>nodes. The lumped mass matrix, the damping matrix, the external</p><p>force and the internal force are expressed as</p><p>M =</p><p>∑∫</p><p>Ve</p><p>�NTNdV, (2.3)</p><p>C =</p><p>∑∫</p><p>Ve</p><p>�NTNdV, (2.4)</p><p>P =</p><p>∑∫</p><p>Ve</p><p>NTbdV +</p><p>∑∫</p><p>Sp</p><p>NTpdS +</p><p>∑∫</p><p>Sc</p><p>NTqdS, (2.5)</p><p>F =</p><p>∑∫</p><p>Ve</p><p>BT�dV, (2.6)</p><p>where � is the material density, � is the damping coefficient, b is</p><p>the volume force, p is the external force and q is the contact force. B</p><p>contains the derivatives of the shape function and � is the Cauchy</p><p>stress.</p><p>Generally, we define C = ˛M, where ̨ is the damping coeffi-</p><p>cient defined in [21]. For a detailed discussion of this coefficient,</p><p>we refer the reader to [21]. Thus, Eq. (2.2) is changed into a number</p><p>of independent equations:</p><p>Mi,iÜi + Ci,iU̇i = Pi − Fi. (2.7)</p><p>2.2. The constitutive model</p><p>The constitutive updating procedure for porous elasto-plastic</p><p>materials proposed by Siddiq et al. [12] is used in this study and</p><p>incorporated in the finite element framework developed by Xu et al.</p><p>[15]. The capability of the FE software has been enhanced to account</p><p>for the porosity and void coalescence effect of the metals in ana-</p><p>lyzing the various manufacturing processes. The constitutive model</p><p>combines the classic von Mises deviatoric plasticity model with the</p><p>volumetric plastic expansion due to the void growth followed by</p><p>damage initiation and evolution due to void coalescence. A short</p><p>review of the void coalescence model for rate-independent porous</p><p>plasticity proposed by Siddiq et al. [12] is presented in this section.</p><p>A few equations that are relevant to the present study are given in</p><p>Appendix A.</p><p>ngineering 37 (2013) 389– 398 391</p><p>r</p><p>u</p><p>A</p><p>w</p><p>s</p><p>r</p><p>c</p><p>m</p><p>i</p><p>t</p><p>F</p><p>w</p><p>w</p><p>t</p><p>p</p><p>r</p><p>w</p><p>m</p><p>c</p><p>o</p><p>2</p><p>s</p><p>T</p><p>a</p><p>A</p><p>w</p><p>s</p><p>g</p><p>{</p><p>d</p><p>v</p><p>i</p><p>S</p><p>i</p><p>F</p><p>b</p><p>�</p><p>m</p><p>W</p><p>s</p><p>p</p><p>c</p><p>i</p><p>a</p><p>c</p><p>c</p><p>J.-p. Xu et al. / Precision E</p><p>We assume that the mechanical response of elasto-plastic mate-</p><p>ials can be represented by a free-energy density per unit of</p><p>ndeformed volume equation that takes the following form:</p><p>= A(F, Fp, �p, �p, �p), (2.8)</p><p>here �p and �p are the effective deviatoric and volumetric plastic</p><p>trains, respectively. �p is defined as the percentage of voids expe-</p><p>iencing coalescence at a given material point and subjected to the</p><p>onstraint of 0 ≤ �p ≤ 1. The constitutive framework is based on the</p><p>ultiplicative decomposition of the total deformation gradient, F,</p><p>nto an elastic part, Fe, and a plastic part, Fp [22]. We assume that</p><p>he plastic deformation rate obeys the following flow rule:</p><p>˙ p</p><p>Fp−1 = �̇pM + �̇pN + ̌�̇pN, (2.9)</p><p>here �̇p ≥ 0, �̇p ≥ 0 and ̌ is a material constant that, together</p><p>ith �p, defines the volumetric strain due to void coalescence. The</p><p>ensors M and N are the directions of the deviatoric and volumetric</p><p>lastic deformation rates [23].</p><p>When the deformation is purely volumetric, the flow rule</p><p>educes to</p><p>d</p><p>dt</p><p>(ln Jp) = ±( ˙�p + ˙ˇ�p), (2.10)</p><p>here the plus sign corresponds to the void expansion and the</p><p>inus sign corresponds to void collapse. Jp is the plastic Jacobian</p><p>orresponding to the void growth and coalescence effects. Details</p><p>n Jp are presented in Appendix A.</p><p>.3. Incremental constitutive updates</p><p>This section briefly describes the main steps involved in the con-</p><p>titutive updating procedure proposed by Ortiz and Stainier [23].</p><p>he free energy density function in Eq. (2.8) is assumed to have an</p><p>dditive structure:</p><p>(F, Fp, �p, �p) = We(Fe) + Wp(�p, �p, �p), (2.11)</p><p>here We(Fe) and Wp(�p, �p, �p) are the elastic strain-energy den-</p><p>ity and the plastic stored energy, respectively.</p><p>The incremental solution procedure assumes that, for a</p><p>iven time interval, [tn, tn+1], the state of the material, i.e.,</p><p>Fn, Fp</p><p>n, �p</p><p>n, �p</p><p>n, �p</p><p>n}, is known at time tn. For a given deformation gra-</p><p>ient, Fn+1, at the time tn+1, our goal is to determine the updated</p><p>alues of the first Piola–Kirchhoff stress tensor, Pn+1, as well as</p><p>nternal variables {Fp</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1} at time tn+1. Following</p><p>iddiq et al. [12], the incremental energy function for a rate-</p><p>ndependent porous plasticity model can be expressed as</p><p>(Fn+1; �p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1, M, N) = We(�e</p><p>n+1) + Wp(�p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1).</p><p>(2.12)</p><p>If we define the effective work of deformation density, Wn(Fn+1),</p><p>y minimizing the left term of Eq. (2.12) with respect to</p><p>p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1, M and N, then the incremental energy function</p><p>ay be expressed in a variational form as</p><p>n(Fn+1) = min</p><p>�p</p><p>n+1</p><p>,�p</p><p>n+1</p><p>,�p</p><p>n+1</p><p>,M,N</p><p>F(Fn+1; �p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1, M, N) (2.13)</p><p>ubjected to constraints, i.e., trM = 0, M · M = 3/2, N = ± I/3 and the</p><p>lastic irreversibility condition (�̇p ≥ 0, �̇p ≥ 0). During the coales-</p><p>ence phase, the minimization of the incremental energy function</p><p>s performed with respect to �p</p><p>n+1 and �p</p><p>n+1. Therefore, the vari-</p><p>ble �p</p><p>n+1 denotes the volumetric plastic strain at the onset of</p><p>oalescence and it is kept constant throughout the void coales-</p><p>ence process. Once the minimization is performed, the values of</p><p>Fig. 3.1. Two workpiece geometries commonly used by mints (a) and the finite</p><p>element assembly (b).</p><p>the internal variables and directions of the plastic flow are updated.</p><p>It can be shown [23] that Wn(Fn+1) acts as a potential for the first</p><p>Piola–Kirchhoff stress, Pn+1, and thus the constitutive update pos-</p><p>sesses an incremental potential structure, i.e.,</p><p>Pn+1 =</p><p>∂Wn(Fn+1; �p</p><p>n+1, �p</p><p>n+1, �p</p><p>n+1, M, N)</p><p>∂Fn+1</p><p>. (2.14)</p><p>The detailed implementation of the constitutive updating based</p><p>on logarithmic strain can be found in [12].</p><p>3. Study of the RFW model and material flow</p><p>In this section, the RFW model is built on the coining surface to</p><p>study the flash lines. Also, the material flow in the coining process</p><p>is studied.</p><p>The old and new geometries of the workpiece and the finite</p><p>element assembly are shown in Fig. 3.1 (these two geometries are</p><p>widely used by mints) and the material parameters used here are</p><p>listed in Table 1. The stroke of the top die is 1.8 mm. The material</p><p>flows are different between the top surface and bottom surface of</p><p>the coin when the top die moves down and the bottom die remains</p><p>392 J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398</p><p>Table 1</p><p>Material parameters.</p><p>E (Pa)</p><p>� (kg/m3) �0 (Pa) εp</p><p>0 n N0 (m−3) a0 (�m)</p><p>8.2 × 1010 0.38 10,500 5 × 107 0.01 3.1 1010 134</p><p>s</p><p>u</p><p>o</p><p>d</p><p>3</p><p>e</p><p>a</p><p>d</p><p>a</p><p>m</p><p>a</p><p>t</p><p>e</p><p>s</p><p>t</p><p>e</p><p>f</p><p>c</p><p>f</p><p>v</p><p>n</p><p>t</p><p>d</p><p>i</p><p>t</p><p>a</p><p>p</p><p>c</p><p>T</p><p>d</p><p>w</p><p>w</p><p>r</p><p>m</p><p>i</p><p>i</p><p>o</p><p>3</p><p>o</p><p>t</p><p>w</p><p>(</p><p>r</p><p>t</p><p>d</p><p>c</p><p>the material in the middle begins to contact the surface of the</p><p>top die and is forced to move down, as shown in Fig. 3.4b at the</p><p>stroke of 0.7 mm. In Fig. 3.4c at the stroke of 0.9 mm, the collar</p><p>starts to act on the workpiece when the material in the rim area</p><p>fc ˛1 ˛2 ̌ εp</p><p>c �p</p><p>c</p><p>0.3 1.0 1.0 1.0 0.55 1.0</p><p>tationary. Similarly, they are different when the bottom die moves</p><p>p and the top die remains stationary. In the present study, we focus</p><p>n the material flow on the top surface when the top die moves</p><p>own.</p><p>.1. The radial friction work model</p><p>There are numerous cavities on the die surface due to exist-</p><p>nce of complex patterns. Therefore, a large number of elements</p><p>re needed to describe these patterns in the FE simulations. These</p><p>ie elements ensure that the variations in the normal vectors of</p><p>djacent elements are small. Hence, when the node of an element</p><p>eshed on the workpiece</p><p>slides across two die elements, the aver-</p><p>ge value of the two normal vectors is accurate enough to calculate</p><p>he frictional force. In addition, the incremental time step in the</p><p>mployed dynamic explicit algorithm is small enough to keep the</p><p>liding on one element in many cases, except for the case in which</p><p>he initial position of the node is very close to the boundary of an</p><p>lement.</p><p>The Coulomb friction model is introduced before the radial</p><p>riction work model. The increment in friction force of node i is</p><p>alculated by the following Coulomb friction law:</p><p>ci =</p><p>{</p><p>−</p><p>(fi.n)t ‖vr‖ > 0</p><p>0 ‖vr‖ = 0</p><p>(3.1)</p><p>r = vw − vd (3.2)</p><p>is the unit normal vector of the die element; t is the unit vec-</p><p>or on the plane of the die element, representing the displacement</p><p>irection of node i; fi is the increment in the nodal force of node i</p><p>n the current step n; vr, vw and vd are the relative sliding velocity,</p><p>he velocity of the node i and the velocity of the die, respectively;</p><p>nd</p><p>is the coefficient of friction.</p><p>To measure the radial work done by friction between the work-</p><p>iece and dies, an RFW model is built on the top surface of the</p><p>oin. It should be noted that this model is limited to circular coins.</p><p>he work is the product of the radial friction vector and the radial</p><p>isplacement vector for one node and expressed as (see Fig. 3.2)</p><p>n+1 = wn + fn</p><p>r Un</p><p>r , (3.3)</p><p>here wn, fn</p><p>r , and Un</p><p>r are the radial friction work, increment in nodal</p><p>adial frictional force and increment in radial displacement incre-</p><p>ent in the nth incremental step for each node, respectively. wn+1</p><p>s the work in the (n + 1)th incremental step. The work of each node</p><p>s accumulated in each step. We utilize this model in the prediction</p><p>f distribution of the flash lines as detailed in Section 3.3.</p><p>.2. Material flow in the coining process</p><p>The material flow in coining is very different from other types</p><p>f forming processes. The main reason for this difference is due</p><p>o the numerous cavities on the die surface as well as gaps. Here,</p><p>e introduce three kinds of gaps in coining, as shown in Fig. 3.3</p><p>xoz plane). Gap 1 represents the space between the dies and the</p><p>im area of the workpiece. Gap 2 represents the space between the</p><p>op/bottom die and the collar. Gap 3 represents the cavities on the</p><p>ie surface. Once gaps 1 and 3 are completely filled, a satisfactory</p><p>oin would be obtained. However, filling these two gaps is relevant</p><p>Fig. 3.2. The radial friction work model.</p><p>to the size of gap 2. As discussed by Xu [17], 0.1 mm of the value of</p><p>one side of gap 2 is chosen here, suggesting that the radius of the</p><p>collar is 19.9 mm (see Fig. 3.1a, the outer radius of the workpiece is</p><p>19.8 mm).</p><p>We first study the material flow on the top surface of the coin</p><p>in the coining process by using the old geometry in Fig. 3.1a. The</p><p>velocities of nodes on the xoz plane shown in Fig. 3.3 are plot-</p><p>ted in Figs. 3.4a–c and 3.5a–c, in which the length of the arrows</p><p>shows the magnitude of the velocity. At the stroke of 0.3 mm shown</p><p>in Fig. 3.4a, the material in the surface region, Q1(7.8 ≤ r ≤ 19.8)</p><p>flows toward the rim area and the material in the middle region,</p><p>Q2(0 ≤ r ≤ 7.8), moves toward the top die. It is easy to understand</p><p>why the material moves up in region Q2. In Fig. 3.4a, only the</p><p>material in the rim area contacts the top and bottom dies while</p><p>the material in the middle area contacts nothing. Once the top</p><p>die moves down, the material in the middle moves up under the</p><p>action of the two dies. With the further movement of the top die,</p><p>Fig. 3.3. Three gaps on the xoz plane.</p><p>J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398 393</p><p>on th</p><p>r</p><p>I</p><p>o</p><p>m</p><p>s</p><p>t</p><p>c</p><p>t</p><p>Fig. 3.4. Plots of the velocities of the nodes</p><p>eaches the collar. The material in region Q2 flows into the inside.</p><p>n addition, the material in the inside half region, Q31(17 ≤ r ≤ 18.5),</p><p>f the rim region, Q3(17 ≤ r ≤ 19.9), changes the direction of its</p><p>ovement and moves into the inside while the material in the out-</p><p>ide half region, Q32(18.5 ≤ r ≤ 19.9), of the rim region, continues</p><p>o move outside. Similarly, the material in region Q12(4 ≤ r ≤ 17)</p><p>hanges the direction of its movement. These observations indicate</p><p>hat</p><p>Fig. 3.5. Plots of the velocities of the nodes on th</p><p>e section xoz plane using the old geometry.</p><p>(1) Initially, the material flows into the rim region under the action</p><p>of top and bottom dies.</p><p>(2) The material begins to fill the cavities of the dies once the collar</p><p>acts on the workpiece.</p><p>Similar observations were made by Fabbri [9,10] in finite ele-</p><p>ment simulations. The filling continues until the stroke reaches</p><p>1.8 mm. These phenomena are shown in Fig. 3.5a–c. With the</p><p>e section xoz plane using the old geometry.</p><p>394 J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398</p><p>f old g</p><p>d</p><p>t</p><p>s</p><p>c</p><p>s</p><p>t</p><p>c</p><p>c</p><p>m</p><p>d</p><p>d</p><p>3</p><p>o</p><p>i</p><p>Fig. 3.6. Flash lines in cases o</p><p>eeper movement of the top die, the filling of the rim area and</p><p>he cavities of the dies continues.</p><p>Region Q31(17 ≤ r ≤ 18.5) deserves more attention for two rea-</p><p>ons. The first reason is that some of the material in this region</p><p>ontacts the top die throughout the coining process. The other rea-</p><p>on is the change in the direction of the flow of the material in</p><p>his region. We suggest that the flash lines are scratches left on the</p><p>oin’s surface due to the long contact time with the top die and the</p><p>hange in the direction of the flow when the material in region Q31</p><p>oves on the top surface. Fig. 3.6a presents images of flash line</p><p>efects. Our simulation results agree well with the experimental</p><p>ata and predict that the flash lines are in region Q31.</p><p>.3. Prediction of distribution of the flash lines</p><p>Based on the RFW model built on the coin surface, the contour</p><p>f the work on the coin surface at the 1.8 mm stroke is plotted</p><p>n Fig. 3.7a. The corresponding deformed shapes of the coin from</p><p>eometry and new geometry.</p><p>the simulation and experiment at the 1.8 mm stroke are plotted in</p><p>Fig. 3.8. The figures show that the filling of the cavities and rim space</p><p>is completed. Fig. 3.7a shows that the large RFW values gather in</p><p>the (17.46 ≤ r ≤ 18.68) region. In this region, the possibility of the</p><p>defect would be greater than in the other zones shown in Fig. 3.6a.</p><p>4. Solution for mitigating flash line defects</p><p>From the above analysis, we draw the conclusion that flash</p><p>line defects can be mitigated by redesigning the geometry of the</p><p>workpiece, which can thus reduce the RFW and avoid the change</p><p>of displacement direction in region Q31. We have used a new</p><p>geometry as shown in Fig. 3.1a. However, the geometry of the</p><p>dies and the material parameters are unchanged in the follow-</p><p>ing analysis. The material flows in this case are demonstrated</p><p>in Figs. 4.1a–c and 4.2a–c. At the 0.3 mm stroke, the material</p><p>flows into region Q2 remain the same in both cases of the old</p><p>and new geometries while the flow in the region Q1 is different.</p><p>J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398 395</p><p>l frict</p><p>A</p><p>t</p><p>s</p><p>r</p><p>w</p><p>Fig. 3.7. Contours of the radia</p><p>s we can see in Fig. 4.1a, the material in region 7.8 ≤ r ≤ 17 flows</p><p>oward the inside and continues in the same direction in the sub-</p><p>equent coining process. At the 0.7 mm stroke, the material in</p><p>egion Q3 flows outside for the new geometry case (see Fig. 4.1b)</p><p>hile the material in the left half of Q3 flows inside and that</p><p>Fig. 3.8. Deformed sha</p><p>ion work on the coin surface.</p><p>in the right half flows outside for the old geometry case (see</p><p>Fig. 3.4b). As expected, the flow directions of the material in region</p><p>Q31 change during the coining process. With the further move-</p><p>ment of the top die, the material flows to fill the cavities and rim</p><p>space, as shown in Fig. 4.2a–c. It can be seen in Fig. 3.7b that the</p><p>pes of the coin.</p><p>396 J.-p. Xu et al. / Precision Engineering 37 (2013) 389– 398</p><p>des on</p><p>m</p><p>g</p><p>o</p><p>(</p><p>c</p><p>r</p><p>Fig. 4.1. Plots of the velocities of the no</p><p>aximum value of RFW reduces from 1.06 J to 0.56 J when the new</p><p>eometry is used. We consider several nodes on the line AB to</p><p>bserve the variations of nodes in the region with large</p><p>RFW values</p><p>Fig. 3.7a). In Fig. 4.3, we find that the RFW values decrease, espe-</p><p>ially in the Q31 region, after the rim geometry of the workpiece is</p><p>edesigned.</p><p>Fig. 4.2. Plots of the velocities of the nodes on</p><p>the xoz plane using the new geometry.</p><p>We note that the material flows on the bottom surface</p><p>of the coin are different from those on the top surface</p><p>between Figs. 3.4, 3.5, 4.1 and 4.2. We suggest that this dif-</p><p>ference is caused by the movement of the top die. As we</p><p>know, the top die first compresses the top surface of the</p><p>workpiece and then the forces are spread to the bottom die to</p><p>the xoz plane using the new geometry.</p><p>J.-p. Xu et al. / Precision Enginee</p><p>s</p><p>e</p><p>t</p><p>t</p><p>5</p><p>l</p><p>c</p><p>1</p><p>2</p><p>3</p><p>4</p><p>5</p><p>2</p><p>where</p><p>and � are the shear and bulk modulus, respectively. �e,dev</p><p>Fig. 4.3. RFW values on the line AB.</p><p>hape the bottom surface of the workpiece. The material flows are</p><p>xchanged if the bottom die moves up and the top die remains sta-</p><p>ionary. The result when both dies move toward the workpiece is</p><p>he subject of a future study.</p><p>. Conclusions</p><p>This study focuses on the prediction of the distribution of flash</p><p>ines by building an RFW model on the coin surface. Five important</p><p>onclusions can be drawn from this analysis:</p><p>At the beginning of the coining process, whether the material</p><p>flows toward the outside or the inside under the action of the</p><p>top and bottom dies depends on the geometry of the rim of the</p><p>workpiece. The material then begins to fill the cavities of dies</p><p>once the collar contacts with the workpiece.</p><p>The flow direction of the edge material contacting the top die dur-</p><p>ing the process changes before the rim area is filled completely.</p><p>These changes generate the scratches, like flash lines, left on the</p><p>surface of the coin.</p><p>By building a radial friction work model on the top surface of</p><p>the coin, we can predict the distribution of the flash lines. Our</p><p>results agree well with experimental data. Also, one solution to</p><p>mitigating these defects is to redesign the geometry of the rim. As</p><p>expected, the flow direction of the material in region Q31 remains</p><p>unvaried and the maximum radial friction work is reduced. In the</p><p>mean time, the mitigation of the defects is verified by experimen-</p><p>tal observations.</p><p>Any redesign of the geometry should make sure that the material</p><p>in region Q31 flows inside once the die acts on the workpiece.</p><p>Otherwise, the redesign is not satisfactory. In practice, such</p><p>redesigning and re-coining may take 3–6 months because the</p><p>dies used in the rimming process prior to the coining process</p><p>must be redesigned. However, it might take only a few hours</p><p>to simulate the re-coining process by using our FE framework</p><p>because there is no need to simulate the rimming process.</p><p>The material flows on the top and bottom surfaces of the coin are</p><p>different due to the direction of the movement of the positive die.</p><p>It will be interesting to study the material flow with the move-</p><p>ment of bottom die or movements of both the top and bottom</p><p>dies toward the workpiece.</p><p>ring 37 (2013) 389– 398 397</p><p>Acknowledgment</p><p>This work was fully funded by KAUST baseline research funds.</p><p>Appendix A.</p><p>A.1. The void coalescence model</p><p>In the deformed configuration, the void volume fraction (or</p><p>porosity), f ∈ R, is defined as the total volume of voids per unit</p><p>of the current volume, that is,</p><p>Jpf = N0</p><p>4�a3</p><p>3</p><p>, (A.1)</p><p>where N0 is the initial void density and a is a mean radius of the</p><p>void. The coalescence model is based on the following assumptions.</p><p>(1) The void coalescence starts at a critical porosity, fc, which is</p><p>regarded as a material property. (2) The void volume fraction can</p><p>be expressed as the sum of void growth and coalescence effects, i.e.,</p><p>f = fgrowth + fcoal, with fgrowth and fcoal representing the void growth</p><p>and coalescence void volume fractions, respectively. (3) The void</p><p>coalescence process is simulated by assuming an equivalent void</p><p>growth model but using an artificially increased void radius, ã.</p><p>Using these assumptions, the porosity contribution for void growth</p><p>and coalescence can be expressed as</p><p>Jpfgrowth = (1 − �p)N0</p><p>4�a3</p><p>3</p><p>, Jpfcoal = (�p)N0</p><p>4�ã3</p><p>3</p><p>, (A.2)</p><p>with(</p><p>ã</p><p>a</p><p>)3</p><p>= 1 + ˛1(f − fc)˛2 , (A.3)</p><p>where ˛1 and ˛2 are material parameters. The current void volume</p><p>fraction can be written using Eq. (A.2) as</p><p>Jpf = N0</p><p>4�a3</p><p>3</p><p>� (A.4)</p><p>with</p><p>� = (1 − �p) + �p</p><p>(</p><p>ã</p><p>a</p><p>)3</p><p>. (A.5)</p><p>with f0 as the initial local volume fraction of the voids, the plastic</p><p>Jacobian and the current void volume fraction can be related by</p><p>Jp = 1 − f0</p><p>1 − f</p><p>. (A.6)</p><p>A.2. Free energy functions</p><p>To fulfill the requirement of material frame indifference, We</p><p>must depend on Fe through the elastic right Cauchy–Green ten-</p><p>sor, Ce. We can write the elastic strain-energy density in terms of</p><p>Hencky’s logarithmic strain measure, i.e., �e = 1/2 log(Ce) and conse-</p><p>quently We(Fe)� We(�e). The use of the logarithmic strain measure</p><p>allows the development of finite strain constitutive models in a</p><p>small strain framework by operating purely at the level of kine-</p><p>matics (see [24]).</p><p>For isotropic materials, the elastic strain-energy density can</p><p>be decoupled into deviatoric and volumetric parts and can be</p><p>expressed as follows:</p><p>We(�e) =</p><p>(�e,dev) · (�e,dev) + 1</p><p>�tr(�e)2, (A.7)</p><p>is the deviatoric part of a logarithmic strain tensor. Similarly, the</p><p>plastic stored energy can be modeled by assuming an additive</p><p>3 nginee</p><p>d</p><p>m</p><p>i</p><p>W</p><p>w</p><p>l</p><p>W</p><p>w</p><p>r</p><p>f</p><p>W</p><p>W</p><p>w</p><p>g</p><p>g</p><p>w</p><p>t</p><p>d</p><p>v</p><p>t</p><p>c</p><p>t</p><p>R</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>[</p><p>ity with multiplicative kinematics. Engineering with Computers 1992;9:</p><p>255–63.</p><p>[25] Ortiz M, Molinari A. Effect of strain-hardening and rate sensitivity on the</p><p>98 J.-p. Xu et al. / Precision E</p><p>ecomposition of deviatoric and volumetric components. Further-</p><p>ore, the volumetric part of the plastic stored-energy can be split</p><p>nto two contributions due to void growth and coalescence.</p><p>p(�p, �p, �p) = Wp,dev(�p) + Wp,vol</p><p>growth</p><p>(�p, �p) + Wp,vol</p><p>coal</p><p>(�p, �p),</p><p>(A.8)</p><p>here the deviatoric part, Wp,dev, is given by conventional power-</p><p>aw hardening, i.e.,</p><p>p,dev(�p) = n�0�p</p><p>0</p><p>n + 1</p><p>(</p><p>1 + �p</p><p>�p</p><p>0</p><p>) n+1</p><p>n</p><p>, (A.9)</p><p>here n, �0 and �p</p><p>0 are the hardening exponent, yield stress and</p><p>eference deviatoric plastic strain, respectively.</p><p>Following Ortiz and Molinari [25], the volumetric strain-energy</p><p>unction for void growth and coalescence can be written as:</p><p>p,vol</p><p>growth</p><p>(�p, �p) = n�0�p</p><p>0</p><p>n + 1</p><p>(1 − �p)</p><p>(</p><p>N0</p><p>4�a3</p><p>3</p><p>)</p><p>g(�p, �p) (A.10)</p><p>p,vol</p><p>coal</p><p>(�p, �p) = n�0�p</p><p>0</p><p>n + 1</p><p>(�p)</p><p>(</p><p>N0</p><p>4�ã3</p><p>3</p><p>)</p><p>g̃(�p, �p), (A.11)</p><p>here</p><p>(�p, �p) =</p><p>∫ 1/f</p><p>1</p><p>(</p><p>1 + 2</p><p>3�p</p><p>0</p><p>log</p><p>x</p><p>x − 1 + (�f0)/(f0 + e(�p+ˇ�p) − 1)</p><p>)(n+1)/n</p><p>dx</p><p>(A.12)</p><p>˜(�p, �p) =</p><p>∫ 1/fã</p><p>1</p><p>(</p><p>1 + 2</p><p>3�p</p><p>0</p><p>log</p><p>x</p><p>x − 1 + (�f0/(ã/a)3)/(f0 + e(�p+ˇ�p) − 1)</p><p>)(n+1)/n</p><p>dx,</p><p>(A.13)</p><p>here f0 = N0</p><p>4</p><p>3 �a3</p><p>0. 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Journal of Applied Mechanics –</p><p>Transactions of the ASME 1992;59:48–53.</p><p>A novel method to alleviate flash-line defects in coining process</p><p>1 Introduction</p><p>2 Integration of the variational constitutive model with the dynamic explicit finite element framework</p><p>2.1 Dynamic equilibrium equation</p><p>2.2 The constitutive model</p><p>2.3 Incremental constitutive updates</p><p>3 Study of the RFW model and material flow</p><p>3.1 The radial friction work model</p><p>3.2 Material flow in the coining process</p><p>3.3 Prediction of distribution of the flash lines</p><p>4 Solution for mitigating flash line defects</p><p>5 Conclusions</p><p>Acknowledgment</p><p>A.1 The void coalescence model</p><p>A.2 Free energy functions</p><p>References</p><p>References</p>