TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007 Trang 49 THE SOFTENING IN PLASTIC DEFORMATION OF METAL Truong Tich Thien University of Technology, VNU-HCM (Manuscript received on December14 th , 2006; Manuscript received on June 28 h , 2007) ABSTRACT: In the plastic deformation stage of metal, work hardening always goes together with softening. The nucleation, growth of small internal voids or cavities according to plastic increasing is the microscopic mechanism of softening. The voids are nucleated at the particle-matrix interface due to the agglutinate loss or the particle crack when the strain reaches critical value. The growth of voids will then occur in company with the increase of plastic deformation. The influence of void growth on material forming behaviour will be considered by softening parameter of porous material. Key words: hardening, constitutive softening, nucleation, growth, porous material. 1. INTRODUCTION The nucleation, growth and coalescence of small internal voids or cavities are the microscopic mechanism of ductile fracture in cold forming processes of metal. The microvoids are nucleated under the tensile loading state at the impurities and hard particles in the ductile metal. After nucleated microvoids, they will be grown due to plastic deformation and coalesced together in order to create the microscopic ductile fracture when the critical state is reached. The nucleation and growth of small internal voids or cavities are interpreted as the reason of the strain softening of material. So, the strain hardening and the strain softening of material are two phenomena occurring simultaneously during the plastic deformation of materials. At first the strength of material increases since the strain hardening, but the material will be degraded due to the growth of microvoids. This induces the strength degradation and the stress−strain relationship will be shown by the curve with the negative slope. The initial shapes of micro-voids are multiform and complex. On the other hand, their distribution is random and difficult to determine. For the feasibility of analysis model, two initial assumptions were proposed. Firstly, the initial shapes of micro-voids are supposed the cylinder with circular cross section in two-dimensional problems and the sphere in three-dimensional problems (fig. 3). Finally, these voids are uniformly arrayed in material. The chief goal of this paper is to examine the growth of voids and the strain softening of structure inside the ductile metal with the initial spherical voids and uniform distribution in cold forming processes of metal. The numerical results are obtained by two models: the analytical model and the finite element model. There is a good agreement between the results of proposed model and experiments. 2. COMPUTING MODELS 2.1. The analytical model 2.1.1. Nucleation of voids The impurities and hard particles always exist in technical ductile metals (fig.1) and the concentrated stress at these loci will be the reason to form micro-voids (fig.2). These voids are nucleated at the particle-matrix interface due to the agglutinate loss or the particle crack. Science & Technology Development, Vol 10, No.06 - 2007 Trang 50 abc ,,lll 000 2.1.2. Analytical model for growth of voids Mc.Clintock (1968) developed firstly a model for growth of cylindrical void in strain hardening materials. An isolated cylindrical void (with longitudinal axis c, semi-axes a and b) will be changed according to () ()() ab eab 0eM 31 n 3 R ln sinh R21n 2 2 ⎡⎤ −σ+σ ⎛⎞ εε+ε =+ ⎢⎥ ⎜⎟ −σ ⎢⎥ ⎝⎠ ⎣⎦ (1) For a material cell with several series of cylindrical voids (fig.4), interaction of neighbouring void must be introduced in the model of void growth. At a uniform void distribution (initial void distances), the growth parameters in radial direction a and b are defined according to 0 a ca 0a a F a = l l and 0 b cb 0b b F b = l l (2) Under all-round tension, voids are grown increasingly faster than the forced strain, according to the equations obtained from Levy-von Mises F F i i g g u u r r e e 2 2 . . T T h h e e c c o o n n c c e e n n t t r r a a t t e e d d s s t t r r e e s s s s a a t t t t w w o o p p o o l l e e s s o o f f h h a a r r d d p p a a r r t t i i c c l l e e F F i i g g u u r r e e 3 3 . . T T y y p p e e s s o o f f v v o o i i d d s s i i n n m m o o d d e e l l . . F F i i g g u u r r e e 1 1 . . T T y y p p e e s s o o f f m m a a n n g g a a n n e e s s e e s s u u l l p p h h i i d d e e p p a a r r t t i i c c l l e e s s i i n n s s t t e e e e l l . . E E n n l l a a r r g g e e 4 4 1 1 5 5 E E n n l l a a r r g g e e 2 2 0 0 0 0 0 0 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007 Trang 51 () () () () ab ba ca e eM eM n dlnF sinh d n ⎧ ⎡⎤ σ+σ − ⎫ σ−σ ⎪ =+ε ⎢⎥ ⎨⎬ −σσ ⎢⎥ ⎭ ⎪ ⎣⎦ ⎩ 31 33 21 2 4 (3a) () () () () ab ab cb e eM eM n dlnF sinh d n ⎧ ⎡⎤ σ+σ − ⎫ σ−σ ⎪ =+ε ⎢⎥ ⎨⎬ −σσ ⎢⎥ ⎭ ⎪ ⎣⎦ ⎩ 31 33 21 2 4 (3b) where σ eM means the equivalent stress of material matrix. The void initiation and effects of necking are not involved in the fracture criterion of Mc.Clintock. For this reason, the predicted fracture strain of a material is normally larger than the experimental value. Thus, an improvement of Mc.Clintock is necessary. Nguyen Luong Dung modified the original Mc.Clintock model by adding a second model, fig.4c, to the original model, fig.4b, with ab ab aabb and ∗∗ σ+σ σ+σ ⎛⎞ ⎛⎞ σ=− −σ σ=− −σ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ 22 (4) The growth parameters in radial direction a and b of modified model are now defined according to () * a ca a a 0 R Fexp2 R =ε−ε (5a) () * b cb b b 0 R Fexp2 R =ε−ε (5b) The accumulated damage rates of modified model in radial direction a and b are approximately given as () () ()() ab ab ca ecae eM eM 31 n 33 dlnF sinh d fd 21 n 2 4 ⎧⎫ ⎡⎤ −σ+σ σ−σ ⎪⎪ =+ε=ε ⎢⎥ ⎨⎬ −σσ ⎢⎥ ⎪⎪ ⎣⎦ ⎩⎭ (6a) () () ()() ab ba cb ecbe eM eM 31 n 33 dlnF sinh d fd 21 n 2 4 ⎧⎫ ⎡⎤ −σ+σ σ−σ ⎪⎪ =+ε=ε ⎢⎥ ⎨⎬ −σσ ⎢⎥ ⎪⎪ ⎣⎦ ⎩⎭ (6b) F F i i g g u u r r e e 4 4 . . M M o o d d i i f f i i e e d d m m o o d d e e l l . . 2 2 b b σ a σ b l a l b 2a σ σ c c a a ) ) c R 0 σ c σ r b b ) ) c a b R 0 * b σ * a σ c c ) ) = + Science & Technology Development, Vol 10, No.06 - 2007 Trang 52 The other expressions for d(lnF ab ) and d(lnF ac ) or d(lnF ba ) and d(lnF bc ) will be obtained in the same way as equality (6). In case of a material cell containing spherical voids as fig.3, the growth of a single spherical void, fig.5, is taken into account in order to analyze the fracture initiation. The rate of change of shape may be obtained if the spherical surfaces concentric with the void assume to become ellipsoids. The accumulated damage rate of the momentary semi-axis a is extrapolated from (6a) and (6b) for a cylindrical void by means of a superposition method, fig.6, and is given as (7a). Generally, the accumulated damage rate of the momentary semi-axis i is given as (7b). () () ()() () ()() () () () ab ab a eM eM ac ac e eM eM abc bc eM eM 31 n 2 2 33 d lnF sinh 21 n 2 4 31 n 2 2 33 sin h d 21 n 2 4 31 n 31 n 3 sinh cosh 1n 4 4 ⎡ ⎧⎫ ⎡⎤ −σ+σ σ−σ ⎪⎪ ⎢ =++ ⎢⎥ ⎨⎬ −σσ ⎢ ⎢⎥ ⎪⎪ ⎣⎦ ⎩⎭ ⎣ ⎤ ⎧⎫ ⎡⎤ −σ+σ σ−σ ⎪⎪ ⎥ ++ε= ⎢⎥ ⎨⎬ −σσ ⎥ ⎢⎥ ⎪⎪ ⎣⎦ ⎩⎭ ⎦ ⎡ ⎧⎫ ⎡⎤⎡⎤ −− σ+σ+σ σ−σ ⎪⎪ ⎢ =+ ⎢⎥⎢⎥ ⎨⎬ −σ σ ⎢ ⎢⎥⎢⎥ ⎪⎪ ⎣⎦⎣⎦ ⎩⎭ ⎣ + abc eae eM 3 dfd 4 ⎤ σ−σ−σ ε= ε ⎥ σ ⎦ (7a) () () () () ijk jk i eM eM ijk eie eM 31 n 31 n 3 dlnF sinh cosh 1n 4 4 dfd ⎡ ⎧⎫ ⎡⎤⎡⎤ σ+σ+σ σ−σ −− ⎪⎪ ⎢ =+ ⎢⎥⎢⎥ ⎨⎬ −σ σ ⎢ ⎢⎥⎢⎥ ⎪⎪ ⎣⎦⎣⎦ ⎩⎭ ⎣ σ−σ−σ ⎤ +η ε = ε ⎥ σ ⎦ (7b) Axis b Axis c F F i i g g u u r r e e 6 6 . . G G r r o o w w t t h h o o f f e e l l l l i i p p s s o o i i d d v v o o i i d d . . F F i i g g u u r r e e 5 5 . . A A c c e e l l l l o o f f m m a a t t e e r r i i a a l l . . b a 2R 0 0 a l 0 b l 0 c l c TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007 Trang 53 where i i 0 i R 3 ln F ln and R2 ⎛⎞ =η= ⎜⎟ ⎝⎠ for the isolated void, 0 ii i 0 ii R3 ln F ln and R4 ⎛⎞ = η= ⎜⎟ ⎝⎠ l l for void in porous media. The accumulated damage of the momentary semi-axis i due to void growth is given as () () ( ) () () e N e e N e ijk ii eM jk ijk eie eM eM 31 n 3 AlnF sinh 1n 4 31 n cosh d f d 4 ε ε ε ε ⎡ ⎛ ⎧⎫ σ+σ+σ − ⎪⎪ ⎢ ⎜ == ⎨⎬ −σ ⎢⎜ ⎪⎪ ⎩⎭ ⎝ ⎣ ⎤ ⎫ ⎧⎫ σ−σ σ−σ−σ − ⎪⎪⎪ ⎥ ⋅+ηε=ε ⎨⎬⎬ σσ ⎥ ⎪⎪⎪ ⎩⎭ ⎭ ⎦ ∫ ∫ (8) The influence of void growth on material forming behaviour was considered by softening parameter of porous material σ e /σ eM . The general form of yield function for porous material is given by TRUONG Tich Thien () 2 ee ij eM eM eM ,,f BcoshA C0 ⎛⎞ ⎛ ⎞ σσ φσ σ = + − = ⎜⎟ ⎜ ⎟ σσ ⎝⎠ ⎝ ⎠ (9) where σ eM is the equivalent stress of matrix material (no voids), the factors A, B, C depend on the porosity f, state of applied stress and material property, according to the formulas ( ) 2 m A 1.616 0.866n S ; B 2.5f; C 1 1.625f=− = =+ (10) with S m = σ m /σ e . The concentrated plastic deformation appears at critical state before neighbouring voids touch together. So, the coalescence of neighbouring voids or ductile fracture will be occurred in a plane perpendicular to the maximum growth direction i of void if the accumulated damage of the momentary semi-axis i satisfies the following condition iif AA ln , f ∗ ⎛⎞ π ==β β≤ ⎜⎟ ⎝⎠ 3 0 14 1 23 (11) The process of micro ductile fracture prediction is shown in the flowchart of figure 7. Science & Technology Development, Vol 10, No.06 - 2007 Trang 54 2.2. The finite element model For the symmetry, the FEM analysis model only includes one-sixteenth of material cell (fig.8). F F i i g g u u r r e e 7 7 . . F F l l o o w w c c h h a a r r t t o o f f f f r r a a c c t t u u r r e e p p r r e e d d i i c c t t i i o o n n a a t t a a p p o o s s i i t t i i o o n n . . S S t t a a r r t t N o Yes I I n n p p u u t t d d a a t t a a : : * * M M a a t t e e r r i i a a l l p p r r o o p p e e r r t t i i e e s s : : n n , , f f o o , , ε ε e e N N . . * * L L o o a a d d s s t t a a t t e e : : S S 1 1 , , S S 2 2 , , S S 3 3 * * S S t t r r a a i i n n i i n n c c r r e e m m e e n n t t : : Δ Δ ε ε e e * * f f a a c c t t o o r r : : η η q q c c < < l l n n F F m m a a x x Result output: •Text file •Graphic file Stop °Define: i Fln °q c = max ijk ln F ,ln F ,ln F ⎡ ⎤ ⎣ ⎦ °Update porosity f. °Re-compute stress, strain •Define ε e = ε e + Δ ε e •Define A, B, C •Define softening of material e eM σ σ f f = = f f o o ; ; q q c c = = 0 0 ; ; ε ε e e = = ε ε e e N N l l n n F F m m a a x x = = β β l l n n 3 0 14 23f ⎛⎞ π ⎜⎟ ⎜⎟ ⎝⎠ TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007 Trang 55 Table 1. Load cases applied on cell Load case me m Sσσ= 3e 3 S σ σ= 2e 2 S σ σ= 1e 1 S σ σ= Axial load 1/3 1.0 0.0 0.0 Triaxial Load 1.25 1.9166 0.9166 0.9166 High triaxial Load 3.0 3.666 2.666 2.666 Two initial porous cases f 0 = 1% and f 0 = 10%, two types of material n = 0.1 and n = 0.2 and three load cases were computed (table 1). Figure 9a.Equivalent strain distribution o f model. n = 0.2 o f 0 = 0.1 Figure 9b. Equivalent stress distribution of model. n = 0.2 f 0 = 0.1 F F i i g g u u r r e e 8 8 . . F F E E M M M M o o d d e e l l a a ) ) M M a a t t e e r r i i a a l l c c e e l l l l 1 1 / / 1 1 6 6 m m a a t t e e r r i i a a l l c c e e l l l l b b ) ) F F E E M M m m e e s s h h Science & Technology Development, Vol 10, No.06 - 2007 Trang 56 Figure 10. Porous variation according to different yield functions in triaxial load. 1.0 2.0 3.0 4.0 5.0 6.0 0 0.05 0.1 0.15 0.2 0.25 0.3 Equivalent strain f / fo f/f0-Lemaitre f/f0-Gurson f/f0-Dung f/f0-Thien f/f0-Finite f 0 = 0.01 n = 0.2 ε e Figure 11a. Material softening according to different yield functions in triaxial load. 0.80 0.85 0.90 0.95 1.00 0.0 0.1 0.1 0.2 0.2 0.3 0.3 Equi val e nt s tr ai n Le mait re Gu rs o n Dung Thien Finite ε e f 0 = 0.01 n = 0.2 σ e / σ eM Figure 11b. Material softening according to different yield functions in high triaxial load. 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 Equivalent strain Lemait re Gu rs o n Dung Thien Finit e σ e / σ eM f 0 = 0.1 n = 0.2 ε e TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007 Trang 57 Table 2. The fracture strain ε ef Steel 1015 1045 1090 Load Yield function Uni- Axia- lity Tri- Axia- lity High Tri- Axia- lity Uni- Axia- lity Tri- Axia- lity High Tri- Axia- lity Uni- Axia- lity Tri- Axia- lity High Tri- Axia- lity Lemaitre 1.38 0.615 0.155 0.89 0.395 0.095 0.73 0.325 0.08 Gurson 1.39 0.65 0.405 0.89 0.46 0.35 0.73 0.4 0.33 Dung 1.39 0.66 0.42 0.91 0.49 0.425 0.75 0.45 0.425 Thien 1.39 0.655 0.415 0.9 0.475 0.39 0.74 0.425 0.375 Experiment 1.4 0.91 0.63 Factor β 1 1 0.85 The strain and stress results from FEM for a material sustained high triaxial load was shown on fig. 9a and fig. 9b. The porous growth and material softening results from FEM were shown and compared with ones from analysis on fig.10, fig. 11. For every steel, the computational program exported the accumulated damage (fig.12) and fracture strains (table 2). 3. CONCLUSION The developed model of the void growth takes into account both macro- and microscopic factors. This model is appropriate for the variation of the shape of ellipsoidal voids in a plastically deformed medium in cold forming processes of metal (T < 0.4T c ). The growth and coalescence of voids depend on temperature of material, hydrostatic stress of load, … Under superposition of hydrostatic stress (negative stress), the growth of voids will be obstructed or delayed and the coalescence of void will not occur. This is an important method in order to avoid micro fracture in metal forming. The void growth due to plastic deformation causes the softening and increases the accumulated damage of material. There is a good agreement between the results of proposed model and experiments. Thus, this predictive process gets promising future in metal forming. SỰ MỀM HÓA CỦA KIM LOẠI TRONG BIẾN DẠNG DẺO Trương Tích Thiện Trường Đại học Bách khoa, ĐHQG-HCM TÓM TẮT: Trong giai đoạn biến dạng dẻo của kim loại, sự tái bền (biến cứng) luôn đi cùng cùng với sự biến mềm. Sự hình thành, tăng trưởng của các lỗ hổng vi mô tương ứng với sự gia tăng biến dạng dẻo là cơ chế vi mô của sự biến mềm. Các lỗ hổng được hình thành ở bề mặt hạt-mạng do mất đi sự dính kết hay do nứt hạt khi biến dạng dẻo đạt đến giá trị giới hạn. Tiếp theo sự tăng trưởng của lỗ sẽ xảy ra trong điều kiện biền dạng dẻo gia tăng. Ảnh hưởng của sự tăng trưởng lỗ hổng sẽ được khảo sát bởi thông số biền mềm của vậ t liệu xốp. Science & Technology Development, Vol 10, No.06 - 2007 Trang 58 REFERENCES [1]. W. F. Chen, D. J. Han, Plasticity for Structure Engineers, Springer-Verlag- New York - Berlin - Heidelberg - London - Paris – Tokyo, (1988). [2]. B. Dodd and Y. Bai, Ductile Fracture and Ductility, Academic Press, London (1987). [3]. E. Doege, H. M. Nolkemper and I. Saeed, Fliesskurvenatlas metallischer Werkstoffe, Hanser Verlag, Muenchen (1986). [4]. N. L. Dung, Fortschr Ber. VDI Reihe 2 Nr. 175, VDI-Verlag, Dusseldorf (1989). [5]. T. T. Thien, The Model for Ductile Fracture Prediction in Metal Forming, Doctor Thesis, Ho Chi Minh City University of Technology, (07/2001). [6]. Truong Tich Thien, Vu Cong Hoa, A Process of Micro Ductile Fracture Prediction for Metal, Proceedings of International Conference for Mechanical and Automotive Technologies (ICMAT) 2005, Chonbuk National University, Korea, June 1 ~ 3, (2005). [7]. M. J. Worswick and R. J. Pick, J. Mech. Phys. Solid 38, 601 (1990). . are interpreted as the reason of the strain softening of material. So, the strain hardening and the strain softening of material are two phenomena occurring simultaneously during the plastic deformation. ABSTRACT: In the plastic deformation stage of metal, work hardening always goes together with softening. The nucleation, growth of small internal voids or cavities according to plastic increasing. growth of voids will then occur in company with the increase of plastic deformation. The influence of void growth on material forming behaviour will be considered by softening parameter of porous