PREDICTION OF METAL PLASTICITY DURING THE METAL FORMING PROCESS. Y.E. Beygelzimer (DonSTU, Ukraine), D.V. Orlov (DonSTU, Ukraine)

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1 PREDICTION OF METAL PLASTICITY DURING THE METAL FORMING PROCESS Y.E. Beygelzimer (DonSTU, Ukraine), D.V. Orlov (DonSTU, Ukraine) ABSTRACT Te matematical model of plastic deformation of structurally inomogeneous materials wit defects of microvoid-type as been described. Te model as been used for metal plasticity investigation wen сold rolling of tin metal seet. For tis goal te value of porosity (defectiveness) of metal wen te rolling as been got and te limit of seet s metal deep-drawing ability ad been determined. It permits to get metod for investigation of influence of rolling parameters to plasticity inde of seet metal. Te development of metal forming tecnologies sould be based on te matematical models describing microfracture of metal at deformation and taking into account te pressure effect of tis process. An adequate description of te microfracture is possible in te framework of te continuum concepts. In tis case te consideration involves te magnitude of porosity tat is te total relative volume of microdefects []. Tus, te present option is embodied in defining pysical relations of te continuum teory, comprising material porosity in terms of inner variable. Te furter reduction of te problem can be fulfilled wit te elp of assumption of te flow teory in terms of wic pysical relations are defined by loading function. Te principal relationsips of te flow teory ave te form [] f e! ij = λ ; () σ ij were σ ij and!e ij are te tensor components of stresses and rates of plastic deformation, respectively; ƒ is te loading function; λ is te Lagrange factor. Relationsip () is te matematical epression of te gradient condition. Te epression for loading function can be designed on te basis of te pysical model reflecting te main features of te deformation mecanisms and fracture of material. Te following pysical model is suggested in te article [3,4]. Te material consists of interconnected structural elements; Plastic deformation of material can be realized by means of joint coordinated deformation of its structural elements, more over its beavior is defined by te ability of elements to accommodate to eac oter; Te ability to be accommodated is determined by te plastic deformation mecanisms acting at tis or tat moment. If tey provide arbitrary deformation of te structural elements (e.g., five-slip systems work), complete accommodation is possible. Oterwise accommodation is only partial. As a result, gaps (or microvoids) appear between te elements (if at te beginning te elements were closely adjusted to eac oter), wic results in loosening of te material. If, on te oter and, microvoids were present before te deformation, tey may disappear under certain conditions since te structural elements are able to adjust to eac oter. On te basis of te analysis of te suggested pysical model te following results are acieved in te [3]: epression for loading function and pysical equations of plasticity teory of structurally inomogeneous materials wit defects of microvoids-type. Tose equations are located in te table below weres for comparison Mises plasticity teory equations, describing te plastic deformation of unstructured material are also enlisted.

2 In te pysical equations of structurally inomogeneous material te parameter α - coefficient of inner friction is inerent. According to [3] it is quantitative measure of separate structural elements' ability to accommodate to eac oter. In te case wen complete adaptation of te elements to eac oter is possible α =. Te value of α grows wit te increase of a number of restrictions to te joint plastic deformation. Tat is, te less efficient are te mecanisms of plastic deformation of te structural elements, te iger is α. Te pysical investigations sow tat te growt of ydrostatic pressure gives rise to inclusion of a few cannels of plastic deformation. Tere is te number of critical pressures, wic leads to activate new deformation mecanisms. Tus, it follows tat te value of α sould decrease wit te increase of pressure p. And it sould be noted tat in intermediate pressures (i.e., tose once between te critical points) α does not depend on p. Table. Te pysical equations of plasticity teory of structurally inomogeneous materials. Pysical equations for structurally inomogeneous material Pysical equations for unstructured material (according to Mises) Comments σ τ f = + ( θ)( k ασ) f = τ k loading function ψθ ( ) ϕθ ( ) σ τ + = ( θ)( k ασ ) τ = k ψθ ( ) ϕθ ( ) condition of yielding e! τ σ = γ! + α( θ)( k ασ) ϕθ ( ) ψθ ( )!e = gradient condition! e! e! ij ij = ( ij ij ) 3 γ τ σ σ i j criteria of macrofracture of θ = θ c -- te material dτ criteria of instability of te dγ -- material and localisation of plastic deformation n ( θ ) n Here: ψθ ( ) = m, ϕθ ( ) = ( θ), θ is a porosity, σ = σikik,!! 6aθ 3 e = e ik ik, τ = σik σik σik σik, γ! = e! ik e! ik e! ik e! ik ; k , α, a, m, n are material parameters. In order to simulate te process of fracture and localisation of deformation pysical equations of structurally inomogeneous materials are supplemented wit te criteria of bot macrofracture and stability of te material (see te table above). System of pysical equation of Mises plasticity teory increased by te equilibrium of continuous body equations permits to investigate plastic deformation liberal processes to define te strengt-stress parameters of material. Te equal abilities are provided wit te system of pysical equations of te structurally inomogeneous material. Neverteless in addition to te above mentioned te stated system permits to analyse te canges of porosity (defectiveness) of

3 material and to define te areas of macroscopic fracture and localisation of deformation and to investigate ydrostatic pressure influence upon te beaviour of material under te deformation. In tis paper we ave used plasticity teory of structurally inomogeneous materials for investigation of metal plasticity at сold rolling of tin metal seet. For tis purpose we will get te porosity (defectiveness) metal value wen te rolling and following metal deep drawing by ig-pressure liquid. After tis, we will determine te limit of deep drawing ability for criteria of macrofracture of material. It permits to get te metod for investigation of influence of rolling parameters to plasticity inde of seet metal. We get a kinetic equation for microporosity θ using te equation from te line 3 of te table. Taking into account tat for compact materials θ<< and α<< [3] from te equation in te line 3 of te table one can obtain: dθ = α + 6aθχ, () dγ were χ = σ k, γ - sear deformation intensity. Te solution of rolling problem [5] was used for calculation χ and γ. Puting on results of work [5] to equation () and doing all matematics transformations, we obtain te net epressions: dθ 6a( ξ ) θ θ α = + 3a + n d, (3) dθ 6aη ( ξ + ) θ θ α = 3a + < n d η were = µ tg( β ) β - roll bite angle; µ is a surface friction coefficient; η =, were is an initial seet tickness, is a final seet tickness; is a current dimensionless seet tickness; n is a dimensionless seet tickness for neutral cross-section, wic is deter- ( ξ ) η (te line from above is meaning quanti- mined by te formula: + + n = η ξ + ties, wic ave been referred to initial seet tickness ); ξ ξ strip tension respectively [5]. If te letters are absent tenξ ξ, are coefficients for entry and eit, will be equivalent to. Te differential equations (3) wit initial condition θ=θ for = ave been investigated wit numerical metods. Calculating diagrams of dependence of porosity value on te current of relative tickness of a metal seet ave been sown in Fig.. θ ,6,7,8,9, Fig.. Dependence of te porosity s value from te current value of te relative eigt of a metal seet. Te calculation is based on te following parameters: α= -3 ; a= - ; θ = -3 ; ξ =; ε=.5 (,); ε=.375 (3,4); µ=. (); µ=.5 (-4); ξ = (-3); ξ =.8 (4). A porosity variation up to te neutral cross-section is solid lines, after it is discontinuous lines.

4 Here one can see tat porosity value of metal migt bot increase monotonously in te direction of te rolling and to manifest te maimum in te region of neutral cross-section. Te diagrams sow tat increase coefficient of friction leads to lowering porosity value after rolling. It can be eplained by te fact tat te value of ydrostatic pressure increases as well as te parameter does. Strain increase leads to qualitative modification of metal loosening caracter; nonmonotonous dependence θ of appears. It is eplained by competing of te processes of loosening and ealing of metal. Maimum of stress contact value near neutral cross-section is leads to predominance ealing processes over porous formation processes in tis region. In case of strip tension te porosity value is strongly increased. It is te result of lowering of ydrostatic pressure value. Tere is a row of different plasticity indees of te seet metal, eac of wic is determined in te special tecnology tests. In tis paper we ll simulate te beaviour of te seet metal for te test by sceme of deep drawing wit ig-pressure liquid. Te sceme of tis test as been sown in Fig.. Cange of metal porosity for te test will be investigated wit te equation (). Te χ parameter for te test was got in te paper [6]. Criterion of a fracture is taken in te form of line 5 of te table. L H stem; - concentrating laying; 3 - liquid (pressure Р ); 4 - container; 5 - testing model; 6 -liquid (pressure Р ); 7 - matri. D Н - strip tickness; L - ole dept; D - ole diameter. Fig. Sceme of deep drawing wit ig-pressure liquid. Substituting parameter of paper [6] into epression () and integrating it, we ll obtain equation to definite porosity transformation during te diapragm deep drawing wit ig-pressure liquid: θ ( ) ( 3α + a )( r + ) r a θ 3α =, (4) a were θ is porosity of te metal seet after te rolling; r is current relative ole dept, wic is obtained by te formula: r = r; is a current ole dept; r is a radius of matri ole. Finding from (4) for θ = θ c, we obtain te epression of maimum ole s dept ma wen metal fracture takes place. Quantity of ma is inde of seet metal plasticity. In Fig. 3 te calculated dependence ma on ξ as been sown in te case of flat rolling. We can see tat te increase of tension between mill stands leads to decrease of dept of spere ole during te diapragm deep drawing, but tat dependence as nonmonotonously caracter. Tis dependence can be applied in designing of te process of flat rolling at wic te coice

5 of tension between mill stands. Te magnitude of te latter is determined, first of all, from reasons of stability of rolling, flatness of a strip. But te influence of tension between mill stands on tougness of metal also sould be taken into consideration., mm 3,,4,6,8,8,6,4,3,4 σ /σ S Fig. 3. Influence of tension between mill stands on te dept of spere ole during te diapragm deep drawing wit ig-pressure liquid. σ /σ S is entry strip tension. REFERENCES. V.L. Kolmogorov: Stresses, deformations, fracture, Metallurgija, Moscow, 97 (in Russian).. D.D. Ivlev and G.I. Bykovtsev, Teory of Hardening Plastic Body (Nauka, 97). 3. Ya.E. Beigelzimer et al., Engineering. Fracture Mec. 48 (994), page Ya.E. Beigelzimer et al., Engineering. Mec. (995) page A.J. Tselikov, G.S. Nikitin, S.E. Rockotyan: A Teory of longitudinal rolling, Metallurgija, Moscow, 98 (in Russian). 6. A.A. Bogatov, O.I. Mijiritsky, S.V. Smirnov: Plasticity resource during te metal forming. Metallurgija, Moscow, 984 (in Russian).