It is important to study the microscopic deformation behavior of inhomogeneous material, for most engineering materials are inhomogeneous. The aim of the present study is to clarify by numerical analysis some features of microscopic plastic strain distributions, the mean flow stress and the material factors affecting on it. The rigid-plastic solution is important not only for plastic deformation problems with large strain, but also for creep deformation problems through the plastic analogy in the creep analysis. The effects of material parameter and loading conditions on the deformation behavior of the material are examined and discussed based on the result of calculation. The effects of the aspect ratio of the inhomogeneous regions on the deformation mode are studied. The patterns of the strain concentration and the averaged flow stress of the inhomogeneous material are also discussed. The results of rigid-plastic material are compared with those of the elastic material

Abe, Takeji

Nagao, Makoto

Nagayama, Noriyuki

Ohta, Tomoyuki

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering, Okayama University, Vol.36, No.2, pp.7 -14, March 2002Numerical Study on Deformation Behavior of Rigid-PlasticInhomogeneous Material Using Three-Dimensional ModelsTakeji ABE·, Makoto NAGAO, Tomoyuki ORTAGraduate School of Natural Science and Technology,Okayama University, Tsushima-Naka 3-1-1, Okayama 700-8530, Japan.and Noriyuki NAGAYAMATechnological Research Center of Okayama Prefecture.Raga 5301, Okayama 701-1221(Received Jan. 25, 2002 )It is important to study the microscopic deformation behavior ofinhomogeneous material, for most engineering materials areinhomogeneous. The aim of the present study is to clarify by numericalanalysis some features of microscopic plastic strain distributions, themean flow stress and the material factors affecting on it. Therigid-plastic solution is important not only for plastic deformationproblems with large strain, but also for creep deformation problemsthrough the plastic analogy in the creep analysis. The effects of materialparameter and loading conditions on the deformation behavior of thematerial are examined and discussed based on the result of calculation.The effects of the aspect ratio of the inhomogeneous regions on thedeformation mode are studied. The patterns of the strain concentration andthe averaged flow stress of the inhomogeneous material are also discussed.The results of rigid-plastic material are compared with those of the elasticmaterial.Key Words Plasticity, Deformation, Inhomogeneous Material,Strain Concentration Coefficient, Rigid-Plastic FEM1. INTRODUCTIONIn order to clarify the deformation behavior of inhomogeneous material, it isnecessary to study the distribution of stress or strain, the influence of the geometry ofthe constituents and the relation between local microscopic deformation and the globaldeformation behavior of the material. In the present analysis, two representative three-• E-mail abe@mech.okayama-u.ac.jp78 Takeji ABE et al. MEM.FAC.ENG.OKA.UNI. Vo1.36. No.2dimensional models of inhomogeneous material are adopted. One is that two kinds ofgrains are placed regularly. The other is that one grain is imbedded in the surroundingmatrix. The uniaxial deformation of the model material is analyzed with therigid-plastic finite element method. It is expected that the characteristic feature of theplastic deformation of inhomogeneous material is well simulated by the rigid-plasticfinite element method [I].2. METHOD OF NUMERICAL ANALYSISThe three-dimensional finite element mesh used in the analysis is 8 X 8 X 8 elements(512 elements and 2673 nodes) for Model A, while 7 X 7 X 7 elements (343 elements and1856 nodes) for Model B where the central one element is assumed to be the embeddedgrain. The three-dimensional cubic element with 20 nodes is employed. The Gaussianpoints in a element are chosen as eight for the strain components, while one for thevolumetric strain, that is, the reduced integral is used. The displacements are given asthe boundary condition, which correspond to the uniaxial tension in z-direction.Namely, the strain 1.0% in z-direction and - 0.5% in x- and y-directions are given,respectively, considering the symmetric configuration of the models. Two models ofinhomogeneous material used in the numerical simulation are shown in Fig. 3; (l) twokinds of gains with different yield stress are assumed to be placed regularly, as shownwith + and - signs in Fig. 3(a), and (2) a grain is imbedded in the surrounding matrixas shown in Fig. 3(b). The calculated area OABC is shown respectively in the figures.The aspect ratio R of the model is expressed as follows,R=a/c , (1)where a = OA = AD and c = OB are in x-, y-directions and in z-direction, respectively.The yield stresses of two grains are assumed to bewhere ,p (0 ~,p < 1) is a parameter of inhomogeneity and eTo is the averaged yieldstress of the material. The yield stress ratio is given as follows.The penalty method is used to satisfy the volume constancy in the rigid-finiteelement analysis. The following functional <1> is taken as minimum for the givenboundary conditions.<1>= lu&dv+~a 18/dv-1TTUth, (4)where (j is the equivalent stress, '& is the equivalent plastic strain rate, T is theboundary conditions at the surface S of the volume V, &v is the volumetric strain anda is the penalty number (a=107). In the following, strain is used instead of strainrate considering the change during a unit time.The personal computer (DOS/V), an original Fortran program and Visual FortranMarch 2002 Deformation Behavior of Rigid-Plastic Inhomogeneous Material 9software (Compaq) are used in the programming and the numerical calculation.BrL--t------rf-----t--r1r'(a) Model A: regularly placed grainsBrL-----t----(C/----+.---...", Dz 0tL:(b) Model B: grain imbedded in matrixFig.l Models of inhomogeneous material3. RESULTS OF CACULATION AND DISCUSSIONIn order to evaluate the obtained straincoefficients are defined as follows [1,2].A- = {E~q }- ,Geqdistribution, the strain constraintwhere A~ is the strain concentration coefficient for an arbitrary point, while A+, A- ,Ag and Am are the averaged strain concentration coefficients for + , -, g and m grainsshown in Fig. I. Geq , {Eeq }+, {Eeq }-, {Eeq}g' {EeqL and 'i~q are the equivalent plasticstrain at an arbitrary point, the averaged equivalent plastic strain for the grain + , -(Model A), g, m (Model B) and the whole material.Fig. 2 shows the relation between the strain concentration coefficient and theposition in an element, including that near the grain boundary. The aspect ratio istaken as R = 1.0, and ¢J = 0.2 in Eq. (3) which corresponds to the yield stress ratioa=lf/ /lfy - =1.5. Figs. 2 (a) and (b) show the strain concentration coefficient in x-direction, while (c) is that in z -direction.Fig. 3 shows the relation between the averaged strain concentration coefficients A+ ,A- and the aspect ratio R of the grain for Model A. The values of A+ and A-comes close to I when R is small, which is due to the constraint of deformation in the1.50.510 Takeji ABE et al.2.0 r------~---------,i I«C 1.5 CQltl'0'u1 Ii:~<: 1.0 <:0~ ."!!!C CQl8 0<:<: 08 0.5 00 yIb = 0.08ICC b. JA>=0.19 ~'!! 0 y.t>=0.31US <> y.t> = 0."I•yJb = 0.940.0 I0.5 1.0x-<:OOl'dinate ( xla )(a) zlc= 0.062.0MEM.FAC.ENG.OKA.UNI. Vo1.36. No.22.0 ,-------------,--------.,1.0 r----==:=--------:::;n-:\-~--~=~Io yJb = 0.08L b. yJb=0.19o yJb = 0.31<> yJb=O...• yJb=O.94O.0 '-----"------::-~'------'------'-------,-J0.5 1.0x-eoordinate ( xla )(b) zic = 0.441C 1.5tl'0Ii:8c 1.00!cBc8 05c"!USO.a '-------'-----'----'--'--::-0."=5----'-L-----'-------'--1.0z-coordinate ( zic )(c) xla =0.06Fig. 2. Distribution of strain concentration coefficient Aeq in Model A( R = 1.0, <p = 0.2 )loading (z-) direction at the grain boundary. When R is large, it again comes close to1 , which is attributed to the constraint of deformation in the perpendicular direction tothe loading axis, namely in x- and y-directions as discussed later. It is clearly shownthat the deformation behavior of polycrystalline metals is affected by the shape ofgrains, which is resulted from the mutual constraint of deformation between grains .. Fig. 4 shows the comparison between the present three-dimensional case and thetwo-dimensional plane stress case reported previously [3]. The result for the planestress case is shown with the broken line. It is seen that the difference in strain in +and - regions in the three-dimensional case is small as compared with the plane stresscase. This shows that the constraint of deformation is much severe in thethree-dimensional model.March 2002 Deformation Behavior of Rigid·Plastic Inhomogeneous Material 112.0 ,........----------------,1.50.5---. ¢ = 0.1-¢=0.2--- 4J =0.310.00.0 '-----~~~~~--'---~-~~~~0.1 1.0Aspect ratio RFig. 3 Relation between averaged strain concentration coefficients A + ,A -and aspect ratio RI~2.0 ,---------------,+• A+o A- 9"--0-----0.,''0..E Plane stress " --'0._.~ ,0'1 I"\-=.-=::--.:..o-'--v-'~'"a1.0~_"",__:-_-.-~I -------...B -. ••--c: 'e,. ,/f"'-8 -_.. .'c:~ 0.0'----_~~~~~_>_L._~'"___'___'__~ .............tJ) 0.1 1.0 10Aspect ratio RFig. 4. Relation between averaged strain concentration coefficients A + ,A -and aspect ratio R (¢ =0.2 )Fig. 5 shows the relation between strain concentration coefficients and aspect ratiofor the three-dimensional rigid-plastic deformation as well as the three-dimensionalelastic deformations [cP = 0.2]. In the elastic case, Young's moduli for the + and -regions are defined, similarly to Eq. (2), as follows.Again, the values of A+ and A- come close to I when R is small, which is due to theconstraint of deformation in the loading (z-) direction at the grain boundary. When Ris large, it comes close to I , which is attributed to the constraint of deformation in x- andy-directions. The latter appears severely in the plastic deformation as well as the elasticdeformation with Poisson's ratio v ~ 0.5, where the condition of the volume constancy isstrictly kept. On the other hand, the constraint is weak in the elastic deformation withsmall value of Poisson's ratio 'V ~ 0.3.12 Takeji ABE et al. MEM.FAC.ENG.OKA.UNI. Vo1.36, No.2v =0.499v=0.3- plastic-- elastic-cG)'0i5 1.0~;[;:;:::3gc8c8Ic(+ - 1.5 r------;------------------,• A+c( 0 A-cg 0.5'---~--'-.........~~.L.J.----'-~-----L~ ..........."'"""""'en 0.1 1.0 10Aspect ratio RRelation between strain concentration coefficients and aspect ratio forthree-dimensional rigid-plastic and elastic deformations [ 4> = 0.2 ]Fig. 52.0 r---------------, 2.0,-----------------.,-r/J=-0.1---- r/J=-0.2-- r/J=-0.3O:Am.: Au0.50.0 ~_~~~~__L_~~~~..............J0.1 1.0 10.0Aspect ratio R1.5_::-=~1.0~===l~-__o__c)_o__<D__o_-__cr-oo10.0-r/J=0.1---- r/J = 0.2-- r/J=0.31.0Aspect ratio RO:Am.: Au1.5c!..~.fIigc 0.5(a) Hard central grain (b) Soft central grainFig. 6 Relation between averaged strain concentration and aspect ratioFig. 6 shows the relation between the averaged strain concentration coefficient andthe aspect ratio R for Model B shown in Fig. 3(b). In Fig. 6, A g is the averaged strainconcentration coefficient for the central grain, while Am is that for the matrix, wherethe parameter of anisotropy is taken as 4> =0.1. 0.2 and 0.3, that is, the yield stress ratioa = 1.22. 1.50 and 1.86, respectively. Fig. 6(a) and (b) show the results for the caseswhere the central grain is harder and softer than the matrix, respectively.It is seen from Figs. 3 and 6 that the relation between the averaged strainconcentration coefficient and the aspect ratio for the Model A shown in Fig. 3 is similarto that for Model B shown in Fig. 6, though their absolute values are different.The value of the strain concentration coefficient is close to 0 at R=': 0.1. which isMarch 2002 Deformation Behavior of Rigid-Plastic Inhomogeneous Material 13due to the mutual constraint in tensile (z-) direction. The value also reduces at R~ 10,which is due to the constraint in the perpendicular (x- and y-) directions to the tensiledirection, as discussed analytically in the previous paper [4].4. CONCLUSIONThe characteristic feature of plastic deformation of inhomogeneous material wasanalyzed with the rigid-plastic finite element method. Two models of inhomogeneousmaterial were used in the numerical simulation. One is that two kinds of gains withdifferent yield stress are placed regularly, and the other is that a grain is imbedded inthe surrounding matrix. The strain distribution was calculated and the averagebehavior was represented with the strain concentration coefficient. The main resultsobtained are as follows.(I) The relation between the averaged strain concentration coefficients and the aspectratio for both models are similar, though their absolute values are different. Theresults for the three-dimensional plastic deformation are compared with those forthe two-dimensional plastic deformation and the three-dimensional elasticdeformation.(2) The value of strain concentration coefficient of the three-dimensional plasticmodels decreases at small aspect ratio of the models, which is due to the mutualconstraint of deformation between grains in the loading direction.(3) The value of strain concentration coefficient of the three-dimensional plasticmodels decreases at large aspect ratio of the models, which is due to the mutualconstraint of deformation between grains in the perpendicular direction to theloading direction.(4) The constraint between gains during plastic deformation is much severe in thethree-dimensional case than that in the two-dimensional case.The authors are indebted to the Science Promotion Program presented by theMinistry of Education and Science. Fundamental Research (B)(2) 11450045,1999-2001 ]REFERENCES[1] T. Abe, N. Fujiyoshi and N. Nagayama, IUTAM Symposium on Micro- andMacrostructural Aspects of Thermo - plasticity, Eds. O. T Bruhns. and E. Stein,(1998), pp.417-426, Kluwer.[2] T. Abe, R. Namikoshi, N. Nagayama and Y. Takano, Memoirs of Faculty ofEngineering, Okayama University, Vol. 33, No.1 (1998), pp.5-17.[3] T. Abe, S. Nagaki and M. Furuno, Trans. JSME, Vol. 51A, No. 468 (1985-8) ,p.2042 (in Japanese).[4] T. Abe and T. Miyake, Mechanical Properties of Advanced Engineering Materials(IMMM200 1), pp. 79-86, (2001), Mie University Press.

Numerical Study on Deformation Behavior of Rigid-Plastic Inhomogeneous Material Using Three-Dimensional Models

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