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Modelling grain growth evolution and necking in superplastic blow-forming
International Journal of Mechanical Sciences 43 (2001) 595}609
Modelling grain growth evolution and necking in superplastic blow-forming J. Lin *, F.P.E. Dunne School of Manufacturing and Mechanical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford, OXI 3PJ UK Received 2 August 1999; received in revised form 3 April 2000
Abstract A stability analysis is carried out to investigate necking in superplastic materials characterised by the sinh-law constitutive equation e NJsinh(bp). The e!ects of load and the strain rate sensitivity parameter b on necking are quantitatively studied and a necking map is obtained for conditions of uniaxial loading. Finite element simulations of a superplastic blow-forming process are carried out in order to investigate both non-uniform thinning and grain size distribution which result from the use of the sinh-law constitutive equation. The pressure cycle required to ensure a target maximum strain rate is not exceeded in the material is obtained. The e!ects of strain rate and the magnitude of the parameter b on the grain size and through-thickness strain distributions for the formed part are investigated. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Superplastic forming; Non-uniform thinning; Grain growth; Gas blow-forming
1. Introduction Superplastic materials can undergo extensive tensile plastic deformation prior to failure, typically achieving elongations of 500% or more [1]. Superplastic forming has been used widely, especially in the aerospace industry, for the production of a range of very light and structurally complex-shaped components [2,3]. One of the limiting constraints in superplastic forming is the development of localised necking, ultimately leading to failure. This phenomenon has been considered in both experimental and theoretical studies [4}9], and can result from the
* Corresponding author. Tel.: #44-0121-414-3317; fax: #44-0121-414-3958. E-mail address: [email protected] (J. Lin). 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 0 ) 0 0 0 5 5 - 2
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inhomogeneity of microstructure [4,9] and from the stress concentrations generated by the geometric features of complex-shaped components [7,8,10]. Necking or thinning is a critical factor for the superplastic forming of 3D structural components. It is well known that strain rate sensitivity is important in determining the tensile ductility of superplastic materials [4,11}13]. When strain rate sensitivity is low, the increase in stress at the neck enhances the necking process leading to failure and a low elongation to fracture. On the other hand, when strain rate sensitivity is high, the strain rate increases slowly due to the increased stress in the neck region, and as a result of this, the neck forms more gradually. The tensile ductility, therefore, increases with an increasing strain rate sensitivity. A model capable of predicting the strain rate sensitivity of superplastic materials, and its dependence on process variables, is of value. Backofen et al. [14] "rst investigated strain rate sensitivity and a quantitative index of strain rate sensitivity was represented within a power-law constitutive equation as follows: p"Ke K,
(1)
where p is the true #ow stress, e the true strain rate, K a material constant and m the strain rate sensitivity, which could then be obtained as follows: * ln p . m" * ln e
(2)
The strain rate sensitivity parameter can be determined from either a series of constant strain rate tests or strain rate jump experimental data, which have been studied by Kim and Dunne [15]. Research has been carried out in investigating the dependence of necking behaviour on the strain rate sensitivity parameter m [4,11}14,16]. The ideal value of m for a superplastic alloy deforming according to power law creep is m"1. That is, for this value of m, an irregular testpiece would maintain its irregularities during deformation, rather than continue to localise deformation at an irregularity. Alloys with total elongation '2000% have high m-values, typically '0.6 [12], when deformation takes place in appropriate temperature and strain rate regimes. Control of strain rate in superplastic forming is therefore of signi"cant importance. Recently, phenomenological constitutive equations [9,15,17] have been proposed for superplastic deformation which have been shown to be capable of capturing stress}strain, and stress}strain rate behaviour, and furthermore, the dependence of strain rate sensitivity on strain rate and average grain size. In addition, they are capable of modelling correctly the strain hardening due to static and dynamic grain growth, and the evolution of average grain size. They have been developed for the analysis of heterogeneous microstructures [18] and the resulting inhomogeneous deformation in superplasticity [19]. This paper is concerned with the practical application of the constitutive equations discussed above to the simulation of a realistic superplastic forming process. In the process, it is necessary to control the pressure cycle to maintain appropriate deformation strain rates to maximise strain rate sensitivity. A classical stability analysis is carried out "rst for the phenomenological constitutive equations, which are then implemented into "nite element software and used to investigate process variable control, grain size distributions, and thinning in superplastic blow-forming. In the following section, the constitutive equations are introduced.
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2. Constitutive equations 2.1. Uniaxial sinh-law model Sinh-law-based elastic}viscoplastic constitutive equations have been used by Kim and Dunne [15] for a range of materials deforming superplastically. The equations are e N"a sinh[b(p!R!k)]d\A,
(3)
RQ "(C !c R)"e N", dQ "(a #b "e N ")d\A , p"E(e2!eN),
(4) (5) (6)
where e2 and eN are total and plastic strains respectively, d is average grain size with an initial value of 6.8 lm, R is an isotropic hardening variable, and E ("1000 MPa at 9003C) is Young's modulus. a, b, c, C , c , a , b , c , are material constants, which were determined from uniaxial experimental data for Ti}6Al}4V at 9003C [15], and listed in Table 1. The set of di!erential equations describing superplastic deformation coupled with average grain size can be integrated numerically. Uniaxial stress}strain curves for Ti}6Al}4V at 9003C are shown in Fig. 1, which have been computed using the material data given in Table 1, but with the three di!erent values of b, namely b"0.08, 0.1135 and 0.16, and at constant strain rate 1;10\ s\. The graphs show the strain rate sensitivity to the material parameter b, which plays a similar role, in the sinh-law used here, to the strain rate sensitivity parameter, m, used in a power-law representation. 2.2. Multiaxial sinh-law superplastic material model In a manner similar to that for creep deformation [19], the above uniaxial sinh-law model can be generalised by consideration of a dissipation potential function. First consider the strain rate equation (3) without the hardening and grain growth variables which then reduces to e N"a sinh(bp).
(7)
Eq. (7) can be generalised for multiaxial conditions by assuming an energy dissipation rate potential of the form ("(a/b) cosh(bp ), C
(8)
Table 1 Material constants for Ti}6Al}4V at 9003C a
b
c
C
c
a
b
c
0.2418e!6
0.1135
1.0170
3.9736
0.8801
0.2059e!13
0.6899e!9
3.0210
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Fig. 1. Variation of stress}strain relationships, at a strain rate of 10\ s\, with the strain sensitivity parameter b for Ti}6Al}4V at 9003C.
where p "(3S S /2) is e!ective stress and S "p !d p /3 are stress deviators. Assuming C GH GH GH GH GH II normality and the associated #ow rule, the multiaxial relationship is given by
*t 3a S deN GH sinh(bp ). GH "jQ " (9) C *p 2 p dt GH C On reintroduction of the hardening and grain growth variables the e!ective plastic strain rate p for the sinh-law material model can be written as p "a sinh[b(p !R!k)]d\A (10) C and then the set of multiaxial viscoplastic constitutive equations, implemented within a large strain formulation, may be written as DN "3S /(2p )p , GH GH C RQ "(C !c R)p , dQ "(a #b p )d\AM ,
(12)
p "GDC #2jDC , GH GH II
(14)
(11)
(13)
£
in which DN is the rate of plastic deformation, DC the rate of elastic deformation, p the Jaumann GH GH GH rate of Cauchy stress, and G and j are the Lame elasticity constants. The multiaxial constitutive equations have been implemented into the large strain "nite element solver ABAQUS through a user de"ned subroutine CREEP and used to simulate the blow-forming of a 3D structural component. Both grain size e!ects and thinning are examined in the process, but "rst, in the next section, a stability analysis is carried out for the constitutive equations described above. £
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3. Stability of sinh-law material model under uniaxial tension Let A be the cross-sectional area of a uniaxial testpiece and P be the load applied. If incompressibility of the material under superplastic deformation is assumed, the reduction rate of the cross-sectional area of the testpiece can be expressed as
dA P ! "A a sinh b !R!k dt A
d\A.
(15)
Thus the shrinkage rate is a function of the cross-sectional area, A, and the parameter b. Consider a uniaxial testpiece containing necked regions with areas less than that of the bulk testpiece. In these regions, given the nonlinearity of the sinh function, the rate of change of the area for given load, P, is higher than that in the bulk of the testpiece, thus tending to lead to increased necking. Necking and localisation are minimised by removing the dependence of dA/dt on cross-sectional area, A, in Eq. (15). For a given load, P, this is in#uenced by the value of the parameter b, and a sensitivity study to evaluate the dependence of necking on b is therefore useful, since it plays a similar role in determining stability in superplasticity as the strain rate sensitivity parameter, m, in superplastic deformation characterised by power-law creep. 3.1. Ewect of load on necking If the un-necked cross-sectional area is unity, the shrinkage rate history can be obtained through numerically integrating the equation set (3)}(6) and (15). Fig. 2 shows the shrinkage rate against cross-sectional area A at the initial stresses of 5, 15 and 25 MPa. The curves represent the variation of !dA/dt with A at constant load, P. At the low stress level of 5 MPa, !dA/dt remains largely constant as the cross-sectional area, A, decreases. This indicates that the shrinkage rate is approximately independent of the cross-sectional area for this load. An irregular testpiece would therefore tend to maintain its irregularities during the deformation. However, at the higher stress level of 25 MPa, the shrinkage rate increases quickly with decreasing cross-sectional area A, showing that necking takes place rapidly leading to low elongation to failure.
Fig. 2. Shrinkage rates predicted using the sinh-law material model. The constant load computations were carried out at initial stresses of 5, 15 and 25 MPa.
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Fig. 3. E!ect of b on shrinkage rate. The results are predicted using the sinh-law with (solid lines) and without (dashed lines) hardening for b"0.05, 0.11 and 0.16 at constant load with an initial stress 20 MPa.
3.2. Ewect of parameter b on necking Equation set (3)}(6) and (15) has been numerically integrated using a range of values for parameter b, namely, 0.05, 0.11 and 0.16, at constant load with an initial stress of 20 MPa. The variation of !dA/dt with A is shown in Fig. 3. The shrinkage rate !dA/dt remains almost constant with A for b"0.05. As the magnitude of b increases, so the propensity for necking also increases since the shrinkage rate increases quickly with decreasing A. If the equation set is integrated in the absence of isotropic hardening, i.e. RQ "0 in Eq. (4), and hardening due to grain growth, i.e. dQ "0 in Eq. (5), the variations of !dA/dt with A for b"0.05, 0.11 and 0.16 are shown by the broken lines in Fig. 3. It can be seen that the shrinkage rates are higher for a given area than for the corresponding results which include the hardening e!ects. Typically, in a necked region, the stress and hence plastic strain rate are initially higher, leading to increased rates of isotropic and grain growth-induced hardening, thus tending to reduce plastic strain rates for a given stress. This has the e!ect of at least reducing the propensity to develop necks. 3.3. Necking map From Figs. 2 and 3, it can be seen that the shrinkage rate of a testpiece under superplastic deformation is related to both the stress level and the parameter b, which in#uences the plastic deformation rate. With a particular combination of stress level and magnitude of b, the shrinkage rate can remain approximately constant. Parametric studies have been carried out to calculate those combinations of load, P, and parameter b that prevent necking, and the results are shown in the form of a necking map in Fig. 4. The symbols in Fig. 4 show combinations of initial stress level and b which keep !dA/dt approximately constant within the cross-sectional area range 1.0'A'0.4. The straight line "tted through the data on logarithmic scales shown in Fig. 4 di!erentiates the combinations of load and b which do and do not lead to necking. Above the line, necking takes place and lower elongation to failure is expected. For combinations below the line,
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Fig. 4. Map showing the dependence of necking on initial stress and the strain rate sensitivity parameter b. The symbols correspond to combinations of initial stress and b which give approximately constant values of !dA/dt within the region A"1}0.4.
!dA/dt is constant or decreasing in magnitude, so that an irregular testpiece is expected to maintain its initial level of irregularity or become more regular during superplastic deformation. For the case of titanium alloy Ti}6Al}4V undergoing superplastic deformation at 9003C, for which the parameter b takes the value 0.1135 (see Table 1), Fig. 4 shows that for stress level below about 8 MPa, necking is inhibited, but that for loading stresses above this level, necking does occur which will lead to lower obtainable elongations. In the superplastic forming of structural components, necking can be initiated by inhomogeneity of loading, by geometrical features, or by heterogeneity of microstructure. In the next section, the incremental multiaxial material model discussed above, when implemented into "nite element software, is used to simulate the superplastic blow-forming of a rectangular-section box with "llet surface radii. Thinning due to inhomogeneity of loading and geometrical features is investigated. E!ects of initially heterogeneous microstructures are not addressed.
4. Simulation of superplastic blow-forming The "nite element simulation for forming a rectangular-section box, which is made of Ti}6Al}4V alloy and superplastically blow-formed at a constant temperature of 9003C, has been carried out using the commercial FE solver ABAQUS. The superplastic forming process consists of clamping a #at metal sheet against the die, the surface of which forms a cavity in the shape required. Gas pressure is applied to the opposite side of the sheet, forcing it to acquire the die shape. The maximum strain rate over the deforming sheet is controlled to be close to the optimum deformation rate of the material, and this is achieved by varying the applied gas pressure. In order to obtain optimal and reliable convergence it is essential that the rigid die surface representation is smooth. ABAQUS allows for constructing complex-shaped 3D rigid surfaces with BeH zier triangular surface patches on the elements generated over the die surface. If the elements generated over the die surface are too coarse for the curved parts of the surface, the re-constructed smoothed surface does not accurately represent the original rigid surface [10]. However, too many
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Fig. 5. Geometry and FE models for the die surface and the material blank (Dimensions in mm and all "llet radii 40 mm).
triangular patches are a waste of computer resources [20]. The number of patches required to re-de"ne accurately the die surface can be reduced by specifying the normals at every vertex point of the mesh [10]. Due to its symmetry only a quarter of the rectangular box is considered for the "nite element forming simulation, which is shown in Fig. 5. In order to avoid having points `fall-o! a the rigid die surface during the numerical simulation of the forming process, more than a quarter of the die surface is modelled. The de"nition of the die surface for the FE analyses are detailed by Lin et al. [10]. The metal sheet for the quarter of the rectangular box is considered for the analysis and shown in Fig. 5. The original material blank has a uniform thickness of 1.25 mm. Four-noded quadrilateral elements were used for the analysis and 682 thin shell elements and 736 nodes were generated over the quarter-sheet blank. Contact between the rigid die surface and the deformable metal sheet is de"ned by specifying (i) the die surface model and (ii) the contact nodal set for the material mesh. Friction, as a surface interaction property, is related to the contact pair by specifying a friction coe$cient. The value of the friction coe$cient a!ects forming pressure histories and through thickness strain distributions [6,7]. However, the main objectives of the current research are to evaluate the grain size variation and necking due to the strain rate sensitivity parameter (material properties) and the geometric features of components. Thus, a low value of friction coe$cient, k"0.05, for the Coulomb friction model with q "0.5 MPa, is chosen here to reduce the e!ect of the friction on the simulation
results for non-uniform thinning. The analysis can be divided into two stages. The initial application of gas pressure is assumed to occur so quickly that is involves a purely elastic response. The maximum pressure applied at this
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stage is 960 Pa. During the viscoplastic analysis, the ABAQUS parameter CETOL, which is used to control the accuracy of the numerical integration for the constitutive equations, is set to 0.002. In addition to the original constitutive model, the gradient, **p/*p , is required by ABAQUS, so that C implicit integration can be used. A target deformation rate e is de"ned for each simulation. The maximum strain rate, e ,
determined by considering all integration points within the deforming sheet, is chosen as the control variable. The gas pressure is therefore chosen to ensure that this maximum strain rate is equal to the target strain rate during the forming process. This is automatically controlled within the "nite element solver by adjusting the applied gas pressure in the gas blow forming. The forming process is considered to be completed when all the nodes on the deforming sheet are in contact with the die surface model.
5. Modelling results Fig. 6 shows the deformation of the superplastic metal sheet at di!erent stages during the forming process, in which t is the time required for the complete forming of the rectangularD section box. The contours represent the magnitude of e!ective plastic strain rate under uniformly distributed gas pressure. The forming simulation is carried out using the target strain rate e "1;10\ s\. It can be seen that the e!ective strain rate is not uniformly distributed over the
Fig. 6. The forming process and e!ective plastic strain rate "eld plots at t/t +(a) 0.1, (b) 0.6 and (c) 1.0. The D computation was carried out using parameter b"0.1135 and target forming rate e "1;10\ s\.
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Fig. 7. Variation of the maximum strain rate history over the deforming material blank for forming the rectangular box with di!erent target strain rates: e "1;10\ and e "1;10\ s\.
sheet. Before it is in contact with the bottom of the die surface, the highest strain rate occurs near the centre point of the sheet. The corner is the last part to be fully formed (Fig. 6(c)) and the material removed from the corner remains stationary due to friction and geometric e!ects. The thinnest area of the formed part is that near the corner, where tearing is most likely to take place in the superplastic forming process. The superplasticity of a material depends on strain rate; it occurs only within a narrow range of strain rate in which an optimum value exists which is unique to each material for a given temperature [4]. However, due to geometric and loading features, it is often not possible to maintain constant strain rate over the entire component, which has been shown in Fig. 6. Finite element forming simulations were carried out using two di!erent target strain rates; e "1;10\ and 1;10\ s\. The comparison of the maximum strain rate histories over the deforming material sheet and the corresponding target strain rates is shown in Fig. 7. The di!erence between the achieved maximum and the corresponding target strain rate is within 20% for both cases. The gas pressure histories required to maintain the maximum strain rates over the deforming sheet near the target deformation rates are shown in Fig. 8 for both cases. Higher gas pressure is needed for the higher target strain rate forming because of the higher #ow stresses required for a higher strain rate. The gas pressure increases continuously with the forming time because of increasing geometrical and frictional constraint, and in addition, to overcome the material hardening due to grain growth e!ects. Very high gas pressure is required to "ll the corner part of the die, which is the last stage of the forming process. The non-uniform thinning of the formed part has been investigated using di!erent target forming rates. The e!ects of the grain size distribution over the formed part and the strain rate sensitivity parameter b on the non-uniform thinning are addressed in the next sections. 5.1. Ewect of forming rate on grain size and thinning distributions The contour plots in Fig. 9 show the distributions of calculated average grain size for the following three cases: (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and
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Fig. 8. Variation of applied gas pressure history for forming the rectangular box with di!erent target strain rates: e "1;10\ and e "1;10\ s\.
(c) b"0.05, e "1;10\ s\. In each case, the simulation has been continued until the superplastic forming is completed, as de"ned earlier. Initially the average grain size is speci"ed to be 6.8 lm uniformly distributed over the material blank. The non-uniformity of the grain size distribution over the formed part is mainly due to the non-uniform plastic deformation in the forming process. This can be seen by integrating the grain growth rate equation (13): (16) d"[dA >#(c #1)(a t#b p)]A >\, where d is the initial grain size. The grain growth governing equation comprises a contribution from static grain growth, given by a t, and a contribution from deformation enhanced growth, given by b p. In the forming process considered, the static grain growth is the same everywhere, but the deformation enhanced growth depends on strain, and therefore varies spatially within the sheet, resulting in the distribution of average grain sizes shown in Fig. 9. By comparison of the "eld plots (a), e "1;10\ s\, and (b), e "1;10\ s\, in Fig. 9, it can be seen that the average grain size over the formed part is larger for the low forming rate (a) than that for the high forming rate (b). This occurs because the time required to form the part is longer with the lower forming rate, which enables further isothermal grain growth to take place. However, the spatial variation in grain size for the formed part is more signi"cant (*d+1.08 lm) at the higher forming rate (b) than that (*d+0.5 lm) at the lower forming rate (a). Both the static and plastic strain-induced grain growth rates decrease with larger grain size. This was experimentally veri"ed [4] and can be seen to be represented in Eq. (6). The non-uniform grain size distribution a!ects the plastic strain rate "eld, resulting in throughthickness strain distributions in the formed part, which are shown in Fig. 10. The thinnest area is at the corner of the formed box. This is mainly due to the geometric features of the component and die. However, the through-thickness strain distributions are also a!ected by forming rates and grain size distributions. Field plots in Fig. 10(a) and (b) show the variation of the true throughthickness strains at the target forming rates for (a) e "1;10\ and (b) 1;10\ s\, respectively. The maximum di!erences in the through-thickness strains over the formed part are about 1.06 in (a) and 0.84 in (b) for the two target forming rates. The more uniform distribution in
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Fig. 9. Grain size distributions for the formed parts. The computations were carried out using (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and (c) b"0.05, e "1;10\ s\.
through-thickness strain at the higher forming rate, shown in Fig. 10(b) is directly related to the greater non-uniformity of the grain size distribution, shown in Fig. 9(b), which signi"cantly in#uences the plastic strain rates. Fig. 11 shows the through-thickness fraction histories at the corner location of the formed box for the three cases. The through-thickness fraction is de"ned as the ratio of the actual throughthickness to the original thickness of the material blank. The thinning is always more signi"cant for the lower target forming rate, b"0.1135 and e "1;10\ s\, than that for the higher target forming rate, b"0.1135, and e "1;10\ s\, at the same forming time fractions. For the "nal formed product, the thickness at the corner is reduced to about 30% of its original for b"0.1135 and e "1;10\ s\ and 40% for b"0.1135 and e "1;10\ s\. The higher forming rate therefore results in more severe non-uniform distribution in grain size but better through-thickness distribution. 5.2. Ewect of parameter b on grain size and thinning distributions Figs. 9(a) and (c) show the grain size "eld plots computed using the same target strain rate e "1;10\ s\ but with di!erent magnitudes of the parameter b, namely 0.1135 and 0.05. Similar distributions and magnitudes of average grain size have been obtained for the two cases, indicating that they are largely independent of b. Field plots (a) and (c) in Fig. 10 show the through-thickness strains for the formed part using the same target forming rate of e "1;10\ s\, but with di!erent values of b, b"0.1135 and 0.05.
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Fig. 10. Through-thickness strain "elds for the formed parts. The computations were carried out using (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and (c) b"0.05, e "1;10\ s\.
Fig. 11. Variation of through-thickness at the corner of the box with the forming time fractions for two values of strain rate parameter b and two target forming strain rates.
A more uniform through-thickness strain distribution is obtained for the lower value of b. This can also be seen from the through-thickness fraction history at the corner location of the formed box, shown in Fig. 11, in which it can be seen that the thinnest area for the formed part is about 35% of its original thickness for b"0.05 and about 30% for b"0.1135.
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6. Conclusions For a material that can be modelled by a sinh-law constitutive equation, under conditions of uniaxial loading, increasing b enhances necking, as does increasing load. In a necked region, the initially higher plastic strain rates can result in locally increased deformation-enhanced grain growth causing the material to harden, and hence reduce the rate of neck formation. A necking map has been derived showing for which combinations of load and b necking will or will not take place. The boundary between the two is a straight line on log p}log b scales. The sinh-law parameter b plays a similar role in controlling necking as the strain rate sensitivity in power-law creep. In the "nite element simulation of the superplastic blow-forming of a rectangular-section box, a spatial variation of strain rate is obtained. A higher target strain rate tends to lead to greater heterogeneity in grain size distribution which may result in greater variation in the resulting product material properties. However, a higher target strain rate did lead to more uniform thinning in the resulting product. The magnitude of the strain rate sensitivity parameter b in the hyperbolic sine equation has little e!ect on grain size distribution, but does a!ect the thinning distribution. A lower value of b results in a more uniform thickness distribution in the superplastically blow-formed product.
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[14] Backfen WA, Turner IR, Avery DH. Superplasticity in an Al}Zn alloy. ASM Transactions Quarterly 1964; 57:980}90. [15] Kim TW, Dunne FPE. Determination of superplastic constitutive equations and strain rate sensitivity for aerospace alloys. Journal of Aerodynamics Engineering 1997;211:367}80. [16] Zhou M, Dunne FPE. Mechanism-based constitutive equations for the superplastic behavior of a titanium alloy. Journal of Strain Analysis 1996;31(3):187}96. [17] Kim T-W, Dunne FPE. Modelling heterogeneous microstructures in superplasticity. Proceedings of the Royal Society of London A 1999;455:701}18. [18] Dunne FPE, Kim T-W. Inhomogeneous deformation and failure in superplasticity. Proceedings of the Royal Society of London A 1999;455:719}36. [19] Lin J, Hayhurst DR, Dyson BF. A new design of uniaxial testpiece with slit extensometer ridges for improved accuracy of strain measurement. International Journal of Mechanical Sciences 1993;35(1):63}78. [20] Hattangady NH. Coarsening of mesh models for representation of rigid objects in "nite element analysis. International Journal for Numerical Methods in Engineering 1999;44:313}26.
Modelling grain growth evolution and necking in superplastic blow-forming J. Lin *, F.P.E. Dunne School of Manufacturing and Mechanical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford, OXI 3PJ UK Received 2 August 1999; received in revised form 3 April 2000
Abstract A stability analysis is carried out to investigate necking in superplastic materials characterised by the sinh-law constitutive equation e NJsinh(bp). The e!ects of load and the strain rate sensitivity parameter b on necking are quantitatively studied and a necking map is obtained for conditions of uniaxial loading. Finite element simulations of a superplastic blow-forming process are carried out in order to investigate both non-uniform thinning and grain size distribution which result from the use of the sinh-law constitutive equation. The pressure cycle required to ensure a target maximum strain rate is not exceeded in the material is obtained. The e!ects of strain rate and the magnitude of the parameter b on the grain size and through-thickness strain distributions for the formed part are investigated. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Superplastic forming; Non-uniform thinning; Grain growth; Gas blow-forming
1. Introduction Superplastic materials can undergo extensive tensile plastic deformation prior to failure, typically achieving elongations of 500% or more [1]. Superplastic forming has been used widely, especially in the aerospace industry, for the production of a range of very light and structurally complex-shaped components [2,3]. One of the limiting constraints in superplastic forming is the development of localised necking, ultimately leading to failure. This phenomenon has been considered in both experimental and theoretical studies [4}9], and can result from the
* Corresponding author. Tel.: #44-0121-414-3317; fax: #44-0121-414-3958. E-mail address: [email protected] (J. Lin). 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 0 ) 0 0 0 5 5 - 2
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inhomogeneity of microstructure [4,9] and from the stress concentrations generated by the geometric features of complex-shaped components [7,8,10]. Necking or thinning is a critical factor for the superplastic forming of 3D structural components. It is well known that strain rate sensitivity is important in determining the tensile ductility of superplastic materials [4,11}13]. When strain rate sensitivity is low, the increase in stress at the neck enhances the necking process leading to failure and a low elongation to fracture. On the other hand, when strain rate sensitivity is high, the strain rate increases slowly due to the increased stress in the neck region, and as a result of this, the neck forms more gradually. The tensile ductility, therefore, increases with an increasing strain rate sensitivity. A model capable of predicting the strain rate sensitivity of superplastic materials, and its dependence on process variables, is of value. Backofen et al. [14] "rst investigated strain rate sensitivity and a quantitative index of strain rate sensitivity was represented within a power-law constitutive equation as follows: p"Ke K,
(1)
where p is the true #ow stress, e the true strain rate, K a material constant and m the strain rate sensitivity, which could then be obtained as follows: * ln p . m" * ln e
(2)
The strain rate sensitivity parameter can be determined from either a series of constant strain rate tests or strain rate jump experimental data, which have been studied by Kim and Dunne [15]. Research has been carried out in investigating the dependence of necking behaviour on the strain rate sensitivity parameter m [4,11}14,16]. The ideal value of m for a superplastic alloy deforming according to power law creep is m"1. That is, for this value of m, an irregular testpiece would maintain its irregularities during deformation, rather than continue to localise deformation at an irregularity. Alloys with total elongation '2000% have high m-values, typically '0.6 [12], when deformation takes place in appropriate temperature and strain rate regimes. Control of strain rate in superplastic forming is therefore of signi"cant importance. Recently, phenomenological constitutive equations [9,15,17] have been proposed for superplastic deformation which have been shown to be capable of capturing stress}strain, and stress}strain rate behaviour, and furthermore, the dependence of strain rate sensitivity on strain rate and average grain size. In addition, they are capable of modelling correctly the strain hardening due to static and dynamic grain growth, and the evolution of average grain size. They have been developed for the analysis of heterogeneous microstructures [18] and the resulting inhomogeneous deformation in superplasticity [19]. This paper is concerned with the practical application of the constitutive equations discussed above to the simulation of a realistic superplastic forming process. In the process, it is necessary to control the pressure cycle to maintain appropriate deformation strain rates to maximise strain rate sensitivity. A classical stability analysis is carried out "rst for the phenomenological constitutive equations, which are then implemented into "nite element software and used to investigate process variable control, grain size distributions, and thinning in superplastic blow-forming. In the following section, the constitutive equations are introduced.
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2. Constitutive equations 2.1. Uniaxial sinh-law model Sinh-law-based elastic}viscoplastic constitutive equations have been used by Kim and Dunne [15] for a range of materials deforming superplastically. The equations are e N"a sinh[b(p!R!k)]d\A,
(3)
RQ "(C !c R)"e N", dQ "(a #b "e N ")d\A , p"E(e2!eN),
(4) (5) (6)
where e2 and eN are total and plastic strains respectively, d is average grain size with an initial value of 6.8 lm, R is an isotropic hardening variable, and E ("1000 MPa at 9003C) is Young's modulus. a, b, c, C , c , a , b , c , are material constants, which were determined from uniaxial experimental data for Ti}6Al}4V at 9003C [15], and listed in Table 1. The set of di!erential equations describing superplastic deformation coupled with average grain size can be integrated numerically. Uniaxial stress}strain curves for Ti}6Al}4V at 9003C are shown in Fig. 1, which have been computed using the material data given in Table 1, but with the three di!erent values of b, namely b"0.08, 0.1135 and 0.16, and at constant strain rate 1;10\ s\. The graphs show the strain rate sensitivity to the material parameter b, which plays a similar role, in the sinh-law used here, to the strain rate sensitivity parameter, m, used in a power-law representation. 2.2. Multiaxial sinh-law superplastic material model In a manner similar to that for creep deformation [19], the above uniaxial sinh-law model can be generalised by consideration of a dissipation potential function. First consider the strain rate equation (3) without the hardening and grain growth variables which then reduces to e N"a sinh(bp).
(7)
Eq. (7) can be generalised for multiaxial conditions by assuming an energy dissipation rate potential of the form ("(a/b) cosh(bp ), C
(8)
Table 1 Material constants for Ti}6Al}4V at 9003C a
b
c
C
c
a
b
c
0.2418e!6
0.1135
1.0170
3.9736
0.8801
0.2059e!13
0.6899e!9
3.0210
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Fig. 1. Variation of stress}strain relationships, at a strain rate of 10\ s\, with the strain sensitivity parameter b for Ti}6Al}4V at 9003C.
where p "(3S S /2) is e!ective stress and S "p !d p /3 are stress deviators. Assuming C GH GH GH GH GH II normality and the associated #ow rule, the multiaxial relationship is given by
*t 3a S deN GH sinh(bp ). GH "jQ " (9) C *p 2 p dt GH C On reintroduction of the hardening and grain growth variables the e!ective plastic strain rate p for the sinh-law material model can be written as p "a sinh[b(p !R!k)]d\A (10) C and then the set of multiaxial viscoplastic constitutive equations, implemented within a large strain formulation, may be written as DN "3S /(2p )p , GH GH C RQ "(C !c R)p , dQ "(a #b p )d\AM ,
(12)
p "GDC #2jDC , GH GH II
(14)
(11)
(13)
£
in which DN is the rate of plastic deformation, DC the rate of elastic deformation, p the Jaumann GH GH GH rate of Cauchy stress, and G and j are the Lame elasticity constants. The multiaxial constitutive equations have been implemented into the large strain "nite element solver ABAQUS through a user de"ned subroutine CREEP and used to simulate the blow-forming of a 3D structural component. Both grain size e!ects and thinning are examined in the process, but "rst, in the next section, a stability analysis is carried out for the constitutive equations described above. £
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3. Stability of sinh-law material model under uniaxial tension Let A be the cross-sectional area of a uniaxial testpiece and P be the load applied. If incompressibility of the material under superplastic deformation is assumed, the reduction rate of the cross-sectional area of the testpiece can be expressed as
dA P ! "A a sinh b !R!k dt A
d\A.
(15)
Thus the shrinkage rate is a function of the cross-sectional area, A, and the parameter b. Consider a uniaxial testpiece containing necked regions with areas less than that of the bulk testpiece. In these regions, given the nonlinearity of the sinh function, the rate of change of the area for given load, P, is higher than that in the bulk of the testpiece, thus tending to lead to increased necking. Necking and localisation are minimised by removing the dependence of dA/dt on cross-sectional area, A, in Eq. (15). For a given load, P, this is in#uenced by the value of the parameter b, and a sensitivity study to evaluate the dependence of necking on b is therefore useful, since it plays a similar role in determining stability in superplasticity as the strain rate sensitivity parameter, m, in superplastic deformation characterised by power-law creep. 3.1. Ewect of load on necking If the un-necked cross-sectional area is unity, the shrinkage rate history can be obtained through numerically integrating the equation set (3)}(6) and (15). Fig. 2 shows the shrinkage rate against cross-sectional area A at the initial stresses of 5, 15 and 25 MPa. The curves represent the variation of !dA/dt with A at constant load, P. At the low stress level of 5 MPa, !dA/dt remains largely constant as the cross-sectional area, A, decreases. This indicates that the shrinkage rate is approximately independent of the cross-sectional area for this load. An irregular testpiece would therefore tend to maintain its irregularities during the deformation. However, at the higher stress level of 25 MPa, the shrinkage rate increases quickly with decreasing cross-sectional area A, showing that necking takes place rapidly leading to low elongation to failure.
Fig. 2. Shrinkage rates predicted using the sinh-law material model. The constant load computations were carried out at initial stresses of 5, 15 and 25 MPa.
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Fig. 3. E!ect of b on shrinkage rate. The results are predicted using the sinh-law with (solid lines) and without (dashed lines) hardening for b"0.05, 0.11 and 0.16 at constant load with an initial stress 20 MPa.
3.2. Ewect of parameter b on necking Equation set (3)}(6) and (15) has been numerically integrated using a range of values for parameter b, namely, 0.05, 0.11 and 0.16, at constant load with an initial stress of 20 MPa. The variation of !dA/dt with A is shown in Fig. 3. The shrinkage rate !dA/dt remains almost constant with A for b"0.05. As the magnitude of b increases, so the propensity for necking also increases since the shrinkage rate increases quickly with decreasing A. If the equation set is integrated in the absence of isotropic hardening, i.e. RQ "0 in Eq. (4), and hardening due to grain growth, i.e. dQ "0 in Eq. (5), the variations of !dA/dt with A for b"0.05, 0.11 and 0.16 are shown by the broken lines in Fig. 3. It can be seen that the shrinkage rates are higher for a given area than for the corresponding results which include the hardening e!ects. Typically, in a necked region, the stress and hence plastic strain rate are initially higher, leading to increased rates of isotropic and grain growth-induced hardening, thus tending to reduce plastic strain rates for a given stress. This has the e!ect of at least reducing the propensity to develop necks. 3.3. Necking map From Figs. 2 and 3, it can be seen that the shrinkage rate of a testpiece under superplastic deformation is related to both the stress level and the parameter b, which in#uences the plastic deformation rate. With a particular combination of stress level and magnitude of b, the shrinkage rate can remain approximately constant. Parametric studies have been carried out to calculate those combinations of load, P, and parameter b that prevent necking, and the results are shown in the form of a necking map in Fig. 4. The symbols in Fig. 4 show combinations of initial stress level and b which keep !dA/dt approximately constant within the cross-sectional area range 1.0'A'0.4. The straight line "tted through the data on logarithmic scales shown in Fig. 4 di!erentiates the combinations of load and b which do and do not lead to necking. Above the line, necking takes place and lower elongation to failure is expected. For combinations below the line,
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Fig. 4. Map showing the dependence of necking on initial stress and the strain rate sensitivity parameter b. The symbols correspond to combinations of initial stress and b which give approximately constant values of !dA/dt within the region A"1}0.4.
!dA/dt is constant or decreasing in magnitude, so that an irregular testpiece is expected to maintain its initial level of irregularity or become more regular during superplastic deformation. For the case of titanium alloy Ti}6Al}4V undergoing superplastic deformation at 9003C, for which the parameter b takes the value 0.1135 (see Table 1), Fig. 4 shows that for stress level below about 8 MPa, necking is inhibited, but that for loading stresses above this level, necking does occur which will lead to lower obtainable elongations. In the superplastic forming of structural components, necking can be initiated by inhomogeneity of loading, by geometrical features, or by heterogeneity of microstructure. In the next section, the incremental multiaxial material model discussed above, when implemented into "nite element software, is used to simulate the superplastic blow-forming of a rectangular-section box with "llet surface radii. Thinning due to inhomogeneity of loading and geometrical features is investigated. E!ects of initially heterogeneous microstructures are not addressed.
4. Simulation of superplastic blow-forming The "nite element simulation for forming a rectangular-section box, which is made of Ti}6Al}4V alloy and superplastically blow-formed at a constant temperature of 9003C, has been carried out using the commercial FE solver ABAQUS. The superplastic forming process consists of clamping a #at metal sheet against the die, the surface of which forms a cavity in the shape required. Gas pressure is applied to the opposite side of the sheet, forcing it to acquire the die shape. The maximum strain rate over the deforming sheet is controlled to be close to the optimum deformation rate of the material, and this is achieved by varying the applied gas pressure. In order to obtain optimal and reliable convergence it is essential that the rigid die surface representation is smooth. ABAQUS allows for constructing complex-shaped 3D rigid surfaces with BeH zier triangular surface patches on the elements generated over the die surface. If the elements generated over the die surface are too coarse for the curved parts of the surface, the re-constructed smoothed surface does not accurately represent the original rigid surface [10]. However, too many
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Fig. 5. Geometry and FE models for the die surface and the material blank (Dimensions in mm and all "llet radii 40 mm).
triangular patches are a waste of computer resources [20]. The number of patches required to re-de"ne accurately the die surface can be reduced by specifying the normals at every vertex point of the mesh [10]. Due to its symmetry only a quarter of the rectangular box is considered for the "nite element forming simulation, which is shown in Fig. 5. In order to avoid having points `fall-o! a the rigid die surface during the numerical simulation of the forming process, more than a quarter of the die surface is modelled. The de"nition of the die surface for the FE analyses are detailed by Lin et al. [10]. The metal sheet for the quarter of the rectangular box is considered for the analysis and shown in Fig. 5. The original material blank has a uniform thickness of 1.25 mm. Four-noded quadrilateral elements were used for the analysis and 682 thin shell elements and 736 nodes were generated over the quarter-sheet blank. Contact between the rigid die surface and the deformable metal sheet is de"ned by specifying (i) the die surface model and (ii) the contact nodal set for the material mesh. Friction, as a surface interaction property, is related to the contact pair by specifying a friction coe$cient. The value of the friction coe$cient a!ects forming pressure histories and through thickness strain distributions [6,7]. However, the main objectives of the current research are to evaluate the grain size variation and necking due to the strain rate sensitivity parameter (material properties) and the geometric features of components. Thus, a low value of friction coe$cient, k"0.05, for the Coulomb friction model with q "0.5 MPa, is chosen here to reduce the e!ect of the friction on the simulation
results for non-uniform thinning. The analysis can be divided into two stages. The initial application of gas pressure is assumed to occur so quickly that is involves a purely elastic response. The maximum pressure applied at this
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stage is 960 Pa. During the viscoplastic analysis, the ABAQUS parameter CETOL, which is used to control the accuracy of the numerical integration for the constitutive equations, is set to 0.002. In addition to the original constitutive model, the gradient, **p/*p , is required by ABAQUS, so that C implicit integration can be used. A target deformation rate e is de"ned for each simulation. The maximum strain rate, e ,
determined by considering all integration points within the deforming sheet, is chosen as the control variable. The gas pressure is therefore chosen to ensure that this maximum strain rate is equal to the target strain rate during the forming process. This is automatically controlled within the "nite element solver by adjusting the applied gas pressure in the gas blow forming. The forming process is considered to be completed when all the nodes on the deforming sheet are in contact with the die surface model.
5. Modelling results Fig. 6 shows the deformation of the superplastic metal sheet at di!erent stages during the forming process, in which t is the time required for the complete forming of the rectangularD section box. The contours represent the magnitude of e!ective plastic strain rate under uniformly distributed gas pressure. The forming simulation is carried out using the target strain rate e "1;10\ s\. It can be seen that the e!ective strain rate is not uniformly distributed over the
Fig. 6. The forming process and e!ective plastic strain rate "eld plots at t/t +(a) 0.1, (b) 0.6 and (c) 1.0. The D computation was carried out using parameter b"0.1135 and target forming rate e "1;10\ s\.
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Fig. 7. Variation of the maximum strain rate history over the deforming material blank for forming the rectangular box with di!erent target strain rates: e "1;10\ and e "1;10\ s\.
sheet. Before it is in contact with the bottom of the die surface, the highest strain rate occurs near the centre point of the sheet. The corner is the last part to be fully formed (Fig. 6(c)) and the material removed from the corner remains stationary due to friction and geometric e!ects. The thinnest area of the formed part is that near the corner, where tearing is most likely to take place in the superplastic forming process. The superplasticity of a material depends on strain rate; it occurs only within a narrow range of strain rate in which an optimum value exists which is unique to each material for a given temperature [4]. However, due to geometric and loading features, it is often not possible to maintain constant strain rate over the entire component, which has been shown in Fig. 6. Finite element forming simulations were carried out using two di!erent target strain rates; e "1;10\ and 1;10\ s\. The comparison of the maximum strain rate histories over the deforming material sheet and the corresponding target strain rates is shown in Fig. 7. The di!erence between the achieved maximum and the corresponding target strain rate is within 20% for both cases. The gas pressure histories required to maintain the maximum strain rates over the deforming sheet near the target deformation rates are shown in Fig. 8 for both cases. Higher gas pressure is needed for the higher target strain rate forming because of the higher #ow stresses required for a higher strain rate. The gas pressure increases continuously with the forming time because of increasing geometrical and frictional constraint, and in addition, to overcome the material hardening due to grain growth e!ects. Very high gas pressure is required to "ll the corner part of the die, which is the last stage of the forming process. The non-uniform thinning of the formed part has been investigated using di!erent target forming rates. The e!ects of the grain size distribution over the formed part and the strain rate sensitivity parameter b on the non-uniform thinning are addressed in the next sections. 5.1. Ewect of forming rate on grain size and thinning distributions The contour plots in Fig. 9 show the distributions of calculated average grain size for the following three cases: (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and
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Fig. 8. Variation of applied gas pressure history for forming the rectangular box with di!erent target strain rates: e "1;10\ and e "1;10\ s\.
(c) b"0.05, e "1;10\ s\. In each case, the simulation has been continued until the superplastic forming is completed, as de"ned earlier. Initially the average grain size is speci"ed to be 6.8 lm uniformly distributed over the material blank. The non-uniformity of the grain size distribution over the formed part is mainly due to the non-uniform plastic deformation in the forming process. This can be seen by integrating the grain growth rate equation (13): (16) d"[dA >#(c #1)(a t#b p)]A >\, where d is the initial grain size. The grain growth governing equation comprises a contribution from static grain growth, given by a t, and a contribution from deformation enhanced growth, given by b p. In the forming process considered, the static grain growth is the same everywhere, but the deformation enhanced growth depends on strain, and therefore varies spatially within the sheet, resulting in the distribution of average grain sizes shown in Fig. 9. By comparison of the "eld plots (a), e "1;10\ s\, and (b), e "1;10\ s\, in Fig. 9, it can be seen that the average grain size over the formed part is larger for the low forming rate (a) than that for the high forming rate (b). This occurs because the time required to form the part is longer with the lower forming rate, which enables further isothermal grain growth to take place. However, the spatial variation in grain size for the formed part is more signi"cant (*d+1.08 lm) at the higher forming rate (b) than that (*d+0.5 lm) at the lower forming rate (a). Both the static and plastic strain-induced grain growth rates decrease with larger grain size. This was experimentally veri"ed [4] and can be seen to be represented in Eq. (6). The non-uniform grain size distribution a!ects the plastic strain rate "eld, resulting in throughthickness strain distributions in the formed part, which are shown in Fig. 10. The thinnest area is at the corner of the formed box. This is mainly due to the geometric features of the component and die. However, the through-thickness strain distributions are also a!ected by forming rates and grain size distributions. Field plots in Fig. 10(a) and (b) show the variation of the true throughthickness strains at the target forming rates for (a) e "1;10\ and (b) 1;10\ s\, respectively. The maximum di!erences in the through-thickness strains over the formed part are about 1.06 in (a) and 0.84 in (b) for the two target forming rates. The more uniform distribution in
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Fig. 9. Grain size distributions for the formed parts. The computations were carried out using (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and (c) b"0.05, e "1;10\ s\.
through-thickness strain at the higher forming rate, shown in Fig. 10(b) is directly related to the greater non-uniformity of the grain size distribution, shown in Fig. 9(b), which signi"cantly in#uences the plastic strain rates. Fig. 11 shows the through-thickness fraction histories at the corner location of the formed box for the three cases. The through-thickness fraction is de"ned as the ratio of the actual throughthickness to the original thickness of the material blank. The thinning is always more signi"cant for the lower target forming rate, b"0.1135 and e "1;10\ s\, than that for the higher target forming rate, b"0.1135, and e "1;10\ s\, at the same forming time fractions. For the "nal formed product, the thickness at the corner is reduced to about 30% of its original for b"0.1135 and e "1;10\ s\ and 40% for b"0.1135 and e "1;10\ s\. The higher forming rate therefore results in more severe non-uniform distribution in grain size but better through-thickness distribution. 5.2. Ewect of parameter b on grain size and thinning distributions Figs. 9(a) and (c) show the grain size "eld plots computed using the same target strain rate e "1;10\ s\ but with di!erent magnitudes of the parameter b, namely 0.1135 and 0.05. Similar distributions and magnitudes of average grain size have been obtained for the two cases, indicating that they are largely independent of b. Field plots (a) and (c) in Fig. 10 show the through-thickness strains for the formed part using the same target forming rate of e "1;10\ s\, but with di!erent values of b, b"0.1135 and 0.05.
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Fig. 10. Through-thickness strain "elds for the formed parts. The computations were carried out using (a) b"0.1135, e "1;10\ s\; (b) b"0.1135, e "1;10\ s\ and (c) b"0.05, e "1;10\ s\.
Fig. 11. Variation of through-thickness at the corner of the box with the forming time fractions for two values of strain rate parameter b and two target forming strain rates.
A more uniform through-thickness strain distribution is obtained for the lower value of b. This can also be seen from the through-thickness fraction history at the corner location of the formed box, shown in Fig. 11, in which it can be seen that the thinnest area for the formed part is about 35% of its original thickness for b"0.05 and about 30% for b"0.1135.
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6. Conclusions For a material that can be modelled by a sinh-law constitutive equation, under conditions of uniaxial loading, increasing b enhances necking, as does increasing load. In a necked region, the initially higher plastic strain rates can result in locally increased deformation-enhanced grain growth causing the material to harden, and hence reduce the rate of neck formation. A necking map has been derived showing for which combinations of load and b necking will or will not take place. The boundary between the two is a straight line on log p}log b scales. The sinh-law parameter b plays a similar role in controlling necking as the strain rate sensitivity in power-law creep. In the "nite element simulation of the superplastic blow-forming of a rectangular-section box, a spatial variation of strain rate is obtained. A higher target strain rate tends to lead to greater heterogeneity in grain size distribution which may result in greater variation in the resulting product material properties. However, a higher target strain rate did lead to more uniform thinning in the resulting product. The magnitude of the strain rate sensitivity parameter b in the hyperbolic sine equation has little e!ect on grain size distribution, but does a!ect the thinning distribution. A lower value of b results in a more uniform thickness distribution in the superplastically blow-formed product.
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