A formability evaluation method for sheet metal forming with non-linear strain path change

Abstract

The strain path dependence of forming limit curve (FLC) restricts its practical application for formability evaluation as the real strain path of material point in true industrial forming processes is often non-linear. In this work, the constraint equation for material necking was extended to the loading condition with non-linear strain path change and an analytical approach for forming limit calculation under combined-linear strain path loading with pre-strain was developed. The approach developed in this work was used to predict the forming limit strains after pre-strained in uni-axial tension, plane strain, and bi-axial tension for Aluminium alloy sheet Al 2008-T4. The comparison between the predicted results and the past experimental data showed that the effect of pre-strain on the material FLC can be reflected by the developed approach. Furthermore, a formability evaluation index accounted for the strain path history was put forward and used for the formability evaluation of a drawing reverse-drawing process, in which the blank underwent a continuous strain path change. Based on the measured data of strain path history and the forming limit strains at the initial failure location in specimens, the validity of the proposed formability evaluation index for sheet metal forming with non-linear strain path change was verified.

Introduction

Sheet metal forming is a material plasticity deformation processes, which is widely used in the field of auto-mobile, aerospace and shipbuilding industry for manufacturing parts and components. As a vital index for quality evaluation, forming limit curve (FLC) originally proposed by [1] and [2] finds widespread acceptance in the design and analysis of metal forming processes. Its accuracy is very important to the design of forming processes. Traditional forming limit diagram (FLD) is obtained by experimental test or theoretical calculation on the basis of linear strain path assumption. The best known approaches for the forming limit curve theoretical calculation are the diffuse and local necking criteria for plastic instability proposed by [3] and [4], respectively. Despite Swift’s diffuse and Hill’s local instability criteria, the M-K model proposed by [5] still remains one of the most reliable ways for FLC prediction.

As the true industry forming processes is considered, the real strain path of the formed material point is often non-linear. According to [6], the strain path has a marked effect on the FLC. They pointed out that the FLD obtained under a proportional loading condition is not suitable to the formability evaluation for multi-pass forming processes and concluded that the forming limit of sheet metal is related to the status of stress. Since then, several researches explored the effect of pre-strain on the forming limit ([7–9]). The works of [10, 11] and [12] showed that the groove-based M-K model can predict the effect of pre-strain on the forming limit curve. Arrieux et al. [13] conducted the experiment of forming limit for annealed Titanium alloys sheet and obtained the forming limit strains after pre-strained in uni-axial and bi-axial tension. Based on the recorded strain path and isotropic von-Mises yield criterion, a stress-based FLD (FLSD) was obtained by transforming the forming limit strains to the forming limit stresses and plotting them in the frame of minor-major principal stresses space. They found that the material forming limit stress curves under various pre-strain loading conditions are almost coincide. The work of [13] pioneered extending the strain-based forming limit diagram to the stress-based FLD. Since [13], several studies were conducted and similar conclusions were obtained. Sing and Rao [14] proposed a determination method for FLSD by using the material properties and plastic potential. They first determined an elliptic stress curve through the uni-axial tension test data, then obtained the FLSD by linear regression. This method is relatively simple, however, due to its own lack of a solid theoretical foundation, its rationality and applicability need further investigation. Zimniak [15] studied the application of FLSD in the finite element based numerical simulation for sheet metal forming, and indicated that the predicted results by using FLSD is more accurate than the traditional FLD in multi-pass forming processes. However, the assertion about the independence of FLSD on strain path is still lack of direct support and verification from experimental data. By conducting the experiment under a combined axial and internal-pressure loading condition, [16] investigated the forming limit for 5000 series Aluminium alloy tube and concluded that the path dependence of forming limit stresses is strongly affected by the strain hardening behaviour of the material for the given loading paths. As the combined loading condition is considered, in which the strain path is abruptly changed without unloading, [17] observed that the FLSD is also strain path dependent.

In the recent years, the forming limit evaluation and prediction for sheet metal forming with non-proportional loading have aroused more and more attentions, Stoughton and his co-workers [18–20], who have done numerous investigations in the enhanced stress-based forming limit criteria (e.g. the polar effective plastic strain diagram, PEPSD) for sheet metal forming. Another theoretical approach is the modified maximum force criterion (MMFC) proposed by Hora et al. [21, 22], which states that as the loading force reached the maximum value, the actual localization will occur and the strain path will gradually evolve toward plane strain loading. Based on the MMFC, [23] developed an analytical approach for the prediction of FLC after subjected to a combined-linear strain path change. Manopulo et al. [24, 25] used the extended MMFC to predict the localized necking under the non-proportional loading condition. With the experimental data performed by [26] and [27], the generalized forming limit concept proposed by [28] and the modified time dependent evaluation method were applied by [29] to evaluate the formability in a two-stage forming processes with drawing and reverse-drawing.

In this paper, the necking criterion for sheet metal forming was first extended to the loading condition with strain path variation, and an analytical approach for forming limit calculation with bi-linear strain path loading was developed. Then, a formability evaluation index accounted the effect of non-linear strain path change on the forming limit strains was proposed. The analytical method for FLCs prediction was validated by the experimental forming limit strains of Al 2008-T4 under the loading condition with a combined-linear strain path change. Finally, a drawing reverse-drawing test with prescribed continuous non-linear strain path variation was developed and used to verify the accuracy of the formability evaluation index.

Evaluation method for sheet metal forming with strain path change

According to the principle of limit analysis, as each linear strain path loading segment approaches to an indefinitely small part, the whole non-linear strain path can be treated as a combination of many divided single linear strain path loading segments. Thus, the loading condition with strain path variation can be represented by two subsequent linear (bi-linear) strain path.

By using the first order homogeneous property of yield criterion and the associated flow rule, the ratio of minor to major principal strain rates (β = d ε ₂/d ε ₁) and the ratio of principal stresses (α = σ ₂/σ ₁) are related. The strain path of material loading can be represented by either β or α, herein α is used. As the isotropic yield criterion is considered, α is varied from 0 to 1, corresponding to uni-axial tension loading (α = 0) to equal-biaxial tension loading states (α = 1). Bi-linear strain path loading can be viewed as a combination of a linear pre-strained stage and a linear subsequent straining stage, and the equivalent plastic strain should be additive for the two loading stages. The ratio of the principal strain rates and the principal stresses for these two stages are denoted as β ^pr and β, α ^pr and α, respectively.

Necking criterion for sheet metal forming with strain path variation

According to the theory of diffuse necking, the material will be in the state of plastic instability as the loading force reaches the maximum value for a tension specimen, i.e., d F = 0. For the bi-axial tension test, other researchers thought that it is not consist with the reality that d F ₁ = 0 and d F ₂ = 0 are satisfied at the same time during the process of plastic deformation. They proposed that only d F ₁ = 0 is the necessary condition for diffuse necking. Then the constraint (1) must be satisfied:

$$ d\sigma_{1}=\sigma_{1}d\varepsilon_{1} $$

(1)

For the loading condition with strain path variation, differentiating the ratio of the principal stresses, i.e., α = σ ₂/σ ₁, one can obtain:

$$ d\sigma_{2}=\alpha d\sigma_{1}+\sigma_{1} d\alpha $$

(2)

Note that in the theory of Hill’s local necking, the plastic instability will be reached as the strength increment of flow stress due to strain hardening balanced with the geometry softening effect by thickness reduction:

$$ d\sigma_{1}=\sigma_{1}(d\varepsilon_{1}+d\varepsilon_{2}) $$

(3)

Due to the constricted condition with a linear strain path loading, the theory of Hill’s local necking instability assumes that the ratio of principal stresses does not change. And an additional constraint equation should satisfied in the local necking criterion is:

$$ \alpha=\sigma_{2}/\sigma_{1}=d\sigma_{2}/d\sigma_{1} $$

(4)

Rearranging (4) in the following form:

$$ d\sigma_{2}=\alpha d\sigma_{1} $$

(5)

Comparing (5) and (2), it is showed that Eq. 2 reduces to Eq. 5 as the proportional loading condition is satisfied, i.e. the ratio of principal stresses keeps constant value (d α = 0).

Based on the first order homogeneous property of yield criterion and the associated flow rule, the relation between the equivalent stress and the major principal stress can be defined in the following form:

$$ \bar{\sigma}=\sigma_{1}/h(\alpha) $$

(6)

where $\bar {\sigma }$ is the equivalent stress defined by yield criterion. h(α) can be obtained by replacing σ ₁ and σ ₂ with 1 and α in the yield criterion.

$$ h(\alpha)=1/\bar{\sigma}(\sigma_{1}=1,\sigma_{2}=\alpha) $$

(7)

According to the equivalent principle of plastic power:

$$ d\bar{\varepsilon}^{\mathrm{p}}=\frac{\sigma_{1} d\varepsilon_{1}+\sigma_{2} d\varepsilon_{2}}{\bar{\sigma}}=g(\alpha) d\varepsilon_{1} $$

(8)

where $\bar {\varepsilon }^{\mathrm {p}}$ is the equivalent plastic strain.

$$ g(\alpha)=h(\alpha)[1+\alpha\cdot\beta(\alpha)] $$

(9)

As the function of α, β can be obtained by the associated flow rule:

$$ \beta(\alpha):=\frac{d\varepsilon_{2}}{d\varepsilon_{1}}=\frac{\partial\bar{\sigma}/\partial\sigma_{2}}{\partial\bar{\sigma}/\partial\sigma_{1}} $$

(10)

Substitution of constraint equations (1) and (2), associated flow rule into the differentiation of yield function, using the chain rule for the differential operation and eliminating the common term $d\bar {\varepsilon }^{\mathrm {p}}$ leads to:

$$ \bar{\sigma}\frac{1}{g(\alpha)}\left[1+\frac{\beta(\alpha)}{\beta^{\prime}(\alpha)}\frac{1}{1+\alpha\cdot\beta(\alpha)}\frac{\partial\beta}{\partial\varepsilon_{1}}\right]=\frac{d\sigma_{\mathrm{h}}}{d\bar{\varepsilon}^{\mathrm{p}}} $$

(11)

where σ _h is the flow stress defined by the material hardening law. β ^′(α) is the differentiation of β with respect to α.

Equation 11 is the plastic instability criterion for sheet metal forming under the loading condition with continuous strain path variation. From Eq. 11, one can see that this criterion is obtained by extending Hill’s local necking criterion to the loading condition with non-linear strain path change. This criterion eliminated the additional constraint equation of Swift’s diffuse and Hill’s local necking criteria. In the following section, it will be demonstrated that this criterion can be applied in the calculation of forming limit strain for bi-linear strain path loading with pre-strain and its accuracy will also be verified.

Analytical approach for forming limit calculation in bi-linear strain path loading

In order to demonstrate the calculation procedure for forming limit strain under bi-linear strain path loading, herein, the power law for material hardening curve, i.e., $\sigma _{\mathrm {h}}=K(\bar {\varepsilon }^{\mathrm {p}})^{n}$, and the anisotropy yield criterion (12) proposed by [30] are employed.

$$\begin{array}{@{}rcl@{}} f\!:=\!\bar{\sigma}\,-\,\sigma_{\mathrm{h}}\!\!&=&\!\!\left[\!\frac{R_{90}}{1\!\,+\,R_{0}}{\sigma_{1}^{k}}\,+\,\frac{R_{0}}{1\,+\,R_{0}}{\sigma_{2}^{k}}\,+\,\frac{R_{0}R_{90}}{1\!\,+\,R_{0}}(\sigma_{1}\,-\,\sigma_{2})^{k}\!\right]^{1/k}\\-\sigma_{\mathrm{h}}(\bar{\varepsilon}^{\mathrm{p}})&=&0 \end{array} $$

(12)

Insertion of the power law for material hardening behaviour into Eq. 11 results in:

$$ \frac{\bar{\varepsilon}^{\mathrm{p}}}{g(\alpha)}\left[1+\frac{\beta(\alpha)}{\beta^{\prime}(\alpha)}\frac{1}{1+\alpha\cdot\beta(\alpha)}\frac{\partial\beta}{\partial\varepsilon_{1}}\right]=n $$

(13)

Under the loading condition with strain path variation, integrating (8) over strain increment, we obtain:

$$ \bar{\varepsilon}^{\mathrm{p}}=\int{h(\alpha)d\varepsilon_{1}+{\alpha}h(\alpha)d\varepsilon_{2}} $$

(14)

For a linear strain path loading segment, Eq. 14 simplifies to:

$$ \bar{\varepsilon}^{\mathrm{p}}=g(\alpha)\varepsilon_{1} $$

(15)

For bi-linear strain path loading, the equivalent plastic strain is additive and constituted by two parts. One part corresponds to the first pre-strained linear loading procedure, and the other one does to the subsequent linear straining process. Then the total equivalent plastic strain for the bi-linear strain path loading procedure is:

$$\begin{array}{@{}rcl@{}} \bar{\varepsilon}^{\mathrm{p}}&=&{\int}_{0}^{\varepsilon_{1}^{\text{pr}}}{h(\alpha^{\text{pr}})d\varepsilon_{1}}+{\int}_{0}^{\varepsilon_{2}^{\text{pr}}}{\alpha^{\text{pr}}h(\alpha^{\text{pr}})d\varepsilon_{2}}\\&&+{\int}_{\varepsilon_{1}^{\text{pr}}}^{\varepsilon_{1}}{h(\alpha)d\varepsilon_{1}}+{\int}_{\varepsilon_{2}^{\text{pr}}}^{\varepsilon_{2}}{{\alpha}h(\alpha)d\varepsilon_{2}}\\ &=&g(\alpha^{\text{pr}})\varepsilon_{1}^{\text{pr}}+h(\alpha)(\varepsilon_{1}-\varepsilon_{1}^{\text{pr}})+{\alpha}h(\alpha)(\varepsilon_{2}-\varepsilon_{2}^{\text{pr}}) \end{array} $$

(16)

On the other hand

$$ \varepsilon_{2}^{\text{pr}}={\int}_{0}^{\varepsilon_{1}^{\text{pr}}}\beta^{\text{pr}}\mathrm{d}\varepsilon_{1}=\beta^{\text{pr}}\varepsilon_{1}^{\text{pr}} $$

(17)

$$ \varepsilon_{2}\,=\,\!{\int}_{0}^{\varepsilon_{1}}\beta d\varepsilon_{1}\,=\,\!{\int}_{0}^{\varepsilon_{1}^{\text{pr}}}\beta^{\text{pr}}d\varepsilon_{1}\,+\,{\int}_{\varepsilon_{1}^{\text{pr}}}^{\varepsilon_{1}}\beta d\varepsilon_{1}\,=\,\beta^{\text{pr}}\varepsilon_{1}^{\text{pr}}\,+\,\beta(\varepsilon_{1}\!-\varepsilon_{1}^{\text{pr}}) $$

(18)

Substitution of Eqs. 17 and 18 into 16 yields:

$$ \bar{\varepsilon}^{\mathrm{p}}=g(\alpha^{\text{pr}})\varepsilon_{1}^{\text{pr}}+g(\alpha)(\varepsilon_{1}-\varepsilon_{1}^{\text{pr}}) $$

(19)

According to Eq. 18, the ratio of principal strain rates can be written as:

$$ \beta=\frac{\varepsilon_{2}-\varepsilon_{2}^{\text{pr}}}{\varepsilon_{1}-\varepsilon_{1}^{\text{pr}}} $$

(20)

then

$$ \frac{\partial\beta}{\partial\varepsilon_{1}}=-\frac{\varepsilon_{2}-\varepsilon_{2}^{\text{pr}}}{(\varepsilon_{1}-\varepsilon_{1}^{\text{pr}})^{2}}=-\frac{\beta}{\varepsilon_{1}-\varepsilon_{1}^{\text{pr}}} $$

(21)

Insertion of Eqs. 19 and 21 into 13, one can obtain:

$$ \frac{g(\alpha^{\text{pr}})\varepsilon_{1}^{\text{pr}}+g(\alpha)(\varepsilon_{1}-\varepsilon_{1}^{\text{pr}})}{g(\alpha)}\left[1\,-\,\frac{\beta^{2}(\alpha)}{\beta^{\prime}(\alpha)}\frac{1}{1+\alpha\cdot\beta(\alpha)}\frac{1}{\varepsilon_{1}-\varepsilon_{1}^{\text{pr}}}\right]\!\,=\,n $$

(22)

Equation 22 is a quadratic functional equation of $\varepsilon _{1}-\varepsilon _{1}^{\text {pr}}$. According to the physical meaning of the major principal limit strain $\varepsilon _{1}^{*}$, it should be the positive root of Eq. 22. Once the major principal limit strain was determined, the minor principal limit strain $\varepsilon _{2}^{*}$ can be calculated from Eq. 18. The forming limit strains for bi-linear strain path loading were completely determined by above described analytical approach.

When the Logan and Hosford’s anisotropy yield criterion was used, the terms in Eq. 22 were calculated as:

$$ h(\alpha)=\left[\frac{R_{90}}{1+R_{0}}+\frac{R_{0}}{1+R_{0}}\alpha^{k}+\frac{R_{0}R_{90}}{1+R_{0}}(1-\alpha)^{k}\right]^{-1/k} $$

(23)

$$ \beta(\alpha)=\frac{R_{0}[\alpha^{k-1}-R_{90}(1-\alpha)^{k-1}]}{R_{90}[1+R_{0}(1-\alpha)^{k-1}]} $$

(24)

$$ \beta^{\prime}(\alpha)=\frac{R_{0}(k-1)\alpha^{k-2}+R_{0}(k-1)(R_{0}\alpha^{k-2}+R_{90})(1-\alpha)^{k-2}}{R_{90}[1+R_{0}(1-\alpha)^{k-1}]^{2}} $$

(25)

Formability evaluation index for non-linear straining

As the material is strained by non-proportional loading, the forming limit curve in the principal strains space is shifting with the pre-strain, which is assumed to keep unchanged in the traditional FLD. Thus, for sheet metal forming processes with a significant non-linear strain path change, the traditional static FLD is no longer applicable to the formability evaluation.

Based on the results obtained by above analytical approach, a formability evaluation index, F _sp, which accounted the strain path loading history during material deformation, was proposed to evaluate the formability. This index was defined in the following form:

$$ F_{\text{sp}}=\sum\limits_{i}\frac{\Delta\varepsilon_{1}^{\alpha_{i}}}{\varepsilon_{1}^{\alpha_{i}^{*}}} $$

(26)

This index is based on the assumption of accumulation of the loss of material formability during plastic deformation. The non-linear strain path was treated as a combination of several linear strain path segments. When the accumulation of the ratio of major principal strain increment of each linear segment (${\Delta }\varepsilon _{1}^{\alpha _{i}}$) to the forming limit strain at current linear segment ($\varepsilon _{1}^{\alpha _{i}^{*}}$), which can be calculated by the analytical approach described in “Analytical approach for forming limit calculation in bi-linear strain path loading”, reaches a critical value (e.g., 1), the material formability was considered completely exerted, and the state of forming limit was reached. The subsequent loading would cause the material to failure.

The calculation for index F _sp was illustrated in Fig. 1. As a material point subjected to a continuous strain path change, its loading path can be approximated by several linear strain path segments. Each loading stages with linear strain path were represented by α ₁,α ₂,α ₃,…. For the first loading stage α ₁, the value of pre-strain is 0, and the forming limit curve is the traditional FLC, i.e., FLC_0 in Fig. 1. The current value of F _sp is the ratio of major principal strain increment (${\Delta }\varepsilon _{1}^{\alpha _{1}}$) to the forming limit strain ($\varepsilon _{1}^{\alpha _{1}^{*}}$) at the current linear strain path α ₁, i.e., $F_{\text {sp}}={\Delta }\varepsilon _{1}^{\alpha _{1}}/\varepsilon _{1}^{\alpha _{1}^{*}}$. When the material point is strained by the second linear strain path loading stage α ₂, the value of pre-strain is (${\Delta }\varepsilon _{2}^{\alpha _{1}},{\Delta }\varepsilon _{1}^{\alpha _{1}}$), and the forming limit curve moves to the FLC_1 (see Fig. 1). The major principal strain on current forming limit curve (FLC_1) at linear strain path α ₂ is changed to $\varepsilon _{1}^{\alpha _{2}^{*}}$, then $F_{\text {sp}}={\Delta }\varepsilon _{1}^{\alpha _{1}}/\varepsilon _{1}^{\alpha _{1}^{*}}+{\Delta }\varepsilon _{1}^{\alpha _{2}}/\varepsilon _{1}^{\alpha _{2}^{*}}$. For the remain loading stages, F _sp can be deduced by the same way. The formability evaluation index F _sp for material point subjected to the whole continuous strain path change loading condition was now obtained.

Fig. 1

Schematic of the definition and calculation procedure for formability evaluation index

As shown in the definition and calculation procedure for formability evaluation index F _sp, the influence of strain path loading history on the forming limit is reflected by the variation of $\varepsilon _{1}^{\alpha _{i}^{*}}$ in Eq. 26 during plasticity deformation. From Fig. 1, $\varepsilon _{1}^{\alpha _{1}^{*}}$ can be written as:

$$ \varepsilon_{1}^{\alpha_{i}^{*}}={\Delta}\varepsilon_{1}^{\alpha_{i}^{*}}+\sum\nolimits_{j=1}^{i-1}{\Delta}\varepsilon_{1}^{\alpha_{j}} $$

(27)

With the major principal strain increment ${\Delta }\varepsilon _{1}^{\alpha _{i}}$ at current linear strain path α _i , as the strain state of a material point is located lower than the shifted forming limit curve FLC_i, i.e., ${\Delta }\varepsilon _{1}^{\alpha _{i}}<{\Delta }\varepsilon _{1}^{\alpha _{i}^{*}}$, the material deformation was thought to be safe. Substituting (27) into (26), the following inequality can be obtained:

$$\begin{array}{@{}rcl@{}} F_{\text{sp}}\!&=&\sum\limits_{i}\frac{\Delta\varepsilon_{1}^{\alpha_{i}}}{\Delta\varepsilon_{1}^{\alpha_{i}^{*}}+\sum\nolimits_{j=1}^{i-1}{\Delta}\varepsilon_{1}^{\alpha_{j}}}\!<\!\sum\limits_{i}\frac{\Delta\varepsilon_{1}^{\alpha_{i}}}{\Delta\varepsilon_{1}^{\alpha_{i}}+\sum\nolimits_{j=1}^{i-1}{\Delta}\varepsilon_{1}^{\alpha_{j}}}\\\!&=&\sum\limits_{i}\frac{\Delta\varepsilon_{1}^{\alpha_{i}}}{\varepsilon_{1}^{\alpha_{i}}}=1 \end{array} $$

(28)

From Eq. 28, it was shown that F _sp<1 as the material necking is not occurred. Otherwise, as ${\Delta }\varepsilon _{1}^{\alpha _{i}}\geq {\Delta }\varepsilon _{1}^{\alpha _{i}^{*}}$, then F _sp ≥ 1, and the status of material failure was considered to be reached.

In addition, as the strain path of material point keeps linear during the deformation process, i.e., d α = 0, the forming limit curve FLC_i should be static and equals to FLC_0. Once the accumulated major principal strain $\varepsilon _{1}^{\alpha }=\sum {\Delta }\varepsilon _{1}^{\alpha }$ increases to be equal to $\varepsilon _{1}^{\alpha ^{*}}$, the condition for material failure F _sp = 1 is also satisfied and the F _sp = 1 based formability evaluation method is equivalent to the FLD. This means that the formability evaluation index F _sp is an extension of the traditional FLD to the loading condition with continuous non-linear strain path variation considered.

Results and discussion

Material characterization

The material data of Aluminum alloy sheet Al 2008-T4 from the work of Graf and Hosford [11] were used in this work. The material parameters for the Logan and Hosford’s yield criterion and the power law and their values were listed in Table 1. The hardening curve fitted by the experimental data for DP 590 and their comparison were plotted in Fig. 2. It was shown that the predicted result obtained by the power law is agree well with the experimental data.

Table 1 The material parameters for Al 2008-T4 and DP 590

Fig. 2

The comparison between the fitted results by material hardening law and experimental data for DP 590 uni-axial tension test

Influence of pre-strain on the forming limit curve

Using the analytical approach described in “Analytical approach for forming limit calculation in bi-linear strain path loading”, the influence of three type of pre-strains (uni-axial tension, plane strain, and bi-axial tension) on the forming limit curves were predicted, and the comparison between the calculated results with the experimental ones were plotted in Figs. 3–5.

For uni-axial tension pre-strain loading, as shown in Fig. 3, the values of pre-strain are $\varepsilon _{1}^{\text {pr}}=$ 0.05, 0.12, and 0.18. Figure 3 showed that the FLC moves toward top-left direction as the pre-strain increases. The influence of pre-strain on the left hand part of FLC is varying its slope. And the right hand part of FLC moves upward as the pre-strain value increased. The forming limit strain increases with uni-axial tension pre-strain loading, thus the formability of sheet metal is enhanced.

Fig. 3

The predicted FLCs and comparison with the experimental ones normal to the RD after pre-strained in uni-axial tension normal to the RD

Comparing the results obtained by the analytical method proposed in this work with the experimental ones showed that they are agree well with each other when pre-strain is low. When the value of uni-axial tension pre-strain is high, the analytical FLC is less than the experimental data. Quantitatively speaking, the deviation of forming limit strain for subsequent plane strain loading after uni-axial tension pre-strain ${\Delta }\varepsilon _{1}^{\text {pr}}=0.18$ to the experimental data reaches a maximum value, ${\Delta }\varepsilon _{1}^{*}=0.05$. The n value decreases as the equivalent plastic strain increases [11], which is not accounted in the analytical calculation procedure. This factor could be one of the reason why the predicted result deviated from the experimental data. In addition, other influencing factors include the effect of friction boundary condition and normal contact stress on the FLC .

For plane strain pre-strain loading, two pre-strained loading conditions with $\varepsilon _{1}^{\text {pr}}=0.08$ and 0.13 were used, respectively. The predicted results and comparison with the experimental data were plotted in Fig. 4. According to [11], no successful test was made for strain states between plane strain and balanced bi-axial tension, which was reflected by the dashed line for the right hand part of experimental FLC as shown in Fig. 4. Thus, in this case, only the left hand part of experimental FLC after pre-strained in plan strain tension was used as a reference to compare the predicted results and the experimental ones.

Fig. 4

The predicted FLCs and comparison with the experimental ones normal to the RD after pre-strained in near plane strain tension normal to the RD

As shown in Fig. 4, the FLC retracted toward plane strain axis as the pre-strain increased, the analytical FLCs can reflect the effect of plane strain pre-strain on the forming limit. To the left hand part of FLC, the predicted results is agree well with the test data.

For balanced bi-axial tension pre-strain loading, the pre-strain loading condition are $\varepsilon _{1}^{\text {pr}}=0.04$, 0.07, and 0.12. The predicted results were displayed and compared with experimental ones in Fig. 5.

Fig. 5

The predicted FLCs and comparison with the experimental ones normal to the RD after pre-strained in balanced bi-axial tension

When the pre-strain value increased from 0 to 0.07, Fig. 5 showed that the FLC moves toward bottom-right direction after balanced bi-axial pre-strain loading. As the pre-strain value increases, the forming limit strain decreases and the formability of sheet metal becomes poor. The influence of balanced bi-axial tension pre-strain at this interval on the forming limit strain can be predicted by the analytical approach. From the experimental data in the work of Graf and Hosford [11], the FLC shifts up-right as the balanced bi-axial tension pre-strain value increased from 0.07 to 0.12. However, the up shifting effect isn’t reflected by the predicted results from the analytical approach. Quantitatively speaking, the deviation of forming limit strain with subsequent plane strain loading after balanced bi-axial tension pre-strain ${\Delta }\varepsilon _{1}^{\text {pr}}=0.12$ to the experimental data reaches a maximum value, ${\Delta }\varepsilon _{1}^{*}=0.056$.

From the forming limit curves with pre-strain predicted by the analytical approach proposed in our work, it showed that the FLC shifting in the principal strain space for combined linear strain path loading can be captured by this analytical method. The influence of pre-strain on the formability of subsequent loading process can be reflected.

Formability evaluation for a drawing reverse-drawing process

In order to verify the application of formability evaluation index, F _sp, in sheet metal forming processes with significant non-linear strain path, an experimental set-up with drawing reverse-drawing process was developed. The material used in this test is DP 590. The material plasticity anisotropy coefficient and hardening parameters were listed in Table 1.

Figure 6 showed the photograph of the drawing reverse-drawing tools and the diagrammatic sketch for three critical forming positions and two strain path loading stages. The whole forming processes is consists of two continuous strain path segments. The first one is corresponding to a linear pre-strain loading. The subsequent segment is a non-linear reverse-drawing process.

Fig. 6

The experimental setup of drawing reverse-drawing process

Three different size specimens were used in the drawing reverse-drawing process. As shown in Fig. 7, the medium and small size blanks were obtained by cutting the full circle specimen using two circles with a certain distance. The parameters and their value to control the size were given as: R _o = 170mm,R _m = 50mm,L _m = 180mm,R _s = 180mm, and L _s = 80mm.

Fig. 7

The blank shape and size for drawing reverse-drawing process

The drawing reverse-drawing forming test for DP 590 was performed on a 100 tons INSTRON single-action hydraulic press. As shown in Fig. 6, the die and semi-ellipsoidal dome (lower punch) were fixed on the frame by bolt. The frame was linked to the hydraulic piston of the press, the rotation freedom was constrained by contact between the sides of frame and four guide plates. Lower punch with adjustable shim was fixed on the die by two perpendicular key slots. Two perpendicular lines were marked on the top surface of die, which were used to align the specimen to the tools. As the specimen was completely clamped, the vertex of dome was below than the lower surface of specimen. The first straining stage was obtained by drawing the constrained sample as the punch moved down. The second one was performed by reverse-drawing the blank with the semi-ellipsoidal dome. The strain path change was triggered by contact between the lower surface of specimen with the top point of dome (see Position 2 in Fig. 6.). The ratio between the two straining stage was adjusted by increasing or decreasing the numbers of shim, which were placed under the dome. The whole procedure for the drawing reverse-drawing test was given bellow:

• Specimen preparing. In order to get different strain path loading modes and forming limit strain state, the specimens with three different sizes (see Fig. 7) were trimmed by Electrical Discharge Machining (EDM). A random distributed black speckle pattern was sprayed on the top surface of specimen, which was used for the strain measurements .

• Pre-forming of draw bead. When performing the “Pre-forming of draw bead”, the challenges and difficulties lie in the punch force needed to fully formed the draw bead out and the significant degree about the effect of pre-strain on the forming limit. As the punch force of the hydraulic press used for drawing reverse-drawing test is not large enough, the draw bead was preformed on a separate press with a higher tonnage. The convex and concave draw bead were placed on the die and punch tools, respectively. It is noted that the draw bead pre-forming only produced a very small amount of tension pre-strain and springback within the central region for strain measurement, which have an ignorable effect on the forming limit strain in the following drawing reverse-drawing process.

• Drawing reverse-drawing process. The blank was clamped between binder and die by tightening the 15 circular uniformly distributed bolts. The applied torque is large enough to ensure the draw-in of the sheet blank is adequate small. And the signals for press action and CCD camera image acquiring were sent synchronously. This strategy assured the time synchronization for displacement, load and CCD camera image data acquiring. The punch was fixed, and the die with the semi-ellipsoidal dome moved up-forward at the speed of 0.085mm/s. The central region of sheet blank experienced drawing reverse-drawing deformation until the failure was detected. In order to check the repeatability, all tests for each condition were performed at least 3 times.

The process conditions in drawing reverse-drawing forming limit test have a significant effect on the experimental results. These factors should be considered in the test. The contact and friction condition between blank and tools is a very important factor, which affected the failure time and crack region of specimen. In order to avoid the direct contact between blank and tools and reduce the friction force, two layers of Teflon with 0.2mm thickness were used as a contact medium. The elastic module for this material is 280Pa. Its elastic characteristic is excellent even with a 200 % engineering straining. The friction coefficient is about 0.02. This medium can effectively reduce the action of friction during experimental process. Thus it is placed between the acting regions of blank-punch and blank-dome, which need lower coefficient of friction. The other acting regions between blank and tools are direct metal contact and corresponding coefficient of friction is 0.2-0.22. The experimental results showed that the contact and friction condition can make sure the crack occured in the central region for each specimen, which is needed for the CCD camera to acquire sparkle image.

The blank holder force is another factor must be carefully considered in sheet metal forming processes. In the drawing reverse-drawing forming limit test, the blank holder force was enforced by tightening 15 circular uniformly distributed bolts. Herein, the elastic deformation of binder and die is very small relative to blank deformation, so they are assumed to be rigid body. The total holder force is sum of the axial pre-tighten action force of the 15 bolts. According to [31], the relationship between the axial pre-tighten force (F _axl) and the torque (T _tq) for a bolt is given by Eq. 29:

$$ F_{\text{axl}}=\frac{T_{\text{tq}}}{K_{\mathrm{b}}d} $$

(29)

$$ K_{\mathrm{b}}=\frac{d_{\mathrm{m}}}{2d}\frac{\tan\theta_{1}+f_{\mathrm{b}}\sec\theta_{\mathrm{n}}}{1-f_{\mathrm{b}}\tan\theta_{1}\sec\theta_{\mathrm{n}}}+0.625f_{\mathrm{c}} $$

(30)

where d and d _m are the nominal and mean diameter, respectively. θ _n is the thread lead angle, and θ _l is the thread angle between thread surface normal and bolt axis. f is the coefficient of friction and the subscript c stands for collar, considering the friction from the contacts of hexagonal bolt head, nut and washer face with a mean diameter of 1.25d. Based on an analysis for large amount of test data, [31] showed that K _b always approximately equals to 0.2 as the coefficient of friction f _b = f _c = 0.15. In the drawing reverse-drawing forming limit test for DP 590, the applied torque for each bolt is 138Nm and the total blank holder force equals to 407KN.

The formed parts are shown in Fig. 8. It is shown that the failure for the tested sheet blanks are all located in the central region. The design for specimen shape and size is in line with the experimental requirement and the CCD camera can conveniently record the sparkle images during the deformation process.

Fig. 8

Sample formed parts for drawing reverse-drawing process

For each test case, the punch force and die stroke displacement along the Z axial direction were recorded. Due to the clamped blank initially doesn’t contact with the punch, the displacement is not corresponding to the start point of the effective stroke stage for formability evaluation. Thus the moment when the punch force is larger than 100N was treated as the start point for effective travel stage. And the punch force and die travel displacement were all offset by the data at this moment. Figure 9 plotted the punch force-die stoke curves for the drawing reverse-drawing test, in which the curves with solid marks (labelled as S1 in legend) stands for the results as 1 shim with the thickness equals to 5.0mm was used in the test, while the ones with hollow marks (labelled as S4 in legend) stands for 4 shims was used. The curves with square, circle and up triangle marks stands for the results of full, medium and small size blanks, which were labelled as F, M and S in legend, respectively. It was showed in Fig. 9 that the punch force are increased as the displacement increased. The rate of slope for the curves at the initial stage is large. Then the increase rate evolved toward to a constant, and the punch force increase linearly with the displacement. The punch force reached a peak value at the moment when the blank initially necked, and it dropped suddenly as the crack ultimately occupied.

Fig. 9

The punch force -die stroke curve for drawing reverse-drawing process

The traditional FLC for linear strain path loading was obtained by using Marciniak test with circle grid analysis (CGA). By changing the specimen width and lubrication condition, the strain path loading mode and forming limit state can be adjusted. In the FLD test for DP 590, specimens were cut to 178mm length and three different widths (25mm, 90mm and 115mm), and all formed until the crack appeared. All specimens were etched by circle grid with 2.5mm diameter. The major and minor principal strain on the top surface of specimen were automatic measured by CGA method. An empirical formula in [32] was used to fit the experimental FLC data. The measured principal strains for safe circle (hollow down triangle marks) and necked circle (hollow diamond marks), and the fitted FLC were plotted in Fig. 10.

Fig. 10

The strain path and forming limit strain of initial failure point for drawing reverse-drawing process

Based on the sparkle images acquired by CCD camera during the drawing reverse-drawing forming limit test, the principal strains and strain path for the initial failure point on the sheet blank were obtained by using digital image correlation (DIC) and tracking technology. The loading mode and effect of pre-strain on the forming limit were discussed and analysed in the following section.

Figure 10 also showed the strain path variation and the forming limit strains. For full size specimen, the straining mode is almost hold the balanced bi-axial tension loading unchanged, and the forming limit strain is close to the traditional FLC under linear strain path loading condition. As the material instability occurs, it fast evolves towards the plane strain loading mode. The strain path transformation mode for medium size blank is turning from the plane strain loading to bi-axial tension and back to plane strain state. The forming limit strain is located below the traditional FLC. The strain path variation mode for small size blank is transferred from uni-axial tension loading to plane strain and back to uni-axial tension state. Similarly, the forming limit strain is also lower than the strain value at the traditional FLC. The deviation between the forming limit strain and the traditional FLC for medium and small size blanks can be attributed to the effect of non-linear strain path loading mode.

By alternating the numbers of shim mounted below the dome, the initial distance between the vertex of dome and the lower surface of blank can be changed and the contact time between them is accordingly adjusted during the drawing reverse-drawing process. The amount of pre-strain at the moment of strain path transition is also changed. When the shim height decreased, the moment when the lower surface of specimen begins to contact with the dome tool is delayed during reverse-drawing process, and the amount of pre-strain is increased.

When the shim height increased from 5mm (S1) to 20mm (S4), the results comparison for strain path variation and the forming limit strains were plotted in Fig. 10. It is seen clearly that the amount of pre-strain decreased when the shim height increased. The effect of the amount of pre-strain on the strain path change for full size blank is not as remarkable as for medium and small size specimens. The strain path for full size blank is also following the mode of transition from bi-axial tension to plane strain loading. The forming limit strain is ε ₂ = 0.1835, ε ₁ = 0.2596, which is lower than the forming limit strain (ε ₂ = 0.2058, ε ₁ = 0.3311) as shim height equals to 5mm. For medium size blank, the strain path transition mode is kept unchanged. However, the forming limit strain is changed from ε ₂ = 0.0273, ε ₁ = 0.2154 to ε ₂ = 0.0599, ε ₁ = 0.2341. For small size blank, the strain path variation is transferred from the plane strain to bi-axial tension to plane strain loading mode, and the forming limit strain is changed from ε ₂ = −0.0356, ε ₁ = 0.2197 to ε ₂ = 0.0070, ε ₁ = 0.2002. The forming limit strain under the condition with shim height equals to 20mm are all lower than the strain value of traditional FLC.

From above experimental results, it is clearly shown that the forming limit strain under complex strain path loading condition is not located on the traditional FLC. Based on the experimental data of strain path variation and forming limit strain, the formability index proposed in this work was used to evaluate the sheet metal forming process. Figure 11 plotted the evolution of formability index F _sp as a function of the accumulated effective plastic strain. As shown in Fig. 11, for all test cases with different strain path transition loading condition, the value of F _sp corresponding to the forming limit strain for initial failure point on specimens are increased to near 1.0. The results showed that it is feasible to use F _sp as the formability evaluation index for sheet metal forming with non-linear strain path loading.

Fig. 11

The formability evaluation index evolution for drawing reverse-drawing process

As shown in Fig. 10, the strain path change for medium size blank with 5mm shim height is the most complex one among all test cases. And the non-linear characteristic is also the strongest. Thus, the test case under this condition was taken as the example for comparing the accuracy between the traditional FLC and F _sp based formability evaluation method applied in finite element (FE) simulation for sheet metal forming when non-linear strain path loading was considered. It is shown that the limit punch force is 184.5KN for S1_M case (see Fig. 9), and the corresponding die stoke displacement is 28.3mm, at which the blank failure occupied. Thus the die stoke displacement equals to 28.3mm is treated as the termination condition for forming process .

The FE simulation was performed using commercial code LS-DYNA. The forming process was simulated using explicit time integration. Based on the geometry of tools and specimen, the FE model was created. The specimen was modelled by quadrilateral, fully integrated shell elements with five integration points across the thickness. The material parameters and their value listed in Table 1 were input into the material model. The tools were modelled in rigid bodies. The surface-surface contact with Columbus friction model in LS-DYNA was adopted to calculate the tangent contact force in the simulation. The coefficient of friction was set as described in “Formability evaluation for a drawing reverse-drawing process”.

Figure 12 shows the predicted forming state and the distribution of major and minor principal strain in forming limit diagram when the die stroke displacement reaches to 28.3mm. As shown in Fig. 12, the principal strains all located below the traditional forming limit curve. And according to the traditional FLC evaluation method, this means that the blank can be further strained. However, it is observed in the drawing reverse-drawing test that the blank was already cracked on the top of the specimen when the die stoke displacement reaches to 28.3mm. Thus the traditional FLC is no longer suitable and is risky for formability evaluation as the forming process has a strong characteristic of non-linear strain path variation.

Fig. 12

The formability evaluation using traditional FLD for drawing reverse-drawing process

Alternatively, the formability evaluation index proposed in this work was used to evaluate the forming limit for S1_M case. The result was plotted in Fig. 13. It is clearly shown that the material crack is happened at the top region of the blank when the die stroke reaches to 28.3mm, at which the material instability condition is satisfied. The sample failure occurrence time and crack location are agree well with the experimental results.

Fig. 13

The prediction for cracking location of drawing reverse-drawing process

From the comparative study, it was demonstrated that the application of formability evaluation index, F _sp, to evaluate the forming limit in forming process with strong non-linear strain path loading is more advantageous than the traditional FLC.

Conclusions

An analytical approach for forming limit calculation under bi-linear strain path loading condition with pre-strain was developed. The approach developed in this work was used to predict the forming limit for Aluminium alloy sheet Al 2008-T4 under bi-linear strain path loading. It was showed that the effect of pre-strain on the material FLC can be predicted by the developed approach.

A formability evaluation index F _sp accounted for the strain path history was put forward, which can promote the application of more accurate formability evaluation for sheet metal forming with continuous strain path change.

A drawing reverse-drawing forming limit test was developed to verify the proposed formability evaluation index. The strain path variation and forming limit strain of the initial failure location in specimens measured by using DIC technology showed that the representative strain path change in sheet metal forming process were successfully obtained by the developed drawing reverse-drawing test. Based on the experimental data, the validity of the proposed formability evaluation index for sheet metal forming with non-linear strain path change was verified.