New instability model leading the wrinkling at the positive-curvature arc part during the tube hydro-forging process

Abstract

The tube hydroforging process (THFG) is an advanced technology for manufacturing tubular components with complex cross-sections. The positive-curvature arc is one of the most fundamental and difficult-to-form features of complex cross-sections. However, its wrinkling mechanism in the THFG process cannot be explained by the existing theory. This restricts the application of the technology. First, because of the bending deformation caused by the excessive circumferential force, compression instability occurs at the positive-curvature arc part. This results in wrinkling similar to that in the conventional linear part. In addition, owing to the existence of the positive-curvature arc, the circumferential force produces a component force along the vertical direction that causes rigid displacement of the materials. This yields another new instability model: motion instability. The corresponding critical pressures for the two instability models were determined by adopting static method and energy methods respectively. Theoretically, motion instability is dominant in the early stages of compression, whereas compression instability is dominant in the subsequent stages. However, considering actual production, the correlations between the critical pressures of the different parts were compared. The wrinkling of the linear part inhibits the occurrence of compression instability in the positive-curvature arc. Thus, wrinkling of the arc can be caused only by motion instability. Therefore, the critical pressure for motion instability is defined as the critical pressure required for the positive-curvature arc. In addition, a forming window that considers the critical pressure of each part was established successfully.

Appendices

Appendix A: The solution of the energy method

From Eq. (23), it can be observed that the critical pressure $p_{{{\text{cri}}}}^{1}$ is related to K, n, ε_θ, t₀, and L₀. Here, K, n, and t₀ are specified, whereas ε_θ and L₀ are unknown. Therefore, to establish the relationship between the compressive strain ε_θ and $p_{{{\text{cri}}}}^{1}$, it is necessary to first establish the relationship between L₀ and ε_θ.

Cao (1999) indicated that when the energy method was used to solve for the critical pressure, with an increase in the pressure, the wrinkling number increased, whereas the wavelength and wave height decreased. When the wavenumber attained a certain value, wrinkling was considered to be absent. Similar phenomena were observed during THFG. According to Eqs. (8–11), the initial length of the wrinkling plate L0 can be resolved using the wrinkling wavelength L. Then, Fig. 

Fig. 26

Relationship between critical pressure and compressive strain with different wrinkling wavelengths

26 shows the relationship between the critical pressure and compressive strain at different wrinkling wavelengths L. Considering the wavelengths L_n and L_n+1 as examples, the two curves intersect at a point ε_θ,n. When the compressive strain is higher than ε_θ,n, the critical pressure required for the wavelength L_n+1 is higher than that required for the wavelength L_n. Therefore, the final wrinkling wavelength tends to L_n+1. When the compressive strain is smaller than ε_θ,n, the final wrinkling wavelength tends to L_n. For this reason, the transition relationship between L_n+1 and L_n+2 is also identical. To obtain the critical pressure under different compressive strains, assuming that there is a group of increasing sequence L₁, L_2, … L_n, the compressive strain (which is the intersection point of the two adjacent wavelengths) is calculated. It satisfies the expression

$$p_{{{\text{cri,}}n}}^{1} \left( {L_{n} ,\varepsilon_{\theta ,n} } \right) = p_{{{\text{cri}},n + 1}}^{1} \left( {L_{n + 1} ,\varepsilon_{\theta ,n} } \right)$$

(A-1)

After the K and n values are specified, the transition point ε_θ,n for each wavelength can be calculated through computer programming. Subsequently, these are substituted back into Eq. (23) to obtain $p_{{{\text{cri}}}}^{1}$. That is, the relationship between ε_θ and $p_{{{\text{cri}}}}^{1}$ can be established:

$$p_{{{\text{cri}}}}^{1} = f\left( {\varepsilon_{\theta } } \right)$$

(A-2)

It can be observed from Eq. (23) that the $p_{{{\text{cri}}}}^{1}$ obtained is related to only the material properties (K and n). It is not related to the other in-process parameters such as the compression displacement, friction coefficient, and geometric characteristics. Therefore, this study combined Eq. (23) with the stress–strain model to obtain the critical pressure $p_{{{\text{cri}}}}^{1}$ that satisfies the actual production. This is shown in Fig. 

Fig. 27

Calculation flowchart

27.

It can be observed from Fig. 9 that owing to friction, the circumferential strain at point B is the largest for the positive-curvature arc part. Thus, compression instability is most likely to occur at point B. The specific solution process is as follows: First, the material properties (K, n), geometric parameters (r₁, r₂, and θ₁), friction coefficient (μ), and initial thickness (t₀) are input. The variation rule between the compressive strain ε_θ and $p_{{{\text{cri}}}}^{1}$ (from Eq. (23)) is obtained using the energy method. First, the initial pressure $p_{i}$ and circumferential force $F_{\theta ,i}$ are provided. The corresponding compressive strain $\varepsilon_{\theta ,i}^{{}}$ at point B is obtained using Eq. (28), and the required support pressure $p_{{{\text{cri}},i}}^{1}$ is obtained by substituting the strain $\varepsilon_{\theta ,i}^{{}}$ into the variation rule. If $\left| {p_{i} /p_{{{\text{cri}},i}}^{1} - 1} \right| \succ 1{\text{\% }}$, then let $p_{i} { = }p_{{{\text{cri}},i}}^{1}$, and re-substitute $p_{i}$ into the computation cycle to calculate the new strain $\varepsilon_{\theta ,i}^{{}}$ under the circumferential force $F_{\theta ,i}$. When the error between p₀ and $p_{{{\text{cri}},{\text{i}}}}^{1}$ is less than 1%, the computational cycle is terminated. Currently, $p_{{{\text{cri}},{\text{i}}}}^{1}$ represents the required support pressure for a certain $F_{\theta ,i}$ and circumferential strain $\varepsilon_{\theta ,i}^{{}}$. Finally, given a group of increasing sequence $F_{\theta ,1}$, $F_{\theta ,2}$, … $F_{\theta ,i}$, the variation rule between ε_θ and $p_{{{\text{cri}}}}^{1}$ that considers a practical process can be constructed. Fig. 27 shows the corresponding calculation flowchart.

Appendix B: Derivation of E _w

Next, we explain the solution process for Eq. (21). The equation is a triple integral. Therefore, it should be integrated sequentially. In a previous study, Cao (1999) indicated that the bending effective strain of a wrinkling plate under plane strain can be simplified as

$$\overline{{\varepsilon_{w} }} = \alpha \left| {\ln \frac{r}{{r_{u} }}} \right|$$

(B-1)

Here, $\alpha { = }{2 \mathord{\left/ {\vphantom {2 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}$.

Subsequently, according to the Hollomon hardening law $\overline{\sigma } = K\overline{\varepsilon }^{n}$, the bending effective stress of the wrinkling plate can be obtained as

$$\overline{{\sigma_{w} }} = K\alpha^{n} \left| {\ln \frac{r}{{r_{u} }}} \right|^{n}$$

(B-2)

Furthermore, the triple integral in Eq. (21) can be simplified to a double integral:

$$E_{w} = \frac{{K\alpha^{n + 1} }}{n + 1}\iint {\left| {\ln \frac{r}{{r_{u} }}} \right|^{n + 1} drdL_{0} }$$

(B-3)

The double integral (B-3) can be solved using a progressive integral. First, integration is performed along the thickness direction:

$$\begin{gathered} \int_{{r_{i} }}^{{r_{o} }} {\left( {\left| {\ln \frac{r}{{r_{u} }}} \right|} \right)^{n + 1} dr} = \int_{{r_{u} }}^{{r_{o} }} {\left( {\left| {\ln \frac{r}{{r_{u} }}} \right|} \right)^{n + 1} dr} + \int_{{r_{i} }}^{{r_{u} }} {\left( {\left| {\ln \frac{{r_{u} }}{r}} \right|} \right)^{n + 1} dr} \hfill \\ \approx \frac{1}{2}\left( {r_{u} - r_{i} } \right)\left( {\ln \frac{{r_{u} }}{{r_{i} }}} \right)^{n + 1} + \frac{1}{2}\left( {r_{o} - r_{u} } \right)\left( {\ln \frac{{r_{o} }}{{r_{u} }}} \right)^{n + 1} { = }\frac{t}{4}\left( {\left( {\ln \frac{{r_{u} }}{{r_{i} }}} \right)^{n + 1} + \left( {\ln \frac{{r_{o} }}{{r_{u} }}} \right)^{n + 1} } \right) \hfill \\ \end{gathered}$$

(B-4)

As shown in Fig. 

Fig. 28

Schematic of the wrinkling plate

28, r_i and r_o denote the radii of the curvatures of the inside and outside surfaces of the wrinkling plate, respectively. These can be obtained from the curvature of the wrinkling plate as follows:

$$\left\{ \begin{gathered} r_{o} = r_{u} + \frac{{t_{0} }}{2} \hfill \\ r_{i} = r_{u} - \frac{{t_{0} }}{2} \hfill \\ \end{gathered} \right.$$

(B-5)

Meanwhile, the neutral layer r_u can be expressed as

$$r_{u} = \frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{3/2} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}$$

(B-6)

Next, the Taylor expansion is used to solve $\left( {\ln \frac{{r_{u} }}{{r_{i} }}} \right)^{n + 1}$ and $\left( {\ln \frac{{r_{o} }}{{r_{u} }}} \right)^{n + 1}$ in Eq. (B-4).

$$\left( {\ln \frac{{r_{u} }}{{r_{i} }}} \right)^{n + 1} = \left( { - \ln \left( {1 - \frac{{t_{0} }}{{2r_{u} }}} \right)} \right)^{n + 1} \approx \left( {\frac{t}{{2r_{u} }} + \frac{1}{2}\left( {\frac{{t_{0} }}{{2r_{u} }}} \right)^{2} } \right)^{n + 1} \approx \left( {\frac{{t_{0} }}{{2r_{u} }}} \right)^{n + 1} \left( {1 + \frac{{\left( {n + 1} \right)t_{0} }}{{4r_{u} }}} \right)$$

(B-7)

$$\left( {\ln \frac{{r_{o} }}{{r_{u} }}} \right)^{n + 1} = \left( {\ln \left( {1 + \frac{{t_{0} }}{{2r_{u} }}} \right)} \right)^{n + 1} \approx \left( {\frac{t}{{2r_{u} }} - \frac{1}{2}\left( {\frac{{t_{0} }}{{2r_{u} }}} \right)^{2} } \right)^{n + 1} \approx \left( {\frac{{t_{0} }}{{2r_{u} }}} \right)^{n + 1} \left( {1 - \frac{{\left( {n + 1} \right)t_{0} }}{{4r_{u} }}} \right)$$

(B-8)

Therefore, Eq. (B-4) can be expressed as

$$\int_{{r_{i} }}^{{r_{o} }} {\left( {\left| {\ln \frac{r}{{r_{u} }}} \right|} \right)^{n + 1} dr} = \frac{t}{4}2\left( {\frac{{t_{0} }}{{2r_{u} }}} \right)^{n + 1} = \left( \frac{t}{2} \right)^{n + 2} \frac{1}{{r_{u}^{n + 1} }}$$

(B-9)

Therefore, Eq. (21) can be simplified:

$$E_{w} = \frac{{K\alpha^{n + 1} }}{n + 1}\left( {\frac{{t_{0} }}{2}} \right)^{n + 2} \int {\frac{1}{{r_{u}^{n + 1} }}ds}$$

(B-10)

Next, integrating along the direction of the wrinkling length and substituting Eq. (B-6) into

$$\begin{gathered} \int {\frac{1}{{r_{u}^{n + 1} }}ds} = \int {\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n - 1} ds} \approx \int {\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n} \frac{y^{\prime\prime}}{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}ds} \hfill \\ = \int {\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n} \frac{y^{\prime\prime}}{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}\sqrt {1 + y^{^{\prime}{2}} } dx} \hfill \\ = \int {\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n} \frac{1}{{\left( {1 + y^{^{\prime}{2}} } \right)}}dy^{\prime}} \hfill \\ \end{gathered}$$

(B-11)

In a recent study, the cosine function (7) was used to describe the wrinkling shape. Therefore,

$$\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n} = \left( {\frac{{\left( {1 + \frac{1}{4}\delta^{2} m^{2} \sin^{2} mx} \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{{\frac{1}{2}\delta m^{2} \cos mx}} + \frac{{t_{0} }}{2}} \right)^{ - n}$$

(B-12)

Let $z = \sin mx$. Substituting it into Eq. (B-12), we obtain

$$\begin{gathered} \left( {\frac{{\left( {1 + \frac{1}{4}\delta^{2} m^{2} \sin^{2} mx} \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{{\frac{1}{2}\delta m^{2} \cos mx}} + \frac{{t_{0} }}{2}} \right)^{ - n} = \left( {\frac{{\left( {1 + \frac{1}{4}\delta^{2} m^{2} z^{2} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{{\frac{1}{2}\delta m^{2} \sqrt {1 - z^{2} } }} + \frac{{t_{0} }}{2}} \right)^{ - n} \hfill \\ = \left( {\frac{{1 + \frac{3}{8}\delta^{2} m^{2} z^{2} }}{{\frac{1}{2}\delta m^{2} }}\left( {1 + \frac{1}{2}z^{2} } \right) + \frac{{t_{0} }}{2}} \right)^{ - n} \hfill \\ = \left( {\left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right) + \frac{1}{2}z^{2} \left( {\frac{{1 + \frac{3}{8}\delta^{2} m^{2} z^{2} }}{{\frac{1}{2}\delta m^{2} }} + \frac{{\frac{3}{8}\delta^{2} m^{2} }}{{\delta m^{2} }}} \right)} \right)^{ - n} \hfill \\ \approx \left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \hfill \\ \end{gathered}$$

(B-13)

We substitute Eqs. (B-13) back into (B-11). In this study, the upper and lower limits of the integrals in Eq. (B-11) are $\left( {{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-0pt} 2} - r_{1} \sin \theta_{1}^{\prime} ,{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-0pt} 2} + r_{1} \sin \theta_{1}^{\prime} } \right)$. Subsequently, we obtain

$$\begin{gathered} \int {\left( {\frac{{\left( {1 + y^{^{\prime}{2}} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{y^{\prime\prime}} + \frac{{t_{0} }}{2}} \right)^{ - n} \frac{1}{{\left( {1 + y^{^{\prime}{2}} } \right)}}dy^{\prime}} \hfill \\ = \int {\left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \frac{1}{{\left( {1 + y^{^{\prime}{2}} } \right)}}dy^{\prime}} = \left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \int {\frac{1}{{\left( {1 + y^{^{\prime}{2}} } \right)}}dy^{\prime}} \hfill \\ = 4\left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \left. {\arctan y^{\prime}} \right|_{0}^{{{{m\delta } \mathord{\left/ {\vphantom {{m\delta } 2}} \right. \kern-0pt} 2}}} - 2\left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \left. {\arctan y^{\prime}} \right|_{0}^{{{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-0pt} 2} - r_{1} \sin \theta_{1}^{\prime} }} \hfill \\ = \left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \left( {4\arctan \left( {\frac{\delta m}{2}} \right) - 2\arctan \left( {\frac{\delta m}{2}\sin mr_{1} \sin \theta_{1}^{\prime} } \right)} \right) \hfill \\ \end{gathered}$$

(B-14)

Finally,

$$E_{w} = \frac{{K\alpha^{n + 1} }}{n + 1}\left( {\frac{{t_{0} }}{2}} \right)^{n + 2} \left( {\frac{2}{{\delta m^{2} }} + \frac{{t_{0} }}{2}} \right)^{ - n} \left( {4\arctan \left( {\frac{\delta m}{2}} \right) - 2\arctan \left( {\frac{\delta m}{2}\sin mr_{1} \sin \theta_{1}^{\prime} } \right)} \right)$$

(B-15)