2023 Volume 64 Issue 10 Pages 2471-2480
The validity of the Tsai-Hill criterion for a porous aluminum alloy with regularly aligned unidirectional pores was investigated experimentally and numerically. The Tsai-Hill criterion predicts failure in different directions in anisotropic composite material. Compression tests of porous aluminum alloy were performed with five different compression angles of 0, 30, 45, 60, and 90°. The compression angle is the angle between the loading direction and the longitudinal direction of the pore. A numerical analysis of a torsion test of the porous aluminum alloy was also performed to obtain shear strength. Compressive yield strength and equivalent shear strength of the specimen with 0 and 90° in compression angle were utilized in the Tsai-Hill criterion. As a result, the yield strength of the specimen with 30, 45, and 60° in compression angle was successfully predicted with a maximum relative error of 4%. The applicable strain range of the Tsai-Hill criterion was also investigated by altering the yield strength to various offset strengths. The resulted prediction showed a maximum relative error of 10% when the offset strain was 40% or less. Above that offset strain, densification of the porous structure caused a rapid increase in stress, leading to a drastic decrease in prediction accuracy.
Porous metals possess excellent properties not only as lightweight structural materials but also as shock-absorbing materials.1) They have a so-called plateau region under a compressive load where compressive stress remains nearly constant while compressive strain increases. Impacting energy is absorbed by porous metals and transmitting compressive load can be maintained below a certain level when used as shock-absorbing materials. Porous metals with directional pores, such as lotus metals, have the advantage of having high energy absorption per unit volume owing to their high stress due to low-to-medium porosity compared to metal foams.2) This attribute is evident especially when they are compressed in the direction parallel to the longitudinal direction of pore. Our research group has developed methods such as pipe-dipping3,4) and rod-dipping5) to manufacture such porous metals with directional pores. Energy absorption efficiency can be even improved by applying severe plastic deformation prior to its usage and suppressing stress increase during plateau region by work hardening.6)
However, compressive stress generally decreases when the angle between the load axis and the longitudinal direction of pore (compression angle) increases.7) In an application such as a crash box for automobiles, compression angle differs case-by-case; thus, the compressive characteristics of porous metals with directional pores depending on compression angle need to be predicted when they are designed.
Various studies as to the strength of porous metals have been conducted. Balshin revealed the strength of porous metals can be described by a power-law with the porosity and index m that depends on materials and manufacturing processes.8) However, the power-law cannot be applied to unknown structures and materials since index m is derived by empirical rules. Boccaccini et al. showed that index m can be replaced by the stress concentration coefficient K by using the stress concentration model of a plate with a hole.9) Hyun et al. applied the replaced formula to lotus metals and investigated whether the tensile strength and yield strength could be predicted.7,10) However, the results of some materials did not show agreement with the prediction formula, thus it is necessary to create a more accurate prediction formula. Tane et al. focused on the macroscopic deformation using the effective-mean-field method and clarified the porosity dependence and angle dependence of Young’s modulus, electrical conductivity, and yield strength of lotus metals.11–13) However, the angle and shape of the pores in the research were inconsistent, the formula was complicated, and the prediction accuracy was about 20% in relative error. Therefore, it is necessary to establish a simpler and more accurate formula using specimens with unified angles and shapes of pores.
There is an advantage that modeling becomes easier by using porous metals with unified angles and shapes of pores. Tamai et al. performed mechanical analyses for each cell wall of porous metals with regularly aligned unidirectional through-pores.14) These results showed the plateau region is initiated due to the plastic collapse caused by bending, shear, and compression of the cell walls when compressed perpendicular to the longitudinal direction of the pores. Ichikawa et al. revealed that the compressive strength can be expressed by the rule of the mixture of the metal and the pores when compressed similar specimens parallel to the longitudinal direction of the pores.15) In this study, porous metals with regularly aligned unidirectional through-pores are aimed to focus on the deformation of the cell wall. Such porous metals and unidirectionally reinforced composites have in common that they are composed of a base material and a directional component. Therefore, we will apply the failure criteria in composite materials to porous metals since many theories of prediction formulas for the strength have been established.
One of the most widely used failure criteria, the Tsai-Hill criterion, is a formula that extended Mises’s yield criterion to transversely isotropic materials.16,17) The Tsai-Hill criterion is a simplified prediction formula that assumes the plane stress state and the transverse isotropy for the yield criterion. This criterion is a prediction formula for fracture strength that uses the angle between the load direction and the longitudinal direction of reinforced fiber as variables. In addition, the variables required for the criterion are the fracture strength in the fiber direction, the fracture strength perpendicular to the fiber direction, and the shear strength around the fiber direction. Since the Tsai-Hill criterion is based on Mises’s yield criterion, it was investigated that the fracture strength in the Tsai-Hill criterion can be replaced with the yield strength. Therefore, this research investigated whether the yield strength of porous metals can be predicted by using the Tsai-Hill criterion after replacement. The replaced Tsai-Hill criterion is shown below.
| \begin{align} \sigma_{\psi_{\text{int}}} &= \bigg( \frac{\cos^{4}\psi_{\text{int}}}{\sigma_{0}{}^{2}} + \frac{\sin^{4}\psi_{\text{int}}}{\sigma_{90}{}^{2}} - \frac{\sin^{2}\psi_{\text{int}}\cos^{2}\psi_{\text{int}}}{\sigma_{0}{}^{2}} \\ &\quad + \frac{\sin^{2}\psi_{\text{int}}\cos^{2}\psi_{\text{int}}}{S^{2}}\bigg)^{-\frac{1}{2}} \end{align} | (1) |
The objective of this research was to clarify the applicability of strength prediction using the Tsai-Hill criterion to the porous metals with regularly aligned unidirectional through-pores. Experimental compression tests and analytical torsion tests were performed to identify the parameters required for the prediction formula. The torsion test was performed analytically since it is difficult to chuck with a testing machine due to the presence of pores.
A6061 aluminum alloy, which is widely used as a structural material, was used for the porous and non-porous specimens. In addition, the aluminum alloy was annealed at 530°C for 24 h to obtain ductile deformation. The chemical composition of the A6061 aluminum alloy is shown in Table 1, that was obtained from the mill sheet from Nikkei Kambara Co., Ltd. which manufactured the extruded A6061 billet. The compression tests of a non-porous specimen were conducted to obtain material properties. Five cubic non-porous specimens with an edge length of 15 mm were made by machining. The compression surfaces of the specimen were lubricated with molybdenum disulfide. The strain rate in the compression test of non-porous specimens was set to 0.1 min−1, which is the same as that of the porous specimens explained below.
The outer shape of porous specimens was fabricated by cutting in the same way as the non-porous specimens and the pores were formed by drilling. Figure 1 shows the shape and dimensions of the specimen at ψint = 45° as an example. The porous specimens were cubic with an edge length of 25 mm, and circular pores with a diameter of 3 mm were regularly arranged in a hexagonal shape at equal intervals of 4.8 mm. 27 pores were arranged on the specimens with the non-porous skins. On the other hand, the specimens without the non-porous skins were filled with pores so as to remove the non-porous skins. The minimum ratio of specimen size W to pore diameter d defined in the ISO 13314:2011 is 10.18) However, this value is on the safe side for standardization. The original paper used to determine this value reported that Young’s modulus and plastic collapse strength reached a plateau when W/d became larger than 6 and 5, respectively, for closed-cell foams.19,20) The specimens in this research had W/d = 25/3 = 8.33; thus, the mechanical response of the specimens was considered not affected by the size effect. We employed W/d = 25/3 in order to compare the compressive behaviors with the specimens used in our past research,14,21) where W was limited at that time by the diameter of the as-extruded aluminum rod we had.
Schematic diagrams of porous specimen at ψint = 45°. Compression was performed in z-axis and deformation was observed in y-axis. (a) Specimen with the skins, (b) Specimen without the skins, and (c) Definition of compressive angle ψ.
Experimental compression tests were conducted on five types of specimens at ψint = 0, 30, 45, 60, and 90°. Photos of the x-z plane were taken every second and videos of the y-z plane were also taken. Especially, a speckle pattern was added to one of the x-z planes of the specimens before the test so that the surfaces can be analyzed by digital image correlation (DIC). The porosity of the porous specimens was calculated by dividing the volume of the pore by the volume of the cube with an edge of 25 mm. The volume of the pore was calculated by subtracting the volume of the metal part from the volume of the cube with an edge of 25 mm. Here, the volume of the metal part was obtained by using drawings of the porous specimens. Table 2 shows the porosity of porous specimens at each compression angle. From Table 2, the maximum difference in porosity between with and without the skins for each compression angle was 1.8%. In addition, compression tests were repeated three times for each condition to eliminate the effects of individual differences. The strain rate in the compression test was set to 0.1 min−1 in accordance with ISO 13314:2011.18) The lubricant for porous specimens was also the same as that for non-porous specimens.
The compression tests of the porous specimen were simulated for discussion using the finite element method (FEM) with Abaqus/Standard 2018 (Dassault Systèmes SE). The shape and dimensions of the analytical model were the same as those of the specimens in the experiments. Analytical compression tests were performed to measure the normal stress on the x-y and x-z plane and the strength when the compression plane was the x-z and y-z plane on the specimen at ψint = 0°. They were simulated as compression between the upper and lower rigid plates, and the friction coefficient between the model and rigid plates was set to 0.08 as reported by Tanaka et al.22) The friction coefficient of self-contact was also set to 0.08. Elastic gradient E and the relationship between true stress and plastic strain were obtained by the compression tests of the non-porous specimen. Fracture criteria were not used in this research. An appropriate mesh size was selected so as not to change the compressive stress-strain curve and deformation depending on the mesh size. As a result, the approximate global size was set to 0.3 mm. The mesh shape was selected as hexahedron and reduced integration was used to reduce computational costs. The validity of the obtained results was verified based on a comparison of the value of the compressive stress-strain curve and equivalent plastic strain at ψint = 0 and 90° obtained by experiments and FEM.
2.2 Analytical torsion test methodThe torsion test was also simulated using the FEM. The model of the analytical torsion test was a 150 mm long cylinder and was given a displacement of 10 rad (= 573°) around the longitudinal direction of pores. Figure 2 shows the shape and dimensions of the specimen of the analytical torsion test. In the cross-section of the specimen, circular pores with a diameter of 3 mm were arranged regularly to remove the non-porous skins in a hexagonal shape at equal intervals of 4.8 mm on the circle with a diameter of 24 mm. The elastic gradient of the base material E and the relationship between true stress and plastic strain were obtained by the compression tests of the non-porous specimen. The analytical conditions in the torsion test such as friction coefficient and mesh size were the same as those in the analytical compression test.
Cross-section perpendicular to pores of the torsion test specimen.
As for experimental compression tests of porous and non-porous specimens, the maximum stress difference due to the individual difference was 10 MPa when the stress was 380 MPa. Hence, the maximum stress difference was so small as 2.6% by using relative error that the influence of individual differences can be considered very small. For all subsequent figures, one result for each compression angle was shown as a representative. Figure 3 shows the compressive stress-strain curve and true stress-plastic strain curve of the non-porous specimen. The elastic gradient of material E was measured to be 4.63 × 103 MPa as the maximum gradient in a linear region in the stress-strain curve (Fig. 3(a)) using a strain range of 0.1% as the width of calculating the gradient. The linear region does not have to be defined explicitly since the gradient naturally decreases after the elastic region.
Experimental stress-strain curves of the non-porous specimen. (a) Compressive stress-strain curve, and (b) True stress-plastic strain curve.
There is a significant difference between our experimental elastic gradient and the commonly reported Young’s modulus of aluminum alloy (i.e., ≈70 GPa). This difference is presumed to have originated from the testing method. The compressive strain was calculated using the stroke of the crosshead of the universal testing machine. This stroke could contain an error, such as from the mechanical lash of the machine and the contact surface between the specimen and the compression jig. Therefore, the measured elastic gradient was considered an effective value that reflects the experimental setup used in this research rather than the true Young’s modulus of aluminum alloy. This effective value was needed to simulate experimental setups in finite element analysis, which are difficult to model individually in addition to the compressive behavior of porous specimens. The term Young’s modulus was explicitly not used because of the above reason.
The elastic gradient was used for the analysis by the FEM. In addition, the plots every 5% from 0% to 50% in Fig. 3(b) were used as the material properties for the analytical model. Figure 4 shows compressive stress-strain curves which were obtained by experimental compression tests. This figure shows the strength from the beginning to about 50% strain decreases with an increase in the compression angle regardless of the non-porous skins. This is the same tendency as the previous research, that is, the stress level when compressed perpendicular to the longitudinal direction of the pores (ψ = 90°) is higher than compressed parallel to them (ψ = 0°).11) However, when the strain becomes large, the magnitude relationship of the stress level is reversed. This is because the specimen densifies more easily with increasing angles.
Experimental compressive stress-strain curves for each compression angle. (a) With the skins, and (b) Without the skins.
Each curve in Fig. 4 has a region with a very small slope. This region is wider in the results without skins than in that with skins. This is because the specimen with skins had a higher porosity. Plateau end strain, εd, has often been discussed in association with relative density, ρ*/ρ, in previous research. Numerous research showed that εd increases linearly with a decrease in ρ*/ρ.23–25) We applied the following equation describing plateau end strain for closed-cell metal foams proposed by Chan et al.24) to the stress-strain curves of the specimens with ψint = 90° for demonstration.
| \begin{equation} \varepsilon_{\text{d}} = 1 - \alpha_{\text{c}} \rho^{*}/\rho \end{equation} | (2) |
Since there is no obvious yield point, the 0.2% proof stress is treated as the yield stress and shown in Fig. 5. Error bars in Fig. 5 represent the standard errors of each specimen and the plots represent the average value of each compression angle. In addition, Fig. 6(a) shows appearance photos of the x-z plane for the specimen with the skins and equivalent plastic strain distribution from DIC. Figure 6(b) suggests that shear deformation occurs in the porous specimen and the compression angle changes during the compression test.
Relationship between stress and compression angle. (a) With the skins for yield stress, (b) Without the skins for yield stress, (c) Without the skins at an offset strain of 4%, and (d) Without the skins at an offset strain of 6%.
Images taken during the test and their analysis results with the skins. (a) Equivalent plastic strain distribution of the x-z plane, and (b) Appearance of the y-z plane.
The torsional torque T corresponding to the given the rotational displacement θ was obtained by the analytical torsion test. The obtained relationship between θ and T was converted into the relationship between shear strain γ and shear stress τ using the polar moment of inertia in a circular cross-section Ip by eqs. (3) and (4), respectively. The shear stress-strain curve obtained by the FEM is shown in Fig. 7.
| \begin{equation} \gamma = D\theta/L \end{equation} | (3) |
| \begin{equation} \tau = DT/I_{\text{p}} \end{equation} | (4) |
Analytical shear stress-strain curve when the torsion angle around the longitudinal direction of pores was given.
As mentioned above, the results of the analytical compression test will be used in the discussion of the transverse isotropy in Section 4.3. The equivalent plastic strain distribution obtained by FEM was added to Fig. 6(a). Figure 6(a) showed that the equivalent plastic strain distributions at ψint = 0° by DIC were about 15% higher than that by FEM. The actual specimen was not in the plane strain state and was deformed in the y-direction. Since the DIC method did not consider the deformation in the y-direction, the strain distribution value was higher than that of FEM. Therefore the difference in the equivalent plastic strain distributions of about 15% is considered not to be the problem.
Figure 8(a) shows compressive stress-strain curves obtained by the FEM of models without the skins at ψint = 90° and the same model that was compressed in the x-direction. From Fig. 8(a), the two curves were almost the same, that is, the results are almost the same regardless of the compression direction when compressed perpendicular to the pores.
Analytical results of the closeness to the transverse isotropy at ψint = 90° without the skins and the same model that was compressed in the x-direction. (a) Comparison with compressive stresses, and (b) The ratio of stress at each offset strain.
Fracture criteria were not used in this research. Our research group previously performed similar analyses14) where we confirmed a good agreement with experimental results in the stress-strain curve and deformation behavior except for stress drops caused by fracture of cell walls. Fracture of cell walls occurred after 30% in compressive strain in the specimens without the skins experimentally according to the stress-strain curve (Fig. 4(b)) in this research. Discussions in this research utilized the results of FEM analyses up to 24% in compressive strain, where fracture does not affect the compressive behavior yet. Thus, fracture criteria were not necessary in the current discussions.
The prediction accuracy of the yield strength was evaluated to confirm the applicability of eq. (1). First, the yield strengths σ0 and σ90 were obtained by offsetting the initial straight line of the compressive stress-strain curve by 0.2% in strain. Second, the shear yield strength S was also obtained by offsetting in the same method. Then, the yield strength at the initial compression angle $\sigma_{\psi_{\text{int}}}$ was calculated from eq. (1) by using σ0, σ90, and S. Finally, the prediction results at ψint = 30, 45, and 60° obtained from the experimental results at ψint = 0 and 90° were compared with the experimental results. Furthermore, it was investigated whether the stress obtained by setting the offset strain to a value larger than ε = 0.2% could be predicted. Similarly, the prediction accuracy of the offset stress at ψint = 30, 45, and 60° was evaluated. In addition, the causes of the deterioration of accuracy were discussed.
4.1 Prediction of yield stressIn eq. (1), σ0 and σ90 are 0.2% proof stress values at ψint = 0 and 90°, respectively. Furthermore, S is shear yield strength around the longitudinal direction of pores. To obtain the offset shear strain used to obtain the shear strength equivalent to the yield strength in compression, it is necessary to derive the relational expression between the compressive strain and the shear strain at the yield point. Here, the conceptual diagram of the values used for the prediction of yield stress is shown in Fig. 9. If the offset strain is substituted for 0.2%, the value at the intersection is the yield stress.
Illustration of the offset stress calculation process.
First, since the base material of the specimens is a ductile metal, Mises’ shear distortion theory (eq. (5)) is valid for the porous structure at ψint = 0°. Then, Hooke’s laws (eq. (6) and (7)) are also valid by assuming that the yield points of compression and shear are at the elastic regions. Considering these three equations, the relational expression (eq. (8)) between compressive strain and shear strain is valid at the yield point.
| \begin{equation} \sigma_{\text{Y}} = \sqrt{3} S \end{equation} | (5) |
| \begin{equation} \sigma = E_{\text{p}}\varepsilon/100 \end{equation} | (6) |
| \begin{equation} \tau = G\gamma/100 \end{equation} | (7) |
| \begin{equation} \gamma = E_{\text{p}}\varepsilon/(\sqrt{3}G) \end{equation} | (8) |
Using eq. (8) above, the offset strain required for the yield shear strength equivalent to 0.2% proof stress was identified as follows. The elastic gradient of porous structure at ψint = 0° Ep was identified as Ep = 4.03 × 103 MPa from the slope of the initial section in Fig. 4. Since Ep is a value for a structure containing pores, Ep is the average value of a total of six specimens with and without the skins. In addition, the transverse elasticity modulus of porous structure around the longitudinal direction of pores G was also identified as G = 14.2 × 103 MPa from Fig. 7. Substituting Ep, G, and offset strain of compression ε = 0.2% to eq. (8), the offset strain of shear γ equivalent to compression was found to be 0.033%. Consequently, S equivalent to 0.2% proof stress was obtained as S = 20.7 MPa using the above offset strain.
The calculated values of the yield stress obtained by substituting σ0, σ90, and S into eq. (1) were also shown in Fig. 5. Here, both σ0 and σ90 were the average values of 0.2% proof stress obtained experimentally at each angle. The relative error, which is defined as the ratio of the difference between the experimental value and the calculated value to the calculated value, was used as an index to evaluate prediction accuracy. Here, the average value was used as the experimental value. Evaluating the relative error for the specimen with the skins at ψint = 30, 45, and 60°, the average relative error was 1% and the maximum relative error was 2% at ψint = 45°. In addition, evaluating in the same way in the case of a specimen without the skins, the average relative error was 2% and the maximum relative error was 4% at ψint = 45°.
The relative error of the specimen with the skins was smaller than that without the skins. This can be explained by the change in the compression angle during the test described later in Section 4.3.1. Although the prediction accuracy was better strictly in the case of the specimen with the skins, the difference was very small from Fig. 5. Subsequent discussions will focus only on the specimen without the skins since the structure of the specimen is homogeneous.
4.2 Prediction of offset stressIn this section, the offset strain is changed to a larger value than 0.2% and the applicability of the prediction formula will be discussed. The compressive stresses obtained by offsetting the slope of the initial section of the compressive stress-strain curve at ψint are defined as $\sigma_{\text{off} - \psi_{\text{int}}}$. Furthermore, the shear strength corresponding to the arbitrary offset strain is defined as Soff. These newly defined values are also schematically shown in Fig. 9. The offset stress was predicted by replacing $\sigma_{\psi_{\text{int}}}$ and S, which are the values at the yield point, with $\sigma_{\text{off} - \psi_{\text{int}}}$ and Soff, respectively. Equation (1) after substitution is shown below as eq. (9).
| \begin{align} \sigma_{\text{off}-\psi_{\text{int}}} &= \bigg( \frac{\cos^{4}\psi_{\text{int}}}{\sigma_{\text{off}-0}{}^{2}} + \frac{\sin^{4}\psi_{\text{int}}}{\sigma_{\text{off}-90}{}^{2}} - \frac{\sin^{2}\psi_{\text{int}}\cos^{2}\psi_{\text{int}}}{\sigma_{\text{off}-0}{}^{2}}\\ &\quad + \frac{\sin^{2}\psi_{\text{int}}\cos^{2}\psi_{\text{int}}}{S_{\text{off}}{}^{2}}\bigg)^{-\frac{1}{2}} \end{align} | (9) |
The prediction accuracy of the offset stress from the results of experiments and analysis will be evaluated. First, the offset strains in the calculation of $\sigma_{\text{off} - \psi_{\text{int}}}$ were set as every 0.2% from 0.4% to 1%, every 2% from 2% to 10%, and every 10% from 20% to 50%. Second, the offset strains used for Soff equivalent to those for $\sigma_{\text{off} - \psi_{\text{int}}}$ were obtained by utilizing eq. (8). Figure 10(a) shows the relative error, which is defined as the ratio of the difference between the experimental value of $\sigma_{\text{off} - \psi_{\text{int}}}$ and the calculated value to the calculated value, for each offset strain. Error bars in Fig. 10(a) represent the maximum and minimum values of each offset strain and the plots represent the average value of each offset strain.
Prediction accuracy of offset stress for each offset strain in compression without the skins. (a) Experimental results, and (b) Calculated results by using the shear strength obtained by fitting.
The prediction accuracy was the best at the offset strain of 2% in Fig. 10(a). The accuracy deteriorated from the offset strain of 2% and deteriorated rapidly from the offset strain of 40%. In addition, Fig. 10(a) shows the maximum error ratio was 10% when the offset strain was 40% or less. While the prediction accuracy of the offset stress deteriorated compared to that of the yield strength, the prediction accuracy of the offset stress was a maximum relative error ratio of 10% for a wide range. Since eq. (8) is an equation obtained on the assumption of an elastic region, Soff calculated using that equation may not correspond to $\sigma_{\text{off} - \psi_{\text{int}}}$. Therefore, it is necessary to compare the prediction accuracy when the value of Soff is set to the ideal value.
The ideal value of Soff, which is defined as Sf, was obtained by fitting eq. (9) to the average of the experimental values. The fitting method was the least-squares method using the Levenberg-Marquardt method for each offset strain. From eq. (9), the ideal value of $\sigma_{\text{off} - \psi_{\text{int}}}$ for each offset strain was obtained by using Sf. Furthermore, the obtained value of $\sigma_{\text{off} - \psi_{\text{int}}}$ will be compared with the experimental value respectively to confirm the validity of the prediction. Figure 10(b) shows the relative error of $\sigma_{\text{off} - \psi_{\text{int}}}$ in the same way as Fig. 10(a). The maximum error ratio was 5% when the offset strain was 40% or less. As in Fig. 10(a), the prediction accuracy in Fig. 10(b) deteriorated from the offset strain of 40%. Unlike Fig. 10(a), it is considered to be able to predict $\sigma_{\text{off} - \psi_{\text{int}}}$ by Fig. 10(b) with the same accuracy as predicting yield stress. That is, if Soff is an ideal value, it can be considered that eq. (9) can be used when predicting $\sigma_{\text{off} - \psi_{\text{int}}}$.
Although the prediction accuracy of the offset stress obtained by the extended Tsai-Hill criterion has been discussed, it is important to clarify why the prediction accuracy changes concerning changes in offset strain to improve the prediction accuracy. The examination will be conducted in Section 4.3 below.
4.3 Factors causing inaccurate predictionIn this section, what factors caused the predicted values to deviate from the experimental values will be investigated to clarify the applicable range of the prediction formula. First, the increase in compression angle due to compression is likely to be a factor of error considering eq. (9). Second, the Tsai-Hill criterion approximates the plane stress condition, and the validity of plane stress approximation in our model will be discussed. Third, the Tsai-Hill criterion also approximates the transverse isotropy, and the validity of transverse isotropy approximation in our model will be discussed. Fourth, the possibility of buckling of cell walls which decrease the strength will be discussed. Finally, there is a possibility that the strength increases due to the densification of the specimen, and the calculated value becomes larger than the experimental value. These five factors will be discussed.
4.3.1 Increase in the compression angle during compression testFigure 6(b) indicates that the compression angle ψ became larger as compression proceeded in the specimen at ψint = 30, 45, and 60°. Figure 11(a) shows the changes in compression angle during the test. The compression angle ψ was obtained using the image analysis program ImageJ 1.53m. The angle was measured by tracing two center points inside the pore on the image of the y-z plane. An example of measurement is shown in Fig. 11(b). The measurement point was every 2% of compressive strain from 2% to 10% and every 5% of that from 15% to 30%. From Fig. 11(a), the compression angle started to increase from a strain of 2% for the specimen at ψint = 30 and 45°. Here, since the yield point is roughly a strain of 2%, Fig. 5(b) which is a predicted result of the yield strength will be referred to. In Fig. 5(b), since the experimental value at ψint = 30° has an actual compression angle larger than 30°, the plot should be to the right of the position of the figure. A similar result is for the specimen at ψint = 45°. Therefore, it has become clear that the predicted value becomes larger than the experimental value due to the increase in the compression angle as compression proceeded.
Changes in compression angle during the compression test for the specimen without the skins. (a) Measurement results, and (b) An example of measurement of compression angle.
Since the Tsai-Hill criterion assumes the plane stress condition, it is necessary to discuss its effectiveness. The validity of the plane stress condition was confirmed by comparing the normal stresses on the x-y and the y-z plane obtained by the FEM. Here, since the purpose is to confirm the plane stress condition in the two-dimensional plane, the model at ψint = 0° was used as a representative. The normal stresses were compared at two points: ε = 1.2%, which is near the yield point, and 24% when the analysis stopped. The value of the normal stress was 54.5 MPa on the x-y plane and 1.7 MPa on the y-z plane at ε = 1.2%. Since the normal stress on the y-z plane is about 3.2% of that on the x-y plane and is negligibly small, there is no problem in approximating the plane stress state. In addition, the normal stress on the x-y plane was 175.2 MPa and that on the y-z plane was 8.0 MPa at ε = 24%. The ratio of the normal stress on the x-y plane to that on the y-z plane was 4.6%, which is higher than the ratio at the yield point, but the normal stress on the y-z plane is negligibly small. Hence, it is also considered that there is no problem in approximating the plane stress state at the regions with large strain after the yield point.
4.3.3 Transverse isotropySince the Tsai-Hill criterion also assumes transverse isotropy, it is necessary to discuss its effectiveness. Therefore, the validity of the transverse isotropy was investigated by comparing the strength in two directions perpendicular to the longitudinal direction of the pores. Figure 8 shows that the ratio of the stress at ψint = 90° to the stress in the same model that was compressed in the x-direction is used as an index showing the degree of approximation of the transverse isotropy. This stress ratio for each offset strain is shown in Fig. 8(b), which indicates that the stress ratio is the smallest at the offset strain of 2%. Therefore, since the transverse isotropy can be approximated most, it is considered that the accuracy improved at the offset strain of 2% in Fig. 10(a). In addition, the stress value at ψint = 90° is always larger than that at ψint = 90° compressed in the x-direction by Fig. 8(a). Therefore, assuming that it is not the transverse isotropy and considering the calculation process of the Tsai-Hill criterion, it is necessary to subtract a minute value from the denominator of the negative term in eq. (9). In that case, the calculated value is smaller than the value when the transverse isotropy is assumed. Therefore, it has become clear that the predicted value becomes smaller than the experimental value due to the increase in the compression angle as compression proceeded.
4.3.4 Buckling of the cell wallsIn Fig. 6(b), the buckling of the cell walls can be seen in the specimen with ψint = 0° when the strain becomes large. It can be assumed that the buckling started to occur after an offset strain of 4% by comparing Fig. 5(c) and (d). Calculated offset stress for the specimens with ψint = 30, 45, and 60° became larger than the experimental value when the offset strain was 6% (Fig. 5(d)), which is an opposite tendency compared to 4% (Fig. 5(c)). This suggests that offset stress of ψint = 0° decreased due to the buckling of the cell walls, and calculated offset stress decreased accordingly because of the nature of the Tsai-Hill criterion, which is an interpolation between the stress of ψint = 0° and ψint = 90°. It can be also said that the effect of buckling affects rather dominantly on the calculated offset stress more than the effect of an increase in compression angle because the latter should make the experimental values smaller than the calculated values.
4.3.5 Densification of the specimenFinally, Fig. 10(a) shows the accuracy deteriorated sharply after the offset strain of 40%. In Fig. 4(b), the relationship between the magnitude of ψint and the relationship between the magnitude of the stress is consistent. However, the specimen at ψint = 90° is densified first when the strain exceeds 45% and the stress level becomes higher. Therefore, the magnitude relationship of the stress level depending on ψint is reversed due to the densification of the specimen at ψint = 90° when the strain exceeds 45%. Since the Tsai-Hill criterion was an equation that interpolates between the strength at ψint = 0 and 90°, the prediction accuracy deteriorates significantly when the strength at ψint = 90° exceeds that at ψint = 60°. This is the cause of the sharp increase in the relative error after the offset strain of 40% in Fig. 10(a).
4.3.6 Summary of factors and predictable rangeThe predicted value of offset stress becomes larger than the experimental value near the yield point due to the increase in the compression angle as compression proceeded and the closeness to the transverse isotropy. On the other hand, the predicted value becomes smaller than the experimental value depending on the buckling of the cell walls after the strain of roughly 6%. Finally, the predicted value becomes about 20% smaller than the experimental value because of the densification of the specimen when the strain exceeds 45%. Therefore, it can be considered that the range in which the strain is smaller than 45%, which is the beginning of densification of the specimen, is the applicable range of the prediction formula. The prediction accuracy of offset stress within that range was a maximum relative error ratio of 10%.
4.4 Extension to the prediction of compressive stress-strain curvesFinally, it is examined whether the compressive stress-strain curve can be predicted using the offset stress prediction formula (eq. (9)). The offset stresses at ψint = 30, 45, and 60° were calculated at every offset strain of 0.001% for the range up to the offset strain of 50% using eq. (9). This prediction requires the shear stress-strain curve around the longitudinal direction of pores and the compressive stress-strain curves at ψint = 0 and 90°. However, since the amount of strain corresponding to each calculated offset stress is unknown, thus two approximations were used. The first approximation is that the strain amount at 0.1% proof stress is a constant value regardless of ψint. The strain was set as the average of the strain amount of 0.1% proof stress at ψint = 0 and 90°, which is 1.17%. The second approximation is that the curve to be predicted is a straight line up to a strain of 1.07%, which is the value obtained by subtracting the offset strain of 0.1% from the strain at 0.1% proof stress. The curves up to 0.1% proof stress can be predicted by these two approximations and the slope of the initial straight line can be calculated. By obtaining the slope of the initial sections of the curve, the strain amount corresponding to each offset stress can be also obtained.
The resulting compressive stress-strain curve with an offset strain of up to 50% is shown in Fig. 12. From Fig. 12, it can be considered that it is possible to predict with an error ratio of 10% or less for the range up to around 40% of the strain where the specimen at ψint = 90° started to densify. This is consistent with the analysis in Section 4.3. Therefore, it was clarified that the compressive stress-strain curve can be predicted with an error ratio of 10% or less when the offset strain was 40% or less by predicting the offset stress using the shear stress-strain curve around the pores and the compressive stress-strain curves at ψint = 0 and 90°.
Comparison of predicted results and experimental values in compressive stress-strain curves for the specimen without the skins.
The compression angle dependence of the strength of porous metals with regularly aligned unidirectional through-pores was investigated by compression tests and FEM. The following findings were obtained.
The A6061 aluminum alloy used in this study was provided by UACJ Corporation. This study was conducted with the support of a grant-in-aid from (a) The Light Metal Educational Foundation, Inc., (b) Grant-Aid for Young Scientists (Early Bird), Waseda Research Institute for Science and Engineering, (c) JST SPRING, Grant Number JPMJSP2128, and (d) RA(Sawada), Kagami Memorial Institute for Materials Science and Technology, Waseda University.
