Effects of pitch length of perforation on the crease bending characteristics of a polypropylene sheet subjected to indentation of a perforation blade

Abbreviations: A: cutting length in one cycle pitch (Figure 1), B: nicked length in one cycle pitch (Figure 1); C = A + B: pitch length of perforation blade; x: downward (compressive) displacement of the upper holder with perforation blade; F: compressive force applied to the perforation blade with a length (L_B) = 30 mm; W = 24 mm: width of the PP sheet; f = F/W: compressive line force applied to the PP sheet; t: thickness of the PP sheet = 0.5 mm; d₁: indentation depth against the PP sheet at the cutting tip (part A); d₂: indentation depth against the PP sheet at nicked (uncut but scored) zone(part B); d_2R: residual (permanent) thickness of the PP sheet at the nicked zone after scoring; d_2as/t = 1 − d_2R/t: normalized residual indentation depth of the scored zone; t_ip: elapsed time; θ: folding angle of the fixture at the crease stress tester (CST); M: bending moment resistance per unit width (N·mm/mm) measured by the CST; θ₁: CST fixture angle when the first cycle's bending resistance (first starting angle) increases; θ₂: CST fixture angle when the second cycle's bending resistance (second starting angle) increases; M_p1: bending moment resistance at the first inflection point in the first cycle; θ_p1: angle at the first inflection point; C₁ = $\partial M/\partial \theta {|}_{\theta = {\theta }_{1}}$: first stiffness (gradient of the bending moment resistance with an angle at θ = θ₁); C₂ = $\partial M/\partial \theta {|}_{\theta = {\theta }_{2}}$: second stiffness (gradient of the bending moment resistance with an angle at θ = θ₂)

1.   Introduction

Clear cartons are resin packaging containers widely used for packaging cosmetics. Since many resin sheets are transparent, the products can be seen through them, which have a display effect that makes the products more appealing. Transparent resin sheets are used in various applications today. To manufacture packaging containers such as paperboard and/or clear cartons, form-cutting and creasing processes are generally applied to a worksheet in the form-cutting process. Creasing (scoring and folding) is a method in which a score, called a ruled line, is applied to the bent portion, and second the scored part is folded ^[1,2,3]. When creasing a resin sheet, its scored thickness is decreased by applying half cutting or pressing at the bent portion to make it easier to fold ^[4,5].

Concerning the creasing of paperboard, to use a plain-uniform round edge blade is suitable for smoothly folding paperboards ^[6,7,8]. However, since a polypropylene (PP) sheet is not delaminated by the scoring of the round edge blade, only the indented depth and its scored shape determine the relative strength of the bending stiffness at the scored position, whereas to precisely control the scored depth is generally difficult or unstable. This difficulty is caused by the mechanical-pressing principle of the flat-bed die-cutter ^[9,10]. Studies have investigated the usage and fundamental properties ^[11,12,13], and a couple of research articles exist on the cutting strength of PP ^[14,15].

Nagasawa et al. studied the edge profile effects on the crease bending characteristics of a 0.3-mm thick PP sheet scored by a plain-uniform two-line wedge blade (not by a perforation) when varying the scored depth ^[16]. This report revealed that the residual cutting depth (scored depth) sensitively determines the scored profile and the bending moment resistance of the scored part of the PP sheet.

To smoothly process a folding line on a resin sheet such as PP, an alternative processing method is empirically known, compared to a plain-uniform, two-line wedge blade. Namely, a leading crease using a perforation blade (sewing machine blade) is sometimes considered ^[17,18]. This method can easily determine the indentation depth from the full stroke cutting and combine the designed uncut length. The pitch length of the dashed line, and its nicked (scored, half-cut zone) length, seem to affect the crease bending characteristics of the resin sheet scored by a perforation blade. However, a lack of studies exists concerning the analysis of the bending moment response of scored dashed lines using the perforation blade. The perforation blade was made from a 42º single wedge of a 0.71-mm thick steel rule, which had a part of cutting line segments and a part of scoring (half-cut) line segments on a creasing line.

To clarify the bending formability of a 0.5-mm thick PP sheet indented by a perforation blade, the bending moment response was investigated with the folding angle of the fixture under a specified rotational velocity when varying the pitch of the dashed line and the residual indentation (scored) depth against the PP sheet. When the ratio of cutting-to-pitch length was 50%, the pitch length of the blade was varied in a certain range, and several kinds of perforated (dashed) lines were processed on the PP sheet. The depth of the nicked (removed) zone against the cutting tip was set as D = 0.25 mm, which was 50% of the 0.5-mm thickness of the PP sheet. The residual indentation depth of the nicked (scored) zone was varied in a certain range, from 25% to 75% of the PP sheet, to reveal the effects of the sheard profiles of the cutting and half-cut (scored) line segments on the crease bending characteristics when varying the residual indentation depth. Using these scored PP sheets, the bending moment resistance was investigated with the folding angle of the fixture from the bending-free state (~0º) up to 90º.

2.   Materials and experimental methods

2.1.   Mechanical characteristics of PP sheet for scoring

Table 1 shows the in-plane mechanical properties of the 0.5-mm thick PP sheet. The strain rate was set as 0.00417 s⁻¹ (a tensile velocity of V_tns = 0.33 mm/s against an 80-mm clamped distance), and the shape of the PP sheet was set as the JIS K-7127 type-5 (dumbbell type).

The preliminary tensile testing results revealed that when the tensile velocity increased from V_tns = 0.08 up to 8.3 mm/s, the breaking logarithmic strain was less than 0.1 (tends to be brittle, fragile) at V_tns = 8.3 mm/s, whereas it exceeded 1.6 (tends to be ductile) for V_tns < 2.5 mm/s. Moreover, in the case of a 42º wedge indentation into the thickness direction, when choosing an indentation velocity of V = 0.005 mm/s, the in-plane spreading velocity becomes 2Vsin21º = 0.0036 mm/s. Hence, at V = 0.005 mm/s, the in-plane behavior of the PP sheet seems to be ductile, not fragile.

2.2.   Mechanical condition of perforation blades

As the original blade material for scoring the PP sheet, a carbon-tool-steel (JIS-G-4401, SK85), 42º single-wedge blade (a commercial-2-point-Thomson knife) was prepared. The blade had tip and body hardness values of 690 and 465 VHN, respectively. Figure 1 shows the profile of the perforation blade, and some parameters are presented in Table 2. The length of the perforation blade was 30 mm. The pitch length C was a sum of the cutting part A and the nicked (half-cut) part B (tie width). The A-to-C ratio (A/C) was kept as 0.5 (A = B = 0.5C), and C was varied from 0.5–8 mm. The nicked part was machined using wire cutting (EDM) at a depth of D = 0.25 mm. The number n = 24/C was the total number of the one cycle pitch formed on a piece of 24-mm width-PP sheet.

Figure 2 shows the attitude of the perforation blade against a 24-mm width-PP sheet, and a general view of the prepared perforation blades. For the comparison with the pitch length of the perforation blade, a trapezoidal (plain-uniform) blade with a tip thickness of 0.21 mm (a height of 0.25 mm was uniformly removed, and the estimation was 0.25 $\times$ 2tan21º = 0.19 mm) was prepared. According to the online catalog of American Micro Industries Inc. ^[17], a certain fine pitch dashed line is called "micro perforation" in which the teeth per inch (TPI) is larger than 30 ^[17,18]. Namely in this study, C < 0.85 mm is prepared as a "micro perforation". Due to the restriction of the wire cutting method using a 0.2-mm copper wire, the grooved profile of the nicked (removed) zone had a 0.1-mm radius (Figure 1b). Therefore, the round profile of the grooved corners seemed to affect the three-dimensional deformation of the scored zone (part of B), especially for C < 1 mm.

Since the trapezoidal part of the depth D = 0.25 mm was indented to the 0.5-mm thick PP sheet, the cutting and scoring combination was considered as a special specification. Generally, some well-known scoring of perforation is considered as D > t = 0.5 mm.

2.3.   Compressive loading procedure of blade against sheet

Figure 3 shows schematics of the scoring (pressing) apparatus schematic for making a perforated (dashed) line on a 0.5-mm thick PP sheet with a width and a length of 24 and 50 mm, respectively. The initial step height of a new cutting tip from the rubber fixture was ~3 mm. When pressing the perforation blade against the PP sheet, the cutting part A was indented to the PP sheet until d₁/t $\approx$ 1, as well as d₂/t $\approx$ 0.5. Generally, since there is an equivalent-spring effect on the blade displacement, the residual (permanent) indentation depth d_2as can be estimated principally with the difference between the blade displacement and device elongation. The real (residual) depth d_2as/t was ~0.3 when considering d₁/t$\approx$1.0 under the real indentation of part A. Herein, the device elongation was not calibrated strictly. Alternatively, the residual indentation depth of the perforation blade d_2as = t − d_2R was calculated from the measured residual thickness d_2R after scoring. When inspecting the effect of the pitch C on the crease formability, named Case 1, the residual indentation depth was fixed as d_2as/t = 1 − d_2R/t $\approx$0.45 (d_2R = 0.27–0.28 mm $\approx$ 0.55t). There was a little insufficient indentation of blade for scoring the bottom layer of the PP sheet. The applied pushing force F was measured by a load cell.

Before indenting the blade, the PP sheet was gently placed on the counter plate without any fixture. When moving the blade downward, the 7-mm$\times$7-mm$\times$50-mm rubber sponge empirically fastened the PP sheet at a 10-mm horizontal distance (Figure 3a). The relationship between the scoring line force F and the displacement of blade holder x against the PP sheet reveals that x increased up x = d₁. The surface level of the PP sheet was defined as x = 0. The rubber reaction force was calibrated with the scoring force F.

When making the scored PP sheets in Case 1 (d_2as/t = 1 − d_2R/t$\approx$0.45), a certain thin uncut layer (Figure 4a) seemed to exist in some samples. or the bottom layer of part A seemed to be cut off, but no gap exists (there seemed to be interference of burrs) after scoring (Figure 4b). Therefore, to discuss the effects of the bottom gap at part A on the bending moment resistance, another condition of indentation depth (Figure 4c) was considered as Case 2. Herein, the residual indentation depth of the scored PP sheet (at part B) was varied in a range of d_2as/t = 1 − d_2R/t = 0.25–0.75 when choosing three patterns of perforation: 0.5$\times$0.5, 3$\times$3 and the plain-uniform trapezoidal.

The indentation velocity of the blade was set as V = 0.005 mm/s, and the sample number of PP sheets was mainly N = 5 for each perforation blade at room temperature and relative humidity of 296 K and 50%, respectively.

2.4.   Bending moment response of scored PP sheet

Figure 5 shows a folding (bending) test method using a crease stress tester CST-J1 ^[19]. The scored side was set up as two patterns: (a) normal folding direction (interior scored notch) and (b) reverse folding direction (exterior scored notch in the folding process from the folding angle θ of θ₁ or θ₂ to 90º). Half of the PP sheet was initially clamped at the scored position by the fixture of the CST-J1, maintaining θ = θ₁. The origin of the folding angle θ = 0 was defined with the attitude of a flat plate. The rotational velocity of the fixture was ω = dθ/dt_ep = 0.2 rps (1.257 rad/s), and the arm length $\ell$ was 10 mm from the scored position to the load cell (rating maximum of 10 N). The bending strain rate $\dot{{ \varepsilon }_{b}}$ at the scored notch is calculated using Eq 1. Herein, Young's modulus of E = 1.25 GPa (Table 1) and a residual thickness of d_2R = 0.55t = 0.275 mm are considered. The rate of bending moment resistance $\dot{M}$ is estimated as a ratio of M_p1 to θ_p1/ω (M_p1ω/θ_p1). This value of bending strain rate is calculated later when obtaining the relationship between M_p1 and θ_p1.

The first starting angle of θ₁ depends on the warpage of the scored PP sheet. As a tendency of the scored PP sheet when d_2as/t = 1 − d_2R/t$\approx$0.45, θ₁ was about −5º in the normal folding, whereas θ₁ was about +5º in the reverse folding. When 0.45 < d_2as/t < 0.75 (a part of Case 2), the amplitude of |θ₁| increased and |θ₁| $\approx$ 15º at d_2as/t = 0.6–0.75.

The folding (bending) test was continuously examined twice until releasing the reaction force of the load cell. For example, in the folding (bending) test of Case 1, when choosing d_2as/t = 1 − d_2R/t$\approx$0.45, seven perforations were examined to measure the bending moment resistance M with the folding angle θ in both the normal and reverse folding directions. Figure 6 illustrates a representative example of the M-θ response when choosing the plain-uniform trapezoidal blade (A = 0, B = 30 mm). To detect some features of the M-θ response, when varying the pitch length C, some representative parameters were defined (extracted) from the response curve illustrated in Figure 6: the first starting angle θ₁, the first inflection point (bending moment resistance M_p1, angle θ_p1), the first stiffness (the first gradient of moment M by θ) C₁, the tracking (rating) bending moment resistance M₉₀ at θ = 90º, the second starting angle θ₂ (the reactional bending moment is zero when the folding angle is released), and the second stiffness C₂ at the second loading. Those parameters were measured for the normal and reverse folding directions, respectively.

According to the previous report on the crease bending characteristics of a scored-0.3-mm-thick PP sheet ^[16], the interior scored contact interference was known from the bending moment resistance in the normal folding. However, the rating bending moment resistance M₉₀ at 90º was not analyzed from aspects of dependency on the residual indentation depth d_2as/t. Some parameters, such as M_p1, C₁, M₉₀ and C₂, were discussed using theoretical models of elastic-limitation and plastic collapse ^[20,21,22]. Since C₁, C₂ and resistance M_p1 were mainly determined by the stress concentration effect at the scored notch, the indentation depth d_2as/t of the scored notch was recognized as a primary control parameter.

3.   Results and discussions

3.1.   Profile of scored PC sheet

Before discussing the bending moment resistance of Case 1 (variable pitch C under d_2as/t = 1 − d_2R/t$\approx$0.45), ascertaining the relationship between the profile of the scored part and the residual indentation depth 1 − d_2R/t is convenient for understanding the contact interference of burrs or uncut layer at the bottom layer. Hence, the profiles of the scored part were first explained in Case 2 (1 − d_2R/t = 0.25–0.75). Figures 7–9 show side views of the scored PP sheets when using the plain-uniform trapezoidal, 0.5$\times$0.5 perforation and 3$\times$3 perforation blades, respectively.

When d_2as/t = 1 − d_2R/t > 0.43–0.47, the bottom layer was remarkably necked at the scored zone, and the warp angle of the PP sheet was about 5º–15º for opening the wedged side (upper side). A narrow range of d_2as/t = 1 − d_2R/t = 0.43–0.47 was collected and used as Case 1. In the case of d_2as/t = 1 − d_2R/t < 0.43, as the cutting zone (part A) was cut off insufficiently, the creasing line comprised two kinds of half-cut grooves (parts A and B).

Figures 10 and 11 show the top (a) and back (b) views of scored the PP sheet when using the 0.5$\times$ 0.5 perforation blade and the 3$\times$3 perforation blade, respectively. The nicked part B (trapezoidal) was relatively spread out compared to the cutting part A, due to the plastic-compressive deformation flow by the trapezoidal tip profile of the blade. The cutting part A was kissed or interfered at the bottom layer for d_2as/t = 1 − d_2R/t < 0.5, whereas it was opened with a certain nonzero gap for d_2as/t = 1 − d_2R/t > 0.5. The width W_c of the cutting zone at part A was measured and arranged in Figure 12. The 3$\times$3 perforation (Figure 12b) shows that the nonzero starting point was detected at d_2as/t = 1 − d_2R/t = 0.5. In this case, the nonzero condition matched d₁/t > 1 geometrically. When d_2as/t = 1 − d_2R/t = 0.76, the spread-out width was calculated as (0.76 mm − 0.5 mm) $\times$2tan21º = 0.1 mm, whereas the measured size of W_c was about 0.08 mm in Figure 12b. Hence, the difference of 0.1 − 0.08 mm = 0.02 mm was estimated as a shrinked width. The 0.5$\times$0.5 perforation in Figure 12b showed that the measured W_c was about +0.02 mm larger than that of the 3$\times$3 perforlation. Thus, the starting point of nonzero was about d_2as/t = 1− d_2R/t = 0.35 in the case of the 0.5$\times$0.5 perforation. Also, the interference of burrs or cutting over at the bottom layer was varied with pitch C.

3.2.   Load response during scoring process

Before seeing the perforation load response, a new virgin blade and the plain-uniform trapezoidal blade of 0.21 mm tip thickness were examined in the cutting test of the 0.5 mm thickness PP sheet. Figure 13a, b show the relationship between the cutting line force f = F/W and the normalized indentation depth d₁/t, d₂/t, when cutting the PP sheet by using the new virgin and the plain-uniform-trapezoidal blades, respectively.

The maximum line force of the new virgin blade was about 10 N/mm at 50% of the thickness t = 0.5 mm, whereas the cutting load by the plain-uniform trapezoidal blade of the 0.21 mm tip thickness showed the yielding resistance of 20–22 N/mm for d₂/t = 0.5–0.9. The drawn tags (a)$-$(g) in Figure 13b corresponded deformation states shown in Figure 7. Since the state of (c) was d_2as/t = 1 − d_2R/t = 0.45, the scoring line force of the plain-uniform trapezoidal blade was estimated at 22 N/mm in Case 1. So far, an equivalent maximum line force of the perforation blade was expected to be 0.5 $\times$ 10 + 0.5$\times$20 = 15 N/mm, because A = B = 0.5C was assumed with respect to the profile of the perforation blade. As the line force f₂ remarkably increased for d₂/t > 1 (including the displacement of underlay and holder device) (Figure 13b), to precisely control the residual indentation depth d_2as becomes difficult using a certain tip thickness of the trapezoidal blade. When considering only d₂/t < 0.9 (d_2as/t < 0.5), the scoring line force is stably estimated as being less than 22 N/mm. Therefore, the loading condition of Case 1 was recognized as a stable scoring state.

Figure 14 shows the relationship between the scoring line force f = F/W and the normalized indentation depth d₁/t, when pushing the PP sheet by the six perforation blades chosen as C = 0.5, 1.0, 2, 4, 6, and 8 mm, respectively. The line force remarkably increased at d₁/t = 0.5–0.6 and culminated at d₁/t = 0.7–0.8. The former (remarkably increasing as a stepped response) was caused by the kissing and starting of the indentation of the nicked zone (part B), whereas the latter passed through the necking state of the cutting zone (part A).

Although the maximum line force varied with the pitch C, its tendency was not monotone with C. The former (remarkably increasing as a stepped response) was weakened and a smooth-peaked response when C < 1 mm (almost being the micro perforation). In Figure 14, the maximum line force was almost 15–20 N/mm, close but exceeded 15 N/mm. Hence, it was found that the prediction of combined cutting resistance from Figures 12 and 13 (described) was close, but slightly smaller than the real maximum peak values. This additional resistance in the real scoring process caused the generation of 3D cutting corners.

3.3.   Effects of pitch length on bending moment resistance

The effects of the pitch length C on the bending moment resistance M were investigated in Case 1 (d_2as/t = 1 − d_2R/t$\approx$ 0.45). Figure 15 shows the representative bending moment resistance M when folding from θ = θ₁ and θ = θ₂ to 90º (double folding) under a rotational velocity of ω = 0.2 rps. The early stage (θ < 40º) of the load response slightly varied with the pitch length. Also, when the bent inside of the scored zone (part B) had no interference, the rating bending moment M₉₀ slightly varied for C$\ge$4 mm. However, when C$\le$2 mm, the bent inside of the scored zone was kissed and the bending moment resistance remarkably increased for 50º < θ < 90º, whereas the bent inside of the scored zone had a certain clearance and the bending moment resistance almost saturated for 50º < θ < 80º when C$\ge$4 mm.

Figure 16 shows two cases: (a) the scored inside was almost kissed state at C = 1 mm, and (b) the clearance existed at C = 6 mm, when the PP sheet was bent to θ = 70º.

To characterize the bending moment resistance of the scored line, the bending moment at the first inflection M_p1, the rating bending moment M₉₀ at θ = 90º, the first and second stiffness values C₁ and C₂, and the second starting angle θ₂ were arranged with the pitch C. Figure 17 shows the bending moment resistance at the first inflection M_p1 and its corresponded angle θ_p1−θ₁ when choosing the six perforation blades (C = 0.5, 1, 2, 4, 6 and 8 mm), and the plain-uniform trapezoidal blade. In Figure 17, blue and brown represent the normal and reverse folding directions, respectively. Since M_p1$\approx$0.3 N·mm/mm in the normal/reverse directions at the six perforation blades, and M_p1 $\approx$ 0.7 N·mm/mm in the reverse direction at the plain-uniform trapezoidal blade, their ratio was about 0.43. Since B/C = 0.5, this result was reasonable, because the cutting part A was half of the width of the PP sheet. The difference between the normal and reverse folding directions was less than 20% of M_p1 at the perforation blades, and less than 15% of M_p1 at the plain-uniform trapezoidal blade. In the case of the plain-uniform trapezoidal blade, the normal M_p1 was stably larger than the reverse M_p1. This trend was probably caused by the stress concentration effect at the scored notch.

The occurrence of the inflection of the bending moment resistance at θ_p1 − θ₁ = 10º–20º was the same tendency as that of the previous report of a 0.3-mm-thick PP sheet by a two-line wedge knife ^[16]. According to the theory of plasticity ^[20,21,22], the bending moment resistance of a unit-width uniform beam at the elastic limitation was estimated as Eq 2, whereas that at the plastic collapse was estimated as Eq 3.

Substituting σ_Y = 31.9 MPa (the tensile yielding stress from Table 1), and d_2R = 0.275 mm (the residual height of scored zone) into Eqs 2 and 3 gives M_el = 31.9$\times$0.275$\times$0.275/6 = 0.4 N·mm/mm, and M_pc = 6M_el/4 = 0.6 N·mm/mm, respectively.

When considering the perforation state of a scored line at B/C = 0.5, the expected bending moment resistance was just half of the plain-uniform scored model, because no bending resistance was assumed at the cutting zone. Hence, the expected resistances are 0.5M_el = 0.2 N·mm/mm and 0.5M_pc = 0.3 N·mm/mm at the six perforation blades. So far, the bending moment resistances of the elastic limitation and plastic collapse were estimated as 67% and 100% of the experimental M_p1 (0.3 N·mm/mm) at the six perforation blades, respectively. They were also estimated as 57% and 86% of the experimental M_p1 (0.7 N·mm/mm) at the plain-uniform trapezoidal blade, respectively. Therefore, the inflection point was mainly caused by the yielding collapse or the plastic bending behavior of the scored zone.

Using Eq 1 and Figure 17, the strain rate of bending deformation at the scored notch is calculated. Watching the uniform crease, as M_p1 = 0.7 N·mm/mm, θ_p1 = 20º, ω = 0.2$\times$360º/s, the bending strain rate $\dot{{ \varepsilon }_{b}}$ was estimated as 0.16 s⁻¹. This folding deformation seems to be a brittle state.

Figure 18 shows the rating bending moment resistance at a folding angle of 90º in the normal and reverse folding directions when using the six perforation blades and the plain-uniform trapezoidal blade for scoring. Herein, the residual indentation depth was set as d_2as/t = 1 − d_2R/t$\approx$ 0.45. Considering the reverse folding, M₉₀$\approx$ 0.7 N·mm/mm at the six perforation blades whereas M₉₀$\approx$1.4 N·mm/mm at the plain-uniform trapezoidal blade, and then the ratio of the former to the latter was 0.5. This trend was reasonable due to the geometrical relation B/C = 0.5.

The normal folding M₉₀ remarkably increased for 0.5 mm < C < 2 mm, whereas the reverse folding M₉₀ was almost constant for 0.5 mm < C < 8 mm. This increment in the normal folding M₉₀ was detected for θ > 50º–70º (Figure 15a), and its mechanism was confirmed as an interior scored kissing (Figure 16). When the interior scored kissing disappeared for the perforation blades (C$\ge$4 mm) and the plain-uniform trapezoidal blade, the normal folding M₉₀ was +0.1 N·mm/mm larger than the reverse folding M₉₀. This small difference of 0.1 N·mm/mm was probably caused by the contact interference of the scored inside.

As for the 90º folding, the bending stress seems to be in the plastic collapse state. Although M₉₀ can be predicted as σ_Y (d_2R)²/4, since the yield stress increases by the work hardening, the plastic collapse bending moment is σ_Y90 (d_2R)²/4. Here, for example, when σ_Y90 is assumed to be σ_B = 42.1 MPa, then M_pc = 42.1$\times$0.275$\times$0.275/4 = 0.8 N·mm/mm. In the reverse folding, since M₉₀ $\approx$ 1.4 N·mm/mm in the experiment, this assumption was insufficient. Due to the work hardening and the large deformation, σ_Y90 appeared to be (1.4/0.8)σ_B = 1.75σ_B = 73.7 MPa.

Evidently, the relative crease strength was well predicted from the design theory of plasticity.

Figure 19a, b show the first and second stiffnesses values (gradients of the bending moment resistance) C₁ and C₂ at the first and second starting angles θ₁ and θ₂ in the normal and reverse folding directions, respectively, when using the six perforation blades and the plain-uniform trapezoidal blade for scoring. The residual indentation depth was set as d_2as/t = 1 − d_2R/t$\approx$ 0.45. Considering the normal folding, C₁ $\approx$ 0.03 N·mm/mm/º when using the six perforation blades, whereas C₁ $\approx$ 0.045 N·mm/mm/º when using the plain-uniform trapezoidal blade. Also, the ratio of the former to the latter was 0.67. This value should be theoretically 0.5 because the cutting part A is expected to experience no bending moment. Hence, the cutting part A excessively increases the bending stiffness in the folding process. This trend is probably caused by some contact interference of burrs or a thin uncut layer at the cutting part A (Figure 4a, b).

Similarly, considering the second normal folding, C₂ $\approx$ 0.018 N·mm/mm/º when using the six perforation blades, whereas C₂ $\approx$ 0.033 N·mm/mm/º when using the plain-uniform trapezoidal blade. The ratio of the former to the latter is 0.55. This value is relatively close to the theoretical value of 0.5.

In the reverse folding, the first stiffness C₁ slightly differed from that of the normal folding, whereas the second stiffness C₂ was stably slightly larger (0.005–0.01 N·mm/mm/º) than that of the normal folding. This difference in C₂ between the normal and reverse folding is probably due to the stress concentration and the work hardening at the outside surface layer.

Figure 20 shows the second starting angle θ₂ in the normal and reverse folding directions when using the six perforation blades and the plain-uniform trapezoidal blade. The reverse folding θ₂ was 5º–10º larger than that of the normal folding. This trend indicates that the spring back in the normal folding exceeds that in the reverse folding. Then, the reverse folding generated larger plastic deformation at the scored zone in the folding process compared to the normal folding. This tendency of θ₂ seemed to be caused by the same reason responsible for the difference in C₂ between the normal and the reverse folding. The stress concentration and the local plastic deformation appeared to be larger/stronger on the outside surface at the scored zone under the reverse folding.

Figure 20 showed that the difference in θ₂ between the normal and the reverse folding was not even with the pitch C. The 2$\times$2 perforation (C = 4 mm) had the minimum difference, whereas the 0.25$\times$0.25 and 4$\times$4 perforations had relatively larger differences. This tendency is similar to the difference in (θ_1p−θ₁) between the normal and the reverse folding (Figure 17b). This trend is a sort of positive correlation. Because when the angle of (θ_1p−θ₁) is smaller, the plastic deformation starts in the earlier position against the bending moment. This phenomenon is principally caused by the high-stress concentration at the bent surface of the scored PP sheet. Consequently, the spring back as 1 − θ₂/90º appears to become smaller (θ₂ increases) when the stress concentration is larger at the bent surface.

When choosing C = 0.5 mm (C/t = 1), the first stiffness C₁ and the bending moment resistance at first inflection M_p1 were almost equal (considerably reducing the bending moment resistance) to that of the other perforation (1 mm < C < 8 mm) in the normal folding, whereas the rating bending moment M₉₀ was almost equal (having a large resistance against the final folding) to that of the plain-uniform trapezoidal blade in the normal folding. Hence, the microperforation (C/t < 1.7) appears to perform differently, compared to others (C/t > 2).

3.4.   Effects of indentation depth on bending moment resistance

As previously reported ^[16], as the residual indentation depth d_2as/t = 1 − d_2R/t increases, the representative three quantities M_p1, C₁ and C₂ decrease. When evenly combining the scored zone (part B) and the cutting zone (part A) as B/C = A/C = 0.5, these quantities should be half that of the plain-uniform trapezoidal blade, because the cutting part A is expected to experience no bending moment. However, as described in Section 3.3, the first and second stiffnesses C₁ and C₂ varied with the pitch C, and they were not always half that of the plain-uniform trapezoidal blade. Therefore, C₁ and C₂ were investigated here in the condition of Case 2: two kinds of perforation and plain-uniform trapezoidal were used in the range of 0.25 < 1 − d_2R/t < 0.75.

Figure 21 shows the dependency of the first stiffness C₁ on the normalized residual indentation depth at the normal and reverse folding directions.

Here, Eqs 4 and 5 are the linear approximation derived from the experimental results of the plain-uniform trapezoidal blade. The ratio of Eq 5 to Eq 4 was about 1.08 for 0.25 < 1 − d_2R/t < 0.75. When the residual indentation depth is shallow (1 − d_2R/t < 0.2), the stiffness C₁ saturates. The upper bound of C₁ is confirmed using the following discussion when the indentation depth is shallow. Regarding the first stiffness C₁, an elementary theory of a cantilever model is useful for determining the order of C₁ as the upper-bound ^[23,24]. Assuming that Young's modulus: E = 1.25 GPa (from Table 1), the span length: $\ell$ = 10 mm, the second moment of area per unit width: I = t³/12, and the height (thickness) of beam: t = 0.5 mm, the upper-bound stiffness per unit width C_1UB is estimated as the deflected angle of the cantilever (Eq 6). Herein, the unit of the rotation is considered as radian.

Using the assumptions described above, C_1UB is calculated as 1250$\times$0.5³/(6$\times$10) $\times$3.14/180 = 0.045 N·mm/mm/º. Figure 21 shows that the plots of the plain-uniform trapezoidal were slightly exceeded 0.045 N·mm/mm/º for 1 − d_2R/t < 0.4. The value of E should be updated in a bending mode test. Also, note that the linearization of Eqs 4 and 5 was limited in the specified range: 0.25 < 1 − d_2R/t < 0.75.

A virtual line of 0.5$\times$Eq 4 and that of 0.5$\times$Eq 5 are drawn in Figure 21a, b, respectively. Here, "0.5$\times$Equation" indicates that a half value of the Equation was calculated. These lines correspond to the stiffness of part B (scored zone).

Figure 21a shows that the stiffness C₁ with the perforation (C = 1 mm, 6 mm) at the normal folding is almost linear between the upper line of Eq 4 and the lower line of 0.5$\times$ Eq 4. Similarly, Figure 21b shows that the stiffness C₁ with the perforation (C = 1 mm, 6 mm) at the reverse folding is almost linear relationship between the upper line of Eq 5 and the lower line of 0.5 $\times$ Eq 5. The stiffnesses C₁ with the two perforations are close to the upper line Eqs 4 and 5 for 1 − d_2R/t < 0.3, whereas they are quite close to the lower line 0.5 $\times$Eqs 4 and 5 for d_2as/t = 1 − d_2R/t > 0.7. When choosing d_2as/t = 1 − d_2R/t$\approx$0.45, C₁ $\approx$ 0.7–0.75 $\times$ Eqs 4 and 5. This result almost matches Figure 19. Also, Figure 12 shows that the width of the cutting zone W_c is nonzero for d_2as/t = 1 − d_2R/t > 0.5, and C₁ is pretty close to the lower line 0.5$\times$Eqs 4 and 5. So far, the contact interference of burrs at the cutting bottom layer disappears for d_2as/t = 1 − d_2R/t > 0.7, and at that time, C₁ is almost half of that of the plain-uniform trapezoidal blade.

Figure 22 shows the dependency of the second stiffness C₂ on the normalized residual indentation depth at the normal and reverse folding directions. Here, Eqs 7 and 8 are the linear approximations derived from the experimental results of the plain-uniform trapezoidal blade. The ratio of Eq 8 to Eq 7 was about 1.1–1.7 for 0.25 < 1 − d_2R/t < 0.75. Evidently, the reverse had a larger stiffness than the normal. Considering the ratios of Eq 4/Eq 7 and Eq 5/Eq 8, C₂ was about 70%–80% of C₁ in the case of the plain-uniform trapezoidal blade. Concerning the perforation blades, C₂ was almost linear between the upper line of Eq 7 and the lower line of 0.5$\times$Eq 7 at the normal folding direction. Also, it was almost a linear relationship between the upper line of Eq 8 and the lower line of 0.5$\times$Eq 8 in the reverse folding direction, respectively. Here, C₂ was close to the lower lines (Figure 22). Namely, through the first folding cycle up to 90º, some contact interference of burrs and/or the thin uncut layer at the cutting zone (part A) appears to be plastically weakened.

As described in Section 3.3, the first inflection of the bending moment resistance M_p1 (Figure 17) was discussed using the plastic collapse by Eq 3 when choosing d_2as/t = 1 − d_2R/t$\approx$0.45. To estimate the M_p1 behavior when changing d_2as/t = 1 − d_2R/t = 0.25–0.75, the relationship between M_p1 and d_2as/t was investigated using two perforation blades (C = 1 mm, 6 mm) and the plain-uniform trapezoidal blade for the normal and reverse folding directions (Figure 23). According to the previous report ^[16], M_p1 was linearly approximated with d_2R/t, although the plastic collapse model was partially discussed ^[20,21,22]. Obviously, the second order polynomial with d_2R/t: Eq 3 and 0.5 $\times$Eq 3 well explains the amplitude of M_p1.

Regarding the rating bending moment resistance at θ = 90º (Figures 15 and 16), since there was a sort of interior kissing of the scored zone in the normal folding, an additional increase in the bending moment resistance occurred at θ > 50º–70º and C$\le$2 mm in the normal folding. Also, the work hardening, and/or contact interference at the scored zone, was remarkable when choosing d_2as/t = 1 − d_2R/t$\approx$ 0.45.

To estimate the behavior of M₉₀ when changing d_2as/t = 1 − d_2R/t = 0.25–0.75, the relationship between M₉₀ and d_2as/t was investigated using two perforation blades (C = 1 mm, 6 mm), and the plain-uniform trapezoidal blade for the normal and reverse folding (Figure 24). Henceforth, the yield stress was assumed to be σ_Y90 = 1.75σ_B = 73.7 MPa in Eq 3. Considering the reverse folding of Figure 24b, the experimental M₉₀ of the plain-uniform trapezoidal blade matched Eq 3, whereas that of the perforation blades (C = 1 mm, 6 mm) matched 0.5$\times$Eq 3. Therefore, the assumption of σ_Y90 = 1.75σ_B based on Eq 3 (a plastic collapse model) is reasonable and useful for predicting the bending moment resistance of scored PP sheets in the reverse folding.

In contrast, considering the normal folding in Figure 24a, in a shallow range of d_2as/t = 1 − d_2R/t$\le$0.45, the experimental M₉₀ is partially predictable using the upper line of Eq 3 and the lower line of 0.5$\times$Eq 3. However, there were several kinds of unmatched behavior to Eq 3 and 0.5 $\times$Eq 3. First, the experimental M₉₀ of the perforation of C = 1 mm was close to that of the plain-uniform trapezoidal and larger than Eq 3 when d_2as/t = 1 − d_2R/t$>$0.45. This trend was due to the additional increase in the bending moment resistance for θ > 50º–70º. Some contact interference at the scored zone appeared to increase the bending resistance. Second, the experimental M₉₀ of the perforation of C = 6 mm was about 70% of that of the plain-uniform trapezoidal blade when d_2as/t = 1 − d_2R/t$>$ 0.45. Some contact interference characteristics at the scored zone increased the bending resistance for θ > 80º – 90º.

Synthetically, the rating bending moment resistance M₉₀ of perforation in the normal folding was unpredictable from some elementary theories for θ > 50º, because the contact interference often occurs at the scored zone (part B). However, the bending moment resistance at the inflection point M_p1 is easily predictable from Eq 3, whereas the first stiffness C₁ is experimentally predictable as a linear relation, which is an intermediate resistance between the plain-uniform trapezoidal model and its 50% model. In the reverse folding, M₉₀ is predictable using the modified Eq 3 and σ_Y90 = 1.75σ_B, although M₉₀ remarkably decreases when d_2as/t = 1 − d_2R/t$>$0.45.

Figure 25 shows the second starting angle θ₂ when changing the residual indentation depth d_2as/t = 1 − d_2R/t for two kinds of perforation blade (C = 1 mm, 6 mm) and the plain-uniform trapezoidal blade. The normal folding θ₂ was almost constant with d_2as/t = 1 − d_2R/t, whereas the reverse folding θ₂ increased with d_2as/t = 1 − d_2R/t. Also, the reverse folding θ₂ exceeded the normal folding θ₂. Comparing the three patterns (perforation), Figure 25 indicates that the fine pitch perforation (C = 1 mm) decreases the spring back (namely, it increases θ₂) in a folding process up to 90º.

Figures 24 and 25 reveal that the deep indentation of the scored zone (part B) contributes to the plastic collapse at the reverse folding of the scored zone, whereas it (deep indentation) accumulates the elastic energy of the contact interference of the scored zone (part B). Also, M₉₀ does not decrease with d_2as/t = 1 − d_2R/t > 0.5.

3.5.   Bent profiles of perforation creasing line

In the bending test by CST-J1, it was challenging to take some bent profiles of desirable scored zones of the scored PP sheet, especially for C < 2 mm. Therefore, some bent profiles of the creasing line were overviewed manually. Figure 26 shows representative general views of bent corners concerning three scored-0.5-mm-thick PP sheets for the normal and reverse folding directions. These specimens were bent from θ₁ to 180º (making a sharp-folding form) and released in a natural, free state. Maintaining a certain bent state again, these bent corners were manually captured by a digital microscope. The specimens were smoothly bent without any cutting-off in the folding motion.

The following features were detected in the images: (i) In the reverse folding, the outside surface of the scored zone was smoothly or continuously cracked with a certain depth in the thickness direction (not split), whereas there was not any large propagation of cracks along the creasing line (in the lateral direction). (ii) In the normal folding, the scored zone was stably kissed at the inside when θ > 50º and the outside was smoothly elongated and rounded. There was no propagation of cracks along the creasing line (in the lateral direction).

4.   Conclusions

To reveal the crease bending characteristics of 0.5-mm-thick PP sheet using some perforation blades in the normal and reverse folding, the effects of an even combination of sharp and nicked-trapezoidal wedge blades on the scoring load response and bending moment resistance at the scored creasing line were experimentally investigated with several perforation-pitch lengths. As the crease bending characteristics, the first and second stiffnesses (gradient of bending moment resistance by folding angle) C₁ and C₂, the rating bending moment resistance at the right angle (θ = 90°) M₉₀, the bending moment resistance at the first inflection point M_p1, and its corresponding angle θ_p1 were discussed concerning the residual indentation depth of the blade against the PP sheet. Through this experiment, the following results were revealed.

(1) When changing the pitch length C of the perforation, the scoring force response remarkably changed at the first half process. When the nicked trapezoidal blade indented the PP sheet in the case of C $\ge$ 2 mm, the load response had a stepped increase, whereas the sharp blade indented the PP sheet in advance (by D = 0.25 mm) and made a monotone increasing response. However, in the case of C $\le$ 1 mm, the load response showed a seamless (not any stepped) increasing response. The peak maximum scoring force of perforation exceeded the arithmetic mean of the sharp uniform wedge and the corresponding scoring force of the uniform nicked (trapezoidal) wedge.

(2) When changing C and keeping the normalized residual indentation depth d_2as/t = 1 − d_2R/t$\approx$0.45, the response of the bending moment resistance M was not significantly changed with the normal and reverse folding in the early stage (θ < 40º), whereas it differed remarkably at the latter half (50º < θ < 90º) between the normal folding and the reverse folding. When C $\le$ 2 mm, the bending moment resistance remarkably increased for 50º < θ < 90º in the normal folding, due to the contact interference of the scored zone. In the scoring process of the perforation, as the wedged width of the cutting part increased with C, the scored width of the scored part was also affected by C, and then the bending moment resistance increased significantly in the latter half, especially for C $\le$ 2 mm.

(3) When changing the normalized residual indentation depth d_2as/t = 1− d_2R/t = 0.25–0.75, the wedged width of the cutting part increased with d_2as/t, whereas the first and second stiffnesses C₁ and C₂ were almost linearly characterized with d_2as/t between the plain-uniform-trapezoidal blade indentation model and its half model (based on A/C = B/C = 0.5). This appeared to be independent of the variance of C.

(4) Bending stiffness parameters such as C₁, C₂, M_p1 and M₉₀ were characterized by the yielding stress σ_Y or its modified/updated yielding stress σ_Y90 and the normalized residual thickness d_2R/t. They were strongly related to the plastic collapse moment. Herein, in the case of normal folding, the behavior of M₉₀ was unpredictable from d_2as/t without considering the contact interference of the scored inside.

(5) The plastic-residual behavior of folding of the creasing line differed between the normal folding and reverse folding due to the difference in stress concentration and the contact interference at the scored zone. The behavior was revealed from some parameters: the angle of the first inflection point θ_1p−θ₁, the second stiffness C₂, the second starting angle θ₂ and the rating bending moment resistance at the right angle M₉₀.

Conflict of interest

All authors declare there is no conflict in this paper.