フジコー技報−tsukuru No. ሣᛮ䛸䛠䜅ቌฦ䛱ᚺງቌฦ᪁ྡྷ౪Ꮛᛮ䜘⩻៎䛝䛥㻃 㻃 㻃 ሣᛮᵋᠺᘟ䛱䛪䛊䛬 ᢇ⾙ゆㄕ㻃 A Study on Plastic Constitutive Law Incorporating Directional Dependence of Plastic Strain Increment on Stress Increment ⌨⌣ኬᏕᕝᏕ㒂 ᶭࢨࢪࢷ࣑ᕝᏕ⛁ ᩅ 㸝ᰬ㸞ἀ⦎࢙ࢾࢷࢴࢠ ༡ኃሒ⛁Ꮥ㸞ᮆྚ ᨼ 7RVKL\DVX6XH\RVKL㻃 㸦㸣ࡢࡋࡴ࡞ ୕Ờὢ ཪ㯖 <XPD8H]X ▩ࡼࡿ࡙࠷ࡾࠉᮞᩩࡡ㜾ఄ⌟㇗௧㜾ࡡሣᛮንᙟࢅ ⮤ິ㌬ࣂࢾࣜࡡᯀᮞᩩࡡࣈࣝࢪຊᕝ࡞࠽ ྱࡳᵋᠺᘟ࡛ࡊ࡙ࡢࠉࡥࡍࡲቌฦ⌦ㄵ࡞ᇱࡘࡂ ࠷࡙ࡢࠉࡊࢂࡷ◒࡛᩷࠷ࡖࡒᠺᙟⰃ΅ࡡࡒࡴࠉ Prandtl-Reuss ์㸝௧ୖࠉJ2F ์㸞࡛ࠉධࡥࡍࡲ⌦ ࢨ࣐࣭ࣖࣝࢨࣘࣤᢇ⾙࡞ࡻࡾῼ⾔ࢂࡿ࡙࠷ࡾࠊ ㄵ࡞ᇱࡘࡂ Henchy ንᙟㄵࡡቌฦᙟ㸝௧ୖࠉJ2D ์㸞 ◒᩷⌟㇗ࡡⰃࡂ▩ࡼࡿࡒౚ࡚࠵ࡾࠉFig. 1 ࡞♟ࡌ ▩ࡼࡿ࡙࠷ࡾ (1) ࠊୌ⯙Ⓩᙆሣᛮゆᯊ࡞ࡢࠉㇿ ༟㍀ᘤᘿንᙟࢅ⩻࠻ࡾ࡛ࠉୌᵕንᙟ㸝Fig. 1(a)㸞ࡡ Ⲭ࣬㝎Ⲭࡡᇱ‵᪺☔࡚࠵ࡾ࡛ࡊ࡙ J2F ์⏕࠷ࡼ ᚃࠉᯀᖕ᪁ྡྷ࡞ࡂࡦࡿฝࡌᣉᩋࡂࡦࡿ㸝Fig. 1(b)㸞ࠉ ࡿ࡙࠷ࡾࠉᠺᙟ㝀⏲ῼ࡞ኣ⏕ࡈࡿ࡙࠷ࡾ ᑻᡜⓏ࡞ንᙟ㞗୯ࡌࡾᑻᡜࡂࡦࡿ㸝Fig. 1(c)㸞 Stören-Rice ⌦ㄵ(2)㸝௧ୖࠉS-R ⌦ㄵ㸞࡞ࡢ J2D ์ ⏍ࡋࡒᚃࠉࡂࡊ࡙ฦ㞫࣬◒᩷㸝Fig. 1(d)㸞 ⏕࠷ࡼࡿࠉJ2F ์ࡻࡽᐁ㥺࡞㎾࠷⤎ᯕࢅ࠻ࡾࡆ ⏍ࡋࡾࠊࡆࡿࡼᠺᙟ㝀⏲࡞ᑊࡌࡾῼࢨ࣐࣭ࣖࣝࢨ ࡛ሒ࿈ࡈࡿ࡙࠷ࡾ(3), ࣘࣤᢇ⾙ࡡ㧏ᗐࡡࡒࡴ࡞ࡢࠉຊᕝᑊ㇗࡛ࡾᮞᩩ ์࡛ J2D ์ࢅࡐࡿࡑࡿ๙ሣᛮᙟᘟ࡚ᘟ(1)࡛ᘟ(2)࡞ ࡡንᙟᣪິࢅ⢥ᗐⰃࡂ⾪⌟࡚ࡀࡾᮞᩩ࣓ࢸࣜࠉࡌ ♟ࡊࠉẒ㍉ࡊ࡙ࡲࡾࠊ ࢂࡔሣᛮᵋᠺᘟᚪこ࡚࠵ࡾࠊ J2F ์㸯 ᵋᠺᘟ࡛ࡢࠉᮞᩩ࡞⏍ࡋࡾᚺງ࡛ࡥࡍࡲࢅ㛭౿ࡘ ࡄࡾᘟ࡚࠵ࡽࠉᙆᛮᵋᠺᘟ࡛ࡊ࡙ Hooke ์Ⰳࡂ (4)ࠊࡆࡡこᅄ࡞ࡗ࠷࡙ࠉJ2F J2D ์㸯 ࡆࡆ࡚ࠉ ࡢሣᛮࡥࡍࡲቌฦࢷࣤࢮࣜࠉ ࡛ ࡢ ⌟೩ᕣᚺງࢷࣤࢮ࡛ࣜࡐࡡቌฦࢷࣤࢮ࡚ࣜ࠵ࡽࠉࡐ ࡡࡢࢪ࣭࢜ࣚ㔖࡚ࠉ ࡛ ࢅࠉ ࡢẒౚ౿ᩐࠉ ࡛ ࡢ┞ᙔᚺງ࡛ࡐࡡቌฦ ࡢࠉ┞ᙔᚺງ-┞ᙔሣᛮ ࡥࡍࡲ᭜⥲ࡡࠉࡐࡿࡑࡿࢬ࢜ࣤࢹ౿ᩐ࡛⥲౿ᩐࢅ ⾪ࡌࠊሣᛮࡥࡍࡲቌฦ᪁ྡྷࡡࡲࢅ⩻࠻ࡒሔྙࠉᘟ(1) ࡡ J2F ์ࡢ⌟೩ᕣᚺງ࡛ሣᛮࡥࡍࡲቌฦࡡ㛭౿ᘟ ࡚࠵ࡽࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷࡢ⌟೩ᕣᚺງ᪁ྡྷ࡞ୌ ⩇Ⓩ࡞Ửᏽࡈࡿࠉᚃ⤾ࡡᚺງቌฦ᪁ྡྷ࡞ࡢධࡂ౪Ꮛ Fig. 1 Illustration as deformation of thin sheet metal with uni-axial tension. ࡊ࠷ࠊᑊࡊ࡙ࠉ々㞟࡚ࡢ࠵ࡾࠉᘟ(2)ࡡ J2D ์ ࡢࠉ➠㡧࡞♟ࡈࡿࡾࡻ࠹࡞ᚃ⤾ࡡᚺງቌฦ࡛ࡡ㛭 10 ౿ࢅࡵྱࢆ࡚࠽ࡽࠉቌฦ᪁ྡྷ᛬ንࡊࡒሔྙ࡞ࡵ㏛ ࣭✒㸞ࠊ 㝮࡚ࡀࡾࡆ࡛ࢂ࠾ࡾࠊࡌࢂࡔࠉFig. 1(c)࡞♟ࡌ ࡻ࠹࡞ࠉୌᵕንᙟ᪁ྡྷ࠾ࡼᑻᡜࡎࢆ᩷ᖈ᪁ྡྷࠉን ࡆࡆ࡚ࠉ ࡢ೩ᕣᚺງቌฦࡡ㡨᪁ྡྷ ᙟ᛬ንࡌࡾࡻ࠹◒᩷⌟㇗ࡡῼ࡞ࡢࠉᚃ⤾ࡡᚺ ࡚࠵ࡽࠉ ງቌฦ᪁ྡྷࢅ⩻៎ࡌࡾࡆ࡛᭯ຝ࡚࠵ࡾ࡛⩻࠻ࡼ ྡྷࢅࠉ ࡿࡾࠊࡐࡆ࡚ࠉሣᛮࡥࡍࡲቌฦ࡞ᚺງቌฦ᪁ྡྷ౪Ꮛ ඔࣂ࣒࣭ࣚࢰ࡚ࠉࡐࡿࡑࡿ ࡡ㛭ᩐ࡚࠵ࡾࠊ ࠽ࡻ ᛮࢅ⩻៎ࡊࡒሣᛮᵋᠺᘟ࠷ࡂࡗ࠾ᥞ᱄ࡈࡿࡒ (5)㹳 ࡦ (7)ࠊᮇ✇࡚ࡢࠉࡐࡡୌࡗ࡚࠵ࡾ࿇ᒁ-⸠์(8), (9)ࡡᴣ ᣲࡿぽ࡛ࡊ࡙ḗᘟ࡞ࡻࡽᏽ⩇ࡈࡿࡾࠊ こ࡞ࡗ࠷࡙㏑ࠉࡈࡼ࡞ⴥ⩽ࡼᵋᠺᘟࡡጂᙔᛮࢅ ᳠ゞࡌࡾࡒࡴ࡞⿿షࡊ࡙࠷ࡾࠉࣀࢺࣞࣆ࢚࣭࣑ᠺ ᙟム㥺ᶭ࡞ࡻࡾᐁチム㥺⨠࡞ࡗ࠷࡙⣺ࡌࡾࠊ 㡨᪁ྡྷ ࡢ㡨᪁ྡྷ ࠾ࡼࡡᣲࡿぽ ࠾ࡼࡡሣᛮࡥࡍࡲቌฦ᪁ ࡢሣᛮࡥࡍࡲቌฦࡡኬࡀࡈࢅ⾪ࡌ↋ḗ ࡢࠉFig. 2 ࠾ࡼࢂ࠾ࡾࡻ࠹࡞㡨᪁ྡྷ | | cos ࠉ ࡞࠽ࡄࡾຊᕝ◫౿ᩐ | | ࠾ࡼࡡ ࠽ࡻࡦ ࡞ࠉ ௧ୖࡡ㛭౿ࢅᏽ⩇ࡌࡾࠊ 㸧㸣࿇ᒁ⸠࡞ࡻࡾᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺ ᘟ ᅸᘇຊᕝ࡞ࡻࡖ࡙⿿㏸ࡈࡿࡾᯀᮞᩩࡢࠉᅸᘇ᪁ ྡྷ࡛ࡐࡡ├ぽ᪁ྡྷ࡚ࠉୌ⯙Ⓩ࡞㜾ఄᚺງࡡᶭ | | | | ౿ᩐ ࠽ࡻࡦ ࡢḗᘟ࡚⾪ࡌࡆ࡛࡚ࡀࡾࠊ ࠉ ᱄ࡈࡿࡒᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ (8) ࡢ➴᪁ᛮ ᑊࡊ࡙ᣉᘿࡈࡿࡒ (9)ࠊIlyushin ࡡ೩ᕣᚺງ✭ 㛣࡞࠽ࡄࡾᴣᛍᅒࢅ Fig. 2 ࡞♟ࡌࠊ ᘟ(4)࠽ࡻࡦ(5)ࢅᘟ(3)࡞ᑙථࡌࡾࡆ࡛࡞ࡻࡽࠉ㔔ࡲ Ⓩᛮ㈹␏ࡾ␏᪁ᛮࢅ♟ࡌ(10)ࠊ࿇ᒁࡼ࡞ࡻࡖ࡙ᥞ ᮞᩩ࡞ࡻࡾࡵࡡ࡚࠵ࡖࡒࠉࡐࡡᚃࠉ␏᪁ᛮᮞᩩ࡞ ᥞ᱄ࡈࡿ࡙࠷ࡾᘟ(3)ࢅࢨ࣐࣭ࣖࣝࢨ࡚ࣘࣤ⏕࠷ࡾ ࡒࡴ࡞ࡢࠉ㡨᪁ྡྷ ࢅරమⓏ࡞Ửᏽࡌࡾᚪこ࠵ ࡾࠊᮇᮮࠉ㡨᪁ྡྷ ࢅỬᏽࡌࡾࡒࡴ࡞ࡢࠉ௴ណࡡ ᚺງ≟ឺ࡞ᑊࡊ࡙ᵕࠍ᪁ྡྷ࡞ᚺງቌฦࢅ࠻ࠉሣ ᛮࡥࡍࡲቌฦ᪁ྡྷ࡛ୌ⮬ࡌࡾ᪁ྡྷࢅῼᏽࡌࡾᐁ㥺 ࢅኣᩐ⾔ࢂࡂ࡙ࡢࡼࡍࠉ⌟ᐁⓏ࡚ࡢ࠷ࠊࡐࡆ ࡚ࠉⴥ⩽ࡼࡢࠉ௴ណࡡᚺງ≟ឺ࡞ᑊࡊ࡙㡨᪁ྡྷ ࢅ ぜᏽࡌࡾ㛭ᩐᙟࢅࠉ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ ࡛ ᩺ࡒ࡞ᏽ⩇ࡊࠉ௴ណࡡࢪ࣭࢜ࣚ㔖 ࢅ⏕࠷࡙ḗᘟ࡚ ᏽ⩇ࡊࡒ(11)ࠊ Fig.2 Geometrical relationship among yield locus in Ilyushin’s stress space, stress increment, plastic strain increment and measured angle at natural direction vector. 㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ ࡡරమⓏ㛭ᩐᙟ࡛ ࡊ࡙ࠉ᭩ࡵ⠾༟࡞␏᪁ᛮࢅ⾪⌟࡚ࡀࡾ Hill ࡡᚺງ ࡡḗᙟᘟࢅ⏕࠷ࠉᖲ㟻ᚺງ≟ឺࢅ௫ᏽࡌࡾ࡛ḗᘟ ࡚⾪⌟ࡈࡿࡾࠊ ௴ណࡡㇿⲬⅤ࡞ᑊࡊࠉ೩ᕣᚺງቌฦ ࡞ᏽࡱࡾሣᛮࡥࡍࡲቌฦ ࡞࠽࠷࡙ࠉࡐࡿࡼࡡ᪁ ྡྷୌ⮬ࡌࡾ᪁ྡྷࢅ㡨᪁ྡྷ ࡊ࡙ᏽ⩇ࡌࡾࠊ㡨᪁ྡྷ 11 ࠾ࡼୌណ 㸝ኬࡀࡈ| ࡛೩ᕣᚺງቌฦ | 㸞࡛ ᘟ(8)ࡢ Hill ࡡḗ␏᪁ᛮ㜾ఄ㛭ᩐ࡛ྜྷᙟᘟ࡚࠵ࡾ ࡞ࡻࡾ ࠉࡐࡿ࡛༇ืࡌࡾࡒࡴ࡞ࠉ࠵࠻࡙ᚺງࡡḗᙟᘟ ᐞࡡࠔ㔔ࡲࡀᖲᆍࠕ࡛ࡊ࡙ࠉ␏᪁ᛮᮞᩩ࡞ᑊࡊ ࡛ࡩࡆ࡛࡞ࡌࡾࠊ ࡙ᣉᘿࡈࡿࡒ࿇ᒁࡼ࡞ࡻࡾᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮ 㜾ఄ㛭ᩐ࡛ྜྷᵕ࡞ࠉLankford ࡡ r ೋ㸝=ᯀᖕࡥࡍࡲ ᵋᠺᘟࡢḗᘟࡡࡻ࠹࡞ᵋ⠇ࡈࡿࡾ㸝ࠔ㸯ࠕࡢࢪ࢜ࣚ 㸤ᯀཉࡥࡍࡲ㸞࠾ࡼỬᏽࡈࡿࡾ౿ᩐ࡚࠵ࡾࠊ⌟ᚺງ 㹳 ࡢࠉHill ࡡḗ␏᪁ᛮ フジコー技報−tsukuru No. ࢷࣤࢮ࡛ࣜሣᛮࡥࡍࡲቌฦࢷࣤࢮࣜࢅ⤎ࡦࡗࡄࡾ 㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜࡡࡻ࠹᩺ࡒ࣎ࢷࣤࢨࣔࣜ 㛭౿࡛ࡊ࡙ࠉ㜾ఄ㛭ᩐࢅሣᛮ࣎ࢷࣤࢨ࡛ࣔࣜࡊࠉ㜾 㛭ᩐࢅ⏕࠷ࡾࡆ࡛᭯⏕࡚࠵ࡾࡆ࡛᳠ゞࡈࡿ࡙ ఄ㛭ᩐ㸝᭜⥲㸞࡞ᑊࡊ࡙ࡐࡡἪ⥲᪁ྡྷሣᛮࡥࡍࡲ ࠷ࡾࠊᘟ(3)࡚♟ࡈࡿࡾ࿇ᒁ-⸠࡞ࡻࡾᚺງቌฦ᪁ ቌฦ᪁ྡྷࢅ♟ࡌ⩻࠻᪁ࢅ㛭㏻Ὦࡿ์(1)࡛࠷࠹ࠉ㡨 ྡྷ౪Ꮛሣᛮᵋᠺᘟࡢࠉ㡨᪁ྡྷ࡛㜾ఄ㛭ᩐࢅวࡽ㞫ࡊ ᪁ྡྷ ࡙ビ౮࡚ࡀࡾࡆ࡛࠾ࡼࠉᠺᙟ㝀⏲ῼ࡞ᑊࡊ࡙ ࡛㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ ࡛ࡡ㛭౿ࡢࠉ 㛭㏻Ὦࡿ์࡛ྜྷᵕࡡ⩻࠻᪁࡚࠵ࡾࠊࡌࢂࡔࠉ㡨᪁ ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ ⢥ᗐྡྷ୕ᙽ࡚ࡀࡾ࡛⩻࠻ࡼࡿࡾࠊ ࡡἪ⥲᪁ྡྷ࡛ࡊ࡙ࠉ㡨᪁ྡྷ ぜᏽࡈࡿࡾࡆ࡛࡞ࡾࠊࡔࡲ࡞ᘟ(1)ࡢࠉ Mises ࡡ㜾ఄ㛭ᩐ ࢅሣᛮ࣎ࢷࣤࢨ࡛ࣔࣜࡌࡾ 㛭㏻Ὦࡿ์ ࡚࠵ࡾࡆ࡛▩ࡼࡿ࡙࠷ ࡾ(1)ࠊ 㸨㸣ሣᛮᵋᠺᘟࡡᐁ㥺ビ౮ࢨࢪࢷ࣑ࡡᵋ⠇ ᐁ㥺࡞ࡻࡖ࡙ᵋᠺᘟࡡ⢥ᗐビ౮ࡷጂᙔᛮࡡ᳠ゞ ࢅ⾔࠹ࡒࡴ࡞ࡢࠉム㥺∞࡞ᑊࡊ࡙ࠉ௴ណࡡንᙟ⤊㊨ ࡷㇿⲬ⤊㊨ࢅ࠻ࡼࡿࡾࡻ࠹⨠ᚪこ࡚࠵ࡾࠊ ࡐࡆ࡚ⴥ⩽ࡼࡢࠉළ⟮ᮞ࡞හᅸㇿⲬ࡛㍀ງࢅ࠻ࡾ ୌ᪁ࠉྚ⏛ࡼ(9)ࡢࠉ➱⩽ࡼ࡛ืࡡ⩻࠻᪁࠾ࡼ㡨᪁ ࡆ࡛࡞ࡻࡽࠉළ࿔᪁ྡྷ࡛㍀᪁ྡྷࡡ᪁ྡྷንᙟྊ⬗ ࢅᏽ⩇ࡊ࡙࠷ࡾࠊᘟ(3)㹳(6)ࢅᖼరᏕⓏ㛭౿ ࣀࢺࣞࣆ࢚࣭࣑ᠺᙟ㸝௧ୖࠉHF ᠺᙟ㸞ࢅ⏕࠷ ྡྷ ࠾ࡼᩒ⌦ࡌࡾࡆ࡛࡞ࡻࡖ࡙ࠉ௴ណࡡࢪ࣭࢜ࣚ㔖λࠉ ࡒᵋᠺᘟࡡᐁ㥺ビ౮ࢨࢪࢷ࣑ࢅࠉโᚒ⣌࠽ࡻࡦ゛ῼ ሣᛮࡥࡍࡲቌฦࡡᚺງቌฦ᪁ྡྷ౪Ꮛᛮࢅ⾪ࡌ᩺ࡒ ⣌ࡡ⿿ష࡞ࡻࡖ࡙ᵋ⠇ࡊ࡙࠷ࡾࠊHF ᠺᙟ࡛ࡢࠉ㔘 ࣂ࣒࣭ࣚࢰ ᆵහഁ࡞ࢬࢴࢹࡊࡒ⟮ᮞ࡞㧏ᅸࡡᾦమࢅὮථࡈࡎ ࢅᑙථࡊࠉḗᘟ࡚⾪⌟ࡊ࡙࠷ࡾࠊ ࡗࡗࠉ㍀᪁ྡྷࡡንనࢅ࠻ࡾࡆ࡛࡚ࠉ⟮ᮞࢅ㔘ᆵහ ഁᙟ≟࡞ೊࢂࡎ࡙ᣉⳚ࣬ᠺᙟࡌࡾຊᕝ᪁Ἢ(14)࡚ࠉ㎾ ᖳࠉ⮤ິ㌬➴ࡡࣆ࣭࣑ࣝ⿿ష࡞⏕࠷ࡼࡿ࡙࠷ࡾࠊ ࡆࡆ࡚ࠉᘟ(9)ࡡ➠ୌ㡧┘ࡢࠉᘟ(10)࡞♟ࡌ Hill ࡡ ᮇ✇࡚ࡢࠉ⌟ᅹ⿿ష୯࡚࠵ࡾࠉ⏤ാฌ⌦ࢅ⏕࠷ࡒ ḗ␏᪁ᛮ㜾ఄ㛭ᩐࢅ࣎ࢷࣤࢨ࡛ࣔࣜࡌࡾᚉᮮࡡ HF ᠺᙟᐁ㥺ࡡ゛ῼἪ࡞ࡗ࠷࡙᳠ゞࡊࡒࡡ࡚⣺ 㛭㏻Ὦࡿ์࡛ྜྷᵕࡡ㛭౿ࢅᑙථࡊ࡙࠷ࡾࠉࣂ࣒ࣚ ࡌࡾࠊ ࣭ࢰ ࡞ࡗ࠷࡙ࡵ᳠ゞࢅこࡌࡾ㛭౿࡛ࡖ࡙࠷ࡾࠊ ࡆࡿࡼᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ࡞࠽ࡄࡾࣂࣚ ࣒࣭ࢰࠉ㡨᪁ྡྷ࠽ࡻࡦ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ࡞㛭ࡊ࡙ ᳠ゞࢅ⾔࠹ࡒࡴ࡞ࠉᐁ㥺࡞ࢂࡾ᪁Ἢ࡛ࡊ࡙ࠉࣜ ࣐ࢼ࣑ᮞࢅᏽࡊࠉ⤎ᬏሣᛮ⌦ㄵ࡞ᇱࡘࡂ᭯㝀こ ⣪ኣ⤎ᬏమ࣓ࢸࣜ(12)࡞ࡻࡾゆᯊ⤎ᯕࢅฺ⏕ࡊࡒᩐ ೋᐁ㥺ࢅ⾔ࡖࡒࠊリ⣵࡞ࡗ࠷࡙ࡢᩝ⊡(11)ࢅཤ↯࠷ ࡒࡓࡀࡒ࠷ࠉᮞᩩࡡࡵࡗ␏᪁ᛮࡡᙫ㡢࡞ࡻࡽ㡨᪁ ྡྷิㇿⲬ᪁ྡྷ࠾ࡼ⛛ິࡌࡾࡆ࡛♟ࡈࡿࠉሣᛮ ࡥࡍࡲቌฦ᪁ྡྷ ⾪ࡌ ࡛ࠉሣᛮࡥࡍࡲቌฦࡡኬࡀࡈࢅ ࡡ㛭ᩐᙟ᳠ゞࡈࡿ࡙࠷ࡾࠊຊ࠻࡙ࠉ㡨᪁ ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ ࠉ㡨᪁ྡྷ ࢅぜᏽࡌࡾ Fig.3 Illustration as experimental apparatus 㛭ᩐ࡛ࡊ࡙ࡡᙲࢅᢰ࠻࡙࠷ࡾࡆ࡛᳠ゞࡈࡿ࡙ ࠷ࡾࠊࡱࡒࠉ㛭㏻Ὦࡿ์࡞ᑊࡊ࡙ࡢࠉ␏᪁ᛮ㜾ఄ㛭 HF ᠺᙟᐁ㥺⨠ࡡᴣこࢅ Fig. 3 ࡞♟ࡌࠊᐁ㥺 ᩐࡡ౿ᩐࢅ㜾ఄᚺງࡡࡲ࡚Ửᏽࡊࡒሔྙ࡞ࡢࠉሣᛮ ⨠ࡢᘤᘿᅸ⦨ム㥺ᶭࢅ࣭࣊ࢪ࡛ࡊ࡙࠽ࡽࠉࢦ࣭࣍ࢨ ࣎ࢷࣤࢨ࡛ࣔࣜࡊ࡙ࡡᙲࢅᯕࡒࡌࡆ࡛ࡢ࡚ࡀࡍࠉ ࣛࣤࢱ࣎ࣤࣈ࡞࡙ᶭἔࢅ㏞ࡾࡆ࡛࡚හᅸㇿⲬࢅ ࠵ࡂࡱ࡚ㇿⲬ࣬㝎Ⲭࡡึ᩷ᇱ‵࡛ࡾࡵࡡ࡚㜾ఄึ ࠻ࠉム㥺ᶭᮇమ࡞ཱིࡽࡄࡒࢦ࣭࣓࣭࣍ࢰࡻࡖ࡙ ᏽࡡࡲ࡞⏕࠷ࡾࡆ࡛࡛ࡊࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷࡢࠉ ㍀ንనࢅ࠻ࡾᵋ㏸࡛ࡖ࡙࠷ࡾࠊム㥺∞࡞ࡢࠉิ 12 =210[mm] ࠉ ⫏ ཉ ࡼḗඔⓏ࡞ንࡌࡾ゛ῼⅤࢅᤍᤂࡊࠉᐁᗑᵾ =2[mm]ࡡ⣟࣐ࣜࢼ࣑ᘤᢜළ⟮㸝A1070㸞ࢅ ḿࡌࡾ᪁Ἢࢅ⤄ࡲ㎰ࢆࡓࡥࡍࡲ⟤ฝࣈࣞࢡ࣑ࣚࢅ ⏕ࡊࠉࡐࡡ⾪㟻࡞ࡢ Fig. 4 ࡞♟ࡌ࣏࣭࢞ࣤࢡࢅ᪃ షᠺࡊࠉࡐࡡ᳠ゞࢅ⾔ࡖࡒࠊࡔࡲ࡞ୌ⯙Ⓩ CCD ࡊ࡙࠵ࡾࠊහഁ࡛አഁࡡ 4 Ⅴ࡚᭯㝀こ⣪Ἢ࡞࠽ࡄ ࣒࢜ࣚࡡ 1 ⏤⣪ࡢ㛏᪁ᙟ࡚࠵ࡾࡒࡴࠉ⦢ᶋࡡḿ⢥ ࡾᄿ⟿Ⅴࢮࣂ࣒ࣚࢹࣛࢴࢠこ⣪࡛ࡲࡌࡆ࡛ ᗐ࡞ᕣฝࡾࠉ⏕࠷ࡒ NI ♣ࡡ CCD ࣒࢜ࣚࡢ 1 ࡚ࠉ㍀᪁ྡྷࡥࡍࡲ࠽ࡻࡦළ࿔᪁ྡྷࡥࡍࡲࢅ⟤ฝࡌࡾࠊ ⏤⣪ḿ᪁ᙟ࡚࠵ࡾࡒࡴࠉࡐࡡࡻ࠹ᕣࡢ↋࠷࡛࠷ โ ᚒ ࣬ ゛ ῼ ࡢ ࠉ Fig. 5 ࡞ ♟ ࡌ NATIONAL ࠹≁ᚡࢅ᭯ࡊ࡙࠷ࡾࠊ⟤ฝࡈࡿࡒළ࿔᪁ྡྷࡥࡍࡲ INSTRUMENTS ♣㸝௧ୖࠉNI ♣㸞ࡡ LabVIEW ࠾ࡼịࡴࡒム㥺∞├ᙼ࡛ࠉᐁ㝷ࡡ゛ῼೋ࡛ࡡẒ㍉ࢅ ࠽ࡻࡦࡐࡡࣀ࣭ࢺ࢙⩄࡞ࡻࡖ࡙ᵋᠺࡈࡿࡾࠊࡥ Fig. 6 ࡞♟ࡌࠊ⦢㍀࡞ム㥺∞├ᙼ d [mm]ࢅࠉᶋ㍀ ࡍࡲࡢࠉCCD ࣒࢜ࣚ࠾ࡼᚋࡼࡿࡾ⏤ാࢅ NI ♣ࡡ ࡞ࡢᐁ㥺࡞こࡊࡒ㛣ࢅ 1 ࡛ࡊࡒྙࢅ⾪ࡊ࡙࠷ Vision Assistant ࢅ⏕࠷࡙ฌ⌦ࡊࡒ⤎ᯕ࠾ࡼ⟤ฝࡊࠉ ࡾࠊᐁ゛ῼࡢࠉ⟤ฝࡈࡿࡒ ᚺງࡢࠉᅸງንᥦჹࠉⲬ㔔ንᥦჹ࠽ࡻࡦࣛࢼࢣ࣭ 0.19 ࡡ࡞ࠉࡐࡿࡑࡿහᅸࢅୌ㝎Ⲭࡊ࡙⾔ࡖࡒࠊ ࢩ࠾ࡼᚋࡼࡿࡾ゛ῼೋࢅ⏕࠷࡙⟤ฝࡊ࡙࠷ࡾࠊ ළ࿔᪁ྡྷࡥࡍࡲ ࠾ࡼ⟤ฝࡊࡒム㥺∞├ᙼࡢࠉᐁ አᙼ =32[mm] ࠉ 㛏 ࡈ 0.05ࠉ0.1ࠉ0.15ࠉ ῼೋ࡛ୌ⮬ࡊ࡙࠷ࡾࡆ࡛࠾ࡼࠉషᠺࡊࡒࡥࡍࡲ ⟤ฝࣈࣞࢡ࣑ࣚࡢࠉᣉ⟮࡞ࡻࡾⅤ㊝㞫ንࡡḿ Fig.4 Making pattern on a specimen Specimen diameter d [mm] ࠹ࡱࡂ⾔ࢂࡿ࡙࠷ࡾ࡛࠷࠻ࡾࠊ 40 Calculated (by εθ) Actual measurement 38 36 34 32 0.0 Initial diameter d0=32.0[mm] 0.2 0.4 0.6 0.8 1.0 Time rate Fig.6 Comparison of specimen diameter d calculated by circumferential strain and actual measurement Fig.5 Measurement and Control Systems for experimental apparatus 㸩㸣ᐁ㥺ビ౮ࢨࢪࢷ࣑ࡡ⏤ാฌ⌦࡞ࡻࡾ゛ῼ⤎ᯕ ࡛⢥ᗐ ᐁ㥺୯ࠉFig. 4 ࡞♟ࡌࠉ᭜⋙ࢅᣚࡗム㥺∞⾪㟻࡞ ᪃ࡈࡿࡒⅤࡡንనࢅ⏤ാฌ⌦࡞ࡻࡖ࡙゛ῼࡊ࡙࠷ ࡾࠉහᅸㇿⲬ࡞ఔ࠹ᣉ⟮ࡡࡒࡴࠉᅖᏽࡊ࡙࠷ࡾ CCD ࣒࡛࢜ࣚム㥺∞⾪㟻࡛ࡡ㊝㞫㎾ࡂࡽࠉ Ⅴ㊝㞫ንࡌࡾࠊࡐࡆ࡚ࠉ᧔ࡼࡿࡒḗඔ⏤ാ࠾ 13 ḗ࡞ࠉᚺງ⟤ฝ࡞ᚪこ᭜⋙༖ᙼࡡ゛ῼࢅࠉよ ᘟࡡࣛࢼࢣ࣭ࢩ࠾ࡼ CCD ࣒࢜ࣚ࡞ࡻࡾ⏤ാฌ⌦ ࡞ࡻࡾ⟤ฝἪ࡞ን᭞ࡊࡒࡡ࡚ࠉࡐࡡ⢥ᗐ࡞ࡗ࠷࡙᳠ ゞࢅ⾔ࡖࡒࠊࣛࢼࢣ࣭ࢩ࡞ࡻࡾ゛ῼ࡚ࡢࠉム㥺∞ ࿔ᅑ࡚ࡡ⨠ࡡཱིࡽᅂࡊᝇࡂࠉࡱࡒ◒᩷ࡊࡒ㝷࡞ ᅸງ፳మࣛࢼࢣ࣭ࢩ࡞࠾࠾ࡾࡆ࡛⩻࠻ࡼࡿ ࡾࠊࡈࡼ࡞㍀ንనࢅ࠻ࡒ㝷ࠉム㥺∞ࢲࣔࢴࢠ㒂࡞ よࡌࡾࡆ࡛ᏽࡈࡿࡾࡒࡴࠉ⏤ാฌ⌦゛ῼ࡞ን ᭞ࡌࡾࡆ࡛ࢅ⩻࠻ࡒࠊ⏤ാ゛ῼⅤࡢࠉࣛࢼࢣ࣭ࢩ ࡛ྜྷࡋ 3 Ⅴࡡన⨠࡛ࠉࡐࡡ㛣࡞ 2 Ⅴࢅ㏛ຊࡊࡒ゛ 5 フジコー技報−tsukuru No. Ⅴ࡚࠵ࡽࠉ⏤ാฌ⌦ࢅ⏕࠷࡙゛ῼࡊࡒࡆࡿࡼࡡⅤࡡ ࣒࢜ࣚࡡన⨠ࠉ゛ῼⅤᩐࢅቌࡷࡌ➴ࡡ᳠ゞㄚ㢗ࢅṟ ንన㔖࡞ࡗ࠷࡙ࠉ᭩ᑚἪࢅ⏕࠷࡙⟤ฝࡊࡒࠊࡐ ࡌ⤎ᯕ࡛ࡖࡒࠊ ࡿࡑࡿ⟤ฝࡈࡿࡒ᭜⋙༖ᙼࡡẒ㍉ࢅ Fig. 7 ࡞♟ࡌࠊ Radius of Curvature R [mm] 1000 ᮇ⨠ࡡ┘ᵾࡢࠉ゛ῼ⤎ᯕ࡞ࡻࡽᚋࡼࡿࡒ㍀᪁ྡྷ ࡥࡍࡲ࠽ࡻࡦළ࿔᪁ྡྷࡥࡍࡲࢅࠉሣᛮᵋᠺᘟࡡ㐲⏕ Linear gauge Processing image ࡈࡿࡒᠺᙟ㝀⏲⥲ᅒ (15) 㸝 Forming Limit Diagram㸯௧ୖࠉFLD㸞࡛Ẓ㍉ࡊࠉᵋᠺᘟࡡጂᙔᛮ 800 ࡡビ౮࠽ࡻࡦ᳠ゞ࡞⏕࠷ࡾࡆ࡛࡚࠵ࡾࠊࡊ࠾ࡊࠉࢨ 600 ࢪࢷ࣑⿿ష୯ࡡࡒࡴࠉFLD ࡢࠉୌ⯙Ⓩ◒᩷㝀⏲ ῼ࡞ኣ⏕ࡈࡿ࡙࠷ࡾ S-R ⌦ㄵࢅ⏕࠷࡙᳠ゞࢅ⾔ 400 ࡖࡒࠊࡈࡼ࡞ࠉ⿿ష୯ࡡࢨࢪࢷ࣑࡚ࡢ㍀ງ㛭౿ࡡโ ᚒᮅᏰᠺࡡࡒࡴࠉහᅸㇿⲬࡡࡲࡡᐁ㥺⤎ᯕ࡚࠵ࡾࠊ ࣏࣭࢞ࣤࢡࡡහഁࡡ 4 Ⅴࢅ⏕࠷࡙⟤ฝࡊࡒࡥࡍࡲ 200 ೋࢅ Inner ࡛ࡊࠉአഁࢅ⏕࠷࡙⟤ฝࡊࡒࡵࡡࢅ 0 0.7 0.8 0.9 1.0 Time rate Fig.7 Comparison of radius of curvature R calculated by linear gauge and processing image Outer ࡛ࡊ࡙ࠉࡐࡿࡼࡡࡥࡍࡲᒓṌࢅ Fig. 8 ࡞♟ࡌࠊ ⦢㍀࡞㍀᪁ྡྷࡥࡍࡲ ࠉᶋ㍀࡞ළ࿔᪁ྡྷࡥࡍࡲ ࡛ ࡊ࡙࠷ࡾࠊහᅸㇿⲬࡊ࠾࠻࡙࠷࠷ࡒࡴࠉࡥࡍࡲ ᒓṌࡢළ࿔᪁ྡྷࡥࡍࡲ ㍀୕࡞㐼⛛ࡊ࡙࠷ࡾࡆ࡛ ࢂ࠾ࡾࠊᯗහࡡᣉኬᅒࢅずࡾ࡛ࠉළ࿔᪁ྡྷࡥࡍࡲ ᐁ㥺㛜ጙ࠾ࡼ༖ࡣࡱ࡚ࡢࠉ゛ῼೋẒ㍉ⓏᏭᏽࡊ࡙ =0.15 ㎾࠾ࡼ㍀᪁ྡྷࡥࡍࡲ ኬࡀࡂࡖ࡙࠷ ࠷࠾ࡖࡒࡒࡴࠉᐁ㥺ᚃ༖࡞ࡗ࠷࡙᳠ゞࢅ⾔ࡖ࡙࠷ ࡾࠉࡆࡿࡢහᅸㇿⲬ࡞ࡻࡾᣉ⟮ࡡ㝷ࠉንᙟࡢළ࿔ ࡾࠊ⏤ാฌ⌦࠾ࡼ⟤ฝࡈࡿࡒ᭜⋙༖ᙼ࡞⣵࠾࠷୕ୖ ᪁ྡྷࡡࡲࡓࡄ࡚ࡂࠉ᭜⋙ࢅᣚࡗࡆ࡛࡚㍀᪁ྡྷ࡞ᘤ ንິずࡼࡿࡾࠉࡆࡿࡢᐁ㥺㛜ጙ࡞Ửᏽࡊࡒ ᘿንᙟ⏍ࡋࡾࡒࡴ࡚࠵ࡾࠊ㍀ንనࢅโᚒ࡚ࡀࡾࡻ ೋฌ⌦࡞⏕࠷ࡾ⏤ാࡊࡀ࠷ೋࠉᐄአ࠾ࡼࡡක㔖 ࠹࡞ࢨࢪࢷ࣑ᵋ⠇࡚ࡀࡿࡣࠉᖲ㟻ࡥࡍࡲ≟ឺ࡚ࡡ ን࡞ࡻࡖ࡙㐲ḿೋ࡚ࡂࡾࡆ࡛ཋᅄ࡛ࡊ࡙ ንᙟࢅ⥌ᣚࡌࡾࡆ࡛ྊ⬗࡛ࡽࠉࡥࡍࡲᒓṌࡢළ ⩻࠻ࡼࡿࡾࠊࡊ࠾ࡊࠉ⏤ാฌ⌦࠾ࡼ᭩ᑚἪ࡞ࡻ ࿔᪁ྡྷࡥࡍࡲ ࡖ࡙ᚋࡼࡿࡒ᭜⋙༖ᙼࡢࠉࣛࢼࢣ࣭ࢩ࡛ྜྷᵕࡡല S-R ⌦ㄵ࡛ࡡᕣ࡞㛭ࡊ࡙ࡢࠉ⌦ㄵೋࡡ゛⟤࡞ຊᕝ◫ ྡྷࢅ♟ࡊ࡙࠷ࡾࡆ࡛ࢂ࠾ࡾࠊᚃࠉක㔖ࡷࠉCCD ᣞᩐ n=0.26 ࡛ࡊࡒࠉࡆࡿࡢ࣐ࣜࢼ࣑ᮞ࡞ ㍀୕ฦᕱࡌࡾࡆ࡛ྊ⬗࡛ࡾࠊ Fig.8 Strain path diagram with FLD calculated by S-R theory 14 ࠽ࡄࡾୌ⯙Ⓩೋࢅ⏕࠷࡙࠽ࡽࠉᚃࠉᐁ㥺࡚⏕ ࡛ࢅ⣺ࡊࡒࠊᚃࡵࠉ◂✪ᐄᡜᒌᏕ⏍࡛භ࡞ࠉᐁ ࡊ࡙࠷ࡾム㥺∞ࢅ⏕࠷ࡒᘤᘿム㥺➴ࢅ⾔࠹ࡆ࡛࡚ n 㥺ビ౮ࡼࡦ࡞ᠺᙟ㝀⏲ῼࡷᙆሣᛮゆᯊࡡ ೋࢅỬᏽࡊ࡙࠷ࡂᚪこ࠵ࡾࠊࡱࡒࠉS-R ⌦ㄵࡢ➴ ࢨ࣐࣭ࣖࣝࢨࣘࣤࡡ㐲⏕ࢅ┘Ⓩ࡞ᵋᠺᘟ࡞㛭ࡌ ᪁ᛮᮞᩩࢅ๑ᥞ࡛ࡊ࡙࠷ࡾࡒࡴࠉᅂ⏕࠷࡙࠷ࡾム ࡾཱིࡽ⤄ࡲࢅ㐅ࡴࡾࡗࡵࡽ࡚࠵ࡾࠊ 㥺∞ࡡ␏᪁ᛮࡡᙫ㡢࠷࡛ࡢゕ࠻࠷ࠊࡈࡼ࡞ࠉ Fig. 9 ࡞♟ࡌහᅸ p[MPa]࡛ළ࿔᪁ྡྷࡥࡍࡲ ࡛ࡡ ཤ⩻ᩝ⊡ ⥲ᅒ࡚ࡢࠉළ࿔᪁ྡྷࡥࡍࡲ =0.04 ㎾࠾ࡼ◒᩷ࡱ 1) ᚃ⸠ᏟࠉሣᛮᏕࠉࢤࣞࢻ♣ࠉ㸝1982㸞㸣 ࡚ࡡ㛣࡞ࠉහᅸን 7.0㹳7.7[MPa]࡛≻ࡂࠉୌẴ 2) Stören, S. and Rice, J.R., J. Mech. Phys. ࡞ንᙟ㐅ࢆ࡚࠷ࡾࡆ࡛ࢂ࠾ࡾࠊᅂࡢහᅸㇿⲬ ࢅ࠻ࡾᶭἔࡡ㏞ࡽ㔖ୌᏽ㏷ᗐ࡚࠵ࡖࡒࡒࡴࠉ ንᙟ࡛භ࡞㏞ࡽ㔖ࢅโᚒ࡚ࡀࡾࡻ࠹ࢨࢪࢷ࣑ࡡ 1222-1229㸣 (1948). 8 Internal pressure p [MPa] 3) ⸠ ⪨ ୌ ࠉ ሣ ᛮ ࡛ ຊ ᕝ ࠉ 28-323ࠉ㸝 1987 㸞ࠉ 4) Stowell, E.Z., NACA Tech. Note, No.1556, ᵋ⠇ᚪこ࡚࠵ࡾ࡛⩻࠻ࡾࠊ 5) Christoffersen, J. and Hutchinson, J.W., J. Mech. Phys. Solids, 27, (1979), 465-487. 6 6) ᚃ⸠Ꮯࠉᶭㄵ Aࠉ47-424ࠉ 㸝1981㸞ࠉ1389-1397㸣 7) ᚃ⸠Ꮯࠉᶭㄵ Aࠉ48-428ࠉ㸝1982㸞ࠉ493-501㸣 8) ࿇ᒁᏬ❮ࠉ⸠⪨ୌࠉᶭㄵ Aࠉ54-504ࠉ 㸝1988㸞ࠉ 4 1617-1622㸣 2 0 Inner pattern Outer pattern 0.1 0.2 Circumferential strain εθ Fig.9 Internal pressure p - circumferential strain Curves 㸪㸣࠽ࢂࡽ࡞ ᮇ✇࡚ࡢࠉ␏᪁ᛮᮞᩩ࡞㐲⏕ࡈࡿࡒ࿇ᒁ-⸠࡞ ࡻࡖ࡙ᥞ᱄ࡈࡿࡒᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ࡞ ࡗ࠷࡙⣺ࡊࡒࠊ࿇ᒁ-⸠์ࡢࠉཿⓏᵋᠺᘟ࡞ ࡲࡼࡿࡾࠉ㜾ఄ㛭ᩐ࡛ሣᛮࡥࡍࡲቌฦ᪁ྡྷࢅỬᏽࡌ ࡾሣᛮ࣎ࢷࣤࢨࣔࣜ㛭ᩐྜྷୌ࡚࠵ࡾ㛭㏻Ὦࡿ์ ࡛ࡢ␏ࡽࠉ㝎Ⲭ࣬ㇿⲬึᏽࢅ⾔࠹ࡀ㜾ఄ㛭ᩐ࡛ ሣᛮࡥࡍࡲቌฦ᪁ྡྷࢅỬᏽࡌࡾ㛭ᩐ࡛ࢅฦ㞫ࡊ࡙ ビ౮࡚ࡀࡾࡆ࡛࠾ࡼࠉ⤊㊨᛬ንࢅఔ࠹ࡻ࠹ᠺᙟ㝀 ⏲ῼ࡞ᑊࡊ࡙⢥ᗐྡྷ୕ᙽ࡚ࡀࡾࠊࡱࡒࠉ ◂✪ᐄ࡞࡙⿿ష୯ࡡᵋᠺᘟࡡᐁ㥺ビ౮ࢨࢪࢷ࣑࡞ ࡗ࠷࡙ࡢࠉ㍀ງ࡛හᅸㇿⲬࡡโᚒ⣌ࡡᵋ⠇᛬ࡿ ࡾࡵࡡࡡࠉᣉ⟮࡞ఔ࠹Ⅴ㊝㞫ḿᶭ⬗Ὡ࠾ࡈࡿࠉ ⏤ാฌ⌦࡞ࡻࡾ゛ῼ࡛ࡥࡍࡲ⟤ฝྊ⬗࡚࠵ࡾࡆ 15 Solids, 23, (1975), 421-441. 9) ྚ⏛ࠉ⸠⪨ୌࠉᰡᒜᖶࠉ⮳⏛ᮿ⏠ࠉሣᛮ ࡛ຊᕝࠉ42-488ࠉ㸝2001㸞ࠉ919-923㸣 10) Hill, R., Proc. R. Soc. Lond., Ser. A, 193, (1948), 281-297. 11) ᮆྚᨼࠉ࿇ᒁᏬ❮ࠉ⸠⪨ୌࠉᐋᇖΰᏳࠉᶭ ㄵ Aࠉ67-663ࠉ㸝2001㸞ࠉ1754-1759㸣 12) Goya, M. and Ito, K., JSME Int. J., Ser. A, 40-2, (1997), 169-178. 13) ୕Ờὢཪ㯖ࠉ⌨⌣ኬᏕኬᏕ㝌ಞኃㄵᩝࠉ 㸝2011㸞㸣 14) Ῑ⃕ᏽගࠉሣᛮ࡛ຊᕝࠉ45-527ࠉ 㸝2004㸞ࠉ974-978㸣 15) ᶣᮇᾀࠉሣᛮ࡛ຊᕝࠉ41-473ࠉ 㸝2000㸞ࠉ527-534㸣
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