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 ofPlastic Strain Increment on Stress Increment⌨⌣ኬᏕᕝᏕ㒂ᶭ᲌ࢨࢪࢷ࣑ᕝᏕ⛁ ෶ᩅ᤭㸝ᰬ㸞ἀ⦎࢙ࢾࢷࢴࢠ༡ኃ ᝗ሒ⛁Ꮥ㸞ᮆྚ ᨼ᜜       7RVKL\DVX6XH\RVKL㻃㸦㸣ࡢࡋࡴ࡞  ୕Ờὢ ཪ㯖    <XPD8H]X▩ࡼࡿ࡙࠷ࡾ࠿ࠉᮞᩩࡡ㜾ఄ⌟㇗௧㜾ࡡሣᛮንᙟࢅ

⮤ິ㌬ࣂࢾࣜ࡝࡜ࡡ⷟ᯀᮞᩩࡡࣈࣝࢪຊᕝ࡞࠽ྱࡳᵋᠺᘟ࡛ࡊ࡙ࡢࠉࡥࡍࡲቌฦ⌦ㄵ࡞ᇱࡘࡂ

࠷࡙ࡢࠉࡊࢂࡷ◒࡛᩷࠷ࡖࡒᠺᙟ୘Ⰳ఩΅ࡡࡒࡴࠉ

Prandtl-Reuss ์㸝௧ୖࠉJ2F ์㸞࡛ࠉධࡥࡍࡲ⌦ࢨ࣐࣭ࣖࣝࢨࣘࣤᢇ⾙࡞ࡻࡾ஢ῼ࠿⾔ࢂࡿ࡙࠷ࡾࠊ

ㄵ࡞ᇱࡘࡂ Henchy ንᙟㄵࡡቌฦᙟ㸝௧ୖࠉJ2D ์㸞◒᩷⌟㇗ࡡⰃࡂ▩ࡼࡿࡒౚ࡚࠵ࡾ࠿ࠉFig. 1 ࡞♟ࡌ࠿▩ࡼࡿ࡙࠷ࡾ (1) ࠊୌ⯙Ⓩ࡝ᙆሣᛮゆᯊ࡞ࡢࠉㇿ༟㍀ᘤᘿንᙟࢅ⩻࠻ࡾ࡛ࠉୌᵕንᙟ㸝Fig. 1(a)㸞ࡡⲬ࣬㝎Ⲭࡡᇱ‵࠿᪺☔࡚࠵ࡾ࡛ࡊ࡙ J2F ์࠿⏕࠷ࡼᚃࠉᯀᖕ᪁ྡྷ࡞ࡂࡦࡿฝࡌᣉᩋࡂࡦࡿ㸝Fig. 1(b)㸞ࠉࡿ࡙࠷ࡾ࠿ࠉᠺᙟ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 ์⏍ࡋࡒᚃࠉ࡮࡜࡝ࡂࡊ࡙ฦ㞫࣬◒᩷㸝Fig. 1(d)㸞࠿࠿⏕࠷ࡼࡿࠉJ2F ์ࡻࡽᐁ㥺࡞㎾࠷⤎ᯕࢅ୙࠻ࡾࡆ⏍ࡋࡾࠊࡆࡿࡼᠺᙟ㝀⏲࡞ᑊࡌࡾ஢ῼࢨ࣐࣭ࣖࣝࢨ

࡛࠿ሒ࿈ࡈࡿ࡙࠷ࡾ(3), (4)ࠊࡆࡡこᅄ࡞ࡗ࠷࡙ࠉJ2Fࣘࣤᢇ⾙ࡡ㧏ᗐ໩ࡡࡒࡴ࡞ࡢࠉຊᕝᑊ㇗࡛࡝ࡾᮞᩩ

์࡛ J2D ์ࢅࡐࡿࡑࡿ๙ሣᛮᙟᘟ࡚ᘟ(1)࡛ᘟ(2)࡞

ࡡንᙟᣪິࢅ⢥ᗐⰃࡂ⾪⌟࡚ࡀࡾᮞᩩ࣓ࢸࣜࠉࡌ࡝♟ࡊࠉẒ㍉ࡊ࡙ࡲࡾࠊ

ࢂࡔሣᛮᵋᠺᘟ࠿ᚪこ࡚࠵ࡾࠊJ2F ์㸯ᵋᠺᘟ࡛ࡢࠉᮞᩩ࡞⏍ࡋࡾᚺງ࡛ࡥࡍࡲࢅ㛭౿ࡘࡄࡾᘟ࡚࠵ࡽࠉᙆᛮᵋᠺᘟ࡛ࡊ࡙ Hooke ์࠿Ⰳࡂ

J2D ์㸯ࡆࡆ࡚ࠉ ࡢሣᛮࡥࡍࡲቌฦࢷࣤࢮࣜࠉ ࡛ ࡢ⌟೩ᕣᚺງࢷࣤࢮ࡛ࣜࡐࡡቌฦࢷࣤࢮ࡚ࣜ࠵ࡽࠉࡐࡡ௙ࡢࢪ࣭࢜ࣚ㔖࡚ࠉ ࡛ࡢ┞ᙔᚺງ࡛ࡐࡡቌฦࢅࠉࡢࠉ┞ᙔᚺງ-┞ᙔሣᛮࡢẒౚ౿ᩐࠉ ࡛ࡥࡍࡲ᭜⥲ࡡࠉࡐࡿࡑࡿࢬ࢜ࣤࢹ౿ᩐ࡛᥃⥲౿ᩐࢅ⾪ࡌࠊሣᛮࡥࡍࡲቌฦ᪁ྡྷࡡࡲࢅ⩻࠻ࡒሔྙࠉᘟ(1)ࡡ J2F ์ࡢ⌟೩ᕣᚺງ࡛ሣᛮࡥࡍࡲቌฦࡡ㛭౿ᘟ࡚࠵ࡽࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷࡢ⌟೩ᕣᚺງ᪁ྡྷ࡞ୌ⩇Ⓩ࡞Ửᏽࡈࡿࠉᚃ⤾ࡡᚺງቌฦ᪁ྡྷ࡞ࡢධࡂ౪Ꮛ
Fig. 1 Illustration as deformation of thin sheet
metal with uni-axial tension.

ࡊ࡝࠷ࠊᑊࡊ࡙ࠉ々㞟࡚ࡢ࠵ࡾ࠿ࠉᘟ(2)ࡡ J2D ์ࡢࠉ➠஦㡧࡞♟ࡈࡿࡾࡻ࠹࡞ᚃ⤾ࡡᚺງቌฦ࡛ࡡ㛭10౿ࢅࡵྱࢆ࡚࠽ࡽࠉቌฦ᪁ྡྷ࠿᛬ንࡊࡒሔྙ࡞ࡵ㏛࣭✒㸞ࠊ㝮࡚ࡀࡾࡆ࡛࠿ࢂ࠾ࡾࠊࡌ࡝ࢂࡔࠉFig. 1(c)࡞♟ࡌ ࡻ࠹࡞ࠉୌᵕንᙟ᪁ྡྷ࠾ࡼᑻᡜࡎࢆ᩷ᖈ᪁ྡྷ࡫ࠉንࡆࡆ࡚ࠉ ࡢ೩ᕣᚺງቌฦࡡ㡨᪁ྡྷᙟ࠿᛬ንࡌࡾࡻ࠹࡝◒᩷⌟㇗ࡡ஢ῼ࡞ࡢࠉᚃ⤾ࡡᚺ࡚࠵ࡽࠉງቌฦ᪁ྡྷࢅ⩻៎ࡌࡾࡆ࡛࠿᭯ຝ࡚࠵ࡾ࡛⩻࠻ࡼྡྷࢅࠉࡿࡾࠊࡐࡆ࡚ࠉሣᛮࡥࡍࡲቌฦ࡞ᚺງቌฦ᪁ྡྷ౪Ꮛඔࣂ࣒࣭ࣚࢰ࡚ࠉࡐࡿࡑࡿ ࡡ㛭ᩐ࡚࠵ࡾࠊ ࠽ࡻᛮࢅ⩻៎ࡊࡒሣᛮᵋᠺᘟ࠿࠷ࡂࡗ࠾ᥞ᱄ࡈࡿࡒ (5)㹳ࡦ(7)ࠊᮇ✇࡚ࡢࠉࡐࡡୌࡗ࡚࠵ࡾ࿇ᒁ-௿⸠์(8), (9)ࡡᴣ

ᣲࡿぽ࡛ࡊ࡙ḗᘟ࡞ࡻࡽᏽ⩇ࡈࡿࡾࠊこ࡞ࡗ࠷࡙㏑࡬ࠉࡈࡼ࡞ⴥ⩽ࡼ࠿ᵋᠺᘟࡡጂᙔᛮࢅ
᳠ゞࡌࡾࡒࡴ࡞⿿షࡊ࡙࠷ࡾࠉࣀ࢕ࢺࣞࣆ࢚࣭࣑ᠺᙟム㥺ᶭ࡞ࡻࡾᐁチム㥺⿞⨠࡞ࡗ࠷࡙⣺௒ࡌࡾࠊ㡨᪁ྡྷࡢ㡨᪁ྡྷ

࠾ࡼࡡᣲࡿぽ࠾ࡼࡡሣᛮࡥࡍࡲቌฦ᪁

ࡢሣᛮࡥࡍࡲቌฦࡡኬࡀࡈࢅ⾪ࡌ↋ḗ
ࡢࠉFig. 2 ࠾ࡼࢂ࠾ࡾࡻ࠹࡞㡨᪁ྡྷ||cosࠉ

࡞࠽ࡄࡾຊᕝ◫໩౿ᩐ|࠾ࡼࡡ ࠽ࡻࡦ

࡞ࠉ|௧ୖࡡ㛭౿ࢅᏽ⩇ࡌࡾࠊ㸧㸣࿇ᒁ௿⸠࡞ࡻࡾᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ
ᅸᘇຊᕝ࡞ࡻࡖ࡙⿿㏸ࡈࡿࡾ⷟ᯀᮞᩩࡢࠉᅸᘇ᪁
ྡྷ࡛ࡐࡡ├ぽ᪁ྡྷ࡚ࠉୌ⯙Ⓩ࡞㜾ఄᚺງ࡝࡜ࡡᶭ᲌



||||

 

ᘟ(4)࠽ࡻࡦ(5)ࢅᘟ(3)࡞ᑙථࡌࡾࡆ࡛࡞ࡻࡽࠉ㔔ࡲ౿ᩐ࠽ࡻࡦࡢḗᘟ࡚⾪ࡌࡆ࡛࠿࡚ࡀࡾࠊ

Ⓩᛮ㈹࠿␏࡝ࡾ␏᪁ᛮࢅ♟ࡌ(10)ࠊ࿇ᒁࡼ࡞ࡻࡖ࡙ᥞ

ࠉ᱄ࡈࡿࡒᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ (8) ࡢ➴᪁ᛮ ᮞᩩ࡞ࡻࡾࡵࡡ࡚࠵ࡖࡒ࠿ࠉࡐࡡᚃࠉ␏᪁ᛮᮞᩩ࡞

ᥞ᱄ࡈࡿ࡙࠷ࡾᘟ(3)ࢅࢨ࣐࣭ࣖࣝࢨ࡚ࣘࣤ⏕࠷ࡾᑊࡊ࡙ᣉᘿ࠿࡝ࡈࡿࡒ (9)ࠊIlyushin ࡡ೩ᕣᚺງ✭ࡒࡴ࡞ࡢࠉ㡨᪁ྡྷࢅරమⓏ࡞Ửᏽࡌࡾᚪこ࠿࠵㛣࡞࠽ࡄࡾᴣᛍᅒࢅ Fig. 2 ࡞♟ࡌࠊࡾࠊᮇᮮࠉ㡨᪁ྡྷࢅỬᏽࡌࡾࡒࡴ࡞ࡢࠉ௴ណࡡᚺງ≟ឺ࡞ᑊࡊ࡙ᵕࠍ࡝᪁ྡྷ࡞ᚺງቌฦࢅ୙࠻ࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷ࡛ୌ⮬ࡌࡾ᪁ྡྷࢅῼᏽࡌࡾᐁ㥺ࢅኣᩐ⾔ࢂ࡝ࡂ࡙ࡢ࡝ࡼࡍࠉ⌟ᐁⓏ࡚ࡢ࡝࠷ࠊࡐࡆ࡚ࠉⴥ⩽ࡼࡢࠉ௴ណࡡᚺງ≟ឺ࡞ᑊࡊ࡙㡨᪁ྡྷ

ࢅぜᏽࡌࡾ㛭ᩐᙟࢅࠉ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ࡛᩺ࡒ࡞ᏽ⩇ࡊࠉ௴ណࡡࢪ࣭࢜ࣚ㔖 ࢅ⏕࠷࡙ḗᘟ࡚ᏽ⩇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ᩐᙟ࡛ࡊ࡙ࠉ᭩ࡵ⠾༟࡞␏᪁ᛮࢅ⾪⌟࡚ࡀࡾ Hill ࡡᚺງࡡ஦ḗᙟᘟࢅ⏕࠷ࠉᖲ㟻ᚺງ≟ឺࢅ௫ᏽࡌࡾ࡛ḗᘟ࡚⾪⌟ࡈࡿࡾࠊ௴ណࡡㇿⲬⅤ࡞ᑊࡊࠉ೩ᕣᚺງቌฦ࡞ᏽࡱࡾሣᛮࡥࡍࡲቌฦ࡞࠽࠷࡙ࠉࡐࡿࡼࡡ᪁ྡྷ࠿ୌ⮬ࡌࡾ᪁ྡྷࢅ㡨᪁ྡྷ㸝ኬࡀࡈ|ࡊ࡙ᏽ⩇ࡌࡾࠊ㡨᪁ྡྷ11࠾ࡼୌណ࡛೩ᕣᚺງቌฦ|                      㸞࡛ᘟ(8)ࡢ Hill ࡡ஦ḗ␏᪁ᛮ㜾ఄ㛭ᩐ࡛ྜྷᙟᘟ࡚࠵ࡾ࡞ࡻࡾ࠿ࠉࡐࡿ࡛༇ืࡌࡾࡒࡴ࡞ࠉ࠵࠻࡙ᚺງࡡ஦ḗᙟᘟᐞ୙ࡡࠔ㔔ࡲ௛ࡀᖲᆍࠕ࡛ࡊ࡙ࠉ␏᪁ᛮᮞᩩ࡞ᑊࡊ࡛࿣ࡩࡆ࡛࡞ࡌࡾࠊ࡙ᣉᘿࡈࡿࡒ࿇ᒁࡼ࡞ࡻࡾᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮ㜾ఄ㛭ᩐ࡛ྜྷᵕ࡞ࠉLankford ࡡ r ೋ㸝=ᯀᖕࡥࡍࡲᵋᠺᘟࡢḗᘟࡡࡻ࠹࡞ᵋ⠇ࡈࡿࡾ㸝ࠔ㸯ࠕࡢࢪ࢜ࣚ㸤ᯀཉࡥࡍࡲ㸞࠾ࡼỬᏽࡈࡿࡾ౿ᩐ࡚࠵ࡾࠊ⌟ᚺງ㹳ࡢࠉHill ࡡ஦ḗ␏᪁ᛮフジコー技報−tsukuru No. ࢷࣤࢮ࡛ࣜሣᛮࡥࡍࡲቌฦࢷࣤࢮࣜࢅ⤎ࡦࡗࡄࡾ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜࡡࡻ࠹࡝᩺ࡒ࡝࣎ࢷࣤࢨࣔࣜ㛭౿࡛ࡊ࡙ࠉ㜾ఄ㛭ᩐࢅሣᛮ࣎ࢷࣤࢨ࡛ࣔࣜࡊࠉ㜾㛭ᩐࢅ⏕࠷ࡾࡆ࡛࠿᭯⏕࡚࠵ࡾࡆ࡛࠿᳠ゞࡈࡿ࡙ఄ㛭ᩐ㸝᭜⥲㸞࡞ᑊࡊ࡙ࡐࡡἪ⥲᪁ྡྷ࠿ሣᛮࡥࡍࡲ

࠷ࡾࠊᘟ(3)࡚♟ࡈࡿࡾ࿇ᒁ-௿⸠࡞ࡻࡾᚺງቌฦ᪁ቌฦ᪁ྡྷࢅ♟ࡌ⩻࠻᪁ࢅ㛭㏻Ὦࡿ์(1)࡛࠷࠹࠿ࠉ㡨ྡྷ౪Ꮛሣᛮᵋᠺᘟࡢࠉ㡨᪁ྡྷ࡛㜾ఄ㛭ᩐࢅวࡽ㞫ࡊ᪁ྡྷ࡙ビ౮࡚ࡀࡾࡆ࡛࠾ࡼࠉᠺᙟ㝀⏲஢ῼ࡝࡜࡞ᑊࡊ࡙࡛㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ࡛ࡡ㛭౿ࡢࠉ

㛭㏻Ὦࡿ์࡛ྜྷᵕࡡ⩻࠻᪁࡚࠵ࡾࠊࡌ࡝ࢂࡔࠉ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ⢥ᗐྡྷ୕࠿᭿ᙽ࡚ࡀࡾ࡛⩻࠻ࡼࡿࡾࠊࡡἪ⥲᪁ྡྷ࡛ࡊ࡙ࠉ㡨᪁ྡྷ

࠿ぜᏽࡈࡿࡾࡆ࡛࡞࡝ࡾࠊࡔ࡝ࡲ࡞ᘟ(1)ࡢࠉ
Mises ࡡ㜾ఄ㛭ᩐࢅሣᛮ࣎ࢷࣤࢨ࡛ࣔࣜࡌࡾ

㛭㏻Ὦࡿ์࡚࠵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こ࡚࠵ࡾࠊࡐࡆ࡚ⴥ⩽ࡼࡢࠉළ⟮ᮞ࡞හᅸㇿⲬ࡛㍀ງࢅ୙࠻ࡾୌ᪁ࠉྚ⏛ࡼ(9)ࡢࠉ➱⩽ࡼ࡛ืࡡ⩻࠻᪁࠾ࡼ㡨᪁ࡆ࡛࡞ࡻࡽࠉළ࿔᪁ྡྷ࡛㍀᪁ྡྷࡡ஦᪁ྡྷንᙟ࠿ྊ⬗ࢅᏽ⩇ࡊ࡙࠷ࡾࠊᘟ(3)㹳(6)ࢅᖼరᏕⓏ㛭౿࡝࡝ࣀ࢕ࢺࣞࣆ࢚࣭࣑ᠺᙟ㸝௧ୖࠉHF ᠺᙟ㸞ࢅ⏕࠷࡜࠾ࡼᩒ⌦ࡌࡾࡆ࡛࡞ࡻࡖ࡙ࠉ௴ណࡡࢪ࣭࢜ࣚ㔖λࠉࡒᵋᠺᘟࡡᐁ㥺ビ౮ࢨࢪࢷ࣑ࢅࠉโᚒ⣌࠽ࡻࡦ゛ῼሣᛮࡥࡍࡲቌฦࡡᚺງቌฦ᪁ྡྷ౪Ꮛᛮࢅ⾪ࡌ᩺ࡒ⣌ࡡ⿿ష࡞ࡻࡖ࡙ᵋ⠇ࡊ࡙࠷ࡾࠊHF ᠺᙟ࡛ࡢࠉ㔘࡝ࣂ࣒࣭ࣚࢰࢅᑙථࡊࠉḗᘟ࡚⾪⌟ࡊ࡙࠷ࡾࠊᆵහഁ࡞ࢬࢴࢹࡊࡒ⟮ᮞ࡞㧏ᅸࡡᾦమࢅὮථࡈࡎ ࡗࡗࠉ㍀᪁ྡྷࡡንనࢅ୙࠻ࡾࡆ࡛࡚ࠉ⟮ᮞࢅ㔘ᆵහྡྷഁᙟ≟࡞ೊࢂࡎ࡙ᣉⳚ࣬ᠺᙟࡌࡾຊᕝ᪁Ἢ(14)࡚ࠉ㎾ᖳࠉ⮤ິ㌬➴ࡡࣆ࣭࣑ࣝ⿿ష࡝࡜࡞⏕࠷ࡼࡿ࡙࠷ࡾࠊ

ࡆࡆ࡚ࠉᘟ(9)ࡡ➠ୌ㡧┘ࡢࠉᘟ(10)࡞♟ࡌ Hill ࡡᮇ✇࡚ࡢࠉ⌟ᅹ⿿ష୯࡚࠵ࡾࠉ⏤ാฌ⌦ࢅ⏕࠷ࡒ஦ḗ␏᪁ᛮ㜾ఄ㛭ᩐࢅ࣎ࢷࣤࢨ࡛ࣔࣜࡌࡾᚉᮮࡡ

HF ᠺᙟᐁ㥺᫤ࡡ゛ῼἪ࡞ࡗ࠷࡙᳠ゞࡊࡒࡡ࡚⣺௒

㛭㏻Ὦࡿ์࡛ྜྷᵕࡡ㛭౿ࢅᑙථࡊ࡙࠷ࡾ࠿ࠉࣂ࣒ࣚࡌࡾࠊ࣭ࢰ࡞ࡗ࠷࡙ࡵ᳠ゞࢅこࡌࡾ㛭౿࡛࡝ࡖ࡙࠷ࡾࠊࡆࡿࡼᚺງቌฦ᪁ྡྷ౪Ꮛሣᛮᵋᠺᘟ࡞࠽ࡄࡾࣂ࣒࣭ࣚࢰࠉ㡨᪁ྡྷ࠽ࡻࡦ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ࡞㛭ࡊ࡙᳠ゞࢅ⾔࠹ࡒࡴ࡞ࠉᐁ㥺࡞௥ࢂࡾ᪁Ἢ࡛ࡊ࡙ࠉ࢓࣐ࣜࢼ࣑ࢗᮞࢅ᝷ᏽࡊࠉ⤎ᬏሣᛮ⌦ㄵ࡞ᇱࡘࡂ᭯㝀こ⣪ኣ⤎ᬏమ࣓ࢸࣜ(12)࡞ࡻࡾゆᯊ⤎ᯕࢅฺ⏕ࡊࡒᩐೋᐁ㥺ࢅ⾔ࡖࡒࠊリ⣵࡞ࡗ࠷࡙ࡢᩝ⊡(11)ࢅཤ↯࠷ࡒࡓࡀࡒ࠷࠿ࠉᮞᩩࡡࡵࡗ␏᪁ᛮࡡᙫ㡢࡞ࡻࡽ㡨᪁ྡྷ࠿ิ᭿ㇿⲬ᪁ྡྷ࠾ࡼ⛛ິࡌࡾࡆ࡛࠿♟ࡈࡿࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷ⾪ࡌ࡛ࠉሣᛮࡥࡍࡲቌฦࡡኬࡀࡈࢅࡡ㛭ᩐᙟ࠿᳠ゞࡈࡿ࡙࠷ࡾࠊຊ࠻࡙ࠉ㡨᪁ྡྷ࣎ࢷࣤࢨࣔࣜ㛭ᩐ࠿ࠉ㡨᪁ྡྷ

ࢅぜᏽࡌࡾ

Fig.3 Illustration as experimental apparatus㛭ᩐ࡛ࡊ࡙ࡡᙲ๪ࢅᢰ࠻࡙࠷ࡾࡆ࡛࠿᳠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࠽ࡽࠉࢦ࣭࣍ࢨ࣎ࢷࣤࢨ࡛ࣔࣜࡊ࡙ࡡᙲ๪ࢅᯕࡒࡌࡆ࡛ࡢ࡚ࡀࡍࠉࣛࣤࢱ࣎ࣤࣈ࡞࡙ᶭ᲌ἔࢅ㏞ࡾࡆ࡛࡚හᅸㇿⲬࢅ࠵ࡂࡱ࡚ㇿⲬ࣬㝎Ⲭࡡึ᩷ᇱ‵࡛࡝ࡾࡵࡡ࡚㜾ఄึ୙࠻ࠉム㥺ᶭᮇమ࡞ཱིࡽ௛ࡄࡒࢦ࣭࣓࣭࣍ࢰࡻࡖ࡙

ᏽࡡࡲ࡞⏕࠷ࡾࡆ࡛࡛ࡊࠉሣᛮࡥࡍࡲቌฦ᪁ྡྷࡢࠉ㍀ንనࢅ୙࠻ࡾᵋ㏸࡛࡝ࡖ࡙࠷ࡾࠊム㥺∞࡞ࡢࠉิ12

=210[mm] ࠉ ⫏ ཉࡼ୔ḗඔⓏ࡞ን໩ࡌࡾ゛ῼⅤࢅᤍᤂࡊࠉᐁᗑᵾ࡫⿭

=2[mm]ࡡ⣟࢓࣐ࣜࢼ࣑ࢗᘤᢜළ⟮㸝A1070㸞ࢅḿࡌࡾ᪁Ἢࢅ⤄ࡲ㎰ࢆࡓࡥࡍࡲ⟤ฝࣈࣞࢡ࣑ࣚࢅ

౐⏕ࡊࠉࡐࡡ⾪㟻࡞ࡢ Fig. 4 ࡞♟ࡌ࣏࣭࢞ࣤࢡࢅ᪃

షᠺࡊࠉࡐࡡ᳠ゞࢅ⾔ࡖࡒࠊࡔ࡝ࡲ࡞ୌ⯙Ⓩ࡝ CCDࡊ࡙࠵ࡾࠊහഁ࡛አഁࡡ྘ 4 Ⅴ࡚᭯㝀こ⣪Ἢ࡞࠽ࡄ࣒࢜ࣚࡡ 1 ⏤⣪ࡢ㛏᪁ᙟ࡚࠵ࡾࡒࡴࠉ⦢ᶋࡡ⿭ḿ⢥ࡾᄿ⟿Ⅴ࢓࢕ࢮࣂ࣒ࣚࢹࣛࢴࢠこ⣪࡛ࡲ࡝ࡌࡆ࡛

ᗐ࡞ᕣ࠿ฝࡾ࠿ࠉ⏕࠷ࡒ NI ♣ࡡ CCD ࣒࢜ࣚࡢ 1࡚ࠉ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࠽ࡻࡦࡐࡡࣀ࣭ࢺ࢙ࢗ࢓⩄࡞ࡻࡖ࡙ᵋᠺࡈࡿࡾࠊࡥFig. 6 ࡞♟ࡌࠊ⦢㍀࡞ム㥺∞├ᙼ d [mm]ࢅࠉᶋ㍀

ࡍࡲࡢࠉCCD ࣒࢜ࣚ࠾ࡼᚋࡼࡿࡾ⏤ാࢅ NI ♣ࡡ࡞ࡢᐁ㥺࡞こࡊࡒ᫤㛣ࢅ 1 ࡛ࡊࡒ๪ྙࢅ⾪ࡊ࡙࠷

Vision Assistant ࢅ⏕࠷࡙ฌ⌦ࡊࡒ⤎ᯕ࠾ࡼ⟤ฝࡊࠉࡾࠊᐁ゛ῼࡢࠉ⟤ฝࡈࡿࡒᚺງࡢࠉᅸງንᥦჹࠉⲬ㔔ንᥦჹ࠽ࡻࡦࣛࢼ࢓ࢣ࣭

0.19 ࡡ᫤࡞ࠉࡐࡿࡑࡿහᅸࢅୌ᪞㝎Ⲭࡊ࡙⾔ࡖࡒࠊࢩ࠾ࡼᚋࡼࡿࡾ゛ῼೋࢅ⏕࠷࡙⟤ฝࡊ࡙࠷ࡾࠊළ࿔᪁ྡྷࡥࡍࡲ ࠾ࡼ⟤ฝࡊࡒム㥺∞├ᙼࡢࠉ྘ᐁ

᭿አᙼ=32[mm] ࠉ 㛏 ࡈ

0.05ࠉ0.1ࠉ0.15ࠉ

ῼೋ࡛࡮࡯ୌ⮬ࡊ࡙࠷ࡾࡆ࡛࠾ࡼࠉషᠺࡊࡒࡥࡍࡲ
⟤ฝࣈࣞࢡ࣑ࣚࡢࠉᣉ⟮࡞ࡻࡾ↌Ⅴ㊝㞫ን໩ࡡ⿭ḿFig.4 Making pattern on a specimen

Specimen diameter d [mm]࠿࠹ࡱࡂ⾔ࢂࡿ࡙࠷ࡾ࡛࠷࠻ࡾࠊ

40

Calculated (by εθ)Actual measurement

383634320.0

Initial diameterd0=32.0[mm]
0.2

0.4

0.6

0.8

1.0

Time rate
Fig.6 Comparison of specimen diameter dcalculated by circumferential strain
and actual measurementFig.5 Measurement and Control Systems forexperimental apparatus㸩㸣ᐁ㥺ビ౮ࢨࢪࢷ࣑ࡡ⏤ാฌ⌦࡞ࡻࡾ゛ῼ⤎ᯕ࡛⢥ᗐᐁ㥺୯ࠉFig. 4 ࡞♟ࡌࠉ᭜⋙ࢅᣚࡗム㥺∞⾪㟻࡞᪃ࡈࡿࡒⅤࡡንనࢅ⏤ാฌ⌦࡞ࡻࡖ࡙゛ῼࡊ࡙࠷ࡾ࠿ࠉහᅸㇿⲬ࡞ఔ࠹ᣉ⟮ࡡࡒࡴࠉᅖᏽࡊ࡙࠷ࡾ
CCD ࣒࡛࢜ࣚム㥺∞⾪㟻࡛ࡡ㊝㞫࠿㎾ࡂ࡝ࡽࠉ↌Ⅴ㊝㞫࠿ን໩ࡌࡾࠊࡐࡆ࡚ࠉ᧔ࡼࡿࡒ஦ḗඔ⏤ാ࠾

13ḗ࡞ࠉᚺງ⟤ฝ࡞ᚪこ࡝᭜⋙༖ᙼࡡ゛ῼࢅࠉ᥃よᘟࡡࣛࢼ࢓ࢣ࣭ࢩ࠾ࡼ CCD ࣒࢜ࣚ࡞ࡻࡾ⏤ാฌ⌦࡞ࡻࡾ⟤ฝἪ࡞ን᭞ࡊࡒࡡ࡚ࠉࡐࡡ⢥ᗐ࡞ࡗ࠷࡙᳠ゞࢅ⾔ࡖࡒࠊࣛࢼ࢓ࢣ࣭ࢩ࡞ࡻࡾ゛ῼ࡚ࡢࠉム㥺∞࿔ᅑ࡚ࡡ⿞⨠ࡡཱིࡽᅂࡊ࠿ᝇࡂࠉࡱࡒ◒᩷ࡊࡒ㝷࡞ᅸງ፳మ࠿ࣛࢼ࢓ࢣ࣭ࢩ࡞࠾࠾ࡾࡆ࡛࠿⩻࠻ࡼࡿࡾࠊࡈࡼ࡞㍀ንనࢅ୙࠻ࡒ㝷ࠉム㥺∞ࢲࣔࢴࢠ㒂࡞᥃よࡌࡾࡆ࡛࠿᝷ᏽࡈࡿࡾࡒࡴࠉ⏤ാฌ⌦゛ῼ࡞ን᭞ࡌࡾࡆ࡛ࢅ⩻࠻ࡒࠊ⏤ാ゛ῼⅤࡢࠉࣛࢼ࢓ࢣ࣭ࢩ
࡛ྜྷࡋ 3 Ⅴࡡన⨠࡛ࠉࡐࡡ㛣࡞ 2 Ⅴࢅ㏛ຊࡊࡒ゛ 5

フジコー技報−tsukuru No. Ⅴ࡚࠵ࡽࠉ⏤ാฌ⌦ࢅ⏕࠷࡙゛ῼࡊࡒࡆࡿࡼࡡⅤࡡ࣒࢜ࣚࡡన⨠ࠉ゛ῼⅤᩐࢅቌࡷࡌ➴ࡡ᳠ゞㄚ㢗ࢅṟንన㔖࡞ࡗ࠷࡙ࠉ᭩ᑚ஦஋Ἢࢅ⏕࠷࡙⟤ฝࡊࡒࠊࡐࡌ⤎ᯕ࡛࡝ࡖࡒࠊࡿࡑࡿ⟤ฝࡈࡿࡒ᭜⋙༖ᙼࡡẒ㍉ࢅ Fig. 7 ࡞♟ࡌࠊ

Radius of Curvature R [mm]1000ᮇ⿞⨠ࡡ┘ᵾࡢࠉ゛ῼ⤎ᯕ࡞ࡻࡽᚋࡼࡿࡒ㍀᪁ྡྷ
ࡥࡍࡲ࠽ࡻࡦළ࿔᪁ྡྷࡥࡍࡲࢅࠉሣᛮᵋᠺᘟࡡ㐲⏕Linear gauge
Processing imageࡈ ࡿ ࡒ ᠺ ᙟ 㝀 ⏲ ⥲ ᅒ (15) 㸝 Forming Limit
Diagram㸯௧ୖࠉFLD㸞࡛Ẓ㍉ࡊࠉᵋᠺᘟࡡጂᙔᛮ

800

ࡡビ౮࠽ࡻࡦ᳠ゞ࡞⏕࠷ࡾࡆ࡛࡚࠵ࡾࠊࡊ࠾ࡊࠉࢨ
600ࢪࢷ࣑⿿ష୯ࡡࡒࡴࠉFLD ࡢࠉୌ⯙Ⓩ࡝◒᩷㝀⏲஢ῼ࡞ኣ⏕ࡈࡿ࡙࠷ࡾ S-R ⌦ㄵࢅ⏕࠷࡙᳠ゞࢅ⾔400ࡖࡒࠊࡈࡼ࡞ࠉ⿿ష୯ࡡࢨࢪࢷ࣑࡚ࡢ㍀ງ㛭౿ࡡโᚒ࠿ᮅᏰᠺࡡࡒࡴࠉහᅸㇿⲬࡡࡲࡡᐁ㥺⤎ᯕ࡚࠵ࡾࠊ
࣏࣭࢞ࣤࢡࡡහഁࡡ 4 Ⅴࢅ⏕࠷࡙⟤ฝࡊࡒࡥࡍࡲ

200

ೋࢅ Inner ࡛ࡊࠉአഁࢅ⏕࠷࡙⟤ฝࡊࡒࡵࡡࢅ
00.7

0.8

0.9

1.0

Time rate
Fig.7 Comparison of radius of curvature R
calculated by linear gauge andprocessing image

Outer ࡛ࡊ࡙ࠉࡐࡿࡼࡡࡥࡍࡲᒓṌࢅ Fig. 8 ࡞♟ࡌࠊ
⦢㍀࡞㍀᪁ྡྷࡥࡍࡲ ࠉᶋ㍀࡞ළ࿔᪁ྡྷࡥࡍࡲ ࡛ࡊ࡙࠷ࡾࠊහᅸㇿⲬࡊ࠾୙࠻࡙࠷࡝࠷ࡒࡴࠉࡥࡍࡲᒓṌࡢළ࿔᪁ྡྷࡥࡍࡲ㍀୕࡞㐼⛛ࡊ࡙࠷ࡾࡆ࡛

࠿ࢂ࠾ࡾࠊᯗහࡡᣉኬᅒࢅずࡾ࡛ࠉළ࿔᪁ྡྷࡥࡍࡲᐁ㥺㛜ጙ࠾ࡼ༖ࡣࡱ࡚ࡢࠉ゛ῼೋ࠿Ẓ㍉ⓏᏭᏽࡊ࡙=0.15 ௛㎾࠾ࡼ㍀᪁ྡྷࡥࡍࡲ ࠿ኬࡀࡂ࡝ࡖ࡙࠷࠷࡝࠾ࡖࡒࡒࡴࠉᐁ㥺ᚃ༖࡞ࡗ࠷࡙᳠ゞࢅ⾔ࡖ࡙࠷ࡾ࠿ࠉࡆࡿࡢහᅸㇿⲬ࡞ࡻࡾᣉ⟮ࡡ㝷ࠉንᙟࡢළ࿔ࡾࠊ⏤ാฌ⌦࠾ࡼ⟤ฝࡈࡿࡒ᭜⋙༖ᙼ࡞⣵࠾࠷୕ୖ᪁ྡྷࡡࡲࡓࡄ࡚࡝ࡂࠉ᭜⋙ࢅᣚࡗࡆ࡛࡚㍀᪁ྡྷ࡞ᘤንິ࠿ずࡼࡿࡾ࠿ࠉࡆࡿࡢᐁ㥺㛜ጙ᫤࡞Ửᏽࡊࡒ஦ᘿንᙟ࠿⏍ࡋࡾࡒࡴ࡚࠵ࡾࠊ㍀ንనࢅโᚒ࡚ࡀࡾࡻ

ೋ໩ฌ⌦࡞⏕࠷ࡾ⏤ാࡊࡀ࠷ೋ࠿ࠉᐄአ࠾ࡼࡡක㔖࠹࡞ࢨࢪࢷ࣑ᵋ⠇࠿࡚ࡀࡿࡣࠉᖲ㟻ࡥࡍࡲ≟ឺ࡚ࡡን໩࡞ࡻࡖ࡙㐲ḿೋ࡚࡝ࡂ࡝ࡾࡆ࡛࠿ཋᅄ࡛ࡊ࡙ንᙟࢅ⥌ᣚࡌࡾࡆ࡛࠿ྊ⬗࡛࡝ࡽࠉࡥࡍࡲᒓṌࡢළ⩻࠻ࡼࡿࡾࠊࡊ࠾ࡊࠉ⏤ാฌ⌦࠾ࡼ᭩ᑚ஦஋Ἢ࡞ࡻ࿔᪁ྡྷࡥࡍࡲࡖ࡙ᚋࡼࡿࡒ᭜⋙༖ᙼࡢࠉࣛࢼ࢓ࢣ࣭ࢩ࡛ྜྷᵕࡡലS-R ⌦ㄵ࡛ࡡᕣ࡞㛭ࡊ࡙ࡢࠉ⌦ㄵೋࡡ゛⟤࡞ຊᕝ◫ྡྷࢅ♟ࡊ࡙࠷ࡾࡆ࡛࠿ࢂ࠾ࡾࠊ௑ᚃࠉක㔖ࡷࠉCCD

໩ᣞᩐ n=0.26 ࡛ࡊࡒ࠿ࠉࡆࡿࡢ࢓࣐ࣜࢼ࣑ࢗᮞ࡞㍀୕࡫ฦᕱࡌࡾࡆ࡛࠿ྊ⬗࡛࡝ࡾࠊFig.8 Strain path diagram with FLD calculated by S-R theory14࠽ࡄࡾୌ⯙Ⓩ࡝ೋࢅ⏕࠷࡙࠽ࡽࠉ௑ᚃࠉᐁ㥺࡚౐⏕࡛ࢅ⣺௒ࡊࡒࠊ௑ᚃࡵࠉ◂✪ᐄᡜᒌᏕ⏍࡛භ࡞ࠉᐁ

ࡊ࡙࠷ࡾム㥺∞ࢅ⏕࠷ࡒᘤᘿム㥺➴ࢅ⾔࠹ࡆ࡛࡚ n㥺ビ౮࡝ࡼࡦ࡞ᠺᙟ㝀⏲஢ῼࡷᙆሣᛮゆᯊ࡝࡜ࡡೋࢅỬᏽࡊ࡙࠷ࡂᚪこ࠿࠵ࡾࠊࡱࡒࠉS-R ⌦ㄵࡢ➴ࢨ࣐࣭ࣖࣝࢨࣘࣤ࡫ࡡ㐲⏕ࢅ┘Ⓩ࡞ᵋᠺᘟ࡞㛭ࡌ᪁ᛮᮞᩩࢅ๑ᥞ࡛ࡊ࡙࠷ࡾࡒࡴࠉ௑ᅂ⏕࠷࡙࠷ࡾムࡾཱིࡽ⤄ࡲࢅ㐅ࡴࡾࡗࡵࡽ࡚࠵ࡾࠊ

㥺∞ࡡ␏᪁ᛮࡡᙫ㡢࠿࡝࠷࡛ࡢゕ࠻࡝࠷ࠊࡈࡼ࡞ࠉ
Fig. 9 ࡞♟ࡌහᅸ p[MPa]࡛ළ࿔᪁ྡྷࡥࡍࡲ ࡛ࡡཤ⩻ᩝ⊡⥲ᅒ࡚ࡢࠉළ࿔᪁ྡྷࡥࡍࡲ =0.04 ௛㎾࠾ࡼ◒᩷ࡱ1) ᚃ⸠ᏟࠉሣᛮᏕࠉࢤࣞࢻ♣ࠉ㸝1982㸞㸣࡚ࡡ㛣࡞ࠉහᅸን໩࠿ 7.0㹳7.7[MPa]࡛≻ࡂࠉୌẴ2) Stören, S. and Rice, J.R., J. Mech. Phys.

࡞ንᙟ࠿㐅ࢆ࡚࠷ࡾࡆ࡛࠿ࢂ࠾ࡾࠊ௑ᅂࡢහᅸㇿⲬࢅ୙࠻ࡾᶭ᲌ἔࡡ㏞ࡽ㔖࠿ୌᏽ㏷ᗐ࡚࠵ࡖࡒࡒࡴࠉ
ንᙟ࡛භ࡞㏞ࡽ㔖ࢅโᚒ࡚ࡀࡾࡻ࠹࡝ࢨࢪࢷ࣑ࡡ

1222-1229㸣
(1948).

8Internal pressure p [MPa]3) ௿ ⸠ ⪨ ୌ ࠉ ሣ ᛮ ࡛ ຊ ᕝ ࠉ 28-323ࠉ㸝 1987 㸞ࠉ4) Stowell, E.Z., NACA Tech. Note, No.1556,

ᵋ⠇࠿ᚪこ࡚࠵ࡾ࡛⩻࠻ࡾࠊ5) Christoffersen, J. and Hutchinson, J.W., J.

Mech. Phys. Solids, 27, (1979), 465-487.

66) ᚃ⸠Ꮯࠉᶭㄵ Aࠉ47-424ࠉ
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㸝1988㸞ࠉ

41617-1622㸣

20Inner pattern
Outer pattern
0.10.2

Circumferential strain εθFig.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ᩐ࡛ሣᛮࡥࡍࡲቌฦ᪁ྡྷࢅỬᏽࡌࡾሣᛮ࣎ࢷࣤࢨࣔࣜ㛭ᩐ࠿ྜྷୌ࡚࠵ࡾ㛭㏻Ὦࡿ์࡛ࡢ␏࡝ࡽࠉ㝎Ⲭ࣬ㇿⲬึᏽࢅ⾔࠹࡬ࡀ㜾ఄ㛭ᩐ࡛ሣᛮࡥࡍࡲቌฦ᪁ྡྷࢅỬᏽࡌࡾ㛭ᩐ࡛ࢅฦ㞫ࡊ࡙ビ౮࡚ࡀࡾࡆ࡛࠾ࡼࠉ⤊㊨᛬ንࢅఔ࠹ࡻ࠹࡝ᠺᙟ㝀⏲஢ῼ࡝࡜࡞ᑊࡊ࡙⢥ᗐྡྷ୕࠿᭿ᙽ࡚ࡀࡾࠊࡱࡒࠉ
◂✪ᐄ࡞࡙⿿ష୯ࡡᵋᠺᘟࡡᐁ㥺ビ౮ࢨࢪࢷ࣑࡞
ࡗ࠷࡙ࡢࠉ㍀ງ࡛හᅸㇿⲬࡡโᚒ⣌ࡡᵋ⠇࠿᛬࠿ࡿࡾࡵࡡࡡࠉᣉ⟮࡞ఔ࠹↌Ⅴ㊝㞫⿭ḿᶭ⬗࠿Ὡ࠾ࡈࡿࠉ⏤ാฌ⌦࡞ࡻࡾ゛ῼ࡛ࡥࡍࡲ⟤ฝ࠿ྊ⬗࡚࠵ࡾࡆ

15

Solids, 23, (1975), 421-441.

9) ྚ⏛஼ࠉ௿⸠⪨ୌࠉᰡᒜᖶ஁ࠉ⮳⏛ᮿ⏠ࠉሣᛮ
࡛ຊᕝࠉ42-488ࠉ㸝2001㸞ࠉ919-923㸣
10) Hill, R., Proc. R. Soc. Lond., Ser. A, 193,(1948), 281-297.
11) ᮆྚᨼ᜜ࠉ࿇ᒁᏬ❮ࠉ௿⸠⪨ୌࠉᐋᇖΰᏳࠉᶭㄵ 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) ᶣᮇᾀ஦ࠉሣᛮ࡛ຊᕝࠉ41-473ࠉ
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