XFEM ࡢỗ⏝ FEM ゎᯒࢥ࣮ࢻ࡬ࡢᐇ⿦࡜

ᛂ⏝ຊᏛㄽᩥ㞟

Vol.13 (2010

ᖺ

8

᭶) ᅵᮌᏛ఍

XFEM ࡢỗ⏝ FEM ゎᯒࢥ࣮ࢻ࡬ࡢᐇ⿦࡜

ᐇᵓ㐀≀ࡢ⑂ປࡁ⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥ

Implementation of XFEM into a general-purpose FEM code and simulation of fatigue crack propagation in actual structure

ᰘ἟୍ᶞ *࣭㟷஭ᘯᶞ **࣭Ᏹ㒔ᐑᬛ᫛***࣭ᆏ㔝ᫀᘯ ****࣭ኟ⛅⩏ᗈ *****

Kazuki SHIBANUMA, Hiroki AOI, Tomoaki UTSUNOMIYA, Masahiro SAKANO and Yoshihiro NATSUAKI

*Ꮫ⏕ဨ ி㒔኱Ꮫ኱Ꮫ㝔 ᕤᏛ◊✲⛉ ♫఍ᇶ┙ᕤᏛᑓᨷ࣭᪥ᮏᏛ⾡᣺⯆఍≉ู◊✲ဨ-DC 䠄䛈615-8540ி㒔ᕷすி༊ி㒔኱Ꮫ᱇䠅

**Ꮫ⏕ဨி㒔኱Ꮫ኱Ꮫ㝔ᕤᏛ◊✲⛉♫఍ᇶ┙ᕤᏛᑓᨷ 䠄䛈615-8540ி㒔ᕷすி༊ி㒔኱Ꮫ᱇䠅

***ṇ఍ဨ ி㒔኱Ꮫ኱Ꮫ㝔 ᕤᏛ◊✲⛉ ♫఍ᇶ┙ᕤᏛᑓᨷ ෸ᩍᤵ䠄䛈615-8540ி㒔ᕷすி༊ி㒔኱Ꮫ᱇䠅

****ṇ఍ဨ 㛵す኱Ꮫ ⎔ቃ㒔ᕷᕤᏛ㒊㒔ᕷ䝅䝇䝔䝮ᕤᏛ⛉ ᩍᤵ䠄䛈564-8680኱㜰ᗓ྿⏣ᕷᒣᡭ⏫3-3-35䠅

*****ṇ఍ဨ᪥ᮏᶫᱱᘓタ༠఍ 䠄䛈550-0005኱㜰ᕷす༊すᮏ⏫1-8-2୕᫭䝡䝹5㝵䠅

In order to evaluate the behavior of a fatigue crack in the local part of complex and large-scale civil engineering structures such as orthotropic steel deck, we develop a fatigue crack simulation code by implementation of PU-XFEM into the general-purpose FEM analysis software. We have adopted the multiple-nodes to increase the nodal degrees of freedom. This makes possible to implement the PU-XFEM approximation to various software without any changes of the definitions in the convention rules. The performance of the developed code is evaluated through verification of the numerical accuracies of fracture mechanics parameters using basic models.

Then, the developed code is applied to the numerical simulations of the fatigue crack propagation in the orthotropic steel deck specimen using bulb rib, as an example.

It is concluded that the developed PU-XFEM analysis code is useful for the quantitative evaluation of fatigue propagation in actual structure.

Key Words: XFEM, PU-XFEM, implementation, fatigue, crack propagation, actual steel structure

1. ࡣࡌࡵ࡟

⌧ᅾ㸪⤒ᖺࡋࡓ㗰ᶫࡢ㗰ᗋ∧࡟࠾࠸࡚⑂ປࡁ⿣ࡢⓎ⏕ࡀ ከᩘሗ࿌ࡉࢀ࡚࠸ࡿ¹⁾㸬ࡇࢀ࡟ᑐࡋ㸪ᐇᶫ࡟Ⓨ⏕ࡋࡓ⑂ປ ࡁ⿣ࡢ㐍ᒎᣲືࢆᩘ್ゎᯒ࡟ࡼࡾ㸪ᐜ᫆࠿ࡘ㧗⢭ᗘ࡟ࢩ࣑

࣮ࣗࣞࢺࡍࡿࡇ࡜ࡀྍ⬟࡜࡞ࢀࡤ㸪㗰ᶫ࡟࠾ࡅࡿ⑂ປࡁ⿣

ࡢⓎ⏕࣭㐍ᒎࡢ࣓࢝ࢽࢬ࣒ゎ᫂࡜ࡑࡢ⥔ᣢ⟶⌮ࡢྜ⌮໬࡟

ᑐࡋ࡚㠀ᖖ࡟᭷ຠ࡞ᡭẁ࡜࡞ࡾ࠺ࡿ㸬

ࡋ࠿ࡋ㸪ᵓ㐀ゎᯒ࡟ᗈࡃ᥇⏝ࡉࢀ࡚࠸ࡿ᭷㝈せ⣲ἲ 㸦

FEM

㸧ࡣ㸪ᮏ᮶㸪㐃⥆యࢆᑐ㇟࡜ࡋ࡚ᵓ⠏ࡉࢀࡓࡶࡢ࡛

࠶ࡿࡓࡵ㸪

FEMࢆࡁ⿣ゎᯒ࡟㐺⏝ࡍࡿሙྜ࡟ࡣ㸪࣓ࢵࢩࣗ

⏕ᡂ࡟㛵ࡋ࡚௨ୗࡢゎỴࡍ࡭ࡁㄢ㢟ࡀᏑᅾࡍࡿ㸬

(1)

ࡁ⿣࡟

ࡼࡿ୙㐃⥆㠃ࡢࣔࢹࣝ໬ࡢࡓࡵ࡟㸪ࡁ⿣ᙧ≧࡟せ⣲ቃ⏺ࢆ

୍⮴ࡉࡏࡿᚲせࡀ࠶ࡿ㸬

(2)

◚ቯຊᏛࣃ࣓࣮ࣛࢱࢆ⢭ᗘࡼࡃ ホ౯ࡍࡿࡓࡵ࡟㸪ࡁ⿣ඛ➃㏆ഐ࡟࠾࠸࡚せ⣲ࡢ⣽ศ໬ࡸ≉

␗せ⣲²⁾ࡢ౑⏝ࡀᚲせ࡜࡞ࡿ㸬ࡉࡽ࡟㸪

(3)

ࡁ⿣ࡢ㐍ᒎ㐣⛬

ࢆ⪃࠼ࡓሙྜ㸪㐍ᒎ࡟క࠸ୖグ

(1)㸪 (2)ࡢ」㞧࡞࣓ࣜࢵࢩࣗ

ฎ⌮ࢆ⧞ࡾ㏉ࡋ⾜࠺ᚲせࡀ࠶ࡿ㸬ࡇࡢࡓࡵ㸪᪤ ࡢ◊✲࡟

࠾࠸࡚FEMࢆ⏝࠸ࡓ⑂ປࡁ⿣ࡢ㐍ᒎゎᯒࡢᐇ᪋ࡀሗ࿌ࡉ

ࢀ࡚࠸ࡿࡀ^{3), 4), 5)}㸪㗰ᶫࡢ㗰ᗋ∧ࡢࡼ࠺࡞」㞧࠿ࡘ኱つᶍ࡞

ᵓ㐀య࡟ᒁᡤⓗ࡟Ⓨ⏕ࡍࡿ⑂ປࡁ⿣ࡢᣲືࢆ⢭ᗘࡼࡃホ ౯ࡍࡿࡇ࡜ࡣ㸪㧗ᗘ࡞ᕤኵ࡜ࣀ࢘ࣁ࢘ࢆせࡍࡿࡓࡵᐜ࡛᫆

ࡣ࡞࠸㸬

୍᪉㸪㏆ᖺᥦ᱌ࡉࢀࡓᣑᙇ᭷㝈せ⣲ἲ㸦

eXtended FEM;

XFEM㸧

⁶⁾ࡣ㸪

FEMࡢᯟ⤌ࡳࢆ⏝࠸ࡓᩘ್ゎᯒᡭἲ࡛࠶ࡿ

࡟ࡶ㛵ࢃࡽࡎ㸪௵ពࡢᒁᡤⓗ࡞㡿ᇦෆ࡟࠾࠸࡚㸪せ⣲ෆ㒊

࡟୙㐃⥆ᛶࡸ≉␗ᛶࢆྵࡴ㧗ᗘ࡞㏆ఝ㸦࢚ࣥࣜࢵࢳ࣓ࣥࢺ㸧

ࢆ⤌ࡳ㎸ࡴࡇ࡜ࡀྍ⬟࡜࡞ࡿ㸬ࡇࢀ࡟ࡼࡾ㸪ࡇࡢ

XFEM

ࢆ ࡁ⿣ゎᯒ࡟㐺⏝ࡋࡓሙྜ㸪㏻ᖖࡢFEM࡜ࡣ␗࡞ࡾ࣓ࢵࢩࣗ

࡜ࡣ⊂❧࡟ࡁ⿣ࡢ୙㐃⥆㠃ࢆࣔࢹࣝ໬࡛ࡁࡿࡓࡵ㸪㏻ᖖࡢ

FEMࢆ⏝࠸ࡓࡁ⿣㐍ᒎゎᯒ࡟㛵ࡍࡿ࣓ࢵࢩࣗࡢ⏕ᡂ࡟㛵

ࡍࡿㄢ㢟ࡀゎỴࡉࢀ㸪ᐜ᫆࡟ࡑࡢ㐍ᒎᣲືࡢホ౯ࡀྍ⬟࡜

࡞ࡿ㸬ࡋ࠿ࡋ㸪ࡇࡢᚑ᮶ࡢXFEM㏆ఝ࡟࠾࠸࡚㸪⠇Ⅼ༢఩

࡛௜ຍࡉࢀࡿ࢚ࣥࣜࢵࢳ࣓ࣥࢺࡀ㒊ศⓗ࡟ྵࡴせ⣲

㸦Blending Elements; BE㸧࡟࠾࠸࡚㸪ゎᯒ⢭ᗘࡢపୗࡀᣦ᦬

ࡉࢀࡓ^{7), 8)}㸬ࡇࢀ࡟ᑐࡋ࡚㸪ⴭ⪅ࡽࡣ㸪ゎᯒ㡿ᇦ඲యࡢ㏆ఝ

⢭ᗘࢆಖドࡍࡿPUFEMࡢ㏆ఝἲ⁹⁾࡟ᇶ࡙࠸࡚XFEMࡢ෌ᐃ ᘧ໬㸦௨ୗ㸪

PU-XFEM

࡜ࡍࡿ㸧ࢆᥦ᱌ࡋ㸪ୖグࡢ

BE

࡟࠾

ࡅࡿၥ㢟ࢆゎỴࡋࡓ^{10), 11)}㸬

࡜ࡇࢁ࡛㸪㏆ᖺ࡛ࡣከࡃࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢ ᶵ⬟ࡀ඘ᐇࡋ㸪≉࡟ࢯࣇࢺ࢙࢘࢔ࡢࣉ࣭࣏ࣜࢫࢺࣉࣟࢭࢵ

ࢧ࡛ࡣ᭷㝈せ⣲ࣔࢹࣝࡢసᡂࡸィ⟬⤖ᯝࡢฎ⌮ࡀᐜ᫆࠿

ࡘຠ⋡ⓗ࡟ᐇ᪋ྍ⬟࡜࡞ࡾ㸪㠀ᖖ࡟᭷⏝࡞ࡶࡢ࡜࡞ࡗ࡚࠸

ࡿ㸬ࡇࡢࡓࡵ㸪ࡑࢀࡽࢆ฼⏝ࡋࡓ」㞧࡞኱つᶍᵓ㐀≀ࡢᣲ 応用力学論文集 Vol.13, pp.945-956  （2010年8月） 土木学会

ᶓࣜࣈ

ࢹࢵ࢟ࣉ࣮ࣞࢺ

ࣂࣝࣈࣜࣈ

⑂ປࡁ⿣

ື࡟㛵ࡋ࡚ゎᯒⓗ࡞᳨ウࡀᐇ᪋ࡉࢀࡘࡘ࠶ࡿ^{12), 13)}㸬 ࡑࡇ࡛㸪ࡇࡢ኱つᶍᵓ㐀≀ࡢゎᯒ࡟㐺ࡋࡓỗ⏝

FEM

ゎᯒ ࢯࣇࢺ࢙࢘࢔࡟㸪ࡁ⿣ゎᯒ࡟㐺ࡋࡓPU-XFEM㏆ఝࢆ⤌ࡳ

㎸ࡴࡇ࡜࡟ࡼࡾ୧⪅ࡢ฼Ⅼࢆ⤌ࡳྜࢃࡏࡿࡇ࡜ࡀ࡛ࡁࢀ

ࡤ㸪኱つᶍ࠿ࡘ」㞧࡞ᵓ㐀≀࡟Ⓨ⏕ࡍࡿ⑂ປࡁ⿣ࡢホ౯ࢆ

⾜࠺ୖ࡛㸪ᐇ⏝ୖ᭷ຠ࡞ࢶ࣮ࣝ࡜࡞ࡾ࠺ࡿ࡜⪃࠼ࡽࢀࡿ㸬 ࡓࡔࡋ㸪ࡇࡢXFEMࡢỗ⏝FEMゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢᐇ

⿦࡟࠾࠸࡚ࡣ㸪

XFEM

㏆ఝࡢᑟධ࡟ᚲせ࡞⠇Ⅼ⮬⏤ᗘࡢቑ ຍἲࡀ㔜せ࡞ㄢ㢟࡜࡞ࡿ㸬౛࠼ࡤ㸪᭱㏆Ginerࡽ¹⁴⁾ࡣ㸪

2ḟ

ඖᖹ㠃ၥ㢟ࢆᑐ㇟࡜ࡋࡓᚑ᮶ࡢ

XFEM

ࡢỗ⏝

FEM

ゎᯒࢯ

ࣇࢺ࢙࢘࢔࡬ࡢᐇ⿦ἲࢆ♧ࡋ㸪ࢯࣇࢺ࢙࢘࢔ࡢつ⣙࡟࠾࠸

࡚㠃እࡢᅇ㌿ࡸ ᗘ࡜ࡋ࡚ᐃ⩏ࡉࢀࡓ⮬⏤ᗘࢆᙉไⓗ࡟

2

ḟඖᖹ㠃ၥ㢟ࡢᵓ㐀ゎᯒ࡟⏝࠸ࡿ⮬⏤ᗘ࡜ࡋ࡚෌ᐃ⩏ࡋ ࡓ㸬ࡋ࠿ࡋ㸪ࡇࡢࡼ࠺࡟ࢯࣇࢺ࢙࢘࢔ࡢつ⣙࡛ᐃࡵࡽࢀࡓ ᮏ᮶ࡢ⮬⏤ᗘࡢᐃ⩏࡟཯ࡋࡓ᪉ἲࢆ᥇⏝ࡋࡓሙྜ㸪ゎᯒࢯ

ࣇࢺ࢙࢘࢔ࡢࢯ࣮ࢫࢥ࣮ࢻࡀබ㛤ࡉࢀ࡚࠾ࡽࡎࣈࣛࢵࢡ

࣎ࢵࢡࢫ࡜࡞ࡗ࡚࠸ࡿࡓࡵ㸪ࡑࡢጇᙜᛶࢆ᩿ᐃࡍࡿࡇ࡜ࡣ

୍⯡ⓗ࡟ᅔ㞴࡛࠶ࡿ㸬ࡉࡽ࡟㸪つ⣙࡛ᐃࡵࡽࢀࡓチᐜྍ⬟

࡞⠇Ⅼ⮬⏤ᗘᩘࡢไ㝈࡟ࡼࡾ㸪⮬⏤ᗘᩘࡢ኱ᖜ࡞ቑຍࡀᚲ せ࡜࡞ࡿ

3

ḟඖゎᯒ࡞࡝ࡢᣑᙇᛶ࡟ࡶㄢ㢟ࡀṧࡉࢀ࡚࠸ࡿ㸬 ࡋࡓࡀࡗ࡚㸪ࡇࡢXFEMࡢỗ⏝

FEMゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢ

ᐇ⿦᪉ἲࡣ㸪ྜ⌮ᛶ࡟㛵ࡋ࡚ᚲࡎࡋࡶ㐺ษ࡞᪉ἲ࡛࠶ࡿ࡜

ࡣ࠸࠼࡞࠸㸬

௨ୖࡢ⫼ᬒࡼࡾ㸪ᮏ◊✲࡛ࡣ㸪ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢘

࢙࢔

ABAQUS

¹⁵⁾ࢆ࣮࣋ࢫ࡜ࡋ࡚

PU-XFEM

㏆ఝࢆᐇ⿦ࡍ

ࡿࡇ࡜࡛㸪ࢩ࢙ࣝせ⣲࡛ࣔࢹࣝ໬ྍ⬟࡞

3

ḟඖᵓ㐀య࡟Ⓨ

⏕ࡋࡓᯈཌ᪉ྥ㈏㏻ࡁ⿣ࢆᑐ㇟࡜ࡋࡓ⑂ປࡁ⿣㐍ᒎࢩ࣑

࣮ࣗࣞࢩࣙࣥࡀྍ⬟࡜࡞ࡿゎᯒࢥ࣮ࢻࡢ㛤Ⓨࢆ⾜ࡗࡓ㸬≉

࡟㸪⠇Ⅼ⮬⏤ᗘࡢቑຍἲ࡟㛵ࡋ࡚ࡣࢯࣇࢺ࢙࢘࢔ࡢつ⣙࡟

ࡼࡗ࡚ᐃࡵࡽࢀࡓᮏ᮶ࡢ⮬⏤ᗘࡢᐃ⩏ࢆᦆ࡞࠺ࡇ࡜࡞ࡃ㸪

ABAQUS

௨እࡢࢯࣇࢺ࢙࢘࢔࡬ࡶᖜᗈࡃ㐺⏝ྍ⬟࡛࠶ࡾ㸪

ᑗ᮶ⓗ࡞ᣑᙇᛶࢆ⪃៖ࡋࡓ᪉ἲࢆ♧ࡍ㸬

࡞࠾㸪᭱㏆

ABAQUS

࡟࠾࠸࡚

XFEM

ࡀᐇ⿦ࡉࢀ㸪ࡑࢀ

࡟ᇶ࡙ࡃࡁ⿣ゎᯒࡀྍ⬟࡜࡞ࡗࡓ㸬ࡋ࠿ࡋ㸪ࡇࢀࡣ(1)ᚑ᮶ ࡢ

XFEM

ࡢ㏆ఝἲ࡟ᇶ࡙࠸࡚࠾ࡾ㸪

XFEM

ࡢᮏ㉁ⓗ࡞ၥ

㢟࡛࠶ࡿ

BE

ࡢၥ㢟ࡀゎỴࡉࢀ࡚࠸࡞࠸㸪

(2)ࢩ࢙ࣝせ⣲࡟

ࡼࡾࣔࢹࣝ໬ྍ⬟࡞

3

ḟඖᵓ㐀≀࡬ࡢ㐺⏝ࡀ࡛ࡁ࡞࠸㸪࡞

࡝ࡢไ㝈ࡀ࠶ࡾ㸪㗰ᗋ∧࡞࡝」㞧࠿ࡘ኱つᶍ࡞ᐇᵓ㐀≀࡟

Ⓨ⏕ࡋࡓ⑂ປࡁ⿣ࢆホ౯ࡍࡿ࡟࠶ࡓࡾㄢ㢟ࡀ࠶ࡿࡢࡀ⌧

≧࡛࠶ࡿ㸬

2. PU-XFEMࢆ⏝࠸ࡓࡁ⿣ゎᯒࡢࣔࢹࣝ໬

௵ពࡢࡁ⿣ඛ➃㏆ഐࡢኚᙧࡣ㸪࣮ࣔࢻ

I

㸦㛤ཱྀᆺ㸧࣭࣮ࣔ

ࢻ

II

㸦㠃ෆࡏࢇ᩿ᆺ㸧࣭࣮ࣔࢻ

III

㸦㠃እࡏࢇ᩿ᆺ㸧ࡢ

3

✀ 㢮ࡢኚᙧᵝᘧࡢ㔜ࡡྜࢃࡏ࡛⾲⌧ࡉࢀࡿ㸬

ࡋ࠿ࡋ㸪ᅗ

-1

࡟♧ࡍ㗰ᶫࡢࣂࣝࣈࣜࣈ㗰ᗋ∧࡟࠾ࡅࡿࣂ

ࣝࣈࣜࣈ࡜ᶓࣜࣈࡢ஺ᕪ㒊ࡼࡾᶓࣜࣈ࡟Ⓨ⏕ࡋࡓᯈཌ㈏

㏻ࡁ⿣࡞࡝㸪࣮ࣔࢻ

I

࠾ࡼࡧ

II

࡟ࡼࡿ㠃ෆኚᙧ࡜ẚ㍑ࡋ࡚㸪

࣮ࣔࢻ

III

࡟ࡼࡿ㠃እኚᙧࡢᐤ୚ࡣᑠࡉ࠸ሙྜࡀከ࠸㸬ࡑ ࡇ࡛ᮏ◊✲࡛ࡣ㸪ࡁ⿣㏆ഐࡢኚᙧࡀ࣮ࣔࢻ

I

࠾ࡼࡧ࣮ࣔࢻ

II

ࡢ㔜ࡡྜࢃࡏ࡛⾲⌧ྍ⬟࡞㠃ෆኚᙧ࡟ࡼࡾᨭ㓄ࡉࢀ࡚࠸

ࡿࡶࡢ࡜௬ᐃ࡛ࡁ㸪ࢩ࢙ࣝせ⣲࡛ࣔࢹࣝ໬ྍ⬟࡞

3

ḟඖᵓ 㐀యࢆᑐ㇟࡜ࡍࡿ㸬௨ୗ࡛ࡣᖹ㠃ࢩ࢙ࣝせ⣲ࡢᐃᘧ໬࡟࠾

࠸࡚㸪ⴭ⪅ࡽࡀ෌ᐃᘧ໬ࡋࡓ

PU-XFEM

࡟ᇶ࡙ࡃ㏆ఝࢆ㐺

⏝ࡍࡿ㠃ෆኚᙧࡢࣔࢹࣝ໬࡟ࡘ࠸࡚ࡢࡳ㏙࡭ࡿࡶࡢ࡜ࡍ

ࡿ㸬࡞࠾㸪

PU-XFEM

ࡢᐃᘧ໬࡟㛵ࡍࡿヲ⣽ࡣ᪤ ࡢᩥ

⊩¹¹⁾ࢆཧ↷ࡉࢀࡓ࠸㸬ࡲࡓ㸪㠃ෆኚᙧ࡟㛵ࡍࡿᛂຊ̺ࡦࡎ

ࡳ㛵ಀࢆ⾲ࡍᵓᡂ๎ࡣᖹ㠃ᛂຊࢆ௬ᐃࡍࡿࡶࡢ࡜ࡍࡿ㸬

PU-XFEM

࡟ᇶ࡙ࡃࡁ⿣ゎᯒࡢ㠃ෆኚ఩ሙࡢ㏆ఝ

ܝ

ୟ୮

ሺܠሻࡣḟᘧ࡛ᐃ⩏ࡉࢀࡿ㸬

ܝ

ୟ୮

ሺܠሻ ൌ ߮

_଴

ሺܠሻܞ

_଴^ୟ୮

ሺܠሻ ൅ ߮

_஼

ሺܠሻܞ

_஼^ୟ୮

ሺܠሻ (1)

ࡇࡇ࡛㸪

߮

_଴

ሺܠሻ࠾ࡼࡧ ߮

_஼

ሺܠሻࡣ㸪 ߮

_଴

ሺܠሻ ൅ ߮

_஼

ሺܠሻ ൌ ͳࢆ‶

㊊ࡍࡿḟᘧࡢ

Partition of Unity

㸦

PU

㸧࡛࠶ࡿ㸬

߮

_஼

ሺܠሻ ൌ ෍ ߶

_ூ

ሺܠሻ

ூא஼

߮

଴

ሺܠሻ ൌ ͳ െ ߮

_஼

ሺܠሻ

(2)

ࡲࡓ㸪

ܞ

_଴^ୟ୮

ሺܠሻ࠾ࡼࡧ ܞ

_஼^ୟ୮

ሺܠሻࡣ㸪ࡑࢀࡒࢀ PU ߮

଴

ሺܠሻ࠾ࡼ

ࡧ

߮

஼

ሺܠሻࡢࢧ࣏࣮ࢺୖ࡟࠾࠸࡚ḟᘧ࡛ᐃ⩏ࡉࢀࡿ㏻ᖖࡢ

᭷㝈せ⣲㏆ఝ࠾ࡼࡧࡁ⿣ඛ➃㏆ഐ࡟௜ຍࡉࢀࡿ࢚ࣥࣜࢵ

ࢳ࣓ࣥࢺ࡟ᑐᛂࡋࡓ㏆ఝ㛵ᩘ࡛࠶ࡿ㸬

ܞ

_଴^ୟ୮

ሺܠሻ ൌ ෍ ߶

_ூ

ሺܠሻܝ

_ூ

ூ

൅ ෍ ߶

_ூ

ሺܠሻܪሺܠሻ܊

_ூ

ூא௃

ܞ

_஼^ୟ୮

ሺܠሻ ൌ ෍ ߶

_ூ

ሺܠሻ ൭ܝ

_ூ

൅ ෍ ߛ

௞

ሺܠሻ܋

_ூ^௞

ସ

௞ୀଵ

൱

ூ

(3)

ᘧ(2)࠾ࡼࡧ(3)࡟࠾࠸࡚㸪

ܝ

_ூ㸪

܊

_ூ࠾ࡼࡧ

܋

_ூ^௞

ሺ݇ ൌ ͳǡ ǥ ǡͶሻࡣ

⠇Ⅼ⮬⏤ᗘ࡛࠶ࡾ㸪

ܬ

ࡣࡁ⿣୙㐃⥆㠃㏆ഐࡢ⠇Ⅼ㞟ྜ㸪

ܥ

ࡣ ࡁ⿣ඛ➃㏆ഐࡢ⠇Ⅼ㞟ྜ࡛࠶ࡿ㸬ࡲࡓ㸪߶_ூ

ሺܠሻࡣᶆ‽ࡢ᭷

㝈せ⣲㏆ఝ࡛⏝࠸ࡿෆᤄ㛵ᩘ࡛࠶ࡿࡀ㸪ᮏ◊✲࡛ࡣ

4

⠇Ⅼ ᅗ

-1

㗰ᗋ∧࡟࠾࠸࡚ࣂࣝࣈࣜࣈ࡜ᶓࣜࣈࡢ஺ᕪ㒊ࡼࡾ

ᶓࣜࣈ࡟Ⓨ⏕ࡋࡓᯈཌ㈏㏻ࡁ⿣

D ^RRCC :_C :C

⠇Ⅼ㞟ྜ

: C : C' : J : C' and J ࡁ⿣ඛ➃

せ⣲ࢆᵓᡂࡍࡿ཮

1

ḟ㛵ᩘࢆ⏝࠸ࡿ㸬㛵ᩘ

ܪሺܠሻ

࠾ࡼࡧ

ߛ

_௞

ሺܠሻሺ݇ ൌ ͳǡ ǥ ǡͶሻ

ࡣᚑ᮶ࡢ

XFEM

࡛⏝࠸ࡽࢀࡿࡶࡢ࡜

ྠ୍ࡢḟᘧ࡛୚࠼ࡽࢀࡿ࢚ࣥࣜࢵࢳ㛵ᩘ࡛࠶ࡿ㸬

ܪሺܠሻ ൌ ൜ ͳ ‘π

ା

െͳ ‘π

ି

(4)

ߛ

_ଵ

ሺܠሻ ൌ ξݎ …‘• ߠ

ʹ ǡ ߛ

_ଶ

ሺܠሻ ൌ ξݎ •‹ ߠ ʹ ǡ ߛ

ଷ

ሺܠሻ ൌ ξݎ …‘• ߠ

ʹ •‹ ߠ ǡ ߛ

ସ

ሺܠሻ ൌ ξݎ •‹ ߠ ʹ •‹ ߠ

(5)

ࡇࡇ࡛㸪ᘧ

(4)ࡢ࢚ࣥࣜࢵࢳ㛵ᩘ ܪሺܠሻࡣࡁ⿣㠃࡟࠾ࡅࡿኚ

఩ࡢ୙㐃⥆ᛶࢆ⾲ࡍ

Heaviside

ᇶᗏ࡛࠶ࡿ¹⁶⁾㸬ࡲࡓ㸪

π

ା࠾

ࡼࡧ

π

_ିࡣ㸪ࡑࢀࡒࢀ࢚ࣥࣜࢵࢳ㛵ᩘ

ܪሺܠሻࡀ௜ຍࡉࢀࡿ

⠇Ⅼ

ܬࡢࢧ࣏࣮ࢺ π

_௃ୖ࡛ᐃ⩏ࡉࢀࡿ㸪ࡁ⿣୙㐃⥆㠃࡟ᑐ ࡍࡿୖഃ࠾ࡼࡧୗഃࡢ㒊ศ㡿ᇦ࡛࠶ࡿ㸬ࡲࡓ㸪ᘧ(5)ࡢ࢚ࣥ

ࣜࢵࢳ㛵ᩘ

ߛ

௞

ሺܠሻሺ݇ ൌ ͳǡ ǥ ǡͶሻࡣ㸪ࡁ⿣ඛ➃㏆ഐኚ఩ሙࡢ

₞㏆ゎࡢᇶᗏ࡛࠶ࡾ㸪ݎ࠾ࡼࡧ

ߠࡣࡁ⿣ඛ➃ࢆཎⅬ࡜ࡋࡓ

ᴟᗙᶆ⣔࡛࠶ࡿ¹⁷⁾㸬

௨ୖࡢᐃᘧ໬ࡼࡾ㸪

PU-XFEM

࡟ᇶ࡙ࡃ㏆ఝἲ࡟࠾࠸࡚㸪 ࡁ⿣ඛ➃㏆ഐ࡟௜ຍࡉࢀࡿ࢚ࣥࣜࢵࢳ࣓ࣥࢺࡣᚑ᮶ࡢ

XFEM

ࡢࡼ࠺࡞⠇Ⅼ༢఩ࡢ௜ຍ࡟ࡼࡗ࡚࡛ࡣ࡞ࡃ㸪PU

߮

஼

ሺܠሻࡢࢧ࣏࣮ࢺୖࡢ㡿ᇦ࣮࣋ࢫ࡛ᐃ⩏ࡉࢀࡿ㸬

ᮏ◊✲࡛ࡣ㸪ᘧ

(2)࠾ࡼࡧᘧ (3)࡛⏝࠸ࡓ⠇Ⅼ㞟ྜ ܬ࠾ࡼࡧ ܥ

ࢆ௨ୗࡢࡼ࠺࡟ᐃ⩏ࡍࡿ㸬

ܬ ൌ ሼܫ א ܰȁπ

ூ

ת ܦ ് ߶ǡ ܫ ב ܥሽ (6) ܥ ൌ ൛ܫ א ܰหหܠ

ூ

െ ܠ

୲୧୮

ห ൑ ܴ

஼

ൟ (7)

ࡇࡇ࡛㸪

ܠ

ூ࠾ࡼࡧ

ܠ

୲୧୮ࡣࡑࢀࡒࢀ⠇Ⅼ

ܫ࠾ࡼࡧࡁ⿣ඛ➃ࡢ

ᗙᶆ㸪

π

_ூࡣෆᤄ㛵ᩘ

߶

_ூ

ሺܠሻࡢࢧ࣏࣮ࢺ࡛࠶ࡾ㸪 ܦࡣࡁ⿣ࡢ

୙㐃⥆㠃࡛࠶ࡿ㸬࡞࠾㸪⠇Ⅼ㞟ྜ

ܥ

ࡢᐃ⩏༙ᚄ

ܴ

஼ࡣ㸪ධ ຊ್࡜ࡋ࡚ᢅ࠺㸬

ࡲࡓ㸪

PU-XFEM

࡟ᇶ࡙ࡃᐃᘧ໬࡟࠾࠸࡚㸪࢚ࣥࣜࢵࢳ

㛵ᩘ

ߛ

_௞

ሺܠሻሺ݇ ൌ ͳǡ ǥ ǡͶሻ

࡟ᑐᛂࡍࡿ⮬⏤ᗘ

܋

_ூ^௞

ሺ݇ ൌ ͳǡ ǥ ǡͶሻ

ࡣ㸪

PU ߮

஼

ሺܠሻࡢࢧ࣏࣮ࢺ

π_஼ୖࡢࡍ࡭࡚ࡢ⠇Ⅼ࡛

௜ຍࡍࡿᚲせࡀ࠶ࡿ㸬ࡑࡇ࡛㸪ḟᘧ࡟ࡼࡾ⠇Ⅼ㞟ྜ

ܥ

^ᇱࢆ ᐃ⩏ࡍࡿ㸬

ܥ

^ᇱ

ൌ ሼ ܫ א ܰ ȁ π

_ܫ

ת π

_ܥ

് ߶ǡ ܫ ב ܥ ሽ (8)

௨ୖࡢᘧ

(6) ~ (8)࡛ᐃ⩏ࡉࢀࡓ⠇Ⅼ㞟ྜ࡟㛵ࡍࡿࣔࢹࣝ

໬ࡢᶍᘧᅗࢆᅗ-2 ࡟♧ࡍ㸬࡞࠾㸪⠇Ⅼ㞟ྜ

ܬ

࠾ࡼࡧ

ܥ

^ᇱ࡟ ࡣ㸪ඹ㏻㒊ศࡀᏑᅾࡍࡿࡇ࡜࡟ὀពࡉࢀࡓ࠸㸬

PU-XFEM

࡟࠾ࡅࡿྛ⠇Ⅼ࡛᭷ຠ࡜࡞ࡿ⮬⏤ᗘࡣ㸪ᘧ

(6)

~ (8)࡛ᐃ⩏ࡋࡓࡑࢀࡒࢀࡢ⠇Ⅼ㞟ྜ࡟ᑐᛂࡋ࡚ᐃ⩏ࡍࡿ

ࡇ࡜ࡀ࡛ࡁࡿ㸬ࡍ࡞ࢃࡕ㸪࢚ࣥࣜࢵࢳ㛵ᩘ

ܪሺܠሻ

࡟ᑐᛂࡍ

ࡿ⮬⏤ᗘ

܊

_ூࡣ⠇Ⅼ㞟ྜ

ܬ࡟ྵࡲࢀࡿ⠇Ⅼ࡟௜ຍࡉࢀ㸪࢚

ࣥࣜࢵࢳ㛵ᩘ

ߛ

௞

ሺܠሻሺ݇ ൌ ͳǡ ǥ ǡͶሻ

࡟ᑐᛂࡍࡿ⮬⏤ᗘ

܋

_ூ^௞

ሺ݇ ൌ ͳǡ ǥ ǡͶሻࡣ⠇Ⅼ㞟ྜ ܥ

࠶ࡿ࠸ࡣ

ܥ

^ᇱ࡟ྵࡲࢀࡿ⠇

Ⅼ࡟௜ຍࡉࢀࡿ㸬࡞࠾㸪ࡑࢀࡒࢀࡢ⠇Ⅼ㞟ྜ࡟ᑐᛂࡋࡓල యⓗ࡞⠇Ⅼ⮬⏤ᗘᩘ࡟㛵ࡋ࡚ࡣ㸪3❶ࢆཧ↷ࡉࢀࡓ࠸㸬

࡞࠾㸪ࡇࡇ࡛♧ࡋࡓ

PU-XFEM

㏆ఝ࡟㛵ࡍࡿᐃᘧ໬࠾ࡼ

ࡧ஦๓ࡢㄗᕪゎᯒࡢヲ⣽ࡸᇶᮏⓗ࡞ᩘ್ゎᯒ౛ࡣ㸪᪤ ࡢ

ᩥ⊩¹¹⁾ࢆཧ↷ࡉࢀࡓ࠸㸬

3. ỗ⏝ FEM ゎᯒࢯࣇࢺ࢙࢘࢔ ABAQUS ࡬ࡢᐇ⿦

ᮏ◊✲࡛ࡣ㸪௦⾲ⓗ࡞ỗ⏝ࡢ

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢ ࡦ࡜ࡘ࡛࠶ࡿ

ABAQUS

¹⁵⁾ࢆ࣮࣋ࢫ࡜ࡋ࡚㸪

PU-XFEM

㏆ఝ

ࢆᐇ⿦ࡋ㸪

3

ḟඖᵓ㐀య࡟Ⓨ⏕ࡋࡓᯈཌ᪉ྥ㈏㏻ࡁ⿣ࢆᑐ

㇟࡜ࡋࡓ⑂ປࡁ⿣ࡢ㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࡀᐇ᪋ྍ⬟࡜

࡞ࡿゎᯒࢥ࣮ࢻࡢ㛤Ⓨࢆ⾜ࡗࡓ㸬ᮏゎᯒࢥ࣮ࢻࡣ㸪

ABAQUS

࡟௨ୗࡢࣉࣟࢢ࣒ࣛࢆ᪂ࡓ࡟㏣ຍࡍࡿࡇ࡜࡛ᵓ

ᡂࡉࢀࡿ㸬

PRE-PRO:

ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡟

PU-XFEM

㏆ ఝࢆᐇ⿦ࡍࡿࣉࣟࢢ࣒ࣛ

POST-PRO:

ᛂຊᣑ኱ಀᩘࡢ⟬ฟ㸪⑂ປࡁ⿣㐍ᒎ᪉ྥ࣭ఏ

᧛㏿ᗘࡢホ౯࠾ࡼࡧࡁ⿣ᙧ≧ࡢ᭦᪂ࢆ⾜࠺

ࣉࣟࢢ࣒ࣛ

XFEM3DS:

ゎᯒࢥ࣮ࢻ඲యࡢࣇ࣮ࣟࢆไᚚࡍࡿࣉࣟࢢ

࣒ࣛ

ᮏ❶࡛ࡣ㸪ࡲࡎ㸪

3.1

⠇࡟࠾࠸࡚㸪ゎᯒ඲యࡢࣇ࣮ࣟࢆ

♧ࡍ࡜࡜ࡶ࡟㸪ࡑࢀࢆไᚚࡍࡿࣉࣟࢢ࣒ࣛ

XFEM3DS

࡟ 㛵ࡋ࡚ᴫㄝࡍࡿ㸬ḟ࡟㸪3.2 ⠇࡟࠾࠸࡚㸪ࣉࣟࢢ࣒ࣛ

PRE-PRO㸪 3.3

⠇࡟࠾࠸࡚㸪ࣉࣟࢢ࣒ࣛ

POST-PRO

࡟㛵 ࡋ࡚㸪ࡑࢀࡒࢀࡢᴫㄝࡍࡿ㸬ࡇࡇ࡛≉࡟㸪

3.2

⠇࡛ࡣ㸪

XFEM

ࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢᐇ⿦࡟࠾࠸࡚㔜せ࡞ㄢ 㢟࡛࠶ࡿ⠇Ⅼ⮬⏤ᗘࡢቑຍἲ࡟㛵ࡋ࡚㸪

ABAQUS

ࡢつ⣙

࡟࠾࠸࡚ᐃࡵࡽࢀࡓᮏ᮶ࡢ⮬⏤ᗘࡢᐃ⩏ࢆᦆ࡞࠺ࡇ࡜࡞

ࡃ㸪࠿ࡘ

ABAQUS

௨እࡢࢯࣇࢺ࢙࢘࢔࡬ࡶᖜᗈࡃ㐺⏝ྍ

⬟࡛࠶ࡾ㸪ᑗ᮶ⓗ࡞ᣑᙇᛶࢆ⪃៖ࡋࡓ᪉ἲࢆ♧ࡍ㸬

࡞࠾㸪௨ୗࡢグ㏙࡟࠾࠸࡚㸪

ABAQUS

࡛⏝࠸ࡿࡇ࡜ࡀ

ྍ⬟࡞ྛ࢜ࣉࢩࣙࣥࡢヲ⣽ࡣ㸪

ABAQUS User’s Manual

¹⁵⁾

ࢆཧ↷ࡉࢀࡓ࠸㸬

ᅗ-2

PU-XFEM

࡟ᇶ࡙ࡃࡁ⿣࡜⠇Ⅼ㞟ྜࡢࣔࢹࣝ໬

ABAQUS CAE (ᵓ㐀యࡢࣔࢹࣝ໬ INPUT.inpࡢసᡂ)

ᶆ‽ࡢせ⣲๛ᛶ

࣐ࢺࣜࢡࢫ⟬ฟ

඲య⣔๛ᛶ࣐ࢺࣜࢡࢫ⤌ࡳ❧࡚

⤌ࡳ㎸ࡳࢯࣝࣂ࣮࡟ࡼࡿ㐃❧᪉⛬ᘧィ⟬

⮬⏤ᗘࡢゎࡢฟຊXn.dat ኚ఩ሙ࣭ᛂຊሙࡢ

ྍど໬

PRE-PRO

*USER ELEMENT࡟ࡼࡿ

XFEM㏆ఝࡢᑟධ

POST-PRO ᛂຊᣑ኱ಀᩘࡢ⟬ฟ࣭

⑂ປࡁ⿣㐍ᒎ᪉ྥ࣭㏿ᗘࡢホ౯

ABAQUS ࡁ⿣ᙧ≧ࢹ࣮ࢱ CRACK.dat (⧞㏉ࡋィ⟬ẖ࡟᭦᪂)

XFEM㏆ఝࢆ⪃៖ࡋࡓ せ⣲๛ᛶ࣐ࢺࣜࢡࢫ

ㄞࡳ㎸ࡳ

(a)

(b), (f)

(c)

(e)

(h) ABAQUS SOLVER

(d)

3.1 ゎᯒ඲యࡢࣇ࣮ࣟ࡜ࡑࡢไᚚ

ᮏゎᯒࢥ࣮ࢻࡣ㸪ᅗ

-3

࡟♧ࡍࡼ࠺࡟㸪᪤Ꮡࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔

ABAQUS

࡟㸪

PRE-PRO

㸦

ABAQUS

࡟

PU-XFEM

㏆ఝࢆᑟධࡍࡿࣉࣟࢢ࣒ࣛ㸧࠾ࡼࡧ

POST-PRO

㸦ᛂຊᣑ኱ಀᩘࡢ⟬ฟ㸪⑂ປࡁ⿣㐍ᒎ᪉ྥ࣭ఏ᧛㏿ᗘࡢホ ౯࡜ࡁ⿣ᙧ≧ࡢ᭦᪂ࢆ⾜࠺ࣉࣟࢢ࣒ࣛ㸧ࢆ㏣ຍࡋ㸪

XFEM3DS

࡟ࡼࡾゎᯒ඲యࡀไᚚࡉࢀࡿ㸬ᮏゎᯒࢥ࣮ࢻࡢ

ලయⓗ࡞ᡭ㡰ࢆ௨ୗ࡟♧ࡍ㸬

(a) ABAQUS CAE

ࢆ฼⏝ࡋ࡚㸪ࡁ⿣ࢆྵࡲ࡞࠸೺඲

࡞ᵓ㐀యࡢ

FEM

ゎᯒࣔࢹࣝࢆసᡂࡋ㸪඲య⣔๛

ᛶ࣐ࢺࣜࢡࢫࡢ⤌ࡳ❧࡚࠾ࡼࡧ㐃❧᪉⛬ᘧࡢィ

⟬ࢆᐇ⾜ࡍࡿỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢࢯࣝ

ࣂ࣮࡬┤᥋ࢪࣙࣈ࡜ࡋ࡚ᢞධྍ⬟࡞ධຊࣇ࢓࢖

ࣝ㸦INPUT.inp㸧࡬ࡢ᭩ࡁฟࡋࢆ⾜࠺㸬

(b) (a)

࡛సᡂࡉࢀࡓゎᯒࣔࢹࣝ࡟ᗙᶆࢆᩚྜࡉࡏࡓ

ึᮇࡁ⿣ᙧ≧ࢹ࣮ࢱ㸦Crack.dat㸧ࢆᐃ⩏ࡍࡿ㸬

(c) (b)

ࡲ ࡓ ࡣ

(f)

࡛ ᐃ ⩏ ࡋ ࡓ ࡁ ⿣ ᙧ ≧ ࢹ ࣮ ࢱ 㸦Crack.dat㸧ࢆཧ↷ࡋ㸪(a)࡛సᡂࡉࢀࡓධຊࣇ

࢓࢖ࣝ㸦INPUT.inp㸧ࡢ୍㒊ࢆ㸪

PU-XFEM

㏆ఝ

ࢆᑟධࡍࡿࡓࡵ࡟᭩ࡁ᥮࠼㸪᪂ࡓ࡞ධຊࣇ࢓࢖

ࣝ㸦Xn.inp㸪ࡓࡔࡋnࡣゎᯒࢫࢸࢵࣉᩘ㸧ࡢసᡂ ࡀྍ⬟࡞ࣉࣟࢢ࣒ࣛ

PRE-PRO

ࢆᐇ⾜ࡍࡿ㸬

(d) (c)

࡛సᡂࡉࢀࡓ

PU-XFEM

㏆ఝࢆᑟධࡋࡓධຊ

ࣇ࢓࢖ࣝ㸦Xn.inp㸧ࢆ⏝࠸࡚㸪ỗ⏝

FEM

ゎᯒࢯ

ࣇࢺ࢙࢘࢔ࡢࢯࣝࣂ࣮࡬ࢪࣙࣈࡢᢞධࢆ⾜࠺㸬 ࡇࢀ࡟ࡼࡾ㸪඲య⣔๛ᛶ࣐ࢺࣜࢡࢫࡢ⤌ࡳ❧࡚

࠾ࡼࡧ㐃❧᪉⛬ᘧࡢィ⟬ࡀᐇ⾜ࡉࢀ㸪ྛ⠇Ⅼ⮬

⏤ᗘ࡟㛵ࡍࡿᩘ್ゎࢆࢹ࣮ࢱ࡜ࡋ࡚ྵࡴฟຊࣇ

࢓࢖ࣝ㸦Xn.dat㸧ࡀฟຊࡉࢀࡿ㸬

(e) (d)ࡢィ⟬࡟ࡼࡾᚓࡽࢀࡓฟຊࣇ࢓࢖ࣝ㸦Xn.dat㸧

ࢆཧ↷ࡋ㸪ᛂຊᣑ኱ಀᩘࡢ⟬ฟ㸪⑂ປࡁ⿣㐍ᒎ ᪉ ྥ ࣭ ఏ ᧛ ㏿ ᗘ ࡢ ホ ౯ ࡀ ྍ ⬟ ࡞ ࣉ ࣟ ࢢ ࣛ ࣒

POST-PRO

ࢆᐇ⾜ࡍࡿ㸬

⾲-1

ABAQUS

ࡢつ⣙¹⁵⁾࡛ᐃ⩏ࡉࢀࡓ⠇Ⅼ⮬⏤ᗘ

⮬⏤ᗘ␒ྕ ᐃ⩏

1 – 3 x, y, z - ኚ఩㸦඲యᗙᶆ⣔㸧

4 - 6 x, y, z - ㍈ࡲࢃࡾࡢᅇ㌿㸦඲యᗙᶆ⣔㸧

7 ཯ࡾࡢ኱ࡁࡉ㸦㛤᩿㠃ࡣࡾせ⣲࡟ᑐࡋ࡚㸧 8 㛫㝽ᅽ, 㟼Ỉᅽὶయࡢᅽຊ㸪㡢ᅽ

9 㟁఩

10 ࢥࢿࢡࢱᮦᩱὶࢀ㸦㛗ࡉࡢ༢఩㸧 11  ᗘ㸦㉁㔞ᣑᩓゎᯒ࡛ࡣṇつ໬⃰ᗘ㸧

12 ➨2 ᗘ

13 ➨3 ᗘ

14 Etc.

(f) (e)

ࡢ⑂ປࡁ⿣㐍ᒎ᪉ྥࡢホ౯⤖ᯝ࡟ᇶ࡙ࡁ㸪ࡁ

⿣ᙧ≧ࢹ࣮ࢱ㸦Crack.dat㸧ࡢ᭦᪂ࢆ⾜࠺㸬

(g)

ࡁ⿣㛗ࡉࡸᛂຊᣑ኱ಀᩘ➼࡟ࡼࡾᣦᐃࡋࡓゎᯒ

ࡢ⤊஢᮲௳ࢆ‶㊊ࡍࡿࡲ࡛㸪(c) ~ (f)ࡢ⾜⛬ࢆ⧞

ࡾ㏉ࡋᐇ⾜ࡍࡿ㸬

(h)

ᚲせ࡟ᛂࡌ࡚㸪ྍど໬ࢯࣇࢺ࢙࢘࢔

TECPLOT

¹⁸⁾

࡟࡚ㄞࡳ㎸ࡳࡀྍ⬟࡞ࡁ⿣㐍ᒎ࡟క࠺ゎᯒࢫࢸ

ࢵ ࣉ ẖ ࡢ ኚ ఩ ሙ ࣭ ᛂ ຊ ሙ ࡢ ฟ ຊ ࣇ ࢓ ࢖ ࣝ 㸦

DSn.dat

࠾ࡼࡧ

STn.dat

㸧ࢆฟຊࡍࡿ㸬 ࡇࢀ࡟ࡼࡾ㸪

PU-XFEM

㏆ఝࢆỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘

࢔

ABAQUS

࡟ᑟධ࡛ࡁ㸪⑂ປࡁ⿣ࢆᐜ᫆࡟ࣔࢹࣝ໬ࡋ㸪

ࡑࡢ㐍ᒎᣲືࢆࢩ࣑࣮ࣗࣞࢺࡍࡿࡇ࡜ࡀྍ⬟࡜࡞ࡿ㸬

3.2

ỗ⏝FEMゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢPU-XFEM㏆ఝࡢᐇ

⿦

ࡇࡇ࡛ࡣ㸪ᮏゎᯒࢥ࣮ࢻ࡛≉࡟㔜せ࡜࡞ࡿỗ⏝

FEM

ゎ ᯒࢯࣇࢺ࢙࢘࢔࡬ࡢ

PU-XFEM

㏆ఝࡢᐇ⿦࡟㛵ࡋ࡚㏙࡭ࡿ㸬

ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢ

XFEM

㏆ఝࡢᑟධࡣ㸪 ࢯࣇࢺ࢙࢘࢔࡟ᐇ⿦ࡉࢀࡓ௵ពࡢせ⣲ࢆᐃ⩏ࡍࡿᶵ⬟ࢆ

⏝࠸ࡿࡇ࡜࡛ྍ⬟࡜࡞ࡿ㸬౛࠼ࡤ㸪ᮏ◊✲࡛⏝࠸ࡓ

ABAQUS

ࡢሙྜ㸪ࡑࡢయ⣔ࢆ⥔ᣢࡍࡿࡓࡵ࡟ᐃࡵࡽࢀࡓ

つ⣙ࡢ⠊ᅖෆ࡛௵ពࡢせ⣲ࢆᐃ⩏ࡋ㸪ゎᯒ࡟⤌ࡳ㎸ࡴࡇ࡜

ࡀྍ⬟࡞࢜ࣉࢩࣙࣥ

*USER ELEMENT

ࡀ⏝ពࡉࢀ࡚࠸ࡿ㸬 ࡓࡔࡋ㸪ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬

XFEM

㏆ఝࢆᑟ ධࡋࡓせ⣲ࢆᐃ⩏ࡍࡿࡓࡵ࡟ࡣ㸪ࡁ⿣㏆ഐࡢྛ⠇Ⅼ࡛⮬⏤

ᗘᩘࡢቑຍࡀᚲせ࡜࡞ࡾ㸪ࡑࡢᐇ⿦᪉ἲࡀ㔜せ࡞ㄢ㢟࡜࡞

ࡿ㸬౛࠼ࡤ

ABAQUS

࡛ࡣ㸪ࡑࡢつ⣙࡟ࡼࡾ⠇Ⅼ༢఩࡛ኚ

఩ሙࢆᵓᡂࡍࡿ⮬⏤ᗘࡢ␒ྕࡀ⾲-1ࡢࡼ࠺࡟つᐃࡉࢀ࡚

࠾ࡾ㸪

XFEM

㏆ఝࡢᑟධ࡟ᚲせ࡞ኚ఩ሙࡢᵓᡂ࡟㛵ࢃࡿ⠇

Ⅼ⮬⏤ᗘᩘࢆ㸪༢⣧࡟ࡣ㏣ຍ࡛ࡁ࡞࠸ࡇ࡜ࡀၥ㢟࡜࡞ࡿ㸬 ࡑࡇ࡛௨ୗ࡛ࡣ㸪ࡲࡎ

(1)

㡯࡟࠾࠸࡚㸪ᮏ◊✲࡛⏝࠸ࡿከ 㔜⠇Ⅼ࡟ࡼࡿ⠇Ⅼ⮬⏤ᗘቑຍἲࢆ♧ࡍ㸬ḟ࡟

(2)㡯࡟࠾࠸࡚㸪

ලయⓗ࡞

PU-XFEM

㏆ఝࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬

ᅗ

-3 ABAQUS

࡟PU-XFEM㏆ఝࢆᐇ⿦ࡍࡿࡇ࡜࡛㛤Ⓨࡋࡓ

⑂ປࡁ⿣㐍ᒎゎᯒࢥ࣮ࢻࡢࣇ࣮ࣟࢳ࣮ࣕࢺ 㸦ゎᯒ඲యࡣXFEM3DS࡟ࡼࡾไᚚࡉࢀࡿ㸧

⾲

-2 XFEM

㏆ఝࢆᑟධࡍࡿせ⣲ࡢ㏣ຍ⠇Ⅼ࠾ࡼࡧ

⮬⏤ᗘ

(a) 4⠇Ⅼᖹᯈࢩ࢙ࣝせ⣲࡟࠾ࡅࡿᶆ‽ࡢ᭷㝈せ⣲㏆ఝ࡟

ᑐᛂࡍࡿ⠇Ⅼᩘ࠾ࡼࡧ⮬⏤ᗘᩘ

⠇Ⅼᩘ ⮬⏤ᗘᩘ

ᶆ‽ࡢ᭷㝈せ⣲㏆ఝ 4 24 (4 × 6) (b) ྛ⠇Ⅼ㞟ྜ࡟ᑐᛂࡍࡿ㏣ຍ⠇Ⅼᩘ࠾ࡼࡧ⮬⏤ᗘࡢቑຍ

㏣ຍ⠇Ⅼᩘ ㏣ຍ⮬⏤ᗘᩘ

ܥ- node + 4 + 12 (4 × 3)

ܥ^ᇱnode + 4 + 12 (4 × 3)

ܬ

-

node + 1 + 3 (1 × 3)

൫ܬ ൅ ܥ^ƍ൯- node + 5 + 15 (5 × 3)

ࡢᐇ⿦ἲࢆ♧ࡍࡓࡵ࡟㸪

ABAQUS

࡟ᑐᛂࡋࡓධຊࣇ࢓࢖

ࣝసᡂࣉࣟࢢ࣒ࣛ

PRE-PRO

ࡢᴫせࢆ♧ࡍ㸬

(1)

ከ㔜⠇Ⅼ࡟ࡼࡿ⠇Ⅼ⮬⏤ᗘࡢቑຍἲ

⠇Ⅼ⮬⏤ᗘࢆቑຍࡉࡏࡿ࡟࠶ࡓࡾ㸪ᑐ㇟࡜࡞ࡿ⠇Ⅼ࡜ྠ

୍ࡢᗙᶆୖ࡟᪂ࡓ࡟」ᩘࡢ⠇Ⅼࢆタ⨨ࡍࡿࡇ࡜࡟ࡼࡾᚲ せ࡜࡞ࡿ⮬⏤ᗘᩘࢆ☜ಖࡍࡿࡇ࡜ࡀྍ⬟࡜࡞ࡿ㸬ࡍ࡞ࢃࡕ㸪

ྠ୍ࡢᗙᶆୖ࡟」ᩘࡢ⠇Ⅼࢆࠕከ㔜⠇Ⅼࠖ࡜ࡋ࡚ᐃ⩏ࡍࡿ

ࡇ࡜࡟ࡼࡾ㸪⤖ᯝⓗ࡟

1

⠇Ⅼ࠶ࡓࡾࡢ⮬⏤ᗘࢆቑຍࡉࡏࡿ

ࡇ࡜ࡀ࡛ࡁࡿ㸬ᮏゎᯒࢥ࣮ࢻ࡛ࡣ㸪

ABAQUS

࡟⿦ഛࡉࢀ

ࡓ࢜ࣉࢩࣙࣥ

*USER ELEMENT

ࢆ⏝࠸࡚

PU-XFEM

㏆ఝ

ࢆᐇ⿦ࡍࡿࡀ㸪ࡇࡢ᪉ἲࢆ⏝࠸ࡿࡇ࡜࡛㸪ࢯࣇࢺ࢙࢘࢔ࡢ つ⣙࡟࠾ࡅࡿ⠇Ⅼ⮬⏤ᗘࡢᐃ⩏࡟཯ࡍࡿࡇ࡜࡞ࡃ⠇Ⅼ⮬

⏤ᗘࢆቑຍࡉࡏࡿࡇ࡜ࡀྍ⬟࡜࡞ࡿ㸬

௨ୗ࡟㸪ලయⓗ࡞⮬⏤ᗘࡢቑຍἲࢆ㏙࡭ࡿ㸬

ᮏ◊✲࡛ࡣ㸪

3

ḟඖၥ㢟ࢆᑐ㇟࡜ࡋ࡚⏝࠸ࡽࢀࡿ⠇Ⅼ࠶

ࡓࡾ

6

⮬⏤ᗘ㸦୪㐍

3

⮬⏤ᗘ㸪ᅇ㌿

3

⮬⏤ᗘ㸧ࢆ᭷ࡋ㸪ྜ

ィ

24

⮬⏤ᗘ࡛ᵓᡂࡉࢀࡿ

4

⠇Ⅼᖹᯈࢩ࢙ࣝせ⣲ࢆᇶ‽ࡢ せ⣲࡜ࡋ࡚⏝࠸ࡿ㸬

ࡇࡇ࡛㸪

2

❶࡛㏙࡭ࡓࡼ࠺࡟㸪ᮏ◊✲ࡢ⠊ᅖ࡛ࡣ

PU-XFEM

㏆ఝࡢᐇ⿦࡟㛵ࡋ࡚㠃እኚᙧࢆ⪃៖ࡋ࡞࠸㸬ࡇ

ࡢࡓࡵ㸪ᅇ㌿⮬⏤ᗘࢆ┬␎ࡋ㸪୪㐍࡟㛵ࡍࡿ

3

⮬⏤ᗘࡢࡳ

ࢆ᭷ࡍࡿ⠇Ⅼࢆ⮬⏤ᗘቑຍࡢࡓࡵࡢ㏣ຍ⠇Ⅼ࡜ࡋ࡚⏝࠸

ࡿ㸬ࡇࢀ࡟ࡼࡾ㸪⾲

-2

࡟♧ࡍࡼ࠺࡟㸪

PU-XFEM

㏆ఝࡢᑟ ධ࡟࠶ࡓࡾ⮬⏤ᗘࡢቑຍࡀᚲせ࡞⠇Ⅼࡣ㸪ᘧ

(6) ~ (8)

࡛ᐃ⩏

ࡋࡓ⠇Ⅼ㞟ྜ㸦ᅗ

-2

ཧ↷㸧࡟ᑐᛂࡍࡿᩘࡢ⠇Ⅼࢆከ㔜⠇Ⅼ

࡜ࡋ࡚㏣ຍࡍࡿࡇ࡜࡛㸪⮬⏤ᗘࢆቑຍࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬

౛࠼ࡤ㸪ᅗ

-4

࡟♧ࡍせ⣲ࢆ⪃࠼ࡓሙྜ㸪せ⣲ࢆᵓᡂࡍࡿ

⠇Ⅼࡢ࠺ࡕ㸪⠇Ⅼ

3

ࡣ⠇Ⅼ㞟ྜ

ܬ

࠾ࡼࡧ⠇Ⅼ㞟ྜ

ܥ

^ƍ࡟ྵࡲ

ࢀࡿࡓࡵ

6

㔜⠇Ⅼ࡜ࡋ࡚ࣔࢹࣝ໬ࡋ㸪⠇Ⅼ

4

ࡀ⠇Ⅼ㞟ྜ

ܬ

࡟

ྵࡲࢀࡿࡇ࡜࠿ࡽ

2

㔜⠇Ⅼ࡜ࡋ࡚ࣔࢹࣝ໬ࡍࡿ㸬ࡇࡢࡓࡵ㸪 ࡇࡢせ⣲ࢆᐃ⩏ࡍࡿࡢ࡟ᚲせ࡞⠇Ⅼᩘ࠾ࡼࡧ⮬⏤ᗘᩘࡣ ࡑࢀࡒࢀ㸪⠇Ⅼᩘ㸸

1 + 1 + 6 + 2 = 10

㸪⮬⏤ᗘᩘ㸸

6 + 6 + 21 + 9 = 42

࡜࡞ࡿ㸬

ᮏゎᯒࢥ࣮ࢻ࡛⏝࠸ࡓ⠇Ⅼ⮬⏤ᗘࡢቑຍἲࡣ㸪௨ୗ࡟♧

⠇Ⅼ㞟ྜ

: C : C' : J : C' and J

3

⠇Ⅼᩘ: 1

⮬⏤ᗘᩘ: 6

せ⣲඲యࡢᐃ⩏

⠇Ⅼᩘ: 10

⮬⏤ᗘᩘ: 42

4

1 2

⠇Ⅼᩘ: 1

⮬⏤ᗘᩘ: 6

⠇Ⅼᩘ: 6

⮬⏤ᗘᩘ: 21

⠇Ⅼᩘ: 2

⮬⏤ᗘᩘ: 9

ࡍ≉ᚩࢆ᭷ࡍࡿ㸬

࣭ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢつ⣙࡟ࡼࡗ࡚ᐃࡵࡽࢀࡓ ᮏ᮶ࡢ⮬⏤ᗘࡢᐃ⩏ࢆᦆ࡞࠺ࡇ࡜ࡀ࡞࠸ࡓࡵ㸪ࣉࣟࢢࣛ

࣒ࡢᐇ⿦࡟࠾࠸࡚ẚ㍑ⓗᐜ᫆࡟

XFEM

㏆ఝࡢࢯࣇࢺ࢘

࢙࢔࡬ࡢᐇ⿦ࡀྍ⬟࡜࡞ࡿ㸬

࣭ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢከࡃࡢᶵ⬟ࢆ⥔ᣢࡋ฼⏝

ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬

࣭௵ពࡢ⠇Ⅼ࡟௵ពࡢ⮬⏤ᗘࢆ㏣ຍྍ⬟࡛࠶ࡿࡇ࡜࠿ࡽ㸪 ࢯࣇࢺ࢙࢘࢔ࡢつ⣙࡟ไ㝈ࡉࢀࡿࡇ࡜࡞ࡃ㸪

3

ḟඖၥ㢟

࡬ࡢᣑᙇ࡞࡝࡟ᚲせ࡜࡞ࡿ⮬⏤ᗘࡢ኱ᖜ࡞ቑຍࢆ㸪ᐜ᫆

࡟㐩ᡂࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬

࣭ABAQUS ࡜࠸࠺ಶูࡢࢯࣇࢺ࢙࢘࢔࡟ࡢࡳᑐᛂࡋࡓࡶ

ࡢ࡛ࡣ࡞ࡃ㸪௚ࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢᛂ⏝

ࡶྍ⬟࡞୍⯡ⓗ࡞⪃࠼᪉ࢆ୚࠼ࡿࡶࡢ࡛࠶ࡿ㸬

(2) PU-XFEM

㏆ఝࡢᐇ⿦ࡢᡭ㡰

⑂ປࡁ⿣ࡣ㸪ᚤᑠኚᙧࢆ௬ᐃࡋࡓᙎᛶᏛࡢ⠊ᅖෆ࡛ࡑࡢ ᣲືࢆホ౯࡛ࡁࡿ࡜⪃࠼ࡽࢀࡿ㸬

ࡇࡢࡓࡵ㸪௵ពࡢせ⣲ࡢᐃ⩏࡟࠾࠸࡚㸪┤᥋ⓗ࡟せ⣲๛

ᛶ࣐ࢺࣜࢡࢫࢆᑟධྍ⬟࡛࠶ࡿሙྜ㸪ᐜ᫆࡟

XFEM

㏆ఝ ࡢᐇ⿦ࡀྍ⬟࡜࡞ࡿ㸬౛࠼ࡤ㸪

ABAQUS

࡟࠾࠸࡚ࡣ㸪࢜

ࣉࢩࣙࣥ

*USER ELEMENT

࡟࠾࠸࡚࢜ࣉࢩࣙࣥ

*MATRIX

ࢆ⏝࠸ࡿࡇ࡜࡛┤᥋ⓗ࡟せ⣲๛ᛶ࣐ࢺࣜࢡࢫࢆᑟධࡍࡿ

ࡇ࡜ࡀ࡛ࡁࡿ㸬

ࡑࡇ࡛㸪

(1)㡯

࡛♧ࡋࡓከ㔜⠇Ⅼ࡟ࡼࡿ⠇Ⅼ⮬⏤ᗘࡢቑຍ ἲࢆ㋃ࡲ࠼㸪

ABAQUS

࡟

PU-XFEM

㏆ఝࢆᑟධࡍࡿࣉࣟ

ࢢ࣒ࣛ

PRE-PRO

࡟ࡼࡿฎ⌮ࡢᐇ⾜ᡭ㡰ࢆ௨ୗ࡟♧ࡍ㸬

(i) ABAQUS CAE

ࢆ⏝࠸࡚సᡂࡋࡓ㸪ࡁ⿣ࢆྵࡲ࡞

࠸

FEM

ゎᯒࣔࢹࣝࡢධຊࣇ࢓࢖ࣝ㸦INPUT.inp㸧

࠿ࡽ㸪ࡍ࡭࡚ࡢ⠇Ⅼᗙᶆ࠾ࡼࡧ⠇Ⅼ-せ⣲㛵ಀࢆ

ㄞࡳ㎸ࡴ㸬ࡲࡓ㸪ࡁ⿣ᙧ≧ࢹ࣮ࢱ㸦CRACK.dat㸧

࠿ࡽ㸪ࡁ⿣ࡢᗙᶆࢹ࣮ࢱࢆㄞࡳ㎸ࡴ㸬

(ii)

ࡁ⿣ᗙᶆ࠾ࡼࡧྛせ⣲ࢆᵓᡂࡍࡿ⠇Ⅼᗙᶆࢆẚ

㍑ࡋ㸪⠇Ⅼ㞟ྜ

ܥ㸪 ܥ

^ƍ࠾ࡼࡧ

ܬࢆࡑࢀࡒࢀᐃ⩏ࡋ㸪

ࡑࢀ࡟ᑐᛂࡋ࡚⾲-2(b)࡟♧ࡍ㏣ຍ⠇Ⅼࢆከ㔜⠇

ᅗ

-4

࢜ࣉࢩࣙࣥ

*USER ELEMENT

࡟ࡼࡾ

XFEM

㏆ఝࢆ

ᑟධࡍࡿせ⣲ࡢ୍౛

Crack Tip

Crack Tip

' a

'a

T

c

Ⅼ࡜ࡋ࡚ᐃ⩏ࡍࡿ㸬

(iii)

せ⣲ࢆᵓᡂࡍࡿ⠇Ⅼࡢ୍㒊ࡀ㸪⠇Ⅼ㞟ྜ

ܥ㸪 ܥ

^ƍ࠾

ࡼࡧ

ܬ

࡟ྵࡲࢀࡿሙྜ㸪࢚ࣥࣜࢵࢳ㛵ᩘࢆ⪃៖ࡋ ࡓせ⣲๛ᛶ࣐ࢺࣜࢡࢫࡢ⟬ฟࢆ⾜࠺㸬࡞࠾㸪せ⣲

ෆ㒊࡟ࡁ⿣ࡢ୙㐃⥆㠃ࢆྵࡴሙྜ㸪✚ศ㡿ᇦࢆศ

๭ࡋ㸪ูࠎ࡟ィ⟬ࢆᐇ⾜ࡍࡿ㸬

(iv) ABAQUS CAE

ࢆ⏝࠸࡚᭩ࡁฟࡉࢀࡓධຊࣇ࢓࢖

ࣝ㸦INPUT.inp㸧࡟࠾࠸࡚㸪

(iii)࡛せ⣲๛ᛶ࣐ࢺࣜ

ࢡࢫࢆ⟬ฟࡋࡓせ⣲࡟㛵ࡋ࡚㸪࢜ࣉࢩࣙࣥ

*USER

ELEMENT

࠾ࡼࡧ*MATRIXࢆ⏝࠸࡚せ⣲๛ᛶ࣐

ࢺࣜࢡࢫࢆ┤᥋ᑟධࡋ㸪ࡑࡢᐃ⩏ࢆ᭩ࡁ᥮࠼ࡿࡇ

࡜࡛㸪᪂ࡓ࡞ධຊࣇ࢓࢖ࣝ㸦Xn.inp㸧ࢆసᡂࡍࡿ㸬 ࡇࡇ࡛సᡂࡉࢀࡓ㸪

XFEM

㏆ఝࢆᑟධࡋࡓධຊࣇ࢓࢖ࣝ

㸦Xn.inp㸧ࡣ㸪ࡑࢀࡢࡳ࡛

ABAQUS

ࡢࢯࣝࣂ࣮࡟ࢪࣙࣈ

࡜ࡋ࡚ᢞධࡋ㸪ྛ⠇Ⅼ⮬⏤ᗘ࡟ᑐᛂࡍࡿᩘ್ゎࢆᚓࡿࡇ࡜

ࡀ࡛ࡁࡿ㸬

3.3

ᛂຊᣑ኱ಀᩘࡢ⟬ฟ࡜⑂ປࡁ⿣㐍ᒎ᪉ྥ࣭㏿ᗘࡢホ౯

⑂ປࡁ⿣ࡢ㐍ᒎゎᯒ࡟ࡣ㸪ᛂຊᣑ኱ಀᩘࡢホ౯࠾ࡼࡧࡑ

ࢀ࡟ᇶ࡙ࡃ⑂ປࡁ⿣ࡢ㐍ᒎᣲືࡢࣔࢹࣝ໬ࡀᚲせ࡜࡞ࡿ㸬 ࡑࡢࡓࡵ㸪ᮏゎᯒࢥ࣮ࢻ࡛ࡣ㸪ࡑࢀࡽࢆᐇ⾜ྍ⬟࡞ࣉࣟࢢ

࣒ࣛ

POST-PRO

ࡢ㏣ຍࢆ⾜࠺㸬௨ୗ࡛ࡣ㸪

POST-PRO

࡛

⏝࠸ࡽࢀࡿᛂຊᣑ኱ಀᩘࡢホ౯ἲ࡜⑂ປࡁ⿣ࡢ㐍ᒎᇶ‽

࡟㛵ࡋ࡚ࡢᴫせࢆ♧ࡍ㸬

ᛂຊᣑ኱ಀᩘࡢホ౯ࡣ㸪

XFEM

ࢆ⏝࠸ࡓ

2

ḟඖࡁ⿣ゎᯒ

࡟࠾࠸࡚᭱ࡶ୍⯡ⓗ࡟⏝࠸ࡽࢀ࡚࠸ࡿM✚ศἲ㸦┦஫✚

ศἲ㸧¹⁹⁾࡟ࡼࡾ⾜࠺㸬M✚ศࡢ⿕✚ศ㛵ᩘ࡟ྵࡲࢀࡿᛂຊ࣭

ࡦࡎࡳሙࡣ㸪⠇Ⅼ⮬⏤ᗘ࡟ᑐᛂࡋࡓᩘ್ゎࢆ⏝࠸࡚෌ᵓᡂ ࡍࡿ㸬࡞࠾㸪ᮏゎᯒࢥ࣮ࢻ࡛ࡣ㸪ᒅᢡࡲࡓࡣ᭤⥺ࡁ⿣࡬ࡢ 㐺⏝࡟᭷ຠ࡞ಟṇM✚ศἲ^{19), 20)}ࢆ⏝࠸ࡿ㸬

࡞࠾㸪

PU-XFEM

ࢆ⏝࠸ࡓሙྜ㸪⠇Ⅼ㞟ྜ

ܥ

ࡢᐃ⩏༙ᚄ

ܴ

_஼࠾ࡼࡧM✚ศホ౯㡿ᇦࡢᐃ⩏༙ᚄ

ܴ

_ிࢆ㸪࡜ࡶ࡟ࡁ⿣ඛ

➃ࢆྵࡴせ⣲࡟ᑐࡋ࡚᭱ᑠ㝈ࡢ⠊ᅖ࡛ࣔࢹࣝ໬ࡍࡿࡇ࡜

ࡀྍ⬟࡜࡞ࡿ¹¹⁾㸬ࡇࡢࡓࡵ㸪

ܴ

_஼࠾ࡼࡧ

ܴ

_ிࢆḟᘧ࡛♧ࡍࡼ

࠺࡟㸪

ͳǤͲ݄

࠶ࡿ࠸ࡣࡁ⿣ඛ➃ࢆྵࡴせ⣲ࢆᵓᡂࡍࡿྛ⠇

Ⅼ࡜ࡁ⿣ඛ➃࡜ࡢ㊥㞳࠸ࡎࢀ࠿ࡢ᭱኱್࡛ᐃ⩏ࡍࡿ㸬

ܴ

_஼

ൌ ܴ

_ி

ൌ ƒš ቄͳǤͲ݄ǡ ƒš

ூ

൛หܠ

_ூ

െ ܠ

_୲୧୮

หൟቅ (9)

ࡇࡇ࡛㸪ᘧ(9)ࡢ

ܫ࠾ࡼࡧ ݄ࡣ㸪ࡑࢀࡒࢀࡁ⿣ඛ➃ࡀ఩⨨ࡍ

ࡿせ⣲ࢆᵓᡂࡍࡿ⠇Ⅼ࠾ࡼࡧせ⣲㎶ࡢᖹᆒ㛗ࡉ࡛࠶ࡿ㸬

ࡁ⿣ࡢ㐍ᒎ᪉ྥ࠾ࡼࡧఏ᧛㏿ᗘ࡟ࡼࡾࣔࢹࣝ໬ࡉࢀࡿ

⑂ປࡁ⿣ࡢ㐍ᒎᣲືࡣ㸪M✚ศἲ࡟ࡼࡾ⟬ฟࡋࡓᛂຊᣑ኱

ಀᩘ࡟ᇶ࡙ࡃ⤒㦂๎ࢆ⏝࠸࡚ࣔࢹࣝ໬ྍ⬟࡛࠶ࡿࡇ࡜ࡀ

▱ࡽࢀ࡚࠸ࡿ㸬

ࡲࡎ㸪ࡁ⿣ࡢ㐍ᒎ᪉ྥࡢุ᩿ᇶ‽࡜ࡋ࡚㸪໭ᕝࡽ࡟ࡼࡾ

ᥦ᱌ࡉࢀࡓࡁ⿣ඛ➃ࡢ᥋⥺ᛂຊࡢ᣺ࢀᖜ

߂ߪ

ఏࡀ᭱኱࡜࡞

ࡿ᪉ྥ

ߠ

_௖࡟ࡁ⿣ࡀ㐍ᒎࡍࡿ࡜௬ᐃࡋࡓḟᘧࡢ᭱኱᥋⥺ᛂ ຊ⠊ᅖᇶ‽²¹⁾࡟ᚑ࠺ࡶࡢ࡜ࡍࡿ㸬

ߠ

௖

ൌ ʹ –ƒ

^ିଵ

ቐ ͳ Ͷ ቌ ߂ܭ

୍

߂ܭ

୍୍

േ ඨ൬ ߂ܭ

୍

߂ܭ

୍୍

൰

ଶ

൅ ͺቍቑ (10)

ࡇࡢ᭱኱᥋⥺ᛂຊ⠊ᅖᇶ‽࡟㛵ࡍࡿᐇ㦂ⓗ࡞ጇᙜᛶࡢ᳨

ドࡣ㸪኱㊰ࡽ࡟ࡼࡾ㧗ᙇຊ㗰ࡢṧ␃ᛂຊሙ࡟࠾ࡅࡿ⑂ປࡁ

⿣ࢆᑐ㇟࡟ᐇ᪋ࡉࢀ࡚࠸ࡿ²²⁾㸬

ࡲࡓ㸪࣮ࣔࢻ

I࣭ II

ΰྜ࣮ࣔࢻ࡟࠾ࡅࡿ➼౯ᛂຊᣑ኱ಀ

ᩘ⠊ᅖ

߂ܭ

ࢆḟᘧ࡛ᐃ⩏ࡍࡿ²³⁾㸬

߂ܭ ൌ ߂ܭ

_୍

…‘•

^ଷ

൬ ߠ

௖

ʹ ൰ െ ͵߂ܭ

_୍୍

…‘•

^ଶ

൬ ߠ

௖

ʹ ൰ •‹ ൬ ߠ

௖

ʹ ൰ (11)

ࡇࡢ➼౯ᛂຊᣑ኱ಀᩘ

߂ܭࢆ⏝࠸࡚㸪⑂ປࡁ⿣ࡢఏ᧛㏿ᗘ

ࡣ㸪ḟᘧࡢᛂຊᣑ኱ಀᩘ⠊ᅖࡢୗ㝈⏺

߂ܭ

௧௛ࢆ⪃៖ࡋࡓࣃ

ࣜࢫ๎࡟ᚑ࠺ࡶࡢ࡜௬ᐃࡍࡿ㸬

݀ܽ

݀ܰ ൌ ܥሼ߂ܭ

^௠

െ ሺ߂ܭ

௧௛

ሻ

^௠

ሽ (12)

ࡇࡇ࡛

ܥ࠾ࡼࡧ ݉ࡣᮦᩱᐃᩘ࡛࠶ࡿ㸬ᮏ◊✲࡛ࡣ᪥ᮏ㗰

ᵓ㐀༠఍㸪㗰ᵓ㐀≀ࡢ⑂ປタィᣦ㔪࣭ྠゎㄝࡢᖹᆒタィ᭤

⥺࡟ᇶ࡙ࡁ㸪ࡑࢀࡒࢀࡢᐃᩘࢆ

ܥ ൌ ͳǤͷͲ ൈ ͳͲ

^ିଵଵ㸪

݉ ൌ ʹǤ͹ͷ

㸪

߂ܭ

௧௛

ൌ ʹǤͻMPaξ

࡜ࡋ²⁴⁾㸪⑂ປࡁ⿣㛢ཱྀ⌧㇟

ࡢᙳ㡪ࡣ⪃៖ࡋ࡞࠸ࡶࡢ࡜ࡍࡿ㸬

ᮏゎᯒࢥ࣮ࢻ࡛ࡣ㸪ᅗ-5࡟♧ࡍࡼ࠺࡟ྛィ⟬ࢫࢸࢵࣉ࡟

࠾࠸࡚㸪࠶ࡽ࠿ࡌࡵᣦᐃࡋࡓࡁ⿣ࡢ㐍ᒎ㛗ࡉ

߂ܽ࡟ࡼࡾ㸪

ࡁ⿣ᙧ≧ࢹ࣮ࢱࢆ᭦᪂ࡍࡿ㸬ࡲࡓ㸪ᛂຊᣑ኱ಀᩘ⠊ᅖ

߂ܭ

ࡢ㐍ᒎ๓ᚋࡢᩘ್ゎࢆ⥺ᙧ⿵㛫࡟ࡼࡾ㏆ఝࡍࡿࡇ࡜࡛㸪ᘧ

(12)

ࡼࡾ㍕Ⲵᅇᩘ

ܰ

ࢆ㏫⟬ࡍࡿ㸬

ᘧ

(10)࠾ࡼࡧᘧ (12)࡟♧ࡍ㐍ᒎ๎ࡣ㸪ࡍ࡭࡚ᛂຊᣑ኱ಀᩘ

⠊ᅖ

߂ܭ

୍࠾ࡼࡧ

߂ܭ

୍୍࡟ᇶ࡙ࡁつᐃࡋ࡚࠾ࡾ㸪ᵓ㐀య࡟⏕

ࡌࡿṧ␃ᛂຊࡸṚⲴ㔜࡞࡝ࡢᖹᆒᛂຊࡢᙳ㡪ࢆ⪃៖ࡋ࡚

࠸࡞࠸ࡀ㸪ࡇࢀ࡟ࡼࡾゎᯒ᮲௳ࡀ༢⣧໬ࡉࢀ㸪ᐜ᫆࡟㐍ᒎ ᣲືࡢࣔࢹࣝ໬ࡀྍ⬟࡜࡞ࡿ㸬

4. ゎᯒࢥ࣮ࢻࡢ᳨ド

1

❶࡟㏙࡭ࡓࡼ࠺࡟㸪ᮏ◊✲ࡣᐇᵓ㐀࡬ࡢ㐺⏝ࢆ┠ⓗ࡜

ࡋ࡚

PU-XFEM

ࡢỗ⏝

FEM

ゎᯒࢥ࣮ࢻ࡬ࡢᐇ⿦ࢆ♧ࡍࡶ

ᅗ

-5

ࡁ⿣㐍ᒎࡢࣔࢹࣝ໬

W = 1

L = 16 a = 3.5

W = 7

ࡢ࡛࠶ࡿ㸬ࡇࢀ࡟ᑐࡋ㸪᪤ ࡢ◊✲ᡂᯝ^{10), 11)}࡛ࡣPU-XFEM

ࢆ⏝࠸ࡓሙྜࡢᐇᵓ㐀≀࡬ࡢ㐺⏝๓࡟ᚲせ࡜࡞ࡿ᭷㝈ᯈ

ࢆᑐ㇟࡜ࡋࡓ◚ቯຊᏛࣃ࣓࣮ࣛࢱࡢゎᯒ⢭ᗘ࡟㛵ࡍࡿ༑

ศ࡞ホ౯ࡀ♧ࡉࢀ࡚࠸࡞࠸㸬ࡑࡇ࡛ᮏ❶࡛ࡣ㸪ᐇᵓ㐀≀࡟

Ⓨ⏕ࡋࡓ⑂ປࡁ⿣ࡢ㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࢆᐇ᪋ࡍࡿ๓

࡟㸪༢⣧࡞᭷㝈ᯈࣔࢹࣝࢆᑐ㇟࡜ࡋ࡚

PU-XFEM

㏆ఝࢆᐇ

⿦ࡋࡓゎᯒࢥ࣮ࢻ࡟㛵ࡍࡿᇶᮏⓗ࡞ゎᯒ⢭ᗘࡢホ౯ࢆ⾜

࠺㸬

ලయⓗ࡟ࡣ㸪

4.1

⠇࡟࠾࠸࡚ΰྜ◚ቯ࣮ࣔࢻ

I

࠾ࡼࡧ

II

ࡢᛂຊᣑ኱ಀᩘࡢゎᯒゎࡀ᪤▱࡛࠶ࡿࣔࢹࣝ㸪4.2⠇࡟࠾

࠸࡚ࡁ⿣㐍ᒎ㐣⛬ࡢᛂຊᣑ኱ಀᩘࡢゎᯒゎࡀ᪤▱࡛࠶ࡿ

ࣔࢹࣝࢆࡑࢀࡒࢀᑐ㇟࡜ࡋ࡚㸪௦⾲ⓗ࡞ỗ⏝

FEM

ゎᯒࢯ

ࣇࢺ࢙࢘࢔ࡢࡦ࡜ࡘ࡛࠶ࡿ

ABAQUS

ࢆ࣮࣋ࢫ࡜ࡋ࡚㛤Ⓨ

ࡋࡓゎᯒࢥ࣮ࢻ࡟ࡼࡿ◚ቯຊᏛࣃ࣓࣮ࣛࢱࡢᩘ್ゎࡢゎ ᯒ⢭ᗘࢆ㸪ゎᯒゎ࠾ࡼࡧⴭ⪅ࡽࡢ๓ሗ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮

ࢻࡢᩘ್ゎ࡜ẚ㍑ࡍࡿࡇ࡜᳨࡛ドࡍࡿ㸬

࡞࠾㸪

3

❶࡛ᐇ⿦᪉ἲࢆ♧ࡋࡓゎᯒࢥ࣮ࢻࡣࢩ࢙ࣝせ⣲

࡛ࣔࢹࣝ໬ྍ⬟࡞

3

ḟඖᵓ㐀≀࡟࠾ࡅࡿᯈཌ㈏㏻ࡁ⿣ࡢ㐍 ᒎゎᯒࢆᐇ᪋ྍ⬟࡞ࡶࡢ࡛࠶ࡿࡀ㸪ᮏ❶࡛ᐇ᪋ࡍࡿ᳨ドࡣ ᖹ㠃ࡦࡎࡳࢆ௬ᐃࡋࡓ

2

ḟඖၥ㢟ࢆᢅ࠺㸬

4.1 ΰྜ◚ቯ࣮ࣔࢻࡢ◚ቯຊᏛࣃ࣓࣮ࣛࢱゎᯒ⢭ᗘ

ࡇࡇ࡛ࡣ㸪ᅗ

-6

࡟♧ࡍΰྜ࣮ࣔࢻ

I

࠾ࡼࡧ

II

ࡢᛂຊᣑ኱

ಀᩘࡢゎᯒゎࡀ᪤▱࡞ࡁ⿣ࡢ࠶ࡿᖏᯈࡢࡏࢇ᩿ࣔࢹࣝ

㸦Model-1㸧ࢆᑐ㇟࡜ࡋ࡚㸪◚ቯຊᏛࣃ࣓࣮ࣛࢱࡢᩘ್ゎ ࡢ⢭ᗘホ౯ࢆ⾜࠺㸬

ᣦᶆ࡜ࡋ࡚⏝࠸ࡿ◚ቯຊᏛࣃ࣓࣮ࣛࢱࡣ㸪ᛂຊᣑ኱ಀᩘ

ܭ

୍࠾ࡼࡧ

ܭ

୍୍࡜㸪ᘧ

(12)

࡟♧ࡋࡓఏ᧛㏿ᗘ

݀ܽȀ݀ܰ

ࢆᢅ࠺㸬 ࡓࡔࡋ㸪ఏ᧛㏿ᗘ

݀ܽȀ݀ܰࡢホ౯࡟࠾࠸࡚ࡣ㸪ࣔࢹࣝࡢୖ

➃࡟㈇Ⲵࡍࡿࡏࢇ᩿ᛂຊ

߬

ࢆࡏࢇ᩿ᛂຊ⠊ᅖ

߂߬

࡟⨨ࡁ᥮

࠼ࡿࡶࡢ࡜ࡋ㸪ᛂຊᣑ኱ಀᩘ⠊ᅖࡢୗ㝈⏺

߂ܭ

_௧௛ࢆ༑ศ࡟

ᑠࡉ࠸ࡶࡢ࡜ࡋ࡚⪃៖ࡋ࡞࠸㸬

ᩘ್ゎࡢㄗᕪホ౯࡟⏝࠸ࡿᛂຊᣑ኱ಀᩘ

ܭ

_୍࠾ࡼࡧ

ܭ

_୍୍

ࡢゎᯒゎࢆ௨ୗ࡟♧ࡍ¹⁹⁾㸬

ܭ

୍

ൌ ͵ͶǤͲ

ܭ

୍୍

ൌ ͶǤͷͷ (13)

ゎᯒࣔࢹࣝࡣ᭷㝈ࡢᑍἲࢆ᭷ࡍࡿࡓࡵ㸪ゎᯒ⢭ᗘࡢホ౯ ࡣࡁ⿣㛗ࡉ

ܽ

ࢆᇶ‽࡜ࡋ࡚ṇつ໬ࡋࡓせ⣲ࢧ࢖ࢬ

݄

㸦௨ୗ㸪

݄Ȁܽ

࡜ࡍࡿ㸧ࢆࣃ࣓࣮ࣛࢱ࡜ࡋ࡚⾜࠺㸬⠇Ⅼ㞟ྜ

ܥ

ࡢᐃ⩏

༙ᚄ

ܴ

஼࠾ࡼࡧM✚ศホ౯㡿ᇦࡢᐃ⩏༙ᚄ ܴிࡣ㸪ᘧ(9)ࡢ ᐃ⩏࡟ᚑ࠺ࡶࡢ࡜ࡍࡿ㸬࡞࠾㸪ᮏ᳨ド࡛ࡣ㸪ᚑ᮶ࡢ

XFEM

¹⁶⁾

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

1.0E-02 1.0E-01

Error (%)

Normalized Mesh Size: h/a std. XFEM (Present) PU-XFEM (Present) std. XFEM (Reference ) PU-XFEM (Reference )

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

1.0E-02 1.0E-01

Error (%)

Normalized Mesh Size: h/a std. XFEM (Present) PU-XFEM (Present) std. XFEM (Reference ) PU-XFEM (Reference )

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

1.0E-02 1.0E-01

Error (%)

Normalized Mesh Size: h/a std. XFEM (Present) PU-XFEM (Present) std. XFEM (Reference ) PU-XFEM (Reference )

ᅗ

-6

⦕ࡁ⿣ࢆ᭷ࡍࡿ᭷㝈ࡢᖏᯈࣔࢹࣝ㸦

Model-1

㸧

(a)

ᛂຊᣑ኱ಀᩘ

ܭ

୍

(b)

ᛂຊᣑ኱ಀᩘ

ܭ

୍୍

(c)

⑂ປࡁ⿣ఏ᧛㏿ᗘ

݀ܽȀ݀ܰ

ᅗ-7

◚ቯຊᏛࣃ࣓࣮ࣛࢱࡢゎᯒ⢭ᗘ㸦

Model-1

㸧

11) 11) 11)

11) 11)

11)

a S = 4W

W P

ࢆ⏝࠸ࡓሙྜ࡜ࡢゎᯒ⢭ᗘࡢẚ㍑ࡶ♧ࡍ㸬 ௨ୖࡢ᮲௳࡟ᇶ࡙࠸ࡓゎᯒ⤖ᯝࢆᅗ-7࡟♧ࡍ㸬

ࡲࡎ㸪ᅗ

-7(a)࡟♧ࡍᛂຊᣑ኱ಀᩘ ܭ

_୍࡟㛵ࡍࡿゎᯒ⤖ᯝ

ࡼࡾ㸪

PU-XFEM

ࢆᐇ⿦ࡋࡓᮏゎᯒࢥ࣮ࢻ࡟ࡼࡿᩘ್ゎࡣ㸪

݄Ȁܽࡀ༑ศ࡟ᑠࡉ࠸ሙྜ࡛ゎᯒゎ࡜ࡢㄗᕪ0.5%⛬ᗘ࡜࡞

ࡾ㸪

݄Ȁܽ

ࡀ኱ࡁ࠸

0.3

⛬ᗘࡢሙྜ࡛ゎᯒゎ࡜ࡢㄗᕪ2.0%࡜

࡞ࡗࡓ㸬ⴭ⪅ࡽࡢ๓ሗ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻ࡜ࡢẚ㍑࡛ࡣ㸪 ୧⪅ࡢᩘ್ゎࡢᕪ␗ࡣ

݄Ȁܽ

ࡀ኱ࡁ࠸

0.3

⛬ᗘࡢሙྜ࡛

1.0%⛬ᗘ࡜࡞ࡗࡓࡀ㸪 ݄Ȁܽࡀ༑ศ࡟ᑠࡉ࠸ሙྜ࡟࠾࠸࡚࡯

ࡰ୍⮴ࡋ㸪୧⪅ࡣྠᵝࡢഴྥࢆ♧ࡋࡓ㸬࡞࠾㸪

݄Ȁܽ

ࡀ኱ࡁ

࠸ሙྜ࡟࠾ࡅࡿ୧⪅ࡢ

1.0%⛬ᗘࡢᕪ␗ࡢཎᅉ࡜ࡋ࡚ࡣ㸪

⠇Ⅼ⮬⏤ᗘࡢቑຍࡀ୙せ࡞せ⣲ࡢ㏆ఝἲ࡟࠾࠸࡚㸪ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡛ࡣ༢⣧࡞ᅄ㎶ᙧ཮୍ḟせ⣲࡛ࡣ

࡞ࡃ㸪㠀㐺ྜせ⣲࡞࡝ࡢ㧗ᛶ⬟᭷㝈せ⣲²⁵⁾ࡀ⤌ࡳ㎸ࡲࢀ࡚

࠸ࡿࡓࡵ࡛࠶ࡿ࡜⪃࠼ࡽࢀࡿ㸬࡞࠾㸪ᚑ᮶ࡢ

XFEM

ࢆ⏝

࠸ࡓሙྜ࡛ࡣ㸪

݄Ȁܽ

ࡀ኱ࡁ࠸ሙྜ࡟ᩘ್ゎࡣ኱ࡁࡃ㐣ᑠホ ౯࡜࡞ࡾ㸪≉࡟

݄Ȁܽࡀ 0.3

⛬ᗘࡢሙྜ࡟ㄗᕪࡣ5.5%࡜࡞

ࡗࡓ㸬ࡲࡓ㸪

݄Ȁܽ

ࡀ༑ศ࡟ᑠࡉ࠸ሙྜ࡛ࡣᩘ್ゎࡣ3.0%

௨ୖࡢ㐣኱ホ౯࡜࡞ࡾ㸪ᩘ್ゎࡀṇゎ࡟཰᮰ࡋ࡞࠸⤖ᯝ࡜

࡞ࡗࡓ㸬ࡇࢀ࡟ࡼࡾ㸪ᚑ᮶ࡢ

XFEM

࡜ẚ㍑ࡋࡓሙྜࡢ

PU-XFEM

ࡢ᭷ຠᛶࡀ♧ࡉࢀࡓ㸬

ᅗ

-7(b)࡟♧ࡍᛂຊᣑ኱ಀᩘ ܭ

୍୍࡟㛵ࡍࡿゎᯒ⤖ᯝ࡛ࡣ㸪

PU-XFEM

ࢆᐇ⿦ࡋࡓᮏゎᯒࢥ࣮ࢻ࡟ࡼࡿᩘ್ゎࡣ㸪ᮏ᳨

ドࡢ⠊ᅖෆ࡛ゎᯒゎ࡜ࡢㄗᕪࡣr0.5%௨ෆࡢ⢭ᗘ࡛࠶ࡾ㸪 ⴭ⪅ࡽࡢ๓ሗ ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻࡢᩘ್ゎ࡜ྠᵝࡢഴ

ྥࢆ♧ࡋࡓ㸬ࡲࡓ㸪ᚑ᮶ࡢ

XFEM

ࢆ⏝࠸ࡓሙྜ㸪ᩘ್ゎ ࡣᖖ࡟㐣኱ホ౯࡜࡞ࡾ㸪≉࡟

݄Ȁܽࡀ༑ศ࡟ᑠࡉ࠸ሙྜ࡛ᩘ

್ゎࡣ4.0%௨ୖࡢㄗᕪࢆ᭷ࡍࡿ⤖ᯝ࡜࡞ࡗࡓ㸬

ࡉࡽ࡟㸪ᅗ-7(c)࡟♧ࡍ⑂ປࡁ⿣ࡢఏ᧛㏿ᗘ࡟㛵ࡍࡿゎᯒ

⤖ᯝ࡛ࡣ㸪ᛂຊᣑ኱ಀᩘ

ܭ

୍࡟㛵ࡍࡿゎᯒ⤖ᯝ࡜ྠᵝࡢഴ

ྥࡀ♧ࡉࢀࡓ㸬ලయⓗ࡟ࡣ㸪PU-XFEMࢆᐇ⿦ࡋࡓᮏゎᯒ ࢥ࣮ࢻ࡟ࡼࡿᩘ್ゎࡣ㸪ゎᯒゎ࡜ࡢㄗᕪࡀ% ~ .5%

࡜࡞ࡾ㸪ⴭ⪅ࡽࡢ๓ሗ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻ࡟ࡼࡿᩘ್ゎ

࡜ྠᵝࡢഴྥࢆ♧ࡋࡓ㸬ࡲࡓ㸪ゎᯒゎ࡜ࡢㄗᕪࡀ% ~ 8.3%࡜࡞ࡾゎᯒ⢭ᗘࡀ

݄Ȁܽ

࡟኱ࡁࡃ౫Ꮡࡍࡿᚑ᮶ࡢ

XFEM

࡜ẚ࡭㸪

PU-XFEM

ࢆ⏝࠸ࡓሙྜ࡛ࡣ㸪ゎᯒ⢭ᗘࡢ

Ᏻᐃᛶ࡟኱ࡁ࡞฼Ⅼࡀ࠶ࡿࡇ࡜ࡀ♧ࡉࢀࡓ㸬

௨ୖࡢ◚ቯຊᏛࣃ࣓࣮ࣛࢱࡢゎᯒ⢭ᗘ࡟㛵ࡍࡿホ౯ࡼ

ࡾ㸪

PU-XFEM

㏆ఝࢆᐇ⿦ࡋࡓゎᯒࢥ࣮ࢻࡢ᭷ຠᛶࡀ☜ㄆ

ࢆࡉࢀࡓ㸬

࡞࠾㸪᭱㏆

ABAQUS

࡟ᐇ⿦ࡉࢀࡓ

XFEM

ࡣᮏ❶ࡢ᳨ド

࡛⏝࠸ࡓᚑ᮶ࡢ

XFEM

࡜ྠᵝࡢᐃᘧ໬࡟ᇶ࡙ࡃࡶࡢ࡜⪃

࠼ࡽࢀࡿ㸬ࡋ࠿ࡋ㸪ᮏ❶ࡢゎᯒ⤖ᯝࡀ

ABAQUS

࡟ᐇ⿦ࡉ

ࢀࡓ

XFEM

ࡢゎᯒ⢭ᗘࢆཝᐦ࡟ಖドࡍࡿࡶࡢ࡛ࡣ࡞࠸ࡇ

࡜࡟ὀពࡉࢀࡓ࠸㸬

4.2 ⑂ປࡁ⿣㐍ᒎ㐣⛬ࡢゎᯒ⢭ᗘ

ࡇࡇ࡛ࡣ㸪ᅗ-8࡟♧ࡍࡁ⿣ࡢ࠶ࡿᖏᯈࡢ

3

Ⅼ᭤ࡆࣔࢹࣝ

㸦Model-2㸧ࢆᑐ㇟࡜ࡋ࡚㸪⑂ປࡁ⿣ࡢ㐍ᒎ㐣⛬࡟࠾ࡅࡿ

◚ቯຊᏛࣃ࣓࣮ࣛࢱ࠾ࡼࡧࡑࢀ࡟ᇶ࡙ࡃ⑂ປࡁ⿣ఏ᧛㏿

ᗘࡢᩘ್ゎ࡟㛵ࡍࡿゎᯒ⢭ᗘࡢ᳨ドࢆ⾜࠺㸬

⑂ປࡁ⿣ఏ᧛㏿ᗘࡢホ౯࡟࠾࠸࡚ࡣ㸪๓⠇࡜ྠᵝ㸪ࣔࢹ

ࣝ࡟㈇ⲴࡍࡿⲴ㔜

ܲࢆ ߂ܲ࡟⨨ࡁ᥮࠼ࡓࡶࡢ࡜ࡋ㸪ᛂຊᣑ

኱ಀᩘ⠊ᅖࡢୗ㝈⏺

߂ܭ

௧௛ࢆ༑ศ࡟ᑠࡉ࠸ࡶࡢ࡜ࡋ࡚⪃៖

ࡋ࡞࠸㸬

ᛂຊᣑ኱ಀᩘ

ܭ

୍ࡢゎᯒゎࢆḟᘧ࡟♧ࡍ²⁶⁾㸬

ܭ

୍

ൌ ߪ

଴

ξߨܽܨሺܽȀܹሻ

ܨሺߦሻ ൌ ͳǤͲͻͲ െ ͳǤ͹͵ͷߦ ൅ ͺǤʹͲߦ^ଶെ ͳͶǤͳͺߦ^ଷ൅ ͳͶǤͷ͹ߦ^ସ

(14)

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Error (%)

Normalized Crack Length: a/W PU-XFEM (Present)

PU-XFEM (Reference )

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0E+00 5.0E+11 1.0E+12 1.5E+12

Normalized Crack Length: a/W

Cycle of Loads: N PU-XFEM (Present) PU-XFEM (Reference )

ᅗ

-8

ࡁ⿣ࡢ࠶ࡿ

3

Ⅼ᭤ࡆࣔࢹࣝ㸦Model-2㸧

ᅗ

-9

ࡁ⿣㐍ᒎ࡟క࠺ᛂຊᣑ኱ಀᩘ

ܭ

୍ࡢゎᯒ⢭ᗘ 㸦Model-2㸧

ᅗ

-10

㍕Ⲵᅇᩘ࡜ࡁ⿣㛗ࡉࡢ㛵ಀ㸦

Model-2

㸧

11) 11)

ࣂࣝࣈࣜࣈ㗰ᗋ∧

ᶍᨃヨ㦂య

㍕Ⲵ఩⨨

ࢫࢺࢵࣉ࣮࣍ࣝ

⑂ປࡁ⿣

࣎ࣝࢺᏍ

ࡇࡇ࡛㸪ߦ ൌ ܽȀܹ㸪

ߪ

_଴

ൌ ͵ܵܲȀʹܹ

^ଶ࡛࠶ࡾ㸪ࡇࡢゎᯒゎ ࡣ

ܽȀܹ ൑ ͲǤ͸

࡟ᑐࡋṇゎ࠿ࡽࡢㄗᕪ

0.2%

௨ୗ࡜ࡉࢀࡿ㸬

ゎᯒࣔࢹࣝࡣ⠇Ⅼᩘ

3720㸪せ⣲ᩘ 3570

࡜ࡋ㸪ึᮇࡁ⿣

㛗ࡉࡣ

ܽ

଴

ൌ ͲǤͲͷܹ

࡜ࡋࡓ㸬ࡲࡓ㸪⠇Ⅼ㞟ྜ

ܥ

ࡢᐃ⩏༙

ᚄ

ܴ

_஼࠾ࡼࡧM✚ศホ౯㡿ᇦࡢᐃ⩏༙ᚄ

ܴ

_ிࡣ㸪

4.1

⠇࡜

ྠᵝᘧ

(9)

ࡢᐃ⩏࡟ᚑ࠺ࡶࡢ࡜ࡍࡿ㸬

௨ୖࡢ᮲௳࡟ᇶ࡙ࡃ㸪ࡁ⿣㐍ᒎ࡟క࠺ᛂຊᣑ኱ಀᩘ

ܭ

_୍ ࡢゎᯒ⢭ᗘࢆᅗ-9࡟㸪㍕Ⲵᅇᩘ࡜ࡁ⿣㛗ࡉࡢ㛵ಀࢆᅗ

-10

࡟♧ࡍ㸬

ࡇࢀࡽࡢゎᯒ⤖ᯝࡼࡾ㸪ᮏ◊✲࡛㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻ࠾

ࡼࡧⴭ⪅ࡽࡢ๓ሗ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻ࡟㛵ࡋ࡚㸪ࡑࢀࡒ

ࢀࡢᩘ್ゎࡣ㠀ᖖ࡟Ⰻ࠸୍⮴ࢆ♧ࡋࡓ㸬

ࡲࡓ㸪ᅗ

-9

࡟♧ࡍᛂຊᣑ኱ಀᩘ

ܭ

_୍࡟㛵ࡋ࡚㸪

ܽȀܹ ൑ ͲǤ͸

ࡢ⠊ᅖ࡟࠾ࡅࡿゎᯒゎ࡜ᮏ◊✲࡛㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢ

ᩘ್ゎࡢẚ㍑ࡼࡾ㸪㐍ᒎᗎ┙ࡢ

ܽȀܹ ൌ ͲǤͲͷࡢሙྜ࡛ㄗᕪ 1.6%

࡜࡞ࡾ㸪

ܽȀܹ ൒ ͲǤͳ

ࡢᗈ࠸⠊ᅖ࡛ㄗᕪ

0.9%

௨ୗ࡜࡞

ࡗࡓ㸬

௨ୖࡢホ౯ࡼࡾ㸪ᮏ◊✲࡛㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢࡁ⿣㐍 ᒎ㐣⛬࡟࠾ࡅࡿゎᯒ⢭ᗘࡀ᫂ࡽ࠿࡜࡞ࡗࡓ㸬ࡲࡓ㸪ⴭ⪅ࡽ

ࡢ๓ሗ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻ࡟ࡼࡿᩘ್ゎ࡜ࡢẚ㍑࡟ࡼ

ࡾ㸪ᮏ◊✲࡟࠾ࡅࡿ

PU-XFEM

㏆ఝࡢᐇ⿦᪉ἲࢆྵࡵ㸪㛤

Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢ᭷ຠᛶࡀ☜ㄆࡉࢀࡓ㸬

5. ⑂ປࡁ⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࡢ㐺⏝౛

²⁷⁾

㏆ᖺ㸪㗰ᗋ∧ᶫᱱࡢࣂࣝࣈࣜࣈ࡜ᶓࣜࣈࡢ஺ᕪ㒊࡛ከᩘ

ࡢ⑂ປࡁ⿣ࡢⓎ⏕ࡀሗ࿌ࡉࢀ࡚࠸ࡿ㸬ࡑࢀࡽࡢࡁ⿣ࡢ࠺ࡕ㸪

≉࡟ᶓࣜࣈࡢࢫࣜࢵࢺୗ㒊ࡢ⁐᥋㒊ୗ➃࠿ࡽᶓࣜࣈෆ㒊

࡬࡜㐍ᒎࡍࡿࡶࡢࡀ඲ᦆയᩘࡢ኱㒊ศࢆ༨ࡵ࡚࠸ࡿ㸦ᅗ

-1

ཧ↷㸧²⁸⁾㸬ࡋ࠿ࡋ㸪ࡇࡢ✀ࡢ⑂ປࡁ⿣ࡢ㐍ᒎᣲືࡢ᥎ᐃࡣ

୍⯡࡟ᐜ࡛᫆ࡣ࡞࠸㸬

ࡑࡇ࡛ᮏ❶࡛ࡣ㸪㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢ㐺⏝౛࡜ࡋ࡚㸪 ᅗ-11࡟♧ࡍࣂࣝࣈࣜࣈ㗰ᗋ∧ࡢᐇࢫࢣ࣮ࣝᶍᨃヨ㦂యࢆ

⏝࠸ࡓ⑂ປヨ㦂 ²⁸⁾ࢆᑐ㇟࡟⑂ປࡁ⿣ࡢ㐍ᒎࢩ࣑࣮ࣗࣞࢩ

ࣙࣥࢆ⾜࠸ᐇ㦂⤖ᯝ࡜ࡢẚ㍑⤖ᯝࢆ♧ࡍ㸬

ヨ㦂యࡣᶓࣜࣈ3ᮏ࠾ࡼࡧࣂࣝࣈࣜࣈ5ᮏࢆ㓄⨨ࡋ࡚࠾

ࡾ㸪୰ኸࡢᶓࣜࣈ࡜ᕥ➃ࡢࣂࣝࣈࣜࣈࡢ஺ᕪ㒊㸦ᅗ-13(b)

࠾ࡼࡧ(c)ཧ↷㸧࡟࠾࠸࡚ᅗ-12࡟♧ࡍ㐍ᒎ⤒㊰ࡢ⑂ປࡁ⿣

ࡀ☜ㄆࡉࢀࡓ㸬⑂ປヨ㦂࡟࠾ࡅࡿ᭱⤊ⓗ࡞ࡁ⿣㛗ࡉࡣ

ܽ =

33.05mm

࡛࠶ࡗࡓ㸬࡞࠾㸪ᅗ

-12

ࡢ෗┿ࡣ⑂ປࡁ⿣ࡢⓎ⏕

㒊఩㏆ഐࢆヨ㦂ᚋ࡟ษࡾฟࡋࡓࡶࡢ࡛࠶㸪ࢫࢺࢵࣉ࣮࣍ࣝ

ࡣ⑂ປヨ㦂ᚋ࡟᪋ࡉࢀࡓࡶࡢ࡛࠶ࡿ㸬ࡲࡓ㸪ࢫࢺࢵࣉ࣮࣍

ࣝࡢ㏆ഐࡢᏍࡣ㸪⑂ປヨ㦂࡟࠾࠸࡚ࢫࢺࢵࣉ࣮࣍ࣝ᪋ᕤᚋ

࡟ᙜ࡚ᯈ⿵ᙉࢆᐇ᪋ࡍࡿࡓࡵࡢ࣎ࣝࢺ✰࡛࠶ࡿ㸬

ゎᯒࣔࢹࣝࡣ㸪

Young

⋡

200GPa

㸪

Poisson

ẚ

0.3

ࡢ⥺ᙧ ᅗ

-13

ࣂࣝࣈࣜࣈ㗰ᗋ∧ᶍᨃヨ㦂యࡢ᭷㝈せ⣲ゎᯒࣔࢹࣝ

(c) ⑂ປࡁ⿣Ⓨ⏕㒊఩ࡢせ⣲ศ๭

㸦(b)ࡢA㒊ᣑ኱㸧

(a) ࢹࢵ࢟ࣉ࣮ࣞࢺࡢせ⣲ศ๭࡜㍕Ⲵ఩⨨ (b) ᶓࣜࣈࡢせ⣲ศ๭࠾ࡼࡧ

⑂ປࡁ⿣Ⓨ⏕㒊఩A 'P

2000mm 622mm

3200mm

A

0 20 40(mm)

ᅗ

-11

ࣂࣝࣈࣜࣈ㗰ᗋ∧ࡢᐇࢫࢣ࣮ࣝᶍᨃヨ㦂య ᅗ

-12

⑂ປヨ㦂࡟ࡼࡿࡁ⿣㐍ᒎ⤒㊰

(a) a0 = 9.30mm (b) a = 15.15mm (c) a = 33.50mm

ᙎᛶయ࡜ࡋ࡚ࣔࢹࣝ໬ࡍࡿ㸬

ABAQUS CAE

࡟ࡼࡾసᡂࡋ ࡓゎᯒࣔࢹࣝࢆᅗ-13 ࡟♧ࡍ㸬ᇶᮏࡢせ⣲ࢧ࢖ࢬࡣ

݄ = 20mm

⛬ᗘ࡜ࡋࡓ㸬ࡓࡔࡋ㸪⑂ປヨ㦂࡟࡚ࡁ⿣ࡢⓎ⏕ࡀ☜

ㄆࡉࢀࡓ୰ኸࡢᶓࣜࣈ࡜ᕥ➃ࡢࣂࣝࣈࣜࣈࡢ஺ᕪ㒊㏆ഐ 㸦ᅗ

-13(b)ࡢ A

㒊㸧࡟࠾࠸࡚ࡣ㸪ᅗ-13(c)࡟♧ࡍࡼ࠺࡟㸪

⑂ປヨ㦂࡛☜ㄆࡉࢀࡓ᭱⤊ⓗ࡞ࡁ⿣඲㛗ࡢ

10

ศࡢ

1

࡛࠶

ࡿ

݄ = 3mm

⛬ᗘࢆせ⣲ࢧ࢖ࢬࡢ᭱ᑠ್࡟タᐃࡋせ⣲ศ๭

ࢆ⾜ࡗࡓ㸬࡞࠾㸪௨ୖࡢࡼ࠺࡞ࡁ⿣ᙧ≧ࢆせ⣲ࡢቃ⏺࡜⊂

❧࡟ࣔࢹࣝ໬ྍ⬟࡛࠶ࡿ

XFEM

ࡢ≉ᚩࢆ⏕࠿ࡋࡓࡁ⿣㏆

ഐ࡟࠾ࡅࡿせ⣲ศ๭ࡣ㸪㏻ᖖࡢ

FEM

ࡢࡁ⿣ゎᯒ࡛⏝࠸ࡽ

ࢀࡿࡁ⿣㏆ഐࡢᚤ⣽࡞せ⣲ศ๭࡜኱ࡁࡃ␗࡞ࡿⅬ࡟ὀព ࡉࢀࡓ࠸㸬

ࡲࡓ㸪ᮏヨ㦂ࡢ㍕Ⲵ఩⨨ࢆᅗ-13(a)࡟ేグࡍࡿ㸬⑂ປヨ 㦂࡛ࡣ㸪᭱⤊ⓗ࡞㍕Ⲵᅇᩘ࡛࠶ࡿ

1.80 × 10

⁶

cycle

࡟⮳ࡿࡲ

࡛㸪ࡁ⿣ࡢ㐍ᒎ࡟క࠸Ⲵ㔜ࢆኚ໬ࡉࡏ࡚࠸ࡿࡀ㸦ᅗ-16ཧ

↷㸧㸪ゎᯒ࡟ࡼࡿࢩ࣑࣮ࣗࣞࢩ࡛ࣙࣥࡣⲴ㔜ࢆ⑂ປヨ㦂࡛

ᚓࡽࢀࡓࡁ⿣㛗ࡉ࡜ᑐᛂࡉࡏࡿࡶࡢ࡜ࡍࡿ㸬

ࢩ࣑࣮ࣗࣞࢩࣙࣥࡢᐇ⾜๓࡟ᐃ⩏ࡀᚲせ࡞ࡁ⿣ᙧ≧ࢹ

࣮ࢱ

CRACK.inp

ࡢึᮇ್࡜ࡋ࡚ࡣ㸪⑂ປヨ㦂ᗎ┙ࡢ㍕Ⲵ

ᅇᩘ

ܰ = 1.0 × 10

⁵

cycle

࡛☜ㄆࡉࢀࡓࡁ⿣㛗ࡉ

ܽ

_଴

= 9.30mm

ࢆ௬ᐃࡋࡓ㸦ᅗ

-14(a)ཧ↷㸧

㸬

௨ୖࡢ᮲௳ࡢࡶ࡜㸪ࣂࣝࣈࣜࣈ㗰ᗋ∧ᶍᨃヨ㦂య࡟ᑐࡍ

ࡿ⑂ປࡁ⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࢆᐇ᪋ࡋࡓ㸬

⑂ປࡁ⿣ࡢ㐍ᒎ⤒㊰࡟㛵ࡍࡿゎᯒ࠾ࡼࡧ⑂ປヨ㦂ࡢ⤖

ᯝࢆẚ㍑ࡍࡿࡓࡵ㸪ࡑࢀࡒࢀࡢ㐍ᒎ⤒㊰ࢆ㔜ࡡྜࢃࡏ࡚ᅗ

-14

࡟♧ࡍ㸬ࡇࢀ࡟ࡼࡾ㸪ࣂࣝࣈࣜࣈ㗰ᗋ∧ࡢᐇࢫࢣ࣮ࣝ

ᶍᨃヨ㦂య࡜࠸࠺」㞧࡞

3

ḟඖᵓ㐀య࡟࠾࠸࡚㸪ᒁᡤⓗ࡞

㡿ᇦ࡟Ⓨ⏕ࡋࡓ⑂ປࡁ⿣ࢆᑐ㇟࡜ࡋ࡚࠸ࡿ࡟ࡶ㛵ࢃࡽࡎ㸪 ゎᯒࢥ࣮ࢻࢆ⏝࠸ࡓ⑂ປࡁ⿣ࡢ㐍ᒎ⤒㊰࡟㛵ࡍࡿゎᯒ⤖

ᯝࡣ㸪⑂ປヨ㦂ࡢᐇ㦂⤖ᯝ࡜㠀ᖖ࡟Ⰻ࠸୍⮴ࢆ♧ࡋࡓ㸬

ࡲࡓ㸪ゎᯒ࡟ࡼࡿᛂຊᣑ኱ಀᩘ⠊ᅖࡢ᥎⛣ࢆᅗ

-15

࡟♧

ࡋ㸪ࡇࢀ࡟ࡼࡾᚓࡽࢀࡓ⑂ປࡁ⿣ࡢఏ᧛㏿ᗘ࡟㛵ࡍࡿ⑂ປ ヨ㦂࡜ࡢẚ㍑⤖ᯝࢆᅗ

-16

࡟♧ࡍ㸬ࡇࢀࡼࡾ㸪⑂ປヨ㦂ࢆ

ᐇ᪋ࡋࡓ⠊ᅖ࡟࠾࠸࡚

2

ಸ⛬ᗘᏳ඲ഃ࡟ホ౯ࡉࢀࡓ㸬ࡇࡇ

࡛㸪⏝࠸ࡓࡁ⿣ఏ᧛㏿ᗘᘧࡀ⁐᥋㒊ࡢ㧗࠸ṧ␃ᛂຊࡢస⏝

ୗ࡛ồࡵࡽࢀࡓࡶࡢ࡛࠶ࡿࡇ࡜ࢆ⪃៖ࡍࢀࡤ²⁴⁾㸪ࡇࡢ⛬ᗘ ࡢᏳ඲ഃࡢホ౯ࡣጇᙜ࡞ࡶࡢ࡜⪃࠼ࡽࢀࡿ㸬ࡇࡢࡓࡵ㸪ᮏ

◊✲࡛㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡣఏ᧛㏿ᗘࡢホ౯࡟㛵ࡋ࡚ࡶ

༑ศ࡞ᐇ⏝ᛶࢆ᭷ࡋ࡚࠸ࡿ࡜⪃࠼ࡽࢀࡿ㸬

࡞࠾㸪ࡁ⿣ࡢ࡞࠸ሙྜ࡜ẚ㍑ࡋ࡚㸪

XFEM

࡟ࡼࡾࡁ⿣ࢆ

ゎᯒࣔࢹࣝ࡟ᑟධࡋࡓሙྜ㸪ゎᯒࣔࢹࣝ඲య࡛ࡢ⮬⏤ᗘᩘ

ࡣ㸪ึᮇࡁ⿣㛗ࡉ

ܽ

଴

= 9.3mm

ࡢሙྜ

0.07%

⛬ᗘቑຍࡋࡓ㸬 ࡇࢀ࡟ᑐࡋ㸪ᮏ᳨ドࡢ⠊ᅖ࡛ࡣ

ABAQUS

࡟ࡼࡗ࡚ฟຊࡉ

ࢀࡓ

CPU

᫬㛫ࡣࡁ⿣ࡢ᭷↓࡟ࡼࡿᕪ␗ࡀᑠࡉࡃ㸪ྠ୍᮲

௳࡟࠾ࡅࡿ

CPU

᫬㛫ࡢࡤࡽࡘࡁࡢ⠊ᅖෆ࡟ྵࡲࢀࡿࡓࡵ㸪 ࡁ⿣ࡢ᭷↓ࡢࡳ࡟ࡼࡿ

CPU

᫬㛫࡬ࡢᙳ㡪ࢆᐃ㔞ⓗ࡟ホ౯

0.0E+00 1.0E+01 2.0E+01 3.0E+01

8.0 16.0 32.0

'K(MPam)

Crack length : a(mm)

Ҁ

8.0 16.0 32.0

8.0 16.0 32.0 64.0 128.0 256.0

Crack length: a(mm)

Cycle of loads : N (™10⁶cycle) Numerical Simulation Fatigue Test

ᅗ

-14

⑂ປヨ㦂࡜ࡢࡁ⿣㐍ᒎ⤒㊰ࡢẚ㍑

[™10⁴ cycle] [kN]

0 < N < 70 140 70 < N < 90 200 90 < N <110 280 110 < N <140 140 140 < N <150 200 150 < N 280

⑂ປヨ㦂࡟࠾ࡅࡿ

Ⲵ㔜ᖜ'Pࡢ᥎⛣

ᅗ

-16

㍕Ⲵᅇᩘ࡜ࡁ⿣㛗ࡉࡢ㛵ಀ

㸦⑂ປヨ㦂࡟࠾ࡅࡿⲴ㔜ᖜࡢ᥎⛣ࢆేグ㸧 ᅗ

-15

➼౯ᛂຊᣑ኱ಀᩘ⠊ᅖ

߂ܭ

ࡢ᥎⛣

29)

ࡍࡿࡇ࡜ࡣᅔ㞴࡛࠶ࡗࡓ㸬ࡍ࡞ࢃࡕ㸪ᮏ◊✲࡛⾜ࡗࡓ

XFEM

ࡢỗ⏝

FEM

ࢯࣇࢺ࢙࢘࢔࡬ࡢᐇ⿦᪉ἲࡣ㸪㐣๫࡞

ィ⟬᫬㛫ࡢቑຍࢆ⏕ࡌࡿࡇ࡜ࡣ࡞ࡃ㸪ィ⟬᫬㛫ࡢほⅬ࠿ࡽ

ࡶࡑࡢ᭷ຠᛶࡀ♧၀ࡉࢀࡓ㸬

6. ⤖ㄽ

ᮏ◊✲࡛ࡣ㸪௦⾲ⓗ࡞ỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔ࡢࡦ

࡜ࡘ࡛࠶ࡿ

ABAQUS

ࢆ࣮࣋ࢫ࡜ࡋ࡚㸪௨ୗࡢࣉࣟࢢ࣒ࣛ

ࢆ㏣ຍࡍࡿࡇ࡜࡟ࡼࡾ㸪ࢩ࢙ࣝせ⣲࡛ࣔࢹࣝ໬࡛ࡁࡿᐇᵓ 㐀≀ࡢ⑂ປࡁ⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࡀྍ⬟࡜࡞ࡿゎᯒ ࢥ࣮ࢻࡢ㛤Ⓨࢆ⾜ࡗࡓ㸬

(1) ࢯࣇࢺ࢙࢘࢔࡟ PU-XFEM

㏆ఝࢆᑟධࡍࡿࣉࣟ

ࢢ࣒ࣛ

(2) ᛂຊᣑ኱ಀᩘࡢ⟬ฟ㸪⑂ປࡁ⿣㐍ᒎ᪉ྥ࣭ఏ᧛

㏿ᗘࡢホ౯࠾ࡼࡧࡁ⿣ᙧ≧ࡢ᭦᪂ࢆ⾜࠺ࣉࣟࢢ

࣒ࣛ

(3) ゎᯒࢥ࣮ࢻ඲యࡢࣇ࣮ࣟࢆไᚚࡍࡿࣉࣟࢢ࣒ࣛ

ࡇࡇ࡛㸪≉࡟ከ㔜⠇Ⅼࢆᐃ⩏ࡍࡿࡇ࡜࡛

ABAQUS

ࡢつ⣙

࡟ࡼࡗ࡚ᐃࡵࡽࢀࡓᮏ᮶ࡢ⮬⏤ᗘࡢᐃ⩏ࢆᦆ࡞࠺ࡇ࡜࡞

ࡃ㸪

XFEM

㏆ఝࡢᐇ⿦࡟࠾࠸࡚㔜せ࡞ㄢ㢟࡜࡞ࡿࡁ⿣㏆ഐ ࡢྛ⠇Ⅼ࡟࠾ࡅࡿ⮬⏤ᗘࡢቑຍࢆྍ⬟࡜ࡋࡓ㸬ࡲࡓ㸪ࡇࡢ

᪉ἲࡣ

ABAQUS

࡜࠸࠺ಶูࡢࢯࣇࢺ࢙࢘࢔࡟ࡢࡳᑐᛂࡋ

ࡓࡶࡢ࡛ࡣ࡞ࡃ㸪௚ࡢỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢᛂ

⏝ࡶྍ⬟࡞୍⯡ⓗ࡞⪃࠼᪉ࢆ୚࠼ࡿࡶࡢ࡛࠶ࡿ㸬

PU-XFEM

㏆ఝࢆᐇ⿦ࡋࡓゎᯒࢥ࣮ࢻ࡟㛵ࡋ࡚㸪ࡑࡢᇶ

ᮏⓗ࡞ゎᯒ⢭ᗘࡢホ౯ࢆ⾜࠺ࡓࡵ࡟㸪ᛂຊᣑ኱ಀᩘࡢゎᯒ ゎࡀ᪤▱࡛࠶ࡿᇶᮏⓗ࡞ࣔࢹࣝࢆᑐ㇟࡜ࡋࡓ◚ቯຊᏛࣃ

࣓࣮ࣛࢱࡢᩘ್ゎࡢ⢭ᗘホ౯ࢆ⾜ࡗࡓ㸬ࡲࡓ㸪ࡑࡢ㝿㸪ⴭ

⪅ࡽࡢ๓ሗ ¹¹⁾࡛⏝࠸ࡓゎᯒࢥ࣮ࢻ࡟ࡼࡿゎᯒ⤖ᯝ࡜ࡢẚ

㍑ࡶྠ᫬࡟⾜ࡗࡓ㸬ࡑࡢ⤖ᯝ㸪ᮏ◊✲࡛♧ࡋࡓỗ⏝

FEM

ゎᯒࢯࣇࢺ࢙࢘࢔࡬ࡢ

PU-XFEM

㏆ఝࡢᐇ⿦ἲࢆྵࡵ㸪ゎ ᯒࢥ࣮ࢻࡢ᭷ຠᛶࡀ☜ㄆࡉࢀࡓ㸬

㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢ㐺⏝౛࡜ࡋ࡚㸪╔┠㒊఩࡟㛵ࡋ࡚

ᐇࢫࢣ࣮ࣝࡢࣂࣝࣈࣜࣈ㗰ᗋ∧ᶍᨃヨ㦂యࡢ⑂ປヨ㦂ࢆ

ᑐ㇟࡜ࡋࡓ⑂ປࡁ⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࢆᐇ᪋ࡋ㸪⑂ປ ヨ㦂࡟ࡼࡿᐇ㦂⤖ᯝ࡜ࡢẚ㍑ࢆ⾜ࡗࡓ㸬ࡑࡢ⤖ᯝ㸪」㞧࡞

3

ḟඖᵓ㐀య࡟࠾࠸࡚㸪ᒁᡤⓗ࡞㡿ᇦ࡟Ⓨ⏕ࡋࡓ⑂ປࡁ⿣

ࢆᑐ㇟࡜ࡋ࡚࠸ࡿ࡟ࡶ㛵ࢃࡽࡎ㸪ࡑࡢ㐍ᒎ⤒㊰ࢆ⢭ᗘⰋࡃ

෌⌧ࡍࡿࡇ࡜ࡀ࡛ࡁ㸪ఏ᧛㏿ᗘࡢホ౯࡟㛵ࡋ࡚ࡶ༑ศ࡞ᐇ

⏝ᛶࢆ᭷ࡋ࡚࠸ࡿࡇ࡜ࡀ♧ࡉࢀࡓ㸬

௨ୖࡼࡾ㸪ᮏ◊✲࡟࠾࠸࡚PU-XFEMࢆỗ⏝

FEM

ゎᯒ ࢯࣇࢺ࢙࢘࢔࡬ᐇ⿦ࡍࡿࡇ࡜࡛㛤Ⓨࡋࡓゎᯒࢥ࣮ࢻࡢ⑂

ປࡁ⿣ࡢ㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥࢶ࣮ࣝ࡜ࡋ࡚ࡢ᭷ຠᛶࡀ

♧ࡉࢀࡓ㸬

ㅰ㎡㸸ᮏ◊✲ࡣ㸪᪂㒔ᕷ♫఍ᢏ⾡⼥ྜ๰㐀◊✲఍࡛ࡢࣉࣟ

ࢪ࢙ࢡࢺࠕ㗰ᶫࡢ⑂ປட⿣㐍ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥᡭἲࡢ㛤

Ⓨ࡜ࡑࡢ⥔ᣢ⟶⌮࡬ࡢᛂ⏝࡟㛵ࡍࡿ◊✲ࠖࡢ୍⎔࡜ࡋ࡚⾜

ࢃࢀࡓ㸬ࡲࡓ㸪㜰⚄㧗㏿㐨㊰ᰴᘧ఍♫ࡢᓮ㇂῕Ặ㸪㛵す኱

Ꮫ኱Ꮫ㝔ࡢᒣᒸ኱㍜Ặࢆࡣࡌࡵ㛵ಀ⪅ྛ఩࡟ࡣ㸪㗰ᗋ∧ᶍ ᨃヨ㦂యࡢ⑂ປヨ㦂࡟㛵ࡍࡿ㈨ᩱࢆᥦ౪ࡋ࡚࠸ࡓࡔ࠸ࡓ㸬 ࡇࡇ࡟グࡋ࡚ឤㅰࡢពࢆ⾲ࡍࡿ㸬

ཧ⪃ᩥ⊩

1) ᪥ᮏ㐨㊰༠఍ : 㗰㐨㊰ᶫࡢ⑂ປタィᣦ㔪, 2002.

2) Barsoum, R. S. : On the use of isoparametric finite elements in linear fracture mechanics. International Journal for Numerical Methods in Engineering, Vol.10, pp.25-37, 1976.

3) Taniguchi, T. : Crack propagation analysis in civil engineering structures. Computers and Structures, Vol.41, pp.1293-1303, 1991.

4) ᕷẟ⁠அ, Ᏹ㒔ᐑᬛ᫛ : ᯈཌ㈏㏻ษḞᮦࡢ⑂ປட⿣㐍ᒎゎ ᯒ࡬ࡢ᭷㝈せ⣲ἲࡢ㐺⏝ᛶ࡟ࡘ࠸࡚, ᅵᮌᏛ఍➨61ᅇᖺḟ Ꮫ⾡ㅮ₇఍, ➨I㒊㛛, pp.1175-1176, 2006.

5) ⯓▼࿴㞝, ㇂฼᫭, ุ἞๛ : 㗰ࣉ࣮ࣞࢺ࣮࢞ࢱ࣮ᶫࡢࡁ⿣㐍 ᒎࢩ࣑࣮ࣗࣞࢩࣙࣥ, 㗰ᵓ㐀ᖺḟㄽᩥሗ࿌㞟, Vol.16, pp.453-458, 2008.

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㞟A, Vol.64, No.4, pp. 970-981, 2008.

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22) ኱㊰ΎႹ, ㎷ᫀᏹ, ஂಖྖ㑻, ᑠ㔝჆㞝, ඵᖭ⠜, ᱵ஭೺ྖ : 㧗ᙇຊ㗰ࡢṧ␃ᛂຊሙ࡟࠾ࡅࡿ⑂ປࡁ⿣ࡢఏࡥ᪉ྥ࠾ࡼࡧ ఏࡥᑑ࿨ࡢண , ᶵᲔᏛ఍ㄽᩥ㞟 A, Vol.59, No.562, pp.1429-1436, 1993.

23) ⏣୰ၨ௓, ⛅ᗞ⩏᫂, ຍ⸨ᣅஓ, 㧗ᶫᘯᶞ : ⧞㏉ࡋࡡࡌࡾ࣭ᘬ ᙇ」ྜⲴ㔜ୗ࡛ࡢணࡁ⿣࠿ࡽࡢ⑂ປࡁ⿣㐍ᒎ⤒㊰ࡢண , ᶵ ᲔᏛ఍ㄽᩥ㞟A, Vol.71, No.704, pp.607-614, 2005.

24) ᪥ᮏ㗰ᵓ㐀༠఍ : 㗰ᵓ㐀≀ࡢ⑂ປタィᣦ㔪࣭ྠゎㄝ㸪ᢏሗᇽ ฟ∧

25) ᒣ⏣㈗༤ : 㧗ᛶ⬟᭷㝈せ⣲ἲ㸦ィ⟬ຊᏛࣞࢡࢳ࣮ࣕࢩ࣮ࣜࢬ 9㸧, ୸ၿ, 2007.

26) ᒸᮧᘯஅ : ⥺ᙧ◚ቯຊᏛධ㛛㸦◚ቯຊᏛ࡜ᮦᩱᙉᗘㅮᗙ1㸧, ಸ㢼㤋, 1976.

27) ᰘ἟୍ᶞ, 㟷஭ᘯᶞ, Ᏹ㒔ᐑᬛ᫛, ᆏ㔝ᫀᘯ, ኟ⛅⩏ᗈ : XFEM࡟ࡼࡿ3ḟඖᵓ㐀యࡢᯈཌ㈏㏻⑂ປࡁ⿣㐍ᒎᣲືࡢ ホ౯, 㗰ᵓ㐀ᖺḟㄽᩥሗ࿌㞟, Vol.17, pp.267-274, 2009.

28) ⏣⏿ᬗᏊ, ᒣᮧΎ, ℊ⏣ಙᙪ, ㏕⏣἞⾜, 㓇஭ඃ஧, ᆏ㔝ᫀ

ᘯ : 㗰ᗋ∧ࣂࣝࣈࣜࣈ࡜ᶓࣜࣈ஺ᕪ㒊ࡢ⑂ປᦆയᑐ⟇࡟㛵 ࡍࡿᐇ㦂ⓗ᳨ウ, ᅵᮌᏛ఍➨62ᅇᖺḟᏛ⾡ㅮ₇఍, ➨I㒊㛛, pp.5-6, 2007.

29) ᓮ㇂῕, ᮡᒣ⿱ᶞ, ⏣⏿ᬗᏊ, ㏕⏣἞⾜, ᆏ㔝ᫀᘯ : ࣂࣝࣈ

ࣜࣈ㗰ᗋ∧ࡢ⑂ປᦆയᑐ⟇࡟㛵ࡍࡿᐇാᛂຊィ ࡜⑂ປヨ 㦂, 㗰ᵓ㐀ᖺḟㄽᩥሗ࿌㞟, Vol.17, pp.337-344, 2009.

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2010

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3

᭶

9

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