Composite Failure Model

In Radioss the following composite failure models may be used to describe composite material failure.
  • /FAIL/HASHIN
  • /FAIL/PUCK
  • /FAIL/LAD_DAMA
  • /FAIL/CHANG

A composite material consists of two different materials (matrix and reinforcement fiber). Each material has a different failure behavior. In Radioss it is possible to use different failure models for matrix and fiber in one composite element (for elements with property TYPE11, TYPE16, TYPE17, TYPE51, PCOMPP or TYPE22). For example, you may use /FAIL/HASHIN for fiber failure, /FAIL/PUCK for matrix failure and /FAIL/LAD_DAMA for delamination between layers or plies (if there is more than one layer or plies defined for the composite).

Besides the above typical composite failure models, /FAIL/FLD (used for isotropic brittle composite materials in layers(plies) as in, glass), /FAIL/ENERGY, /FAIL/TBUTCHER and /FAIL/TENSSTRAIN may also be used to describe failure for composite layers(plies).

/FAIL/HASHIN

In HASHIN failure, two primary failure modes are considered.
  • Fiber Mode:

    Composite fails, due to fiber rupture in tension or fiber buckling in compression. So, in /FAIL/HASHIN, tensile/shear fiber mode, compression fiber mode and crush mode are the fiber modes. If direction 1 is the fiber direction, then plane 23 is the predominant failure plane for fiber mode.

  • Matrix Mode:

    Composite fails, due to matrix cracking from the fiber. Failure matrix mode (or shear failure matrix mode) and delamination mode are both matrix modes. The failure plane for matrix mode is parallel to the fiber, and stress σ 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaaaaa@395B@ will not be considered in this mode.



Figure 1. Fiber modes and matrix modes for uni-directional lamina model
Fibers in uni-directional lamina model 1 are only in one direction and in fabric lamina model are in two directions.
Uni-directional Lamina Model Fabric Lamina Model
Damage criteria If D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ =1, then failure.

If 0 D < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ , D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ then no failure.

With D = M a x ( F 1 , F 2 , F 3 , F 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaI XaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilai aadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaa caaI0aaabeaaaOGaayjkaiaawMcaaaaa@4509@

If D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ =1, then failure.

If 0 D < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ , D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ then no failure.

With D = M a x ( F 1 , F 2 , F 3 , F 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaI XaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilai aadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaa caaI0aaabeaaaOGaayjkaiaawMcaaaaa@4509@

Tensile/shear fiber mode F 1 = ( σ 11 σ 1 t ) 2 + ( σ 12 2 + σ 13 2 σ 12 f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaIXaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakiaawMYicaGLQm caaeaacqaHdpWCdaqhaaWcbaGaaGymaaqaaiaadshaaaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaada Wcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGc cqGHRaWkcqaHdpWCdaqhaaWcbaGaaGymaiaaiodaaeaacaaIYaaaaa GcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamOzaaaakmaa CaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaaa@5538@ F 1 = ( σ 11 σ 1 t ) 2 + ( σ 12 2 + σ 13 2 σ a f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaIXaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakiaawMYicaGLQm caaeaacqaHdpWCdaqhaaWcbaGaaGymaaqaaiaadshaaaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaada Wcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGc cqGHRaWkcqaHdpWCdaqhaaWcbaGaaGymaiaaiodaaeaacaaIYaaaaa GcbaGaeq4Wdm3aa0baaSqaaiaadggaaeaacaWGMbaaaOWaaWbaaSqa beaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@54A7@

F 2 = ( σ 22 σ 2 t ) 2 + ( σ 12 2 + σ 23 2 σ b f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaIYaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQm caaeaacqaHdpWCdaqhaaWcbaGaaGOmaaqaaiaadshaaaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaada Wcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGc cqGHRaWkcqaHdpWCdaqhaaWcbaGaaGOmaiaaiodaaeaacaaIYaaaaa GcbaGaeq4Wdm3aa0baaSqaaiaadkgaaeaacaWGMbaaaOWaaWbaaSqa beaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@54AD@

With, σ a f = σ 12 f , σ b f = σ 12 f σ 2 t σ 1 t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaqhaaWcbaGaamyyaaqaaiaadAgaaaGccqGH9aqpcqaH dpWCdaqhaaWcbaGaaGymaiaaikdaaeaacaWGMbaaaOGaaGzaVlaays W7caGGSaGaaGjcVlaaywW7caaMb8Uaeq4Wdm3aa0baaSqaaiaadkga aeaacaWGMbaaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaaigdacaaIYa aabaGaamOzaaaakmaalaaabaGaeq4Wdm3aa0baaSqaaiaaikdaaeaa caWG0baaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaa aaaaa@5AE6@

Compression fiber mode F 2 = ( σ a σ 1 c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOramaaBaaaleaacaaIYaaabeaakiabg2da9maabmaabaWaaSaa aeaadaaadaqaaiabeo8aZnaaBaaaleaacaWGHbaabeaaaOGaayzkJi aawQYiaaqaaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaa kiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMe8oaaa@48FE@

With σ a = σ 11 + σ 22 + σ 33 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaaGjbVlabeo8aZnaaBaaaleaacaWGHbaabeaakiabg2da9iabgkHi Tiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcaaMb8 +aaaWaaeaacqGHsisldaWcaaqaaiabeo8aZnaaBaaaleaacaaIYaGa aGOmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaae qaaaGcbaGaaGOmaaaaaiaawMYicaGLQmcaaaa@515F@

F 3 = ( σ a σ 1 c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaIZaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiabeo8aZnaaBaaaleaacaWGHbaabeaaaOGaayzkJiaawQYiaaqaai abeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaaaaa@4388@

With σ a = σ 11 + σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggaaeqaaOGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaaigdacaaIXaaabeaakiabgUcaRiaaygW7daaadaqaaiabgkHiTi abeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGLQmca aaa@46D3@

F 4 = ( σ b σ 2 c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOramaaBaaaleaacaaI0aaabeaakiabg2da9maabmaabaWaaSaa aeaadaaadaqaaiabeo8aZnaaBaaaleaacaWGIbaabeaaaOGaayzkJi aawQYiaaqaaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaaaa kiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@476B@

With σ b = σ 22 + σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaadkgaaeqaaOGaeyypa0JaeyOeI0Iaeq4W dm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaaygW7daaada qaaiabgkHiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaa wMYicaGLQmcaaaa@4AB6@

Crush mode F 3 = ( p σ c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaIZaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiaadchaaiaawMYicaGLQmcaaeaacqaHdpWCdaWgaaWcbaGaam4yaa qabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@40E2@

With p = σ 11 + σ 22 + σ 33 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iabgkHiTmaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaa beaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccq GHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaG4m aaaaaaa@45C2@

F 5 = ( p σ c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI1aaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiaadchaaiaawMYicaGLQmcaaeaacqaHdpWCdaWgaaWcbaGaam4yaa qabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@40E4@

With p = σ 11 + σ 22 + σ 33 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iabgkHiTmaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaa beaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccq GHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaG4m aaaaaaa@45C2@

Shear failure matrix mode F 6 = ( σ 12 σ 12 m ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI2aaabeaakiabg2da9maabmaabaWaaSaaaeaacqaHdpWC daWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaai aaigdacaaIYaaabaGaamyBaaaaaaaakiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaaaaa@4312@
Failure matrix mode F 4 = ( σ 22 σ 2 t ) 2 + ( σ 23 S 23 ) 2 + ( σ 12 S 12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI0aaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqa aiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQm caaeaacqaHdpWCdaqhaaWcbaGaaGOmaaqaaiaadshaaaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaada Wcaaqaaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaaakeaacaWG tbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaaakiaawIcacaGLPaaada ahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaamaalaaabaGaeq4W dm3aaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofadaWgaaWcba GaaGymaiaaikdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaaaaa@56F7@

Where,

S 12 = σ 12 m + σ 22 tan ϕ S 23 = σ 23 m + σ 22 tan ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtb WaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iabeo8aZnaaDaaa leaacaaIXaGaaGOmaaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgk HiTiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGL QmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaacaWGtbWaaSbaaSqaai aaikdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIYaGa aG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgkHiTiabeo8aZn aaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQmcaciGG0bGa aiyyaiaac6gacqaHvpGzaaaa@5D2F@

Delamination mode F 5 = S d e l 2 [ ( σ 33 σ 3 t ) 2 + ( σ 23 S ˜ 23 ) 2 + ( σ 13 S 13 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI1aaabeaakiabg2da9iaadofadaqhaaWcbaGaamizaiaa dwgacaWGSbaabaGaaGOmaaaakmaadmaabaWaaeWaaeaadaWcaaqaam aaamaabaGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzk JiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIZaaabaGaamiDaaaaaa aakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqa daqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaaO qaaiqadofagaacamaaBaaaleaacaaIYaGaaG4maaqabaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaada Wcaaqaaiabeo8aZnaaBaaaleaacaaIXaGaaG4maaqabaaakeaacaWG tbWaaSbaaSqaaiaaigdacaaIZaaabeaaaaaakiaawIcacaGLPaaada ahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaaaaa@5D97@

Where,

S 13 = σ 13 m + σ 33 tan ϕ S ˜ 23 = σ 23 m + σ 33 tan ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtb WaaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaa leaacaaIXaGaaG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgk HiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGL QmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaaceWGtbGbaGaadaWgaa WcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaa ikdacaaIZaaabaGaamyBaaaakiabgUcaRmaaamaabaGaeyOeI0Iaeq 4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQYiaiGa cshacaGGHbGaaiOBaiabew9aMbaaaa@5D44@

F 7 = S d e l 2 [ ( σ 33 σ 3 t ) 2 + ( σ 23 S 23 ) 2 + ( σ 13 S 13 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI3aaabeaakiabg2da9iaadofadaqhaaWcbaGaamizaiaa dwgacaWGSbaabaGaaGOmaaaakmaadmaabaWaaeWaaeaadaWcaaqaam aaamaabaGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzk JiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIZaaabaGaamiDaaaaaa aakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqa daqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaaO qaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaWaaSaaae aacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaam4uamaa BaaaleaacaaIXaGaaG4maaqabaaaaaGccaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaaGccaGLBbGaayzxaaaaaa@5D8A@

Where,

S 13 = σ 13 m + σ 33 tan ϕ S ˜ 23 = σ 23 m + σ 33 tan ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtb WaaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaa leaacaaIXaGaaG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgk HiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGL QmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaaceWGtbGbaGaadaWgaa WcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaa ikdacaaIZaaabaGaamyBaaaakiabgUcaRmaaamaabaGaeyOeI0Iaeq 4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQYiaiGa cshacaGGHbGaaiOBaiabew9aMbaaaa@5D44@

Note: a = { a i f a > 0 0 i f a < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaaadaqaaiaadggaaiaawMYicaGLQmcacqGH9aqpdaGabaqaauaa beqaceaaaeaacaWGHbGaaGjbVlaaysW7caWGPbGaamOzaiaaysW7ca WGHbGaeyOpa4JaaGimaaqaaiaaicdacaaMe8UaaGjbVlaadMgacaWG MbGaaGjbVlaadggacqGH8aapcaaIWaaaaaGaay5Eaaaaaa@5187@

In /FAIL/HASHIN, material strength σ 1 t , σ 2 t , σ 3 t , σ 1 c , σ 2 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIYaaabaGaamiDaaaakiaacYcacqaHdpWCdaqhaaWcbaGaaG4maa qaaiaadshaaaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG JbaaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaa a@4AF4@ are derived from tension/compression test for composite.

Crush strength σ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadogaaeqaaaaa@38CD@ and fiber shear strength σ 12 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdacaaIYaaabaGaamOzaaaaaaa@3A48@ may be obtained from a quasi-static punch shear test (QS-PST). 6 Crush strength σ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadogaaeqaaaaa@38CD@ from the span to punch ratio (SPR) =0 and fiber shear strength σ 12 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdacaaIYaaabaGaamOzaaaaaaa@3A48@ from SPR=1.1.

ϕ is the Coulomb friction angle. It is observed that composite shear strength is higher if the composite is also under compression (rather than under tension). This is due to the friction between matrix and fiber.

The shear strength is assumed to be proportional to compression stress and is computed as:(1) S 12 = σ 12 m + σ 22 tan ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGa aGymaiaaikdaaeaacaWGTbaaaOGaey4kaSYaaaWaaeaacqGHsislcq aHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLPmIaayPkJaGa ciiDaiaacggacaGGUbGaeqy1dygaaa@498D@


Figure 2.
Friction angle ϕ may be fitted with an Off-Axis compression test with different angles θ (for example, 30 , 45 , 60 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIZaGaaG imamaaCaaaleqabaGaeSigI8gaaOGaaiilaiaaisdacaaI1aWaaWba aSqabeaacqWIyiYBaaGccaGGSaGaaGOnaiaaicdadaahaaWcbeqaai ablIHiVbaaaaa@417F@ , …).


Figure 3.

σ 12 m , σ 13 m , σ 23 m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdacaaIYaaabaGaamyBaaaakiaacYcacqaHdpWCdaqh aaWcbaGaaGymaiaaiodaaeaacaWGTbaaaOGaaiilaiabeo8aZnaaDa aaleaacaaIYaGaaG4maaqaaiaad2gaaaaaaa@4478@ may be derived from a matrix shear test in three directions.

S d e l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGKbGaamyzaiaadYgaaeqaaaaa@39BE@ is the scale factor for delamination criteria. It may be fitted with composite delamination experimental datain order for delamination failure to correlate with the damage area in experiment.

/FAIL/PUCK

In Puck failure, two types of failure are considered.
  • Fiber fracture: composite fails, due to the fiber reaching the tensile or compression strength limit.
  • Inter fiber failure (IFF): composite fails, due to the fiber matrix cracking.
Damage criteria If D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ =1, then failure.

If 0 D < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ , then no failure.

With D = M a x ( e f ( t e n s i l e ) , e f ( c o m p r e s s i o n ) , e f ( M o d e A ) , e f ( M o d e B ) , e f ( M o d e C ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWG MbaabeaakiaacIcacaWG0bGaamyzaiaad6gacaWGZbGaamyAaiaadY gacaWGLbGaaiykaiaacYcacaWGLbWaaSbaaSqaaiaadAgaaeqaaOGa aiikaiaadogacaWGVbGaamyBaiaadchacaWGYbGaamyzaiaadohaca WGZbGaamyAaiaad+gacaWGUbGaaiykaiaacYcacaWGLbWaaSbaaSqa aiaadAgaaeqaaOGaaiikaiaad2eacaWGVbGaamizaiaadwgacaWGbb GaaiykaiaacYcacaWGLbWaaSbaaSqaaiaadAgaaeqaaOGaaiikaiaa d2eacaWGVbGaamizaiaadwgacaWGcbGaaiykaiaacYcacaWGLbWaaS baaSqaaiaadAgaaeqaaOGaaiikaiaad2eacaWGVbGaamizaiaadwga caWGdbGaaiykaaGaayjkaiaawMcaaaaa@6DD7@

Fiber fraction failure Tensile fiber failure mode: σ 11 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH+aGpcaaI Waaaaa@3F79@

e f ( t e n s i l e ) = σ 11 σ 1 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaamiDaiaadwga caWGUbGaam4CaiaadMgacaWGSbGaamyzaaGaayjkaiaawMcaaiabg2 da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqa aiabeo8aZnaaDaaaleaacaaIXaaabaGaamiDaaaaaaaaaa@4C2A@

Compressive fiber failure mode: σ 11 < 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH8aapcaaI Waaaaa@3F75@

e f ( c o m p r e s s i o n ) = | σ 11 | σ 1 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaam4yaiaad+ga caWGTbGaamiCaiaadkhacaWGLbGaam4CaiaadohacaWGPbGaam4Bai aad6gaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaaemGabaGaeq4W dm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOGaay5bSlaawIa7aaqaai abeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaaaa@530F@

Inter fiber failure (IFF) 2 Mode A, if σ 22 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH+aGpcaaI Waaaaa@3F7B@ :

fail_puck_modeA
Figure 4.

e f ( M o d e A ) = 1 σ ¯ 12 [ ( σ ¯ 12 σ 2 t p 12 + ) 2 σ 22 2 + σ 12 2 + p 12 + σ 22 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaamytaiaad+ga caWGKbGaamyzaiaadgeaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai aaigdaaeaacuaHdpWCgaqeamaaBaaaleaacaaIXaGaaGOmaaqabaaa aOWaamWaaeaadaGcaaqaamaabmaabaWaaSaaaeaacuaHdpWCgaqeam aaBaaaleaacaaIXaGaaGOmaaqabaaakeaacqaHdpWCdaqhaaWcbaGa aGOmaaqaaiaadshaaaaaaOGaeyOeI0IaamiCamaaDaaaleaacaaIXa GaaGOmaaqaaiabgUcaRaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaakiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGcdaahaa WcbeqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGymaiaa ikdaaeqaaOWaaWbaaSqabeaacaaIYaaaaaqabaGccqGHRaWkcaWGWb Waa0baaSqaaiaaigdacaaIYaaabaGaey4kaScaaOGaeq4Wdm3aaSba aSqaaiaaikdacaaIYaaabeaaaOGaay5waiaaw2faaaaa@68DA@

Mode C, if σ 22 < 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH8aapcaaI Waaaaa@3F77@ :

fail_puck_modeC
Figure 5.

e f ( M o d e C ) = [ ( σ 12 2 ( 1 + p 22 ) σ ¯ 12 ) 2 + ( σ 22 σ 2 c ) 2 ] ( σ 2 c σ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaamytaiaad+ga caWGKbGaamyzaiaadoeaaiaawIcacaGLPaaacqGH9aqpdaWadiqaam aabmGabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqa aaGcbaGaaGOmaiaacIcacaaIXaGaey4kaSIaamiCamaaDaaaleaaca aIYaGaaGOmaaqaaiabgkHiTaaakiaacMcacuaHdpWCgaqeamaaBaaa leaacaaIXaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabe aacaaIYaaaaOGaey4kaSYaaeWaceaadaWcaaqaaiabeo8aZnaaBaaa leaacaaIYaGaaGOmaaqabaaakeaacqaHdpWCdaqhaaWcbaGaaGOmaa qaaiaadogaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa aaGccaGLBbGaayzxaaWaaeWaceaadaWcaaqaaiabeo8aZnaaDaaale aacaaIYaaabaGaam4yaaaaaOqaaiabgkHiTiabeo8aZnaaBaaaleaa caaIYaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaaaaa@69A4@

Mode B:

fail_puck_modeB
Figure 6.

e f ( M o d e B ) = 1 σ ¯ 12 ( σ 12 2 + ( p 12 σ 22 ) 2 + p 12 σ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaamytaiaad+ga caWGKbGaamyzaiaadkeaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai aaigdaaeaacuaHdpWCgaqeamaaBaaaleaacaaIXaGaaGOmaaqabaaa aOWaaeWaceaadaGcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaa qaaiaaikdaaaGccqGHRaWkdaqadiqaaiaadchadaqhaaWcbaGaaGym aiaaikdaaeaacqGHsislaaGccqaHdpWCdaWgaaWcbaWaaSbaaWqaai aaikdacaaIYaaabeaaaSqabaaakiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaaabeaakiabgUcaRiaadchadaqhaaWcbaGaaGymaiaaik daaeaacqGHsislaaGccqaHdpWCdaWgaaWcbaWaaSbaaWqaaiaaikda caaIYaaabeaaaSqabaaakiaawIcacaGLPaaaaaa@5F9F@

In inter fiber failure, Mode A shows failure under tension in transverse fiber direction (90 degrees to fiber direction), and in this case, shear loading could reduce the failure limit.

If under compression in transverse fiber direction, at first increasing compression will increase composite shear loading (Mode B). If compression continues to increase, then shear loading will decrease (Mode C).

Input Parameters

For fiber fracture failure, you could obtain fiber strengths σ 1 t , σ 1 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIXaaabaGaam4yaaaaaaa@3DE7@ from tension and compression composite tests in the fiber direction.

For inter fiber failure, you could obtain strengths σ 2 t , σ 2 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIYaaabaGaam4yaaaaaaa@3DE9@ from tension and compression composite tests in the transverse fiber direction.

You could obtain shear strength σ ¯ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae badaWgaaWcbaGaaGymaiaaikdaaeqaaaaa@3974@ with a pure shear test ( σ 2 = σ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigda aeqaaOGaeyypa0JaaGimaaaa@3E25@ ).

With σ 2 t , σ 2 c , σ ¯ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIYaaabaGaam4yaaaakiaacYcacuaHdpWCgaqeamaaBaaaleaaca aIXaGaaGOmaaqabaaaaa@4221@ , then p 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIYaGaaGOmaaqaaiabgkHiTaaaaaa@397D@ and p 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIYaGaaGOmaaqaaiabgkHiTaaaaaa@397D@ for Mode C and Mode B may be determined.

With σ 2 t , σ ¯ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiqbeo8aZzaaraWaaSba aSqaaiaaigdacaaIYaaabeaaaaa@3DD3@ and additional tension-shear tests in the transverse fiber direction, p 12 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIXaGaaGOmaaqaaiabgUcaRaaaaaa@3971@ may be determined. The additional tension-shear test in transverse fiber direction could take equal tension-shear (by σ 22 = σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaakiabg2da9iabeo8aZnaaBaaaleaa caaIXaGaaGOmaaqabaaaaa@3DD3@ ) loading.

Then you may derive the fracture curve in σ 22 σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaakiabgkHiTiabeo8aZnaaBaaaleaa caaIXaGaaGOmaaqabaaaaa@3DBA@ plane, as shown below.


Figure 7. IFF fracture curve in σ 22 σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaakiabgkHiTiabeo8aZnaaBaaaleaa caaIXaGaaGOmaaqabaaaaa@3DBA@ plane

For p 12 + , p 12 , p 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiaacYcacaWGWbWaa0ba aSqaaiaaigdacaaIYaaabaGaeyOeI0caaOGaaiilaiaadchadaqhaa WcbaGaaGOmaiaaikdaaeaacqGHsislaaaaaa@41F2@ parameters 3. For carbon fiber composite, use p 12 + = 0.35 , p 12 = 0.3 , p 22 = 0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiabg2da9iaaicdacaGG UaGaaG4maiaaiwdacaGGSaGaamiCamaaDaaaleaacaaIXaGaaGOmaa qaaiabgkHiTaaakiabg2da9iaaicdacaGGUaGaaG4maiaacYcacaWG WbWaa0baaSqaaiaaikdacaaIYaaabaGaeyOeI0caaOGaeyypa0JaaG imaiaac6cacaaIYaaaaa@4C47@ and for glass fiber composite, use p 12 + = 0.3 , p 12 = 0.25 , p 22 = 0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa aaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiabg2da9iaaicdacaGG UaGaaG4maiaacYcacaWGWbWaa0baaSqaaiaaigdacaaIYaaabaGaey OeI0caaOGaeyypa0JaaGimaiaac6cacaaIYaGaaGynaiaacYcacaWG WbWaa0baaSqaaiaaikdacaaIYaaabaGaeyOeI0caaOGaeyypa0JaaG imaiaac6cacaaIYaaaaa@4C46@ .

/FAIL/LAD_DAMA

/FAIL/LAD_DAMA is used to describe delamination between composite layers (damage propagation in the matrix). Assume that layers are connected through a virtual interface (contact).


Figure 8.
For example, if composite is under loading, shown below, the traction σ and displacement in direction 3 δ is as shown in curve.


Figure 9.
The area under the traction versus displacement curve is the absorbed energy by delamination, which is also termed the strain energy of the damage interface. The failure is governed by this strain energy is described below, considering 3 Modes of delamination:(2) E D = 1 2 [ σ 33 2 K 3 ( 1 d 3 ) + σ 33 2 K 3 + σ 32 2 K 2 ( 1 d 2 ) + σ 31 2 K 1 ( 1 d 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGebaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaWaamWaaeaadaWcaaqaamaaamaabaGaeq4Wdm3aaSbaaSqaaiaaio dacaaIZaaabeaaaOGaayzkJiaawQYiamaaCaaaleqabaGaaGOmaaaa aOqaaiaadUeadaWgaaWcbaGaaG4maaqabaGcdaqadaqaaiaaigdacq GHsislcaWGKbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaa aiabgUcaRmaalaaabaWaaaWaaeaacqGHsislcqaHdpWCdaWgaaWcba GaaG4maiaaiodaaeqaaaGccaGLPmIaayPkJaWaaWbaaSqabeaacaaI YaaaaaGcbaGaam4samaaBaaaleaacaaIZaaabeaaaaGccqGHRaWkda Wcaaqaaiabeo8aZnaaBaaaleaacaaIZaGaaGOmaaqabaGcdaahaaWc beqaaiaaikdaaaaakeaacaWGlbWaaSbaaSqaaiaaikdaaeqaaOWaae WaaeaacaaIXaGaeyOeI0IaamizamaaBaaaleaacaaIYaaabeaaaOGa ayjkaiaawMcaaaaacqGHRaWkdaWcaaqaaiabeo8aZnaaBaaaleaaca aIZaGaaGymaaqabaGcdaahaaWcbeqaaiaaikdaaaaakeaacaWGlbWa aSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0Iaamizam aaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaaaaiaawUfacaGL Dbaaaaa@6B73@
Where, σ 33 , σ 32 , σ 31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaiodacaaIZaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa aG4maiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG ymaaqabaaaaa@41A2@ are the stresses shown below in three modes of delamination behavior.

fail_lad-dama_delam1A
Figure 10.
With strain energy of delamination E D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGebaabeaaaaa@37B5@ , the thermodynamic force (contact force of virtual interface), also termed damage energy release rates may be calculated for these three modes:
  • Model I (DCB specimen 5)

    Y d 3 = E D d 3 | σ = c s t = 1 2 σ 33 2 K 3 ( 1 d 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGKbWaaSbaaWqaaiaaiodaaeqaaaWcbeaakiabg2da9maa eiaabaWaaSaaaeaacqGHciITcaWGfbWaaSbaaSqaaiaadseaaeqaaa GcbaGaeyOaIyRaamizamaaBaaaleaacaaIZaaabeaaaaaakiaawIa7 amaaBaaaleaacqaHdpWCcqGH9aqpcaWGJbGaam4CaiaadshaaeqaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaamaaamaa baGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQ YiamaaCaaaleqabaGaaGOmaaaaaOqaaiaadUeadaWgaaWcbaGaaG4m aaqabaGcdaqadaqaaiaaigdacqGHsislcaWGKbWaaSbaaSqaaiaaio daaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaaa@5869@

  • Model II (ENF specimen 5)

    Y d 2 = E D d 2 | σ = c s t = 1 2 σ 32 2 K 2 ( 1 d 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGKbWaaSbaaWqaaiaaikdaaeqaaaWcbeaakiabg2da9maa eiaabaWaaSaaaeaacqGHciITcaWGfbWaaSbaaSqaaiaadseaaeqaaa GcbaGaeyOaIyRaamizamaaBaaaleaacaaIYaaabeaaaaaakiaawIa7 amaaBaaaleaacqaHdpWCcqGH9aqpcaWGJbGaam4CaiaadshaaeqaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabeo8a ZnaaBaaaleaacaaIZaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaa aakeaacaWGlbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaaIXaGa eyOeI0IaamizamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaaaaaa@5694@

  • Model III

    Y d 1 = E D d 1 | σ = c s t = 1 2 σ 31 2 K 1 ( 1 d 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGKbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiabg2da9maa eiaabaWaaSaaaeaacqGHciITcaWGfbWaaSbaaSqaaiaadseaaeqaaa GcbaGaeyOaIyRaamizamaaBaaaleaacaaIXaaabeaaaaaakiaawIa7 amaaBaaaleaacqaHdpWCcqGH9aqpcaWGJbGaam4CaiaadshaaeqaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabeo8a ZnaaBaaaleaacaaIZaGaaGymaaqabaGcdaahaaWcbeqaaiaaikdaaa aakeaacaWGlbWaaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaaIXaGa eyOeI0IaamizamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaaaaaa@568F@

Where, K 3 , K 2 , K 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIZaaabeaakiaacYcacaWGlbWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaadUeadaWgaaWcbaGaaGymaaqabaaaaa@3C92@ are the stiffnesses of the virtual interface, also called interlaminar stiffness. These values may be computed as:(3)
K 3 = 2 E 33 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIZaaabeaakiabg2da9maalaaabaGaaGOmaiaadweadaWg aaWcbaGaaG4maiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFE@
K 2 = 2 G 23 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIYaaabeaakiabg2da9maalaaabaGaaGOmaiaadEeadaWg aaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFE@
K 1 = 2 G 13 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGOmaiaadEeadaWg aaWcbaGaaGymaiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFC@
Where,
t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG0baaaa@39B0@
Thickness of the virtual interface. It may be assumed to be 1/5 layer thickness.
G 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIYaGaaG4maaqabaaaaa@3867@ , G 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIYaGaaG4maaqabaaaaa@3867@ , E 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIYaGaaG4maaqabaaaaa@3867@
From upper or lower layer.
d i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbWaaSbaaSqaaiaadMgaaeqaaaaa@3ABA@
(with i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGPbaaaa@39A5@ =1,2,3), the damage variable.
It has a range from 0~1. It starts to accumulate once the composite reaches Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@ .
An example of Mode I, during traction in direction 3, at the beginning d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaaIZaaabeaaaaa@37C8@ always remains 0 until thermodynamic force Y d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqa baaaaa@3D30@ reaches Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@ (left figure).


Figure 11.

After Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@ is reached, the damage variable starts to increase and when it reaches 1, d 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaaIZaaabeaakiabg2da9iaaigdaaaa@3993@ (thermodynamic force Y d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqa baaaaa@3D30@ at this point then becomes the critical damage Y c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaam4yaaqabaaaaa@3C3A@ ). The composite could be considered as fully delaminated and may be deleted immediately or the stress may be reduced. In Radioss, the option τ max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabes8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@3F0D@ is used to simulate exponential function stress reduction nd the stress at Y c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaam4yaaqabaaaaa@3C3A@ is σ d ( t r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsgaaeqaaOGaaiikaiaadshadaWgaaWcbaGaamOCaaqa baGccaGGPaaaaa@3C58@ (Stress Decrease in Damage).

The relation with thermodynamic force Y d i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaWGPbaabeaaaSqa baaaaa@3D61@ and d i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaaaaa@37F9@ is:
  • If d 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgw MiZkaaigdaaaa@3960@ , then take d = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2 da9iaaigdaaaa@38A0@
  • If d < 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgY da8iaaigdaaaa@389E@ , then d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36DF@ is function of Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36DF@ (damage evaluation law):(4) d = w ( Y ) = Y Y 0 Y c Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadsgacqGH9aqpcaWG3bGaaiikaiaadMfacaGGPaGaeyypa0Za aSaaaeaadaaadiqaamaakaaabaGaamywaaWcbeaakiabgkHiTmaaka aabaGaamywamaaBaaaleaacaaIWaaabeaaaeqaaaGccaGLPmIaayPk JaaabaWaaOaaaeaacaWGzbWaaSbaaSqaaiaadogaaeqaaaqabaGccq GHsisldaGcaaqaaiaadMfadaWgaaWcbaGaaGimaaqabaaabeaaaaaa aa@4AED@

    Y = Y d 3 + γ 1 Y d 1 + γ 2 Y d 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbGaey ypa0JaamywamaaBaaaleaacaWGKbWaaSbaaWqaaiaaiodaaeqaaaWc beaakiabgUcaRiabeo7aNnaaBaaaleaacaaIXaaabeaakiaadMfada WgaaWcbaGaamizamaaBaaameaacaaIXaaabeaaaSqabaGccqGHRaWk cqaHZoWzdaWgaaWcbaGaaGOmaaqabaGccaWGzbWaaSbaaSqaaiaads gadaWgaaadbaGaaGOmaaqabaaaleqaaaaa@4909@ with Y d i | t = sup Y d i | τ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbWaaS baaSqaaiaadsgadaWgaaadbaGaamyAaaqabaaaleqaaOWaaqqaceaa daWgaaWcbaGaamiDaaqabaGccqGH9aqpciGGZbGaaiyDaiaacchaca WGzbWaaSbaaSqaaiaadsgadaWgaaadbaGaamyAaaqabaaaleqaaOWa aqqaceaadaWgaaWcbaGaeqiXdqNaeyizImQaamiDaaqabaaakiaawE a7aaGaay5bSdaaaa@4A9D@

    Here, γ 1 , γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHZoWzda WgaaWcbaGaaGymaaqabaGccaGGSaGaeq4SdC2aaSbaaSqaaiaaikda aeqaaaaa@3D3E@ are scale factors to consider two other delamination modes. This may be validated with experiments (DCB and ENF specimen test 5).

In the example of Mode I, γ 1 , γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHZoWzda WgaaWcbaGaaGymaaqabaGccaGGSaGaeq4SdC2aaSbaaSqaaiaaikda aeqaaaaa@3D3E@ may be 0 as this is pure delamination in direction 3 and then Y = Y d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbGaey ypa0JaamywamaaBaaaleaacaWGKbWaaSbaaWqaaiaaiodaaeqaaaWc beaaaaa@3C33@ , the relation of Y d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqa baaaaa@3D30@ and d 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaaIZaaabeaaaaa@37C8@ is:


Figure 12.
How quickly will the damage variable increase? The damage velocity d ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaaca aaaa@36E9@ (also called damage evaluation law) is computed as:
  • If d = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2 da9iaaigdaaaa@38A0@ , then d ˙ = c o n s t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaaca Gaeyypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshacaGGUaaaaa@3D61@
  • If d < 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgY da8iaaigdaaaa@389E@ , then d ˙ = k a [ 1 exp ( a w ( Y ) d ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaaca Gaeyypa0ZaaSaaaeaacaWGRbaabaGaamyyaaaadaWadaqaaiaaigda cqGHsislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaadggada aadaqaaiaadEhadaqadaqaaiaadMfaaiaawIcacaGLPaaacqGHsisl caWGKbaacaGLPmIaayPkJaaacaGLOaGaayzkaaaacaGLBbGaayzxaa aaaa@4AAF@
k a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGRbaabaGaamyyaaaaaaa@37DD@ is the maximum damage rate, which means minimum duration of the failure phenomenon. Its reciprocal, a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbaabaGaam4Aaaaaaaa@37DD@ is called the characteristic time, which may be obtained with a one-dimension tensile test. 7


Figure 13.
Through tensile sample with different stress to find the minimum time of composite damage Δ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iDaaaa@3856@ , the σ Δ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey OeI0IaeuiLdqKaamiDaaaa@3B06@ curve is vertical asymptote corresponding to the characteristic time a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbaabaGaam4Aaaaaaaa@37DD@ .


Figure 14.
Parameters a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ govern the damage evaluation law. For example, with constant parameter a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ (here a = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbGaeyypa0JaaGymaaaa@3B5E@ ), decreasing values of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ leads to more brittle failure of the composite.


Figure 15.
With constant parameter k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ (here k = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbGaeyypa0JaaGymaaaa@3B5E@ ), increasing values of a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGHbaaaa@399D@ leads to more brittle failure of the composite.


Figure 16.

/FAIL/CHANG

In Chang-Chang failure, two primary failure modes are considered.
  • Fiber mode: composite fails, due to fiber rupture in tension or fiber buckling in compression.
  • Matrix mode: composite fails, due to matrix failure under tension or compression.
This failure criteria is used only for shell elements.
Damage criteria If D = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebGaeyypa0JaaGymaaaa@3B41@ , then failure.

If 0 D < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebaaaa@3980@ , then no failure.

With D = M a x ( e f 2 , e c 2 , e m 2 , e d 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWG MbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacYcacaWGLbWaaSbaaS qaaiaadogaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiilaiaadwga daWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGSa GaamyzamaaBaaaleaacaWGKbaabeaakmaaCaaaleqabaGaaGOmaaaa aOGaayjkaiaawMcaaaaa@4A0D@ .

Fiber breakage Tensile fiber mode σ 11 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOpa4JaaGim aaaa@3DE9@ e f 2 = ( σ 11 σ 1 t ) 2 +β ( σ 12 σ ¯ 12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGMbaabeaakmaaCaaaleqabaGaaGOmaaaa kiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRi abek7aInaabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaGafq4WdmNbaebadaWgaaWcbaGaaGymaiaaikdaae qaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@5256@
Compression fiber mode σ 11 < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyipaWJaaGim aaaa@3DE5@ e c 2 = ( σ 11 σ 1 c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGJbaabeaakmaaCaaaleqabaGaaGOmaaaa kiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWGJbaa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@463B@
Matrix cracking Tensile matrix mode σ 22 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOpa4JaaGim aaaa@3DE9@ e m 2 = ( σ 22 σ 2 t ) 2 + ( σ 12 σ ¯ 12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGTbaabeaakmaaCaaaleqabaGaaGOmaaaa kiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGOmai aaikdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRi aaykW7caaMb8UaaGjcVlaayIW7daqadaqaamaalaaabaGaeq4Wdm3a aSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiqbeo8aZzaaraWaaSbaaS qaaiaaigdacaaIYaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaaaaa@56F6@
Compressive matrix mode σ 22 < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyipaWJaaGim aaaa@3DE5@ e d 2 = ( σ 22 2 σ ¯ 12 ) 2 +[ ( σ 2 c 2 σ ¯ 12 ) 2 1 ] σ 22 σ 2 c + ( σ 12 σ ¯ 12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyzamaaBaaaleaacaWGKbaabeaakmaaCaaaleqabaGaaGOmaaaa kiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGOmai aaikdaaeqaaaGcbaGaaGOmaiqbeo8aZzaaraWaaSbaaSqaaiaaigda caaIYaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaa GccqGHRaWkdaWadaqaamaabmaabaWaaSaaaeaacqaHdpWCdaqhaaWc baGaaGOmaaqaaiaadogaaaaakeaacaaIYaGafq4WdmNbaebadaWgaa WcbaGaaGymaiaaikdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaaigdaaiaawUfacaGLDbaadaWcaaqaai abeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqaHdpWCdaqh aaWcbaGaaGOmaaqaaiaadogaaaaaaOGaey4kaSYaaeWaaeaadaWcaa qaaiabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacuaHdpWC gaqeamaaBaaaleaacaaIXaGaaGOmaaqabaaaaaGccaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaaaa@6754@
Where,
direction 1
Fiber direction.
σ 1 t , σ 1 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIXaaabaGaam4yaaaaaaa@3DE7@
Fiber tensile/compressive strength.
σ 2 t , σ 2 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaa caaIYaaabaGaam4yaaaaaaa@3DE9@
Matrix strength.
Tensile or compressive loading in direction 2 (transverse to direction 1).
σ ¯ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae badaWgaaWcbaGaaGymaiaaikdaaeqaaaaa@3974@
Shear strength in composite ply plane.
β
Shear scale factor, which can be determined experimentally.

Stress Decrease in Damage

After reaching the damage criteria:
  • HASHIN:

    D = M a x ( F 1 , F 2 , F 3 , F 4 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaI XaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilai aadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaa caaI0aaabeaaaOGaayjkaiaawMcaaiabgwMiZkaaigdaaaa@478A@

  • PUCK:

    D = M a x ( e f ( t e n s i l e ) , e f ( c o m p r e s s i o n ) , e f ( M o d e A ) , e f ( M o d e B ) , e f ( M o d e C ) ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWG MbaabeaakiaacIcacaWG0bGaamyzaiaad6gacaWGZbGaamyAaiaadY gacaWGLbGaaiykaiaacYcacaWGLbWaaSbaaSqaaiaadAgaaeqaaOGa aiikaiaadogacaWGVbGaamyBaiaadchacaWGYbGaamyzaiaadohaca WGZbGaamyAaiaad+gacaWGUbGaaiykaiaacYcacaWGLbWaaSbaaSqa aiaadAgaaeqaaOGaaiikaiaad2eacaWGVbGaamizaiaadwgacaWGbb GaaiykaiaacYcacaWGLbWaaSbaaSqaaiaadAgaaeqaaOGaaiikaiaa d2eacaWGVbGaamizaiaadwgacaWGcbGaaiykaiaacYcacaWGLbWaaS baaSqaaiaadAgaaeqaaOGaaiikaiaad2eacaWGVbGaamizaiaadwga caWGdbGaaiykaaGaayjkaiaawMcaaiabgwMiZkaaigdaaaa@7058@

  • LAD_DAMA:

    d 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgw MiZkaaigdaaaa@3960@

  • CHANG:

    D = M a x ( e f 2 , e c 2 , e m 2 , e d 2 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWG MbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacYcacaWGLbWaaSbaaS qaaiaadogaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiilaiaadwga daWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGSa GaamyzamaaBaaaleaacaWGKbaabeaakmaaCaaaleqabaGaaGOmaaaa aOGaayjkaiaawMcaaiabgwMiZkaaigdaaaa@4C8E@

Stresses start decreasing and decrease gradually by using an exponential function to avoid numerical instabilities.(5) σ ( t ) = σ d ( t r ) f ( t )         = σ d ( t r ) exp ( t t r τ max ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHdp GaaiikaiaadshacaGGPaGaeyypa0JaaC4WdmaaBaaaleaacaWGKbaa beaakiaacIcacaWG0bWaaSbaaSqaaiaadkhaaeqaaOGaaiykaiabgw SixlGacAgaciGGOaGaamiDaiaacMcaaeaacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaeyypa0JaaC4WdmaaBaaaleaaca WGKbaabeaakiaacIcacaWG0bWaaSbaaSqaaiaadkhaaeqaaOGaaiyk aiabgwSixlGacwgacaGG4bGaaiiCamaabmGabaGaeyOeI0YaaSaaae aacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaWGYbaabeaaaOqaaiab es8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaaaaOGaayjkai aawMcaaaaaaa@620C@

with, t t r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaaGPaVlaayIW7caWG0bGaeyyzImRaamiDamaaBaaaleaacaWGYbaa beaaaaa@41CD@

The option τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3ABC@ controls how gradually the stress is decreased in damage.


Figure 17.
Where,
σ d ( t r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaaC4WdmaaBaaaleaacaWGKbaabeaakmaabmGabaGaamiDamaaBaaa leaacaWGYbaabeaaaOGaayjkaiaawMcaaaaa@3FF5@
Stress components when damage is reached D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey yzImRaaGymaaaa@3AB2@ .
t r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG0bWaaSbaaSqaaiaadkhaaeqaaaaa@3AD3@
Time of σ d ( t r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaaC4WdmaaBaaaleaacaWGKbaabeaakmaabmGabaGaamiDamaaBaaa leaacaWGYbaabeaaaOGaayjkaiaawMcaaaaa@3FF5@ .
τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3ABC@
Time of dynamic relaxation.
The higher the value of τ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3ABC@ , the slower stress decreases during damage. Normally, it takes 10~20 time step.

References

1 Hashin, Z., "Failure Criteria for Unidirectional Fiber Composites," Journal of Applied Mechanics, Vol. 47, 1980, pp. 329-334.
2 A. Puck, J. Kopp, and M. Knops., “Failure analysis of FRP laminates by means of physically based phenomenological models”. Composites Science Technology, 62. pp. 1633-1662. 2002.
3 A. Puck, J. Kopp, and M. Knops. “Guidelines for the determination of the parameters in Puck's action plane strength criterion”. Composites Science Technology 62. pp. 371-378. 2002.
4 L. Gornet, “Finite Element Damage Prediction of Composite Structures”.
5 Ladevèze, P., Allix, O., Douchin, B., Lévêque, D., “A Computational Method for damage Intensity Prediction in a Laminated Composite Structure”, Computational mechanics—New Trends and Applications In: Idelsohn, S., Oñate E., and Dvorkin E., (eds.) CIMNE, Barcelona, Spain (1998).
6 Gama B.A., Gillespie J.W., Punch Shear Behavior of Composites at Low and High Rates[M]// Fracture of Nano and Engineering Materials and Structures. Springer Netherlands, 2006.
7 Allix, O. & Deü, Jean-François. (1997). Delay-damage modeling for fracture prediction of laminated composites under dynamic loading. Engineering Transactions. 45. 29-46.