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 σ11 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaaa@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@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ =1, then failure.

If 0D<1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ , D MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ then no failure.

With D=Max(F1,F2,F3,F4) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaIXaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaacaaI0aaabeaaaOGaayjkaiaawMcaaaaa@4509@

If D MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ =1, then failure.

If 0D<1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ , D MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ then no failure.

With D=Max(F1,F2,F3,F4) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaIXaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaacaaI0aaabeaaaOGaayjkaiaawMcaaaaa@4509@

Tensile/shear fiber mode F1=(σ11σ1t)2+(σ122+σ132σ12f2) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaIXaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakiaawMYicaGLQmcaaeaacqaHdpWCdaqhaaWcbaGaaGymaaqaaiaadshaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaaGymaiaaiodaaeaacaaIYaaaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamOzaaaakmaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaaa@5538@ F1=(σ11σ1t)2+(σ122+σ132σaf2) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaIXaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakiaawMYicaGLQmcaaeaacqaHdpWCdaqhaaWcbaGaaGymaaqaaiaadshaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaaGymaiaaiodaaeaacaaIYaaaaaGcbaGaeq4Wdm3aa0baaSqaaiaadggaaeaacaWGMbaaaOWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@54A7@

F2=(σ22σ2t)2+(σ122+σ232σbf2) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaIYaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQmcaaeaacqaHdpWCdaqhaaWcbaGaaGOmaaqaaiaadshaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaaGOmaiaaiodaaeaacaaIYaaaaaGcbaGaeq4Wdm3aa0baaSqaaiaadkgaaeaacaWGMbaaaOWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@54AD@

With, σaf=σ12f,σbf=σ12fσ2tσ1t MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacqaHdpWCdaqhaaWcbaGaamyyaaqaaiaadAgaaaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaaGymaiaaikdaaeaacaWGMbaaaOGaaGzaVlaaysW7caGGSaGaaGjcVlaaywW7caaMb8Uaeq4Wdm3aa0baaSqaaiaadkgaaeaacaWGMbaaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamOzaaaakmaalaaabaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaaaaaaa@5AE6@

Compression fiber mode F2=(σaσ1c)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamOramaaBaaaleaacaaIYaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaWGHbaabeaaaOGaayzkJiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMe8oaaa@48FE@

With σa=σ11+σ22+σ332 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaaGjbVlabeo8aZnaaBaaaleaacaWGHbaabeaakiabg2da9iabgkHiTiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcaaMb8+aaaWaaeaacqGHsisldaWcaaqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaGOmaaaaaiaawMYicaGLQmcaaaa@515F@

F3=(σaσ1c)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaIZaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaWGHbaabeaaaOGaayzkJiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@4388@

With σa=σ11+σ33 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadggaaeqaaOGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaaygW7daaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGLQmcaaaa@46D3@

F4=(σbσ2c)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamOramaaBaaaleaacaaI0aaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaWGIbaabeaaaOGaayzkJiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@476B@

With σb=σ22+σ33 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkgaaeqaaOGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaaygW7daaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGLQmcaaaa@4AB6@

Crush mode F3=(pσc)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaIZaaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiaadchaaiaawMYicaGLQmcaaeaacqaHdpWCdaWgaaWcbaGaam4yaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@40E2@

With p=σ11+σ22+σ333 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iabgkHiTmaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaG4maaaaaaa@45C2@

F5=(pσc)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaI1aaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiaadchaaiaawMYicaGLQmcaaeaacqaHdpWCdaWgaaWcbaGaam4yaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@40E4@

With p=σ11+σ22+σ333 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iabgkHiTmaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaG4maaaaaaa@45C2@

Shear failure matrix mode F6=(σ12σ12m)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaI2aaabeaakiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamyBaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@4312@
Failure matrix mode F4=(σ22σ2t)2+(σ23S23)2+(σ12S12)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaI0aaabeaakiabg2da9maabmaabaWaaSaaaeaadaaadaqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQmcaaeaacqaHdpWCdaqhaaWcbaGaaGOmaaqaaiaadshaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@56F7@

Where,

S12=σ12m+σ22tanϕS23=σ23m+σ22tanϕ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaacaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIYaGaaG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawMYicaGLQmcaciGG0bGaaiyyaiaac6gacqaHvpGzaaaa@5D2F@

Delamination mode F5=Sdel2[(σ33σ3t)2+(σ23S˜23)2+(σ13S13)2] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBaaaleaacaaI1aaabeaakiabg2da9iaadofadaqhaaWcbaGaamizaiaadwgacaWGSbaabaGaaGOmaaaakmaadmaabaWaaeWaaeaadaWcaaqaamaaamaabaGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQYiaaqaaiabeo8aZnaaDaaaleaacaaIZaaabaGaamiDaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiqadofagaacamaaBaaaleaacaaIYaGaaG4maaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiabeo8aZnaaBaaaleaacaaIXaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaaaaa@5D97@

Where,

S13=σ13m+σ33tanϕS˜23=σ23m+σ33tanϕ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIXaGaaG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGLQmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaaceWGtbGbaGaadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaaikdacaaIZaaabaGaamyBaaaakiabgUcaRmaaamaabaGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQYiaiGacshacaGGHbGaaiOBaiabew9aMbaaaa@5D44@

F7=Sdel2[(σ33σ3t)2+(σ23S23)2+(σ13S13)2] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5D8A@

Where,

S13=σ13m+σ33tanϕS˜23=σ23m+σ33tanϕ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIXaGaaG4maaqaaiaad2gaaaGccqGHRaWkdaaadaqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaakiaawMYicaGLQmcaciGG0bGaaiyyaiaac6gacqaHvpGzaeaaceWGtbGbaGaadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaaikdacaaIZaaabaGaamyBaaaakiabgUcaRmaaamaabaGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaaaOGaayzkJiaawQYiaiGacshacaGGHbGaaiOBaiabew9aMbaaaa@5D44@

Note: a={aifa>00ifa<0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaadaaadaqaaiaadggaaiaawMYicaGLQmcacqGH9aqpdaGabaqaauaabeqaceaaaeaacaWGHbGaaGjbVlaaysW7caWGPbGaamOzaiaaysW7caWGHbGaeyOpa4JaaGimaaqaaiaaicdacaaMe8UaaGjbVlaadMgacaWGMbGaaGjbVlaadggacqGH8aapcaaIWaaaaaGaay5Eaaaaaa@5187@

In /FAIL/HASHIN, material strength σ1t,σ2t,σ3t,σ1c,σ2c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaamiDaaaakiaacYcacqaHdpWCdaqhaaWcbaGaaG4maaqaaiaadshaaaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWGJbaaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaaa@4AF4@ are derived from tension/compression test for composite.

Crush strength σc MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadogaaeqaaaaa@38CD@ and fiber shear strength σ12f MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamOzaaaaaaa@3A48@ may be obtained from a quasi-static punch shear test (QS-PST). 6 Crush strength σc MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadogaaeqaaaaa@38CD@ from the span to punch ratio (SPR) =0 and fiber shear strength σ12f MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamOzaaaaaaa@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) S12=σ12m+σ22tanϕ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaaGymaiaaikdaaeaacaWGTbaaaOGaey4kaSYaaaWaaeaacqGHsislcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLPmIaayPkJaGaciiDaiaacggacaGGUbGaeqy1dygaaa@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@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIZaGaaGimamaaCaaaleqabaGaeSigI8gaaOGaaiilaiaaisdacaaI1aWaaWbaaSqabeaacqWIyiYBaaGccaGGSaGaaGOnaiaaicdadaahaaWcbeqaaiablIHiVbaaaaa@417F@ , …).


Figure 3.

σ12m,σ13m,σ23m MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaamyBaaaakiaacYcacqaHdpWCdaqhaaWcbaGaaGymaiaaiodaaeaacaWGTbaaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaGaaG4maaqaaiaad2gaaaaaaa@4478@ may be derived from a matrix shear test in three directions.

Sdel MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWGKbGaamyzaiaadYgaaeqaaaaa@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@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ =1, then failure.

If 0D<1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ D MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ , then no failure.

With D=Max(ef(tensile),ef(compression),ef(ModeA),ef(ModeB),ef(ModeC)) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DD7@

Fiber fraction failure Tensile fiber failure mode: σ11>0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH+aGpcaaIWaaaaa@3F79@

ef(tensile)=σ11σ1t MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaamiDaiaadwgacaWGUbGaam4CaiaadMgacaWGSbGaamyzaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabeo8aZnaaDaaaleaacaaIXaaabaGaamiDaaaaaaaaaa@4C2A@

Compressive fiber failure mode: σ11<0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH8aapcaaIWaaaaa@3F75@

ef(compression)=|σ11|σ1c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamyzamaaBaaaleaacaWGMbaabeaakmaabmaabaGaam4yaiaad+gacaWGTbGaamiCaiaadkhacaWGLbGaam4CaiaadohacaWGPbGaam4Baiaad6gaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaaemGabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOGaay5bSlaawIa7aaqaaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaaaaa@530F@

Inter fiber failure (IFF) 2 Mode A, if σ22>0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH+aGpcaaIWaaaaa@3F7B@ :

fail_puck_modeA
Figure 4.

ef(ModeA)=1σ¯12[(σ¯12σ2tp12+)2σ222+σ122+p12+σ22] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=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@68DA@

Mode C, if σ22<0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH8aapcaaIWaaaaa@3F77@ :

fail_puck_modeC
Figure 5.

ef(ModeC)=[(σ122(1+p22)σ¯12)2+(σ22σ2c)2](σ2cσ22) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=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@69A4@

Mode B:

fail_puck_modeB
Figure 6.

ef(ModeB)=1σ¯12(σ122+(p12σ22)2+p12σ22) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=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@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 σ1t,σ1c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaa@3DE7@ from tension and compression composite tests in the fiber direction.

For inter fiber failure, you could obtain strengths σ2t,σ2c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaaa@3DE9@ from tension and compression composite tests in the transverse fiber direction.

You could obtain shear strength σ¯12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaebadaWgaaWcbaGaaGymaiaaikdaaeqaaaaa@3974@ with a pure shear test (σ2=σ1=0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3E25@).

With σ2t,σ2c,σ¯12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaakiaacYcacuaHdpWCgaqeamaaBaaaleaacaaIXaGaaGOmaaqabaaaaa@4221@ , then p22 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIYaGaaGOmaaqaaiabgkHiTaaaaaa@397D@ and p12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIYaGaaGOmaaqaaiabgkHiTaaaaaa@397D@ for Mode C and Mode B may be determined.

With σ2t,σ¯12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiqbeo8aZzaaraWaaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3DD3@ and additional tension-shear tests in the transverse fiber direction, p12+ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIXaGaaGOmaaqaaiabgUcaRaaaaaa@3971@ may be determined. The additional tension-shear test in transverse fiber direction could take equal tension-shear (by σ22=σ12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9iabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaaaaa@3DD3@ ) loading.

Then you may derive the fracture curve in σ22σ12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabgkHiTiabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaaaaa@3DBA@ plane, as shown below.


Figure 7. IFF fracture curve in σ22σ12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabgkHiTiabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaaaaa@3DBA@ plane

For p12+,p12,p22 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiaacYcacaWGWbWaa0baaSqaaiaaigdacaaIYaaabaGaeyOeI0caaOGaaiilaiaadchadaqhaaWcbaGaaGOmaiaaikdaaeaacqGHsislaaaaaa@41F2@ parameters 3. For carbon fiber composite, use p12+=0.35,p12=0.3,p22=0.2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiabg2da9iaaicdacaGGUaGaaG4maiaaiwdacaGGSaGaamiCamaaDaaaleaacaaIXaGaaGOmaaqaaiabgkHiTaaakiabg2da9iaaicdacaGGUaGaaG4maiaacYcacaWGWbWaa0baaSqaaiaaikdacaaIYaaabaGaeyOeI0caaOGaeyypa0JaaGimaiaac6cacaaIYaaaaa@4C47@ and for glass fiber composite, use p12+=0.3,p12=0.25,p22=0.2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaaIXaGaaGOmaaqaaiabgUcaRaaakiabg2da9iaaicdacaGGUaGaaG4maiaacYcacaWGWbWaa0baaSqaaiaaigdacaaIYaaabaGaeyOeI0caaOGaeyypa0JaaGimaiaac6cacaaIYaGaaGynaiaacYcacaWGWbWaa0baaSqaaiaaikdacaaIYaaabaGaeyOeI0caaOGaeyypa0JaaGimaiaac6cacaaIYaaaaa@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) ED=12[σ332K3(1d3)+σ332K3+σ322K2(1d2)+σ312K1(1d1)] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6B73@
Where, σ33,σ32,σ31 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaG4maiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaGymaaqabaaaaa@41A2@ are the stresses shown below in three modes of delamination behavior.

fail_lad-dama_delam1A
Figure 10.
With strain energy of delamination ED MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGebaabeaaaaa@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)

    Yd3=EDd3|σ=cst=12σ332K3(1d3)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5869@

  • Model II (ENF specimen 5)

    Yd2=EDd2|σ=cst=12σ322K2(1d2)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBaaaleaacaWGKbWaaSbaaWqaaiaaikdaaeqaaaWcbeaakiabg2da9maaeiaabaWaaSaaaeaacqGHciITcaWGfbWaaSbaaSqaaiaadseaaeqaaaGcbaGaeyOaIyRaamizamaaBaaaleaacaaIYaaabeaaaaaakiaawIa7amaaBaaaleaacqaHdpWCcqGH9aqpcaWGJbGaam4CaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabeo8aZnaaBaaaleaacaaIZaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaaaakeaacaWGlbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0IaamizamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaaaa@5694@

  • Model III

    Yd1=EDd1|σ=cst=12σ312K1(1d1)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@568F@

Where, K3,K2,K1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBaaaleaacaaIZaaabeaakiaacYcacaWGlbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadUeadaWgaaWcbaGaaGymaaqabaaaaa@3C92@ are the stiffnesses of the virtual interface, also called interlaminar stiffness. These values may be computed as:(3)
K3=2E33t MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBaaaleaacaaIZaaabeaakiabg2da9maalaaabaGaaGOmaiaadweadaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFE@
K2=2G23t MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaaGOmaiaadEeadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFE@
K1=2G13t MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBaaaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGOmaiaadEeadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaamiDaaaaaaa@3CFC@
Where,
t MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWG0baaaa@39B0@
Thickness of the virtual interface. It may be assumed to be 1/5 layer thickness.
G13 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBaaaleaacaaIYaGaaG4maaqabaaaaa@3867@ , G23 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBaaaleaacaaIYaGaaG4maaqabaaaaa@3867@ , E33 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBaaaleaacaaIYaGaaG4maaqabaaaaa@3867@
From upper or lower layer.
di MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGKbWaaSbaaSqaaiaadMgaaeqaaaaa@3ABA@
(with i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ =1,2,3), the damage variable.
It has a range from 0~1. It starts to accumulate once the composite reaches Y0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@.
An example of Mode I, during traction in direction 3, at the beginning d3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaaIZaaabeaaaaa@37C8@ always remains 0 until thermodynamic force Yd3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqabaaaaa@3D30@ reaches Y0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@ (left figure).


Figure 11.

After Y0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@3C0C@ is reached, the damage variable starts to increase and when it reaches 1, d3=1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaaIZaaabeaakiabg2da9iaaigdaaaa@3993@ (thermodynamic force Yd3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqabaaaaa@3D30@ at this point then becomes the critical damage Yc MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaam4yaaqabaaaaa@3C3A@ ). The composite could be considered as fully delaminated and may be deleted immediately or the stress may be reduced. In Radioss, the option τmax MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiabes8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@3F0D@ is used to simulate exponential function stress reduction nd the stress at Yc MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaam4yaaqabaaaaa@3C3A@ is σd(tr) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadsgaaeqaaOGaaiikaiaadshadaWgaaWcbaGaamOCaaqabaGccaGGPaaaaa@3C58@ (Stress Decrease in Damage).

The relation with thermodynamic force Ydi MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaWGPbaabeaaaSqabaaaaa@3D61@ and di MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaWGPbaabeaaaaa@37F9@ is:
  • If d1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgwMiZkaaigdaaaa@3960@ , then take d=1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9iaaigdaaaa@38A0@
  • If d<1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgYda8iaaigdaaaa@389E@ , then d MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36DF@ is function of Y MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36DF@ (damage evaluation law):(4) d=w(Y)=YY0YcY0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadsgacqGH9aqpcaWG3bGaaiikaiaadMfacaGGPaGaeyypa0ZaaSaaaeaadaaadiqaamaakaaabaGaamywaaWcbeaakiabgkHiTmaakaaabaGaamywamaaBaaaleaacaaIWaaabeaaaeqaaaGccaGLPmIaayPkJaaabaWaaOaaaeaacaWGzbWaaSbaaSqaaiaadogaaeqaaaqabaGccqGHsisldaGcaaqaaiaadMfadaWgaaWcbaGaaGimaaqabaaabeaaaaaaaa@4AED@

    Y=Yd3+γ1Yd1+γ2Yd2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbGaeyypa0JaamywamaaBaaaleaacaWGKbWaaSbaaWqaaiaaiodaaeqaaaWcbeaakiabgUcaRiabeo7aNnaaBaaaleaacaaIXaaabeaakiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIXaaabeaaaSqabaGccqGHRaWkcqaHZoWzdaWgaaWcbaGaaGOmaaqabaGccaWGzbWaaSbaaSqaaiaadsgadaWgaaadbaGaaGOmaaqabaaaleqaaaaa@4909@ with Ydi|t=supYdi|τt MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbWaaSbaaSqaaiaadsgadaWgaaadbaGaamyAaaqabaaaleqaaOWaaqqaceaadaWgaaWcbaGaamiDaaqabaGccqGH9aqpciGGZbGaaiyDaiaacchacaWGzbWaaSbaaSqaaiaadsgadaWgaaadbaGaamyAaaqabaaaleqaaOWaaqqaceaadaWgaaWcbaGaeqiXdqNaeyizImQaamiDaaqabaaakiaawEa7aaGaay5bSdaaaa@4A9D@

    Here, γ1,γ2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHZoWzdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4SdC2aaSbaaSqaaiaaikdaaeqaaaaa@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, γ1,γ2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHZoWzdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4SdC2aaSbaaSqaaiaaikdaaeqaaaaa@3D3E@ may be 0 as this is pure delamination in direction 3 and then Y=Yd3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbGaeyypa0JaamywamaaBaaaleaacaWGKbWaaSbaaWqaaiaaiodaaeqaaaWcbeaaaaa@3C33@ , the relation of Yd3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaOqaaiaadMfadaWgaaWcbaGaamizamaaBaaameaacaaIZaaabeaaaSqabaaaaa@3D30@ and d3 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaaIZaaabeaaaaa@37C8@ is:


Figure 12.
How quickly will the damage variable increase? The damage velocity d˙ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaacaaaaa@36E9@ (also called damage evaluation law) is computed as:
  • If d=1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9iaaigdaaaa@38A0@ , then d˙=const. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaacaGaeyypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshacaGGUaaaaa@3D61@
  • If d<1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgYda8iaaigdaaaa@389E@ , then d˙=ka[1exp(aw(Y)d)] MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaacaGaeyypa0ZaaSaaaeaacaWGRbaabaGaamyyaaaadaWadaqaaiaaigdacqGHsislciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaadggadaaadaqaaiaadEhadaqadaqaaiaadMfaaiaawIcacaGLPaaacqGHsislcaWGKbaacaGLPmIaayPkJaaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaa@4AAF@
ka MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGRbaabaGaamyyaaaaaaa@37DD@ is the maximum damage rate, which means minimum duration of the failure phenomenon. Its reciprocal, ak MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaam4Aaaaaaaa@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 Δt MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamiDaaaa@3856@ , the σΔt MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyOeI0IaeuiLdqKaamiDaaaa@3B06@ curve is vertical asymptote corresponding to the characteristic time ak MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaam4Aaaaaaaa@37DD@ .


Figure 14.
Parameters a MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@399D@ and k MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@399D@ govern the damage evaluation law. For example, with constant parameter a MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@399D@ (here a=1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbGaeyypa0JaaGymaaaa@3B5E@ ), decreasing values of k MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@399D@ leads to more brittle failure of the composite.


Figure 15.
With constant parameter k MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@399D@ (here k=1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbGaeyypa0JaaGymaaaa@3B5E@ ), increasing values of a MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGHbaaaa@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@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebGaeyypa0JaaGymaaaa@3B41@ , then failure.

If 0D<1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadseacqGH8aapcaaIXaaaaa@3AEE@ D MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGebaaaa@3980@ , then no failure.

With D=Max(ef2,ec2,em2,ed2) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWGMbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacYcacaWGLbWaaSbaaSqaaiaadogaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiilaiaadwgadaWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGSaGaamyzamaaBaaaleaacaWGKbaabeaakmaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaaa@4A0D@ .

Fiber breakage Tensile fiber mode σ11>0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOpa4JaaGimaaaa@3DE9@ ef2=(σ11σ1t)2+β(σ12σ¯12)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamyzamaaBaaaleaacaWGMbaabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRiabek7aInaabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGafq4WdmNbaebadaWgaaWcbaGaaGymaiaaikdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@5256@
Compression fiber mode σ11<0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyipaWJaaGimaaaa@3DE5@ ec2=(σ11σ1c)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamyzamaaBaaaleaacaWGJbaabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWGJbaaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@463B@
Matrix cracking Tensile matrix mode σ22>0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOpa4JaaGimaaaa@3DE9@ em2=(σ22σ2t)2+(σ12σ¯12)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaamyzamaaBaaaleaacaWGTbaabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaykW7caaMb8UaaGjcVlaayIW7daqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaaigdacaaIYaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@56F6@
Compressive matrix mode σ22<0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyipaWJaaGimaaaa@3DE5@ ed2=(σ222σ¯12)2+[(σ2c2σ¯12)21]σ22σ2c+(σ12σ¯12)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@6754@
Where,
direction 1
Fiber direction.
σ1t,σ1c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIXaaabaGaam4yaaaaaaa@3DE7@
Fiber tensile/compressive strength.
σ2t,σ2c MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0baaSqaaiaaikdaaeaacaWG0baaaOGaaiilaiabeo8aZnaaDaaaleaacaaIYaaabaGaam4yaaaaaaa@3DE9@
Matrix strength.
Tensile or compressive loading in direction 2 (transverse to direction 1).
σ¯12 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaebadaWgaaWcbaGaaGymaiaaikdaaeqaaaaa@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=Max(F1,F2,F3,F4)1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iaad2eacaWGHbGaamiEamaabmaabaGaamOramaaBaaaleaacaaIXaaabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadAeadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOramaaBaaaleaacaaI0aaabeaaaOGaayjkaiaawMcaaiabgwMiZkaaigdaaaa@478A@

  • PUCK:

    D=Max(ef(tensile),ef(compression),ef(ModeA),ef(ModeB),ef(ModeC))1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7058@

  • LAD_DAMA:

    d1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgwMiZkaaigdaaaa@3960@

  • CHANG:

    D=Max(ef2,ec2,em2,ed2)1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iaad2eacaWGHbGaamiEamaabmaabaGaamyzamaaBaaaleaacaWGMbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacYcacaWGLbWaaSbaaSqaaiaadogaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiilaiaadwgadaWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGSaGaamyzamaaBaaaleaacaWGKbaabeaakmaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgwMiZkaaigdaaaa@4C8E@

Stresses start decreasing and decrease gradually by using an exponential function to avoid numerical instabilities.(5) σ(t)=σd(tr)f(t)       =σd(tr)exp(ttrτmax) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHdpGaaiikaiaadshacaGGPaGaeyypa0JaaC4WdmaaBaaaleaacaWGKbaabeaakiaacIcacaWG0bWaaSbaaSqaaiaadkhaaeqaaOGaaiykaiabgwSixlGacAgaciGGOaGaamiDaiaacMcaaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaeyypa0JaaC4WdmaaBaaaleaacaWGKbaabeaakiaacIcacaWG0bWaaSbaaSqaaiaadkhaaeqaaOGaaiykaiabgwSixlGacwgacaGG4bGaaiiCamaabmGabaGaeyOeI0YaaSaaaeaacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaWGYbaabeaaaOqaaiabes8a0naaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaaaaOGaayjkaiaawMcaaaaaaa@620C@

with, ttr MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaaGPaVlaayIW7caWG0bGaeyyzImRaamiDamaaBaaaleaacaWGYbaabeaaaaa@41CD@

The option τmax MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3ABC@ controls how gradually the stress is decreased in damage.


Figure 17.
Where,
σd(tr) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaaC4WdmaaBaaaleaacaWGKbaabeaakmaabmGabaGaamiDamaaBaaaleaacaWGYbaabeaaaOGaayjkaiaawMcaaaaa@3FF5@
Stress components when damage is reached D1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaeyyzImRaaGymaaaa@3AB2@ .
tr MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWG0bWaaSbaaSqaaiaadkhaaeqaaaaa@3AD3@
Time of σd(tr) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbba9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0FirpepesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcbaGaaC4WdmaaBaaaleaacaWGKbaabeaakmaabmGabaGaamiDamaaBaaaleaacaWGYbaabeaaaOGaayjkaiaawMcaaaaa@3FF5@.
τmax MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3ABC@
Time of dynamic relaxation.
The higher the value of τmax MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@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.