RD-E: 4701 Concrete Validation with Kupfer Tests

Concrete validation with Kupfer tests.

Radioss includes the material models LAW24 and LAW81 to model concrete failure modeling under compression and tension. In this example, the simulation results are compared to the experiment data.

ex47_concrete
Figure 1.

Options and Keywords Used

Input Files

The following input file is used in this example:

<install_directory>/hwsolvers/demos/radioss/example/47_concrete_test/Kupfer_tests/

Model Description

A 10 mm concrete cube is modeled using one brick element with the same boundary conditions as the experimental tests.

ex47_concrete
Figure 2. Geometry of the Cube

For stability reasons, 1 element models must use a time step scale factor of 0.1.

Solid properties are:
  • qa = 1.1 and qb= 0.05 (default values)
  • Isolid= 24
  • Iframe= 2 (co-rotational formulation)
  • Istrain= 1 (to post-treat stains)

In this example, two material laws, /MAT/LAW24 and /MAT/LAW81, will be compared to the experiment data.

The following system is used: mm, ms, g, MPa

The material data 1 used:
Concrete Material Law (/MAT/LAW24)
Initial density
0.0022 [ g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4zaaqaaiaac2gacaGGTbWaaWbaaSqabeaacaaIZaaa aaaaaOGaay5waiaaw2faaaaa@3D2B@
Concrete elasticity Young's modulus
E c = 31700 [ MPa ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaadogaaeqaaOGaeyypa0JaaG4maiaaigdacaaI3aGaaGim aiaaicdacaGGBbGaciytaiaaccfacaGGHbGaciyxaaaa@4251@
Poisson's ratio
ν = 0.22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaikdacaaIYaaaaa@3D0A@
Concrete plasticity initial value of hardening parameter
k y = 0.35 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaS baaSqaaiaadMhaaeqaaOGaeyypa0JaaGimaiaac6cacaaIZaGaaGyn aaaa@3D7A@
Concrete plasticity dilatancy factor at yield
α y = 0.6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeg7aHnaaBaaaleaacaWG5baabeaakiabg2da9iabgkHiTiaa icdacaGGUaGaaGOnaaaa@4047@
Concrete plasticity dilatancy factor at failure
α f = 0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeg7aHnaaBaaaleaacaWGMbaabeaakiabg2da9iaaicdacaGG UaGaaGOmaaaa@3F43@
Data Read Kupfer Experimental Data
Concrete uniaxial compression strength
f c =32.22[MPa] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4yaaqabaGccqGH9aqpcaaIZaGaaGOm aiaac6cacaaIYaGaaGOmaiaacUfaciGGnbGaaiiuaiaacggaciGGDb aaaa@4455@
Concrete uniaxial tension strength
0.01 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ , then set f t f c = 0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaliaabaGaamOzamaaBaaaleaacaWG0baabeaaaOqaaiaadAga daWgaaWcbaGaam4yaaqabaaaaOGaeyypa0JaaGimaiaac6cacaaIXa aaaa@40B7@ (Default=0.1 in LAW24)
Concrete biaxial strength
1.15 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ , then set f b f c = 1.15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaliaabaGaamOzamaaBaaaleaacaWGIbaabeaaaOqaaiaadAga daWgaaWcbaGaam4yaaqabaaaaOGaeyypa0JaaGymaiaac6cacaaIXa GaaGynaaaa@4165@
All other parameters can be left as default in LAW24 because the default values are representative of generic concrete materials.
Concrete Material Law (/MAT/LAW81)
Initial density
0.0022 [ g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4zaaqaaiaac2gacaGGTbWaaWbaaSqabeaacaaIZaaa aaaaaOGaay5waiaaw2faaaaa@3D2B@
Bulk modulus
K = E c 3 ( 1 2 ν ) = 18869 .048[MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaey ypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiaadogaaeqaaaGcbaGaaG4m amaabmaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPa aaaaGaeyypa0JaaeymaiaabIdacaqG4aGaaeOnaiaabMdacaqGUaGa aeimaiaabsdacaqG4aGaae4waiaab2eacaqGqbGaaeyyaiaab2faaa a@4D58@
Young's modulus
E c = 31700 [ MPa ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaadogaaeqaaOGaeyypa0JaaG4maiaaigdacaaI3aGaaGim aiaaicdacaGGBbGaciytaiaaccfacaGGHbGaciyxaaaa@4251@
Poisson's ratio
ν = 0.22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaikdacaaIYaaaaa@3D0A@
Shear modulus
G = E c 2 ( 1 + ν ) = 12991 .8[MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey ypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiaadogaaeqaaaGcbaGaaGOm amaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaaaaacq GH9aqpcaqGXaGaaeOmaiaabMdacaqG5aGaaeymaiaab6cacaqG4aGa ae4waiaab2eacaqGqbGaaeyyaiaab2faaaa@4B18@
The following data was calculated by curve fitting the experimental data using the solidThinking Compose script, included with the input files.
Friction angle
ϕ=68 .35 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcq GH9aqpcaqG2aGaaeioaiaab6cacaqGZaGaaeynamaaCaaaleqabaGa eSigI8gaaaaa@3F30@
Ratio
α = 0 .4186898 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqycq GH9aqpcaqGWaGaaeOlaiaabsdacaqGXaGaaeioaiaabAdacaqG4aGa aeyoaiaabIdaaaa@4082@
Cap limit pressure set constant
P b = 0.838 f c =27 [MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaadkgaaeqaaOGaeyypa0JaaGimaiaac6cacaaI4aGaaG4m aiaaiIdacaqGSaGaaeiiaiaaykW7caWGMbWaaSbaaSqaaiaadogaae qaaOGaaeypaiaabkdacaqG3aGaaeiiaiaabUfacaqGnbGaaeiuaiaa bggacaqGDbaaaa@4A0A@
Cap beginning pressure
P a = α P b =0 .351,  f c = 11.305  [MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaadggaaeqaaOGaeyypa0JaeqySdeMaeyyXICTaamiuamaa BaaaleaacaWGIbaabeaakiaab2dacaqGWaGaaeOlaiaabodacaqG1a GaaeymaiaabYcacaqGGaGaaGPaVlaadAgadaWgaaWcbaGaam4yaaqa baGccqGH9aqpcaaIXaGaaGymaiaac6cacaaIZaGaaGimaiaaiwdaca qGGaGaae4waiaab2eacaqGqbGaaeyyaiaab2faaaa@53B2@
Material cohesion set constant
c = 0.169175 f c = 5.4508  [MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbGaey ypa0JaaGimaiaac6cacaaIXaGaaGOnaiaaiMdacaaIXaGaaG4naiaa iwdacaqGSaGaaeiiaiaaykW7caWGMbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaaGynaiaac6cacaaI0aGaaGynaiaaicdacaaI4aGaaeii aiaabUfacaqGnbGaaeiuaiaabggacaqGDbaaaa@4E79@
Note: In this example, stresses are scaled by f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ . A common practice for concrete materials is to define pressures as positive in compression. Stresses are then negative in traction or tension.

Simulation Iterations

The purpose of this example is to compare the simulation results to experimental data from the Kupfer 2 tests.
Table 1. Loading and Failure
Test Principle Stress Triaxiality Failure Stress
T000

Uniaxial tension

σ 1 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ ; σ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ 1/3 0.1 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
C000

Uniaxial compression

σ 1 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE4@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ ; σ 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ -1/3 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
CC00

Biaxial compression

σ 1 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE4@ ; σ 2 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE5@ ; σ 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3CF6@ -2/3 1.15 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
CC01

Compression/Compression

σ 1 = 0.052 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGim aiaac6cacaaI1aGaaGOmaaaa@4010@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE5@ ; σ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE6@ -0.5849 1.22 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
TC01

Compression/Tension

σ 1 = 0.052 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGim aiaac6cacaaI1aGaaGOmaaaa@4010@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE5@ ; σ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE6@ -0.3077 0.8 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
TC02

Compression/Tension

σ 1 = 0.102 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGim aiaac6cacaaI1aGaaGOmaaaa@4010@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE5@ ; σ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE6@ -0.2838 0.6 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
TC03

Compression/Tension

σ 1 = 0.204 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGim aiaac6cacaaI1aGaaGOmaaaa@4010@ ; σ 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE5@ ; σ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeyOeI0IaaGym aaaa@3DE6@ -0.2377 0.35 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@
The triaxiality can be computed using the principal stresses:(1) σ * = σ m σ V M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaCaaaleqabaWdbiaacQcaaaGccqGH9aqpdaWcaaWd aeaapeGaae4Wd8aadaWgaaWcbaWdbiaad2gaa8aabeaaaOqaa8qacq aHdpWCpaWaaSbaaSqaa8qacaWGwbGaamytaaWdaeqaaaaaaaa@4081@

with σ m = p = 1 3 ( σ 1 + σ 2 + σ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaad2gaa8aabeaakiabg2da9iaadcha peGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaiodaaaWaae Waa8aabaWdbiaabo8apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa ey4kaSIaae4Wd8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHRa WkcaqGdpWdamaaBaaaleaapeGaaG4maaWdaeqaaaGcpeGaayjkaiaa wMcaaaaa@4858@ and σ V M = 1 2 [ ( σ 1 σ 2 ) 2 + ( σ 2 σ 3 ) 2 + ( σ 3 σ 1 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Wd8aadaWgaaWcbaWdbiaadAfacaWGnbaapaqabaGcpeGaeyyp a0ZaaOaaa8aabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYa aaamaadmaapaqaa8qadaqadaWdaeaapeGaae4Wd8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacqGHsislcaqGdpWdamaaBaaaleaapeGaaG OmaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaOGaey4kaSYaaeWaa8aabaWdbiaabo8apaWaaSbaaSqaa8qaca aIYaaapaqabaGcpeGaeyOeI0Iaae4Wd8aadaWgaaWcbaWdbiaaioda a8aabeaaaOWdbiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaa aakiabgUcaRmaabmaapaqaa8qacaqGdpWdamaaBaaaleaapeGaaG4m aaWdaeqaaOWdbiabgkHiTiaabo8apaWaaSbaaSqaa8qacaaIXaaapa qabaaak8qacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaa kiaawUfacaGLDbaaaSqabaaaaa@5A1B@ .

Figure 3 shows concrete material experiment failure stress (scaled by f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ ) in the σ VM  versus p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo8aZnaaBaaaleaacaWGwbGaamytaaqabaGccaqGGaqcaaIa aeODaiaabwgacaqGYbGaae4CaiaabwhacaqGZbGccaqGGaGaamiCaa aa@4617@ stress space.


Figure 3. Experiment Data in von Mises/Pressure Curves

Results

Failure Results with LAW24 and LAW81

The failure curve in LAW24 is:(2) r f = 1 a ( b + b 2 a ( σ m c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOCamaaBaaaleaacaWGMbaabeaakiabg2da9maalaaabaGaaGym aaqaaiaadggaaaGaaiikaiaadkgacqGHRaWkdaGcaaqaaiaadkgada ahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbGaaiikaiabeo8aZnaa BaaaleaacaWGTbaabeaakiabgkHiTiaadogacaGGPaaaleqaaaaa@49A7@

With b = 1 2 ( b c + b t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOyaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiaa dkgadaWgaaWcbaGaam4yaaqabaGccqGHRaWkcaWGIbWaaSbaaSqaai aadshaaeqaaOGaaiykaaaa@424C@

Radioss will curve fit Equation 2 using the different strength input f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ , f t f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaliaabaGaamOzamaaBaaaleaacaWG0baabeaaaOqaaiaadAga daWgaaWcbaGaam4yaaqabaaaaaaa@3D80@ , to generate the r f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadkhadaWgaaWcbaGaamOzaaqabaaaaa@3B63@ failure curve (green).


Figure 4. Experiment Data and LAW24 Analytical Data in von Mises/Pressure Curves
The failure curve and yield curve in LAW81 are the same, and can be described in two parts:
  1. p P a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbGaey izImQaamiuamaaBaaaleaacaWGHbaabeaaaaa@3BF9@

    It is linear with p tan ϕ + c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbGaci iDaiaacggacaGGUbGaeqy1dyMaey4kaSIaam4yaaaa@3EC0@

    The failure is σ m tan ( 68.35 ) + 5.4508 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamyBaaqabaGcciGG0bGaaiyyaiaac6gacaGGOaGaaGOn aiaaiIdacaGGUaGaaG4maiaaiwdadaahaaWcbeqaaiablIHiVbaaki aacMcacqGHRaWkcaaI1aGaaiOlaiaaisdacaaI1aGaaGimaiaaiIda aaa@48EA@

  2. P a < p P b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaadggaaeqaaOGaeyipaWJaamiCaiabgsMiJkaadcfadaWg aaWcbaGaamOyaaqabaaaaa@3EEF@ (cap)
    The cap curve is:(3) 1 ( p p a p b p a ) 2 ( p tan ϕ + c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaeyOeI0YaaeWaaeaadaWcaaqaaiaadchacqGHsislcaWGWbWa aSbaaSqaaiaadggaaeqaaaGcbaGaamiCamaaBaaaleaacaWGIbaabe aakiabgkHiTiaadchadaWgaaWcbaGaamyyaaqabaaaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaaqabaGccqGHflY1daqadaqaai aadchaciGG0bGaaiyyaiaac6gacqaHvpGzcqGHRaWkcaWGJbaacaGL OaGaayzkaaaaaa@4E68@
    The failure is:(4) 1 ( σ m 27 2711.305 ) 2 σ m tan( 68.35 )+5.4508 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai aaigdacqGHsisldaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaa d2gaaeqaaOGaeyOeI0IaaGOmaiaaiEdaaeaacaaIYaGaaG4naiabgk HiTiaaigdacaaIXaGaaiOlaiaaiodacaaIWaGaaGynaaaaaiaawIca caGLPaaadaahaaWcbeqaaiaaikdaaaaabeaakiabgwSixlabeo8aZn aaBaaaleaacaWGTbaabeaakiabgwSixlGacshacaGGHbGaaiOBaiaa cIcacaaI2aGaaGioaiaac6cacaaIZaGaaGynamaaCaaaleqabaGaeS igI8gaaOGaaiykaiabgUcaRiaaiwdacaGGUaGaaGinaiaaiwdacaaI WaGaaGioaaaa@5DDF@
These two parts of the failure curve for LAW81 are:


Figure 5. Experiment Data and LAW81 Analytical Data in von Mises/Pressure Curves
The failure results with LAW24 and LAW81 under different loading paths (from Kupfer tests) show as:


Figure 6. LAW24 Analytical and Simulation Results (load path) in von Mises/Pressure Curves


Figure 7. LAW81 Analytical and Simulation Results (load path) in von Mises/Pressure Curves
Comparing LAW24 and LAW81 failure results with experiment data, the LAW81 results are better than LAW24.


Figure 8. Experiment Data and LAW24 Radioss Results in von Mises/Pressure Curves


Figure 9. Experiment Data and LAW81 Radioss Results in von Mises/Pressure Curves
The failure results in LAW81 match the experiment data even in cap region. For the LAW24 results, most are well match the analytical results; except CC00 shows a little bit larger difference with analytical curve, but it is almost the same as the experiment data. The following CC00 stress-strain diagram shows almost the same failure stress.


Figure 10. CC00 Experiment Data and LAW24 Radioss Results

Results for Concrete Tension Tests

Concrete does not support very much load in tension. In LAW24 the uniaxial tensile failure (modeled by stress) and elastic modulus softening behavior is defined by H t , D sup , ε max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamisamaaBaaaleaacaWG0baabeaakiaacYcacaaMi8UaaGPaVlaa dseapaWaaSbaaSqaa8qaciGGZbGaaiyDaiaacchaa8aabeaakiaacY cacqaH1oqzdaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaaa@453D@ . The softening modulus H t = E c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaS baaSqaaiaadshaaeqaaOGaeyypa0JaeyOeI0IaamyramaaBaaaleaa caWGJbaabeaaaaa@3C2B@ (default) for tension is set. The peak for the above curve is at 0.1 where it is defined by f t f c = 0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWccaqaai aadAgadaWgaaWcbaGaamiDaaqabaaakeaacaWGMbWaaSbaaSqaaiaa dogaaeqaaaaakiabg2da9iaaicdacaGGUaGaaGymaaaa@3DC0@ (default) in input.

In LAW81, the same bulk and shear modulus are used for tension and compression. With LAW24 it is possible to use E = ( 1 D sup ) E c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamiramaaBaaaleaaciGGZbGa aiyDaiaacchaaeqaaaGccaGLOaGaayzkaaGaeyyXICTaamyramaaBa aaleaacaWGJbaabeaaaaa@436C@ to represent a residual stiffness in the concrete after softening. This is not possible with LAW81.


Figure 11. Uniaxial Tension T000 Experiment Data and Radioss (LAW24 and LAW81) Results

Conclusion

Under complex loading, the concrete mechanic failure behavior is shown using two Radioss material models LAW24 and LAW81 and results compared to experiments. For LAW24 the default values are a good choice, if no experimental data is available. For LAW81, the material parameters ϕ , c , α , P b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzca GGSaGaam4yaiaacYcacqaHXoqycaGGSaGaamiuamaaBaaaleaacaWG Ibaabeaaaaa@3FAF@ need to be calculated with curve fitting using at least four experimental tests.

References

1 Han, D. J., and Wai-Fah Chen. "A nonuniform hardening plasticity model for concrete materials." Mechanics of materials 4, no. 3-4 (1985): 283-302
2 Kupfer, Helmut B., and Kurt H. Gerstle. "Behavior of concrete under biaxial stresses." Journal of the Engineering Mechanics Division 99, no. 4 (1973): 853-866