Elastic-Plastic Orthotropic Composite Solids

The material LAW14 (COMPSO) in Radioss allows to simulate orthotropic elasticity, Tsai-Wu plasticity with damage, brittle rupture and strain rate effects. The constitutive law applies to only one layer of lamina. Therefore, each layer needs to be modeled by a solid mesh. A layer is characterized by one direction of the fiber or material. The overall behavior is assumed to be elasto-plastic orthotropic.

Direction 1 is the fiber direction, defined with respect to the local reference frame r , s , t as shown in Figure 1.


Figure 1. Local Reference Frame

For the case of unidirectional orthotropy (i.e. E 33 = E 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIZaGaaG4maaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa ikdacaaIYaaabeaaaaa@3BE4@ and G 31 = G 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIZaGaaG4maaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa ikdacaaIYaaabeaaaaa@3BE4@ ) the material LAW53 in Radioss allows to simulate an orthotropic elastic-plastic behavior by using a modified Tsai-Wu criteria.

Linear Elasticity

When the lamina has a purely linear elastic behavior, the stress calculation algorithm:
  • Transform the lamina stress, σ i j ( t ) , and strain rate, d ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@ , from global reference frame to fiber reference frame.
  • Compute lamina stress at time t + Δ t by explicit time integration:
    (1) σ ij ( t+Δt )= σ ij ( t )+ D ijkl d kl Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWG 0bGaey4kaSIaaeiLdiaadshaaiaawIcacaGLPaaacqGH9aqpcqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaey4kaSIaamiramaaBaaaleaacaWGPbGaamOAaiaadU gacaWGSbaabeaakiaayIW7caWGKbWaaSbaaSqaaiaadUgacaWGSbaa beaakiaayIW7caqGuoGaamiDaaaa@5756@
  • Transform the lamina stress, σ i j ( t + Δ t ) , back to global reference frame.
The elastic constitutive matrix C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaaaa@36BE@ of the lamina relates the non-null components of the stress tensor to those of strain tensor:(2) { σ } = [ D ] { ε }
The inverse relation is generally developed in term of the local material axes and nine independent elastic constants:(3) { ε 11 ε 22 ε 33 γ 12 γ 23 γ 31 } = [ 1 E 11 ν 21 E 22 ν 31 E 33 0 0 0 1 E 22 ν 32 E 33 0 0 0 1 E 33 0 0 0 1 2 G 12 0 0 S y m m . 1 2 G 23 0 1 2 G 31 ] { σ 11 σ 22 σ 33 σ 12 σ 23 σ 31 }
Where,
E i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@
Young's modulus
G i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@
Shear modulus
ν i j
Poisson's ratios
γ i j
Strain components due to the distortion


Figure 2. Strain Components and Distortion

Orthotropic Plasticity

Lamina yield surface defined by Tsai-Wu yield criteria is used for each layer:(4) F = f ( W p ) = F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + F 11 σ 1 2 + F 22 σ 2 2 + F 33 σ 3 2 + F 44 σ 12 2 + F 55 σ 23 2 + F 66 σ 31 2 + 2 F 12 σ 1 σ 2 + 2 F 23 σ 2 σ 3 + 2 F 13 σ 1 σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGgb Gaeyypa0JaamOzaiaacIcacaWGxbWaaSbaaSqaaiaadchaaeqaaOGa aiykaiabg2da9iaadAeadaWgaaWcbaGaaGymaaqabaGccqaHdpWCda WgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGgbWaaSbaaSqaaiaaikda aeqaaOGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamOram aaBaaaleaacaaIZaaabeaakiabeo8aZnaaBaaaleaacaaIZaaabeaa kiabgUcaRiaadAeadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeq4Wdm 3aa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamOramaaBaaa leaacaaIYaGaaGOmaaqabaGccqaHdpWCdaqhaaWcbaGaaGOmaaqaai aaikdaaaGccqGHRaWkcaWGgbWaaSbaaSqaaiaaiodacaaIZaaabeaa kiabeo8aZnaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgUcaRiaadA eadaWgaaWcbaGaaGinaiaaisdaaeqaaOGaeq4Wdm3aa0baaSqaaiaa igdacaaIYaaabaGaaGOmaaaaaOqaaiabgUcaRiaadAeadaWgaaWcba GaaGynaiaaiwdaaeqaaOGaeq4Wdm3aa0baaSqaaiaaikdacaaIZaaa baGaaGOmaaaakiabgUcaRiaadAeadaWgaaWcbaGaaGOnaiaaiAdaae qaaOGaeq4Wdm3aa0baaSqaaiaaiodacaaIXaaabaGaaGOmaaaakiab gUcaRiaaikdacaWGgbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabeo 8aZnaaBaaaleaacaaIXaaabeaakiabeo8aZnaaBaaaleaacaaIYaaa beaakiabgUcaRiaaikdacaWGgbWaaSbaaSqaaiaaikdacaaIZaaabe aakiabeo8aZnaaBaaaleaacaaIYaaabeaakiabeo8aZnaaBaaaleaa caaIZaaabeaakiabgUcaRiaaikdacaWGgbWaaSbaaSqaaiaaigdaca aIZaaabeaakiabeo8aZnaaBaaaleaacaaIXaaabeaakiabeo8aZnaa BaaaleaacaaIZaaabeaaaaaa@93C6@

with:

F i = 1 σ i y c + 1 σ i y t ( i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ =1,2,3);

F 11 = 1 σ 1 y c σ 1 y t ; F 22 = 1 σ 2 y c σ 2 y t ; F 33 = 1 σ 3 y c σ 3 y t ;

F 44 = 1 σ 12 y c σ 12 y t ; F 55 = 1 σ 23 y c σ 23 y t ; F 66 = 1 σ 31 y c σ 31 y t ;

F 12 = 1 2 ( F 11 F 22 ) ; F 23 = 1 2 F 22

Where, σ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaamyAaaqabaaaaa@3BB4@ is the yield stress in direction i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ , c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ and t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@ denote respectively for compression and tension. f ( W p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaam4vamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaaaaa @3A71@ represents the yield envelope evolution during work hardening with respect to strain rate effects:(5) f ( W p ) = ( 1 + B W p n ) ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )
Where,
W p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGWbaabeaaaaa@37F3@
Plastic work
B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Hardening parameter
n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Hardening exponent
c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DE@
Strain rate coefficient
f ( W p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaam4vamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaaaaa @3A71@ is limited by a maximum value f max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@39E1@ :(6) f ( W p ) f max = ( σ max σ y ) 2

If the maximum value is reached the material is failed.

In Equation 5, the strain rate effects on the evolution of yield envelope. However, it is also possible to take into account the strain rate ε ˙ effects on the maximum stress σ max as shown in Figure 3.
(a) Strain rate effect on σ max (b) No strain rate effect on σ max




σ = σ y ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

σ max = σ max 0 ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

f max = ( σ max σ y ) 2

σ = σ y ( 1 + c .1 n ( ε ˙ ε ˙ 0 ) )

σ max = σ max 0

Figure 3. Strain Rate Dependency

Unidirectional Orthotropy

LAW 53 in Radioss provides a simple model for unidirectional orthotropic solids with plasticity. The unidirectional orthotropy condition implies:(7) E 33 = E 22 G 31 = G 12
The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, Equation 4) in which:(8) F 12 = 2 ( σ y 45 c ) 2 1 2 ( F 11 + F 22 + F 44 ) + F 1 + F 2 σ y 45 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake GabaaVqiaadAeadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Za aSaaaeaacaaIYaaabaWaaeWaaeaacqaHdpWCdaqhaaWcbaGaamyEaa qaaiaaisdacaaI1aGaam4yaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaGaamOramaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWk caWGgbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaadAeada WgaaWcbaGaaGinaiaaisdaaeqaaaGccaGLOaGaayzkaaGaey4kaSYa aSaaaeaacaWGgbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOram aaBaaaleaacaaIYaaabeaaaOqaaiabeo8aZnaaDaaaleaacaWG5baa baGaaGinaiaaiwdacaWGJbaaaaaaaaa@5D97@
Where, σ 45 y c is yield stress in 45° unidirectional test. The yield stresses in direction 11, 22, 12, 13 and 45° are defined by independent curves obtained by unidirectional tests (Figure 4). The curves give the stress variation in function of a so-called strain ε v :(9) ε ν = 1 ( T r a c e [ ε ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiabe27aUbqabaGccqGH9aqpcaaIXaGaeyOeI0YaaeWaaeaa caWGubGaamOCaiaadggacaWGJbGaamyzamaadmaabaGaeqyTdugaca GLBbGaayzxaaaacaGLOaGaayzkaaaaaa@45E3@


Figure 4. Yield Stress Curve for a Unidirectional Orthotropic Material