Processing math: 69%

/MAT/LAW71

Block Format Keyword This law describes the behavior of superelastic materials. It allows modeling the behavior of the shape memory alloys (such as Nitinol).

The particularity of these materials is that all of the strain is recovered upon unloading even when large deformations are reached. Besides, the material shows a hysteretic response in a complete loading-unloading cycle. The full recovery is due to phase change in the microstructure. The model is based on the work of Auricchio et al. 1997. This law is compatible with beam (/PROP/TYPE18 (INT_BEAM) only), solid and shell elements.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW71/mat_ID/unit_ID
mat_title
ρi                
E υ E_mart        
σASS σASF σSAS σSAF α
EpsL CAS CSA TS_AS TF_AS
TS_SA TF_SA Cp Tini  

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

 
unit_ID Unit Identifier.

(Integer, maximum 10 digits)

 
mat_title Material title.

(Character, maximum 100 characters)

 
ρi Initial density.

(Real)

[kgm3]
E Young's modulus.

(Real)

[Pa]
υ Poisson's ratio.

(Real)

 
E_mart Martensite Young's modulus.

Default = 0.0 (Real)

[Pa]
σASS Material parameter defining the start of phase transformation from austenite to martensite (AS). 1

(Real)

[Pa]
σASF Material parameter defining the end of phase transformation from austenite to martensite (AS). 1

(Real)

[Pa]
σSAS Material parameter defining the start of phase transformation from martensite to austenite (SA). 1

(Real)

[Pa]
σSAF Material parameter defining the end of phase transformation from martensite to austenite (SA). 1

(Real)

[Pa]
α Material parameter measuring the difference in response between tension and compression.

Default = 0 (Real)

 
EpsL Maximum residual strain. 2

(Real)

 
CAS Stress-Temperature rate during loading.

Default = 0 (Real)

[PaK]
CSA Stress-Temperature rate during unloading.

Default = 0 (Real)

[PaK]
TS_AS Reference temperature for start of transformation (AS).

Default = 298K (Real)

[K]
TF_AS Reference temperature for end of transformation (AS).

Default = 298K (Real)

[K]
TS_SA Reference temperature for start of transformation (SA).

Default = 298K (Real)

[K]
TF_SA Reference temperature for end of transformation (SA).

Default = 298K (Real)

[K]
Cp Specific heat capacity.

Default = 1030 (Real)

[JkgK]
Tini Initial temperature.

Default = 360 K (Real)

[K]

Example (Metal)

Comments

  1. If E_mart=0, then Young's modulus is considered constant, equal to E, and not dependent on the phase fraction of the material.
  2. The different stresses σASS , σASF , σSAS and σSAF , defining the start and the end of phase transformation, as well as the residual strain EpsL, correspond to the case of a uniaxial tensile test:

    law71_transformation
    Figure 1.
  3. The parameter α is computed from the initial value of the austenite to martensite phase transformation in tension (σASS)T and compression (σASS)C from the relation.(1)
    α=23(σASS)C(σASS)T(σASS)C+(σASS)T

    When /MAT/LAW71 is used with beam elements, the parameter must be set to α=123 .

  4. The Drucker-Prager type loading function F is introduced using the stress deviator s , the pressure p and the temperature.(2)
    F = s + 3 α p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0ZaauWaaeaacaWGZbaacaGLjWUaayPcSdGaey4kaSIaaG4maiab eg7aHjaadchaaaa@418B@
    Two functions are defined for the start and the final point of transformation from austenite to martensite (A → S) or from martensite to austenite (S → A).
    (A→S) (S →A)
    Start point of transformation F S A S = F R S A S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadofaaeaacaWGbbGaam4uaaaakiabg2da9iaadAeacqGH sislcaWGsbWaa0baaSqaaiaadofaaeaacaWGbbGaam4uaaaaaaa@4118@

    R S A S = σ S A S ( 2 3 + α ) C A S ( T T S A S ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaadofaaeaacaWGbbGaam4uaaaakiabg2da9iabeo8aZnaa DaaaleaacaWGtbaabaGaamyqaiaadofaaaGcdaqadaqaamaakaaaba WaaSaaaeaacaaIYaaabaGaaG4maaaaaSqabaGccqGHRaWkcqaHXoqy aiaawIcacaGLPaaacqGHsislcaWGdbWaaWbaaSqabeaacaWGbbGaam 4uaaaakmaabmaabaGaamivaiabgkHiTiaadsfadaqhaaWcbaGaam4u aaqaaiaadgeacaWGtbaaaaGccaGLOaGaayzkaaaaaa@5079@

    F S S A = F R S S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadofaaeaacaWGtbGaamyqaaaakiabg2da9iaadAeacqGH sislcaWGsbWaa0baaSqaaiaadofaaeaacaWGtbGaamyqaaaaaaa@4118@

    R S S A = σ S S A ( 2 3 + α ) C S A ( T T S S A ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaadofaaeaacaWGtbGaamyqaaaakiabg2da9iabeo8aZnaa DaaaleaacaWGtbaabaGaam4uaiaadgeaaaGcdaqadaqaamaakaaaba WaaSaaaeaacaaIYaaabaGaaG4maaaaaSqabaGccqGHRaWkcqaHXoqy aiaawIcacaGLPaaacqGHsislcaWGdbWaaWbaaSqabeaacaWGtbGaam yqaaaakmaabmaabaGaamivaiabgkHiTiaadsfadaqhaaWcbaGaam4u aaqaaiaadofacaWGbbaaaaGccaGLOaGaayzkaaaaaa@5079@

    Final point of transformation F F A S = F R F A S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadAeaaeaacaWGbbGaam4uaaaakiabg2da9iaadAeacqGH sislcaWGsbWaa0baaSqaaiaadAeaaeaacaWGbbGaam4uaaaaaaa@40FE@

    R F A S = σ F A S ( 2 3 + α ) C A S ( T T F A S ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaadAeaaeaacaWGbbGaam4uaaaakiabg2da9iabeo8aZnaa DaaaleaacaWGgbaabaGaamyqaiaadofaaaGcdaqadaqaamaakaaaba WaaSaaaeaacaaIYaaabaGaaG4maaaaaSqabaGccqGHRaWkcqaHXoqy aiaawIcacaGLPaaacqGHsislcaWGdbWaaWbaaSqabeaacaWGbbGaam 4uaaaakmaabmaabaGaamivaiabgkHiTiaadsfadaqhaaWcbaGaamOr aaqaaiaadgeacaWGtbaaaaGccaGLOaGaayzkaaaaaa@5052@

    F F S A = F R F S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadAeaaeaacaWGtbGaamyqaaaakiabg2da9iaadAeacqGH sislcaWGsbWaa0baaSqaaiaadAeaaeaacaWGtbGaamyqaaaaaaa@40FE@

    R F S A = σ F S A ( 2 3 + α ) C S A ( T T F S A ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaadAeaaeaacaWGtbGaamyqaaaakiabg2da9iabeo8aZnaa DaaaleaacaWGgbaabaGaam4uaiaadgeaaaGcdaqadaqaamaakaaaba WaaSaaaeaacaaIYaaabaGaaG4maaaaaSqabaGccqGHRaWkcqaHXoqy aiaawIcacaGLPaaacqGHsislcaWGdbWaaWbaaSqabeaacaWGtbGaam yqaaaakmaabmaabaGaamivaiabgkHiTiaadsfadaqhaaWcbaGaamOr aaqaaiaadofacaWGbbaaaaGccaGLOaGaayzkaaaaaa@5052@

    Condition F S A S > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadofaaeaacaWGbbGaam4uaaaakiabg6da+iaaicdaaaa@3CA2@

    F F A S < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadAeaaeaacaWGbbGaam4uaaaakiabgYda8iaaicdaaaa@3C91@

    F ˙ > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGgbGbai aacqGH+aGpcaaIWaaaaa@39FE@

    F S S A < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadofaaeaacaWGtbGaamyqaaaakiabgYda8iaaicdaaaa@3C9E@

    F F S A > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadAeaaeaacaWGtbGaamyqaaaakiabg6da+iaaicdaaaa@3C95@

    F ˙ < 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGgbGbai aacqGH8aapcaaIWaaaaa@39FA@

    Evolution equation of martensite During loading:

    X ˙ m = ( 1 X m ) F ˙ F R F A S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbai aadaWgaaWcbaGaamyBaaqabaGccqGH9aqpcaGGOaGaaGymaiabgkHi TiaadIfadaWgaaWcbaGaamyBaaqabaGccaGGPaWaaSaaaeaaceWGgb GbaiaaaeaacaWGgbGaeyOeI0IaamOuamaaDaaaleaacaWGgbaabaGa amyqaiaadofaaaaaaaaa@458B@

    During unloading:

    X ˙ m = X m F ˙ F R F S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbai aadaWgaaWcbaGaamyBaaqabaGccqGH9aqpcaWGybWaaSbaaSqaaiaa d2gaaeqaaOWaaSaaaeaaceWGgbGbaiaaaeaacaWGgbGaeyOeI0Iaam OuamaaDaaaleaacaWGgbaabaGaam4uaiaadgeaaaaaaaaa@428A@

    σ S A S , σ F A S , T S A S , T F A S , α , C A S , σ S S A , σ F S A , T S S A , T F S A , C S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4uaaqaaiaadgeacaWGtbaaaOGaaiilaiabeo8aZnaa DaaaleaacaWGgbaabaGaamyqaiaadofaaaGccaGGSaGaamivamaaDa aaleaacaWGtbaabaGaamyqaiaadofaaaGccaGGSaGaamivamaaDaaa leaacaWGgbaabaGaamyqaiaadofaaaGccaGGSaGaeqySdeMaaiilai aadoeadaahaaWcbeqaaiaadgeacaWGtbaaaOGaaiilaiabeo8aZnaa DaaaleaacaWGtbaabaGaam4uaiaadgeaaaGccaGGSaGaeq4Wdm3aa0 baaSqaaiaadAeaaeaacaWGtbGaamyqaaaakiaacYcacaWGubWaa0ba aSqaaiaadofaaeaacaWGtbGaamyqaaaakiaacYcacaWGubWaa0baaS qaaiaadAeaaeaacaWGtbGaamyqaaaakiaacYcacaWGdbWaaWbaaSqa beaacaWGtbGaamyqaaaaaaa@64BB@ are the material parameters. The conversion of austenite to martensite takes place when above conditions (in table) are verified.

  5. List of Animation output (/ANIM/BRICK/USRI):
    • USR 1= Martensite phase fraction
    • USR 2= Loading function
    • USR 3= Unloading function