G. =, etc.

Transcription

1 Maerial Models υ υ υ υ υ3 υ = l (9..4) he subscris denoe he maerial axes, i.e., υ = υ and = (9..5) i j xi xj ii xi Since l is symmeric υ υ =, ec. (9..6) The vecor of reen-s. Venan srain comonens is =,,,, 3, 3, (9..7) Afer comuing S, we use quaion (8.3) o obain he auchy sress. This model will redic realisic behavior for finie dislacemen and roaions as long as he srains are small. Maerial Model 3: lasic Plasic wih Kinemaic Hardening Isoroic, kinemaic, or a combinaion of isoroic and kinemaic hardening may be obained by varying a arameer, called β beween 0 and. For β equal o 0 and, resecively, kinemaic and isoroic hardening are obained as shown in Figure Krieg and Key [976] formulaed his model and he imlemenaion is based on heir aer. In isoroic hardening, he cener of he yield surface is fixed bu he radius is a funcion of he lasic srain. In kinemaic hardening, he radius of he yield surface is fixed bu he cener ranslaes in he direcion of he lasic srain. Thus he yield condiion is y φ = ξ ξ = 0 (9.3.) 3 9.5

2 Maerial Models LS-DYNA Theory Manual ξ = s α (9.3.) = + β ε (9.3.3) y 0 The co-roaional rae of α is ( ) α = β ε 3 (9.3.4) Hence, n + α = α + α + α Ω + α Ω Δ. (9.3.5) n+ n n n+ n n+ n+ ik kj jk ki Srain rae is accouned for using he ower-symonds [Jones 983] model which scales he yield sress by a srain rae deenden facor ε = + + β ε ( 0 ) y (9.3.6) and are user defined inu consans and ε is he srain rae defined as: ε = ε ε (9.3.7) The curren radius of he yield surface, y, is he sum of he iniial yield srengh, 0, lus he growh β ε, is he lasic hardening modulus = (9.3.8) and ε is he ecive lasic srain d ε = ε ε 3 (9.3.9) 0 9.6

3 Maerial Models Figure lasic-lasic behavior wih isoroic and kinemaic hardening l 0 and l are he undeformed and deformed lengh of uniaxial ension secimen, resecively. The lasic srain rae is he difference beween he oal and elasic (righ suerscri e) srain raes: ε = ε ε (9.3.0) e In he imlemenaion of his maerial model, he deviaoric sresses are udaed elasically, as described for model, bu reeaed here for he sake of clariy: = + Δ ε (9.3.) * n kl kl * n is he rial sress ensor, is he sress ensor from he revious ime se, kl is he elasic angen modulus marix, kl is he incremenal srain ensor. 9.7

4 Maerial Models LS-DYNA Theory Manual and, if he yield funcion is saisfied, nohing else is done. If, however, he yield funcion is violaed, an incremen in lasic srain is comued, he sresses are scaled back o he yield surface, and he yield surface cener is udaed. Le s reresen he rial elasic deviaoric sress sae a n+ * and s = (9.3.) * * * 3 kk ξ = s α. (9.3.3) * * Define he yield funcion, 3 0 for elasic or neural loading φ = ξ ξ = y y > 0 for lasic harding (9.3.4) For lasic hardening hen n+ n n y = + = + Δ ε ε ε ε 3 + (9.3.5) scale back he sress deviaors 3 n+ = ξ (9.3.6) and udae he cener: ( β) n n + α = α + ξ (9.3.7) Plane Sress Plasiciy The lane sress lasiciy oions aly o beams, shells, and hick shells. Since he sresses and srain incremens are ransformed o he lamina coordinae sysem for he consiuive evaluaion, he sress and srain ensors are in he local coordinae sysem. The alicaion of he Jaumann rae o udae he sress ensor allows for he ossibiliy ha he normal sress,, will no be zero. The firs se in udaing he sress ensor is o comue a rial lane sress udae assuming ha he incremenal srains are elasic. In he above, he normal srain incremen Δ ε is relaced by he elasic srain incremen + λ Δ ε + Δ ε = λ+ μ (9.3.8) 9.8

5 Maerial Models λ and μ are Lamé s consans. When he rial sress is wihin he yield surface, he srain incremen is elasic and he sress udae is comleed. Oherwise, for he lasic lane sress case, secan ieraion is used o solve quaion (9.3.6)for he normal srain incremen ( Δ ε ) required o roduce a zero normal sress: i 3 i * ξ = (9.3.9) Here, he suerscri i indicaes he ieraion number. The secan ieraion formula for Δ ε (he suerscri is droed for clariy) is Δ ε = i i i+ i i i i (9.3.0) he wo saring values are obained from he iniial elasic esimae and by assuming a urely lasic incremen, i.e., Δ ε = Δ ε (9.3.) These saring values should bound he acual values of he normal srain incremen. The ieraion rocedure uses he udaed normal sain incremen o udae firs he deviaoric sress and hen he oher quaniies needed o comue he nex esimae of he normal i sress in quaion (9.3.9). The ieraions roceed unil he normal sress is sufficienly small. The convergence crierion requires convergence of he normal srains: i i 4 < 0 i+ (9.3.) Afer convergence, he sress udae is comleed using he relaionshis given in quaions (9.3.6) and (9.3.7) Maerial Model 4: Thermo-lasic-Plasic This model was adaed from he NIKD [Hallquis 979] code. A more comlee descriion of is formulaion is given in he NIKD user s manual. Leing T reresen he emeraure, we comue he elasic co-roaional sress rae as T = ε ε + θ dt (9.4.) kl kl kl 9.9