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Comutational Solid Mechanics Comutational

Cataluña (UPC), Barcelona, Sain International

1 Comutational Solid Mechanics Comutational Plasticity Chater 3. J Plasticity Models C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Cataluña (UPC), Barcelona, Sain International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Sain

2 J Plasticity Models > Contents Contents Contents 1. J rate indeendent lasticity models 1.. J rate deendent lasticity models 1. March 1, 017 Carlos Agelet de Saracibar

3 J Plasticity Models > Contents Contents Contents 1. J rate indeendent lasticity models 1.. J rate deendent lasticity models 1. March 1, 017 Carlos Agelet de Saracibar 3

4 Hyothesis J Plasticity Models > Rate Indeendent Plasticity Models Within the framework of the infinitesimal deformation theory, we introduce the following hyothesis for a J rate-indeendent linear elastic-hardening lasticity model, within the incremental theory of lasticity: H1. Additive slit of the infinitesimal strain tensor e ε= ε + ε H. Set of lastic internal variables E { ξ } : = ε,, ξ March 1, 017 Carlos Agelet de Saracibar 4

5 J Plasticity Models > Rate Indeendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Elastic otential for a linear elastic material model ( e) 1 e e W ε = ε : : ε Elastic otential Constant isotroic elastic constitutive tensor (λ 0, μ>0, κ>0) Elastic otential for isotroic material ( 1 ) : = λ 1 1+ µ = κ 1 1+ µ W ε : = κ tr ε + µ dev ε : devε = κ ( tr ε ) + µ devε e e e e e e March 1, 017 Carlos Agelet de Saracibar 5 3

6 J Plasticity Models > Rate Indeendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Isotroic hardening otential for a linear isotroic hardening material model Π = 1 K ( ξ ) ξ Isotroic hardening otential March 1, 017 Carlos Agelet de Saracibar 6

7 J Plasticity Models > Rate Indeendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Kinematic hardening otential for a linear kinematic hardening material model 1 1 Π ξ = H ξ = H ξ: ξ Kinematic hardening otential 3 3 κ > 0, µ > 0, µ + K + H > March 1, 017 Carlos Agelet de Saracibar 7

8 J Plasticity Models > Rate Indeendent Plasticity Models H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissiation e ( e ξ ) D : = σε : ψ ε,, ξ 0 ε e D : = σε : ψ: ε ψξ ψ: ξ ε e = σ ψ : ε + ψ: ε ψξ ψ: ξ 0 ε ε ε e ξ σ = ψ = : ε = λ tr ε 1+ µ ε = κ tr ε 1+ µ devε ε e q: = ξψ = Kξ, q : = ψ = H ξ 3 D : = σε : + q ξ + q: ξ 0 March 1, 017 Carlos Agelet de Saracibar 8 ξ ξ e e e e e ξ ξ

9 Linear isotroic hardening Nonlinear isotroic hardening J Plasticity Models > Rate Indeendent Plasticity Models q: = ξψ = Kξ (1) Exonential saturation law + linear isotroic hardening () Power law isotroic hardening q: = ξψ : = σ σy 1 ex δξ Kξ m ξψ σy 1 ξ q: = : = k k + March 1, 017 Carlos Agelet de Saracibar 9

10 J Plasticity Models > Rate Indeendent Plasticity Models H5. Sace of admissible stresses, elastic domain, and yield surface. Yield function σ { } σ q q f σ q q σ q σ Y q : =,,,, : = dev 0 { } Y ( ) σ q q f ( σ q q) σ q ( σ q) int : =,,,, : = dev < 0 σ { } σ q q f σ q q σ q σ Y q : =,,,, : = dev = 0 σ March 1, 017 Carlos Agelet de Saracibar 10

Mises yield surface ( σ q) Octahedral lane

11 J Plasticity Models > Rate Indeendent Plasticity Models Yield surface: gometrical reresentation in the sace of rincial stresses σ + σ + σ = 1 3 ( σ q) σ q ( σ ) f, q, : = dev q = 0 cte σ 3 Hydrostatic stress axis σ1 = σ = σ3 Von Mises yield surface ( σ q) Octahedral lane R= 3 Y 3 Y σ σ σ 1 σ 1 March 1, 017 Carlos Agelet de Saracibar 11

12 J Plasticity Models > Rate Indeendent Plasticity Models Yield surface: gometrical reresentation in the sace of rincial stresses ( σ q) σ q ( σ ) f, q, : = dev q = 0 3 Y R 0 = σ 3 σ 3 Y devσ 0 R = 3 ( σ q ) t Y t q t devσ t q = 0 0 σ σ1 σ1 σ March 1, 017 Carlos Agelet de Saracibar 1

13 H6. Plastic flow rule J Plasticity Models > Rate Indeendent Plasticity Models Non-associative lastic flow rule g ( σ, q, q) (, q, ) ( σ, q, q) ξ = γ qg ξ = γ g σ q ε = γ σg σ q q (, q, ) where is a lastic otential defined in the (deviatoric) stress sace. Taking the yield function as lastic otential, the lastic flow rule is said to be associative. March 1, 017 Carlos Agelet de Saracibar 13

14 H6. Plastic flow rule Associative lastic flow rule where that, n J Plasticity Models > Rate Indeendent Plasticity Models (, q, ) ( σ q q) ε = γ σ f σ q = γ n ξ = γ f,, = γ 3 q ξ = γ f σ q = γ n q (, q, ) is the unit outward normal to the yield surface such ( q ) n : = f σ,, q = σ devσ q devσ q tr n= 1: n= 0, dev n= n, n = 1 March 1, 017 Carlos Agelet de Saracibar 14

15 J Plasticity Models > Rate Indeendent Plasticity Models Note that, both the lastic strain tensor and the kinematic hardening tensor defined in the strain sace are deviatoric, ε = γ σ f ( σ, q, q) = γ n, ξ = γ q f ( σ, q, q) = γ n dev ε = ε, dev ξ = ξ dev ε = ε, dev ξ = ξ And, therefore, the kinematic hardening tensor defined in the stress sace is also deviatoric, dev q = q: = ψ = 3 H ξ ξ March 1, 017 Carlos Agelet de Saracibar 15

16 J Plasticity Models > Rate Indeendent Plasticity Models H7. Kuhn-Tucker loading/unloading conditions if γ > 0 then f σ, q, q = 0 Plastic loading if f σ, q, q < 0 then γ = 0 Elastic loading/unloading ( σ q) γ ( σ q) γ 0, f, q, 0, f, q, = 0 March 1, 017 Carlos Agelet de Saracibar 16

17 H8. Plastic consistency condition Plastic loading J Plasticity Models > Rate Indeendent Plasticity Models if f σ, q, q = 0 and γ > 0 then f σ, q, q = 0 if f σ, q, q = 0 and f σ, q, q < 0 then γ = 0 Plastic loading Elastic unloading ( σ q) γ ( σ q) γ ( σ q) if f, q, = 0 then 0, f, q, 0, f, q, = 0 f σ, q, q = 0 and γ > 0 f σ, q, q = 0 March 1, 017 Carlos Agelet de Saracibar 17

18 J Plasticity Models > Rate Indeendent Plasticity Models Plastic loading: lastic consistency condition f = f: σ + fq + f: q = n: σ + 3 q n: q σ q q e = n : : ε 3 K ξ + Hn: ξ 3 = n : : ε n: : ε 3 K ξ + Hn: ξ 3 = n : : ε γ n: : n+ K + Hn: n 3 3 = n : : ε γ n: : n+ K + H = March 1, 017 Carlos Agelet de Saracibar 18 > 0

19 Plastic loading: lastic multilier or lastic consistency arameter Taking into account that J Plasticity Models > Rate Indeendent Plasticity Models The lastic multilier takes the form March 1, 017 Carlos Agelet de Saracibar 19 1 γ = : : + K + H : : n n 3 3 n ε n: 1= 0, n: : n= n: n= 1 ( ) ( ) n: : n= n: λ 1 1+ µ : n= µ n : : ε = n: λ 1 1+ µ : ε = µ n: ε = µ n:devε 1 γ = µ + K + H µ n : devε 3 3

20 Trial deviatoric stress state J Plasticity Models > Rate Indeendent Plasticity Models trial devσ = µ devε The lastic multilier takes the form trial γ = µ + K + H n : devσ trial γ = µ + K + H n : devσ March 1, 017 Carlos Agelet de Saracibar 0

21 J Plasticity Models > Rate Indeendent Plasticity Models Plastic loading Elastic unloading Neutral loading trial γ = µ + K + H n : devσ trial n : dev σ > 0 γ > 0, f = 0, γ f = 0 trial n : devσ < 0 γ = 0, f < 0, γ f = 0 trial n : devσ = 0 γ = 0, f = 0, γ f = 0 1 March 1, 017 Carlos Agelet de Saracibar 1

22 J Plasticity Models > Rate Indeendent Plasticity Models Plastic loading, elastic unloading, and neutral loading σ devσ trial Elastic unloading n σ trial Neutral loading σ trial devσ trial Plastic loading ( σ qf q q ) : = f,, σn : = σ,, q σ devσ trial March 1, 017 Carlos Agelet de Saracibar

23 J Plasticity Models > Rate Indeendent Plasticity Models Reduced lastic dissiation Plastic dissiation rate er unit of volume D = σ: ε + q ξ + q: ξ = γ σ q : n+ ( ) q ( ) dev σ q : n q γ devσ q q = γ + = + ( ) f σ, q, q σy = γ + = γ σ 0 ε 3 Y March 1, 017 Carlos Agelet de Saracibar 3

24 J Plasticity Models > Rate Indeendent Plasticity Models Maximum lastic dissiation rincile Given a strain rate and a rate of the lastic internal variables { ε 0} E { ε ξ ξ } E : =,0,, : =,, the maximum lastic dissiation rincile states that the stress state satisfies DSE D E T (, ) ( T, ) σ where, S: = σ, q, q, T : = π,, { } { } D S, E : = SE, D, E : = TE ( T ) March 1, 017 Carlos Agelet de Saracibar 4

25 Maximum lastic dissiation rincile Maximum lastic dissiation J Plasticity Models > Rate Indeendent Plasticity Models DSE D E T (, ) ( T, ) DS T, E : = S T E 0 T ( ) Constrained minimization roblem S= arg max D, E T ( T ) ( ( T )) S= arg min D, E T σ σ σ σ March 1, 017 Carlos Agelet de Saracibar 5

26 J Plasticity Models > Rate Indeendent Plasticity Models Maximum lastic dissiation rincile Maximum lastic dissiation, given by, ( ( T )) S= arg min D, E T is equivalent to associative lastic flow rule, Kuhn-Tucker loading/unloading conditions, and convexity of the yield surface, given by, E = γ γ 0, f S 0, γ f S = 0 T S f S f T f S T S S f S σ March 1, 017 Carlos Agelet de Saracibar 6

27 J Plasticity Models > Rate Indeendent Plasticity Models Hyothesis H1. Additive slit of the infinitesimal strain tensor H. Set of lastic internal variables H3. Free energy er unit of volume H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced lastic dissiation H5. Sace of admissible stresses, elastic domain, and yield surface. Yield function H6. Associative lastic flow rule H7. Kuhn-Tucker loading/unloading conditions March 1, 017 Carlos Agelet de Saracibar 3

28 J Plasticity Models > Rate Indeendent Plasticity Models Hyothesis H1. Additive slit of the infinitesimal strain tensor H. Set of lastic internal variables H3. Free energy er unit of volume H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced lastic dissiation H5. Sace of admissible stresses, elastic domain, and yield surface. Yield function H6. Maximum lastic dissiation March 1, 017 Carlos Agelet de Saracibar 33

29 J Plasticity Models > Contents Contents Contents 1. J rate indeendent lasticity models 1.. J rate deendent lasticity models 1. March 1, 017 Carlos Agelet de Saracibar 34

30 Hyothesis J Plasticity Models > Rate Deendent Plasticity Models Within the framework of the infinitesimal deformation theory, we introduce the following hyothesis for a J rate-deendent linear elastic-hardening lasticity model, within the incremental theory of lasticity: H1. Additive slit of the infinitesimal strain tensor e ε= ε + ε H. Set of lastic internal variables E { ξ } : = ε,, ξ March 1, 017 Carlos Agelet de Saracibar 35

31 J Plasticity Models > Rate Deendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Elastic otential for a linear elastic material model ( e) 1 e e W ε = ε : : ε Elastic otential Constant isotroic elastic constitutive tensor (λ 0, μ>0, κ>0) 1 : = λ 1 1+ µ = κ 1 1+ µ Elastic otential for isotroic material 1 1 W ε : = κ tr ε + µ dev ε : devε = κ ( tr ε ) + µ devε e e e e e e March 1, 017 Carlos Agelet de Saracibar 36

32 J Plasticity Models > Rate Deendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Isotroic hardening otential for a linear isotroic hardening material model Π = 1 K ( ξ ) ξ Isotroic hardening otential March 1, 017 Carlos Agelet de Saracibar 37

33 J Plasticity Models > Rate Deendent Plasticity Models H3. Free energy er unit of volume ( e ) ( e ε,, ξ : = W ε ) +Π +Π( ξ) ψ ξ ξ Kinematic hardening otential for a linear kinematic hardening material model 1 1 Π ξ = H ξ = H ξ: ξ Kinematic hardening otential 3 3 κ > 0, µ > 0, µ + K + H > March 1, 017 Carlos Agelet de Saracibar 38

34 J Plasticity Models > Rate Deendent Plasticity Models H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissiation e ( e ξ ) D : = σε : ψ ε,, ξ 0 ε e D : = σε : ψ: ε ψξ ψ: ξ ε e = σ ψ : ε + ψ: ε ψξ ψ: ξ 0 ε ε ε e ξ σ = ψ = : ε = λ tr ε 1+ µ ε = κ tr ε 1+ µ devε ε e q: = ξψ = Kξ, q : = ψ = Hξ ξ 3 D : = σε : + q ξ + q: ξ 0 March 1, 017 Carlos Agelet de Saracibar 39 ξ ξ e e e e e ξ

35 J Plasticity Models > Rate Deendent Plasticity Models H5. Elastic domain, lastic domain and yield surface. Yield function σ { } σ q q f σ q q σ q σ Y q : =,,,, : = dev 0 { } Y ( ) σ q q f ( σ q q) σ q ( σ q) ext : =,,,, : = dev > 0 σ { } σ q q f σ q q σ q σ Y q : =,,,, : = dev = 0 σ March 1, 017 Carlos Agelet de Saracibar 40

36 H6. Associative lastic flow rule Associative lastic flow rule where that, n J Plasticity Models > Rate Deendent Plasticity Models (, q, ) ( σ q q) ε = γ σ f σ q = γ n ξ = γ f,, = γ 3 q ξ = γ f σ q = γ n q (, q, ) is the unit outward normal to the yield surface such ( q ) n : = f σ,, q = σ devσ q devσ q tr n= 1: n= 0, dev n= n, n = 1 March 1, 017 Carlos Agelet de Saracibar 41

37 H7. Plastic multilier J Plasticity Models > Rate Deendent Plasticity Models 1 γ = f ( σ, q, q) 0 η Note that, for the non-trivial case of lastic loading the following exression holds, ( σ q) f, q, = ηγ > 0 March 1, 017 Carlos Agelet de Saracibar 4

38 J Plasticity Models > Rate Deendent Plasticity Models Reduced lastic dissiation Plastic dissiation rate er unit of volume D = σ: ε + q ξ + q: ξ = γ σ q : n+ ( ) q ( ) dev σ q : n q γ devσ q q = γ + = + ( ) f σ, q, q σy = γ + σy = γ ηγ + 0 ε March 1, 017 Carlos Agelet de Saracibar 43

39 Perzyna model J Plasticity Models > Rate Deendent Plasticity Models The associative lastic flow rule can be obtained as the solution of an unconstrained minimization roblem, arising from the maximization of a regularized lastic dissiation, given by, ( ( )) η T S= arg min D, E T 1 Dη SE DSE S η ( ) (, : =, ) f March 1, 017 Carlos Agelet de Saracibar 44

40 J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem, arising from the maximization of a regularized lastic dissiation, yields the associative lastic flow rule, ( ( )) η T S= arg min D, E T 1 η = SD SE DSE S η ( ) (, :, ) f S 1 : = E f ( S) f S S =0 η 1 E = f S S S η f March 1, 017 Carlos Agelet de Saracibar 45

41 J Plasticity Models > Rate Deendent Plasticity Models The associative lastic flow rule takes the form, 1 E = f S S S η f (, q, ) ( σ q q) ε = γ σ f σ q = γ n ξ = γ f,, = γ 3 q ξ = γ f σ q = γ n q (, q, ) March 1, 017 Carlos Agelet de Saracibar 46

42 Duvaut-Lions model J Plasticity Models > Rate Deendent Plasticity Models The associative lastic flow rule can be obtained as the solution of an unconstrained minimization roblem, arising from the maximization of a regularized lastic dissiation, given by, ( ( )) τ T S= arg min D, E T 1 Dτ SE DSE S S τ 1 : = DSE, S S* τ C ( ) (, : =, ) Ξ( *) ( ) 1 March 1, 017 Carlos Agelet de Saracibar 47

43 J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem, arising from the maximization of a regularized lastic dissiation, yields the associative lastic flow rule, ( ( )) τ T S= arg min D, E T 1 τ = SD SE DSE S S τ 1 1 : = E C ( S S* ) =0 τ ( ) ( ) 1, :, ( *) S C 1 τ 1 E = C S S * March 1, 017 Carlos Agelet de Saracibar 48

44 Closest-oint-rojection (c) J Plasticity Models > Rate Deendent Plasticity Models The closest-oint-rojection (c) is obtained as the solution of the following constrained minimization roblem, written in terms of the comlementary energy norm as, S * = arg min Ξ( S T* ) T* σ Ξ S S* = S S* = S S* C S S* C Using the Lagrange multiliers method, it can be transformed into the following unconstrained minimization roblem L S* = argmin L S, T*; λ* T* SS, *; λ* : =Ξ S S* + λ* f S* March 1, 017 Carlos Agelet de Saracibar 49

45 Closest-oint-rojection (c) J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem yields the closest-oint-rojection, S* = argmin L S, T*; λ* T* L SS, *; λ* : = Ξ S S* + λ* f S* S* S* S* 1 : = C ( S S* ) + λ* S* f S* = 0 S* = S λ* C S * f S* λ* 0, f S* 0, λ* f S* = 0 March 1, 017 Carlos Agelet de Saracibar 50

46 J Plasticity Models > Rate Deendent Plasticity Models Closest-oint-rojection (c) Geometric interretation S* S σ S S* C 1 March 1, 017 Carlos Agelet de Saracibar 51

47 Closest-oint-rojection (c) J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem yields the closest-oint-rojection, σ* = σ λ* : f σ*, q*, q* = σ λ* : n* = σ λ * µ n* q* σ* S* = S λ* C S * f S* q* = q λ* K f σ*, q*, q* = q λ* K 3 q* = q λ* H q* f σ*, q*, q* = q+ λ* Hn* 3 3 λ* 0, f σ*, q*, q* 0, λ* f σ*, q*, q* = 0 March 1, 017 Carlos Agelet de Saracibar 5

48 Closest-oint-rojection (c) J Plasticity Models > Rate Deendent Plasticity Models Taking into account that the rojection takes lace in the deviatoric sace (octahedral lane), the solution of the unconstrained minimization roblem, defining the closest-ointrojection, yields, dev σ* = dev σ λ * µ n* q* = q λ* K 3 q* = q+ λ* H n* 3 λ* 0, f σ*, q*, q* 0, λ* f σ*, q*, q* = 0 March 1, 017 Carlos Agelet de Saracibar 53

49 Closest-oint-rojection (c) J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem, defining the closest-oint-rojection, yields, ( H ) dev σ* q* = dev σ q λ * µ + n* ( H ) dev σ* q* n* = dev σ qn λ * µ + n* ( ( λ µ H )) 3 devσ qn= dev σ* q* + * + n* dev * * dev λ * µ, * 3 H σ q = σ q + n = n 3 3 March 1, 017 Carlos Agelet de Saracibar 54

50 J Plasticity Models > Rate Deendent Plasticity Models For the non-trivial case, using the Kuhn-Tucker comlementarity conditions for the c, the Lagrange multilier reads, λ* 0, f σ*, q*, q* 0, λ* f σ*, q*, q* = 0 if λ* > 0 then f σ*, q*, q* = dev σ* q* σ q* = 0 f σ*, q*, q* = dev σ q λ * µ + H σ q* ( ) 3 3 ( ) K H Y q = dev σ q λ * µ + + σ ( σ q λ µ ) 3 Y = f,, q * + K + H = ( K H) f ( σ q q) λ * = µ + +,, > Y March 1, 017 Carlos Agelet de Saracibar 55

51 Closest-oint-rojection J Plasticity Models > Rate Deendent Plasticity Models The solution of the unconstrained minimization roblem yields the closest-oint-rojection, S* = S λ* C S * f S* 1 ( K H) f ( q ) σ* = σ µ + + σ,, q µ n ( µ ) ( σ q) q* = q + K + H f, q, K ( ) 1 q* = q+ µ + K + H f ( σ, q, q) Hn March 1, 017 Carlos Agelet de Saracibar 56

52 J Plasticity Models > Rate Deendent Plasticity Models Associative lastic flow rule The associative lastic flow rule takes the form, ε = µ σ σ* = µ + K + H f ( σ, q, q) n 3 3 τ τ 1 ( ) ξ = K q q* = µ + K + H f ( σ, q, q) τ τ 1 1 τ 1 E = C S S 1 1 ( ) 1 ξ = H q q* = µ + K + H f ( σ, q, q) n 3 3 τ 3 τ * March 1, 017 Carlos Agelet de Saracibar 57

53 J Plasticity Models > Rate Deendent Plasticity Models Associative lastic flow rule The associative lastic flow rule can be recast in the form, where the relaxation time takes the form, ε = γ n ξ = γ 3 ξ = γ n ( K H) f ( σ, q, q) f ( σ, q, q) γ = µ + + = 3 3 τ η ( ) 1 τ : = µ + K + H η 3 3 March 1, 017 Carlos Agelet de Saracibar 58

8.7 Associated and Non-associated Flow Rules

8.7 Associated and Non-associated Flow Rules Recall the Levy-Mises flow rule, Eqn. 8.4., d ds (8.7.) The lastic multilier can be determined from the hardening rule. Given the hardening rule one can more