Lecture #5: Introduction to Continuum Mechanics Three-dimensional Rate-independent Plasticity. by Dirk Mohr

Transcription

1 Lecture #5: : Dynamic behavior of material and tructure Introduction to Continuum Mechanic Three-dimenional Rate-indeendent Platicity by Dirk Mohr ETH Zurich, Deartment of Mechanical and Proce Engineering, Chair of Comutational Modeling of Material in Manufacturing 05

2 Introduction to Continuum Mechanic

3 Cauchy tre tenor Suoe that a mechanically loaded body i hyothetically cut into two art. The created hyothetical urface can be decribed by the unit normal vector field nn[x] with the aociated infiniteimal area da. da n t t n da e x The traction vector tt[x] decribe the force er unit area that would need to act on the hyothetical urface nda to enure equilibrium. e 3 3 3

4 Cauchy tre tenor da n t e e x The Cauchy tre tenor [x] rovide the traction vector t that act on the hyothetical urface nda at a oition x (in the current configuration). t σ( nda) From a mathematical oint of view, the above equation define the linear maing of vector in R 3. The oerator i thu called a tenor

5 Cauchy tre tenor For a given et of orthonormal coordinate vector {e, e, e 3 }, we can alo define the tre comonent ij : e i e i t traction vector t acting on unit urface defined by normal vector e j ij σe j ij e j e σe i j jj e j e i σe j jj e σe j e j j 5 5 5

6 Cauchy tre tenor For a given et of orthonormal coordinate vector {e, e, e 3 }, it can alo be ueful to write the tre tenor in matrix notation: { σ} Stre comonent ij e along direction e i acting on urface e j coordinate ytem: e e 3

7 Symmetry of the Cauchy tre tenor Unlike other tenor ued in mechanic, the Cauchy tre tenor i ymmetric, T ij ji σ σ which can be demontrated by evaluating the local equilibrium. In other word, there are only ix indeendent Cauchy tre tenor comonent. Vector notation i therefore alo frequently emloyed, { σ} Sym or σ

8 Change of the tre tenor due to rotation e n t e n ~ Rn ~ t Rt e e Let denote the Cauchy tre tenor in the unrotated configuration which rovide the traction vector t for a given normal vector n. The traction vector after rotating the tre configuration read: ~ T t Rt R( σn) Rσ( R n~ ) T ( RσR ) n~ σn ~ ~ And hence, the Cauchy tre tenor in the rotated configuration read: σ ~ T RσR 8 8 8

9 Princial tree & direction Shear comonent σ I e t σe II I e normal comonent σ I I I rincial tre rincial direction We eek the direction for which the traction vector acting on the urface da ha no hear comonent

10 Princial tree & direction σ σ 0 Non-trivial olution can be found for if σ 0 3 det I I I 0 (characteritic olynomial) The characteritic olynomial i a cubic equation for the rincial tree. It i determined through the tre tenor invariant 3 firt invariant: I tr[ σ] econd invariant: I ( I σ : σ) with σ : σ third invariant: I3 det[ σ] 0 0 0

11 Princial tree & direction Solving the characteritic olynomial yield three olution which are called rincial tree. After ordering, we have Intermediate rinc. tre maximum rinc. tre I II III minimum rinc. tre The correonding orthogonal rincial tre direction { I, II, III } are found after olving σ 0 i i i for i I,.., III i

12 Sectral decomoition (of ymmetric tenor) With the hel of the rincial tree and their direction, the tre tenor may alo be rewritten a σ I I I II II which i called the ectral decomoition of the Cauchy tre tenor. II III III Recall that the tenor roduct of two vector e and e define the linear ma e e ) a e ( e ) In matrix notation, we have ( a 0 { e } e III 0 0 0

13 Stre tenor invariant The value of the rincial tree remain unchanged under rotation. Only the rincial direction will rotate: R σ R T R R R R I I I II II II III R III R III Thi i can alo be exlained by the fact that the value of I, I and I 3 remain unchanged under rotation (that i why thee are called invariant ), e.g. T I tr[ σ ] tr[ R σ R ] Hence the characteritic olynomial remain unchanged a well a it root I, II and III. The rincial tree are therefore alo invariant of the tre tenor

14 Decrition of Motion in 3D A body i conidered a a cloed et of material oint. body in it INITIAL CONFIGURATION u body in it CURRENT CONFIGURATION e 3 e X e x The current oition of a material oint initially located at the oition X i decribed by the function x x[ X, t] 4 4 4

15 : Dynamic behavior of material and tructure Deformation Gradient (3D) The dilacement vector i then given by the difference in oition X X x X u u ], [ ], [ t t X x u The deformation gradient i defined a X X u X X u X X X x X F ], [ ]), [ ( ], [ ], [ t t t t

16 Deformation Gradient (3D) dx dx X x It follow from the definition of the deformation gradient that the change in length and orientation of an infiniteimal vector dx attached to a material oint can be decribed by the linear maing dx F(dX) The deformation gradient i thu alo conidered a a tenor

17 Velocity gradient The time derivative of dilacement gradient i F [ X, t] x[ X, t] tx u[ X, t] tx v[ X, t] X It correond to the atial gradient of the velocity field with reect to the material oint coordinate X in the initial configuration. The atial gradient of the velocity field with reect to the current oition coordinate x i called velocity gradient: v L : x We have the relationhi v v x F X x X LF

18 Rate of deformation tenor A any other non-ymmetric econd-order tenor, the velocity gradient can be decomoed into a ymmetric and kew art: with L D W D : ym[ L] ( L L W : kw[ L] ( L L In mechanic, the ymmetric art of the velocity gradient i tyically called rate of deformation tenor D, while the kew art i called in tenor W. T T ) ) 8 8 8

19 Polar decomoition The deformation gradient F (non-ymmetric tenor) i often decomoed into a rotation tenor R and a ymmetric tretch tenor. T T F RU VR with R ( R ) ( R ) R T U U T V V V U The tenor U i called right tretch tenor, while V i called left tretch tenor 9 9 9

20 Interretation of tretch tenor Left tretch tenor Right tretch tenor F VR F RU F V F R R. Rotation. Stretching U. Stretching. Rotation 0 0 0

21 Logarithmic train tenor A frequently ued deformation meaure in finite train theory i the o-called logarithmic train tenor or Hencky train tenor: ε H ln U 3 i ln[ ]( u i i u It evaluation require the ectral decomoition of the right tretch tenor, 3 U ( u u ) i.e. Uui iui i i i i The value i are called the rincial tretche. The latter may alo be comuted uing the left tretch tenor due to the identity: V RUR T 3 i ( Ru i i i ) Ru i )

22 Three-dimenional Rate-indeendent Platicity

23 3D Kinematic: Incremental roblem F n F t n+ t n V n F n V n INITIAL R n t n R t n+ R n 3 3 3

24 3D kinematic: Incremental roblem Incremental deformation gradient: dx ( ΔF) n dx n F ( ΔF) n F n Incremental rotation R ( ΔR) n R n Incremental left tretch tenor V ΔV ( ΔR V n n ΔR T ) With the above definition in lace, it can be hown that the incremental rotation can be obtained from the olar decomoition of the incremental deformation gradient: T ΔF ΔV (ΔR ) T with ( ΔR )( ΔR ) and ΔV ΔV t 0 F n Fn R n t n t n Vn R n F R t n t n V n

25 Strain rate and total train The rate of deformation tenor i work-conjugate to the Cauchy tre tenor and i thu frequently ued to define the train rate: T v v ε D : x x To obtain a total train meaure, the train rate i integrated on a fixed bai (e.g. initial configuration) and then rotated forward to the bai of the current time t: t [ ] [ ] [ ] [ ] [ ] T T ε t R t R D R d R [ t] 0 In commercial finite element oftware, thi integration i often aroximated by T εn ( ΔR) εn( ΔR) ln( ΔV) In the abence of rotation, the train tenor obtained after integration i the ame a the Hencky train tenor

26 Additive train rate decomoition The train rate i decomoed into an elatic and a latic art, ε ε e ε The correonding algorithmic decomoition of the train increment aociated wit finite time increment t read ε ln( ΔV) ε e ε (*) The above decomoition i an aroximation of the well-etablihed multilicative decomoition of the total deformation gradient, F F e F (**) The aroximation (*) of (**) yield reaonable reult in finite train roblem when the elatic train are mall comared to unity

27 : Dynamic behavior of material and tructure Elatic contitutive equation The linear elatic iotroic contitutive equation read ε e C σ : with C denoting the fourth-order elatic tiffne tenor. For notational convenience, the above tre-train relationhi i rewritten in vector notation e e e e e e E ) )( ( with the Young modulu E and the Poion ratio n. Sym.

28 Equivalent tre definition The yield function i often exreed in term of an equivalent tre, i.e. a calar meaure of the magnitude of the Cauchy tre tenor. The mot widely ued calar meaure in engineering ractice i the von Mie equivalent tre: 3 [ σ ] S : S with the deviatoric tre tenor S dev[ σ] σ tr[ σ] 3 Note that the von Mie equivalent tre i a function of the deviatoric art of the tre tenor only. It i thu reure-indeendent, i.e. it i inenitive to change of the trace of

29 Equivalent tre definition The von Mie equivalent tre i an iotroic function, i.e. it i invariant to rotation of the Cauchy tre tenor: T [ σ] [ R σ R ] for any rotation A an alternative it may alo be exreed a a function of the tre tenor invariant or the rincial tree, e.g. R 3J with J S : S {( I II ) ( I III ) ( II III ) } Von Mie laticity model are therefore alo often called J- laticity model

30 Yield function and urface With the von Mie equivalent tre definition at hand, the yield function i written a: III f [ σ, ] [ σ] k[ ] The yield urface i f [ σ, ] 0 II I

31 Flow rule In 3D, it ha been demontrated that the direction of latic flow i aligned with the outward normal to the yield urface, ε f σ with f 3 S σ σ In other word, the ratio of the comonent of the latic train rate tenor are the ame a the deviatoric tre ratio f σ f 0 ij kl S S ij kl 3 3 3

32 Flow rule The rooed aociated flow rule alo imlie that the latic flow i incomreible (no volume change), 3 tr[ S] tr[ ε ] The magnitude of the latic train rate tenor i controlled by the non-negative latic multilier 0. It i alo called equivalent latic train rate. 0 f σ f

33 Iotroic train hardening The flow tre i exreed a a function of the equivalent latic train, with k [ t] k[ ] dt It control the ize of the elatic domain (diameter of the von Mie cylinder in tre ace). 3 k[ ]

34 Iotroic hardening The ame arametric form for k k[ ] are ued in 3D a in D E E E+0 Hardening aturation 4.00E+0 k k dk k 0, k k0 Q d 3.50E E E+0.50E E+0.00E E+0.50E k A ) S Swift ( 0 n.00e E E+00 k V k Voce Q ex[ ] 0.00E E E+00 k ( ) k V k S

35 Loading/unloading condition The ame loading and unloading condition are ued in 3D a in D: 0 if f 0 0 if f 0 and f 0 0 if f 0 and f

36 Iotroic hardening laticity (3D) - Summary i. Contitutive equation for tre ii. Yield function iii. Flow rule f σ C : ( ε ε ) [ iv. Loading/unloading condition σ, ] [ σ] k[ ] f ε σ 0 if f 0 0 if f 0 and f 0 0 if f 0 and f 0 v. Iotroic hardening law k k[ ] with dt

37 State variable at time t n εn, n : Dynamic behavior of material and tructure Return Maing Algorithm (3D) Alied total train increment Δε Calculate Trial State trial trial σ, n fn trial f n 0 trial f n 0 0 OUTPUT: State variable at time t n+ ε n εn n n Stre at time t n+ σ n σn C : Δε Solve: State variable at time t n+ εn εn ε n n f n [ ] OUTPUT: σ 0 0 σ Stre at time t n+ C : ( Δε Δε n n ) Simlified chematic aume that all tenor variable at time t n have already been uhed forward to the bai at time t n

38 Reading Material for Lecture #5 M.E. Gurtin, E. Fried, L. Anand, The Mechanic and Thermodynamic of Continua, Cambridge Univerity Pre, 00. Abaqu Theory Manual abaqu.ethz.ch:080/v6./df_book/theory.df