Lecture #5: 5-0735: 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
Introduction to Continuum Mechanic
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
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. 4 4 4
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
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} 3 3 3 3 33 3 3 33 3 3 Stre comonent ij e along direction e i acting on urface e j coordinate ytem: e 6 6 6 e 3
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. 3 3 33 or 7 7 7 σ 33 3 3
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
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. 9 9 9
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 33 3 3 3 third invariant: I3 det[ σ] 0 0 0
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
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 0 0 0 0 III 0 0 0
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. 3 3 3
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
5 5 5 5-0735: 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
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. 6 6 6
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 7 7 7 LF
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
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
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
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 )
Three-dimenional Rate-indeendent Platicity
3D Kinematic: Incremental roblem F n F DEFORMED @ t n+ DEFORMED @ t n V n F n V n INITIAL R n ROTATED @ t n R ROTATED @ t n+ R n 3 3 3
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 4 4 4 t 0 F n Fn R n t n t n Vn R n F R t n t n V n
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. 5 5 5
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. 6 6 6
7 7 7 5-0735: 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 3 3 33 3 3 33 0 0 0 0 0 0 0 0 0 0 0 0 ) )( ( with the Young modulu E and the Poion ratio n. Sym.
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. 8 8 8
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. 9 9 9
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 30 30 30
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
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 0 3 3 3
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[ ] 33 33 33
Iotroic hardening The ame arametric form for k k[ ] are ued in 3D a in D. +0 4.00E+0 +0 3.50E+0 +0 3.00E+0 Hardening aturation 4.00E+0 k k dk k 0, k k0 Q d 3.50E+0 3.00E+0 +0.50E+0.50E+0 +0.00E+0.00E+0 +0.50E+0.50E+0 +0 +0 +00 k A ) S Swift ( 0 n.00e+0 5.00E+0 0.00E+00 k V k Voce Q ex[ ] 0.00E+0 5.00E+0 0.00E+00 k ( ) k V k S 34 34 34
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 0 35 35 35
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 36 36 36
State variable at time t n εn, n 5-0735: 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+. 37 37 37
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 38 38 38