Plasticty Theory (5p)

Transcription

1 Cmputatinal Finite Strain Hyper-Elast Plasticty Thery (5p) à General ü Study nn-linear material and structural respnse (due t material as well as gemetrical effects) ü Fundamental principles fl Cntinuum mechanics ü Cmputatinal prcedures fl FE-prcedures fr nn-linear prblem ü Restrictins Structure mdeling: statics f slids Material mdeling: hyperelasticity (part I), hyperelastplasticity (part 2)

2 L1: Curse utline, Kinematics, [1]: 3.4-5: 1. Curse utline, Repetitin, Kinematics, [1]: 3.4-5: Nn-linear defrmatin map, Defrmatin gradient, Right-left Cauchy-Green defrmatin tensrs, Vlume change, Area change (Nansn's frmula), Spectral decmpsitin, Useful lemmas based n spectral decmpsitin. 2. Kinematics cnt'd, Cnservatin principles, [1]: 4.3 Rate f defrmatin: Spatial velcity gradient tensr, Change f vlume, Frmulatins f the strng frm f the mmentum balance, Static principle f virtual wrk, Cnservatin f mass. 3. Alternative representatins f virtual wrk Alternative wrk induced stress representatins, [1]: 4.5, Stress rates [1]: 4.6, Material peratrs representing linearized stress respnse [2]. 4. Cntinuum thermdynamics Cnservatin f energy: isthermal case, Entrpy inequality: Helmhltz free energy frmulatin, Cnstitutive equatins f hyper-elasticity [1]: Cnstitutive equatins f hyperelasticity, cnt'd Incmpressible materials, [1], 5.5, Ischric vlumetric decupling) 6. Discretizatin and FE frmulatin, assignment 1

3 1. Hyperelast-plasticity L1: Curse utline Basic cncepts: Multiplicative decmpsitin, Thermdynamic preliminaries [2], [3]. Cnstitutive equatins f Hyperelast-plasticity: Prttype mdel. 8. Hyperelast-plasticity, cnt'd Numerical Integratin f flw rule: Basic frmulatin, Nn-linear cnstitutive prblem, Newtn-Raphsn prcedure, Applicatin: prttype mdel based n istrpic elastic and plasticity. 9. Discretizatin and FE frmulatin, assignment 2. Curse wrk and examinatin Pertinent assignments, that invlve cmputer implementatin and calculatins, are given. The cre f the curse wrk cncerns the develpment f (MATLAB) cde fr FE-analysis f hyperelasticity and hyperelastplasticity. Cmpleted curse wrk gives 5 credit pints. Literature [1] J. Bnet and R. D. Wd, Nnlinear cntinuum mechanics fr finite element analysis, 1997 [2] R. Larssn, Multiplicative Finite Strain Hyper-Elast Plasticty - Basic Thery and its Relatin t Numerical Methdlgy, U68, Hållfasthetslära, (KOMMER SOM PDF FIL PÅHEMSIDAN) [3] R. Larssn, Lecture ntes.

4 L1. Repetitin, Kinematics, [1]: à Nn-linear defrmatin map (Fundamental map) à Defrmatin gradient à Right-Left Cauchy-Green defrmatin tensrs à Vlume change, [1], 3.7 à Area change (Nansn's frmula), [1], 3.9 à Spectral decmpsitin à Useful lemmas based n spectral decmpsitin

5 à Nn-linear defrmatin map (Fundamental map) Cnsider fundamental defrmatin map frm the implicit nn-linear functin where x = ϕ@x, td X B 0, x B with B = ϕ@b 0 D

6 à Defrmatin gradient Cnsider stretch vectr λ defined as the directinal derivative λ = d dε» ε 0 ϕ@x +εmd = F M where F is the defrmatin gradient F = ϕ X = ϕ X Nte! material unit vectr M B 0, with» M» = 1. Nte! the stretch vectr λ dented the "push-frward" f M B 0. Nte! defrmatin gradient F defines relatin between material (undefrmed) and spatial (defrmed) line element dx and dϕ = dx, i.e. dϕ =dx = F dx (3)

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8 à Right-Left Cauchy-Green defrmatin tensrs Cnsider nw the representatin f the stretch vectr and λ =λm withλ=actual stretch and m B λ 2 = λ λ=m HF t FL M = M C M where C is the right Cauchy-Green defrmatin tensr C = F t F = F t 1 F with 1 B Nte! C is dented the "pull-back" 1. (3) Intrduce als material stretch Λ via the representatin: Λ =λm def = m F The actual stretch is then btained as (4) λ 2 = Λ Λ = m HF F t L m = m b m (5) Nte! b = F F t = left Cauchy-Green defrmatin (r Finger) tensr.

9 à Vlume change, [1], 3.7 Relatin between defrmed and undefrmed vlume elements is given dv = JdV 0 with J = det@fd =λ 1 λ 2 λ 3 à Area change (Nansn's frmula), [1], 3.9 Cnsider material pint in B 0, and x B with B = ϕ@b 0 D. Intrduce the vlumes in terms f an arbitrary line elements such that dv 0 = dx da with da = da N dv = dx da with da = da n (3) Frm basic relatins dx = F dx; dv = JdV 0 (4) fl da = JdA F 1 This relatin is smetimes dented the "Nansn's frmula".

10 à Spectral decmpsitin Cnsider defrmatin gradient represented as C = F t 3 F =... = i=1 λ 2 i N i N i = b = F F t 3 2 =... = i=1 λ i n i n i 3 F = i=1 λ i n i N i 8N i < i=1,2,3 = material unit principal stretch directins 8n i < i=1,2,3 = crrespnding unit spatial principal strecth directins 8λ i < i=1,2,3 are the principal stretches Spectral prperties given by eigenvalue prblems 8HC λ 2 i 1L N i = 0, Hb λ 2 i 1L n i = 0,i = 1, 2, 3< Stretch directins satisfy the rthgnality cnditins N i N j = 1 iff i = j N i N j = 0 iff i j = n i n j = 1 iff i = j n i n j = 0 iff i j = (3) (4)

11 à Useful lemmas based n spectral decmpsitin Lemma: Based n the spectral decmpsitin we have that λ i 2 C = N i N i λ i 2 C = 2 λ i λ i C = N i N i λ i C = 1 2 λ i 1 N i N i (3) Lemma: Based n the spectral decmpsitin we have that J C = 1 2 J C 1 (4) Lemma: We have that: J F = J F t (5)