# NONLINEAR CONTINUUM FORMULATIONS CONTENTS

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1 NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell element Post-buckling of composite panels

2 DESCRIPTIONS OF MOTION t = t > X 3 x 3 X x X Reference configuration Material description x Current configuration x ( X, t), ( X,) X f= f( X, t) Spatial description X X( x, t) f= f( x, t) JN Reddy Continuum Shell Element

3 MATERIAL TIME DERIVATIVE Material description d d f [ f( X, t)] = [ f ( X, t)] = = X fixed dt dt t Spatial description d dx j [ f( x, t)] = [ f( x, t)] + [ f( x, t)] dt t dt x = [ f( x, t)] + v [ f( x, t)] j t x = [ f( x,)] t + v f( x,) t t j j JN Reddy Continuum Shell Element 3

4 LAGRANGIAN DESCRIPTION OF MOTION Incremental descriptions of deformed configurations Total Lagrangian description Updated Lagrangian description JN Reddy Continuum Shell Element 4

5 Kinematics of Deformation x3, X3 Reference configuration X= ( X,) Deformed configuration Particle X, occupying position x at time t : x = ( X, t ) Deformed configuration x, X Particle X, occupying position x at time t x : = ( X, t ) x, X JN Reddy Continuum Elements 5

6 Kinematics of Deformation (continued) x3 X 3 (time t = ) ( X) T dx,, dx = F dx dx = F dx F dx XQ XP Q dx P x Q u P x P uq x X (time t) _ Q dx _ P x X u x X dxdx-dxdx= dxedx - T EF F I T E= u u u u E ij u u j u u X X X X i m m j i i j T E Green strain tensor JN Reddy Continuum Elements 6

7 Definitions MEASURES OF STRAIN u= x x or u i = x i x i u= x x or u i = x i x i u= u u or u i = u i u i E ij = ( u i E ij = Green Incremental Strain Tensor ε ij d x i d x j =( ds) ( ds) = ( E ij E ij ) d x i d x j ( e ij + η ij )d x i d x j + u j x j + u k ) u k x i x i x j ( u i + u j x j + u k ) u k x i x i x j e ij = ( ui x j + u j + u k u k x i x i + x j Linear Green Incremental Strain Tensor e ij = ( ui x j + u k ) u k x i x j η ij = u k u k x i x j Nonlinear Incremental Strain Tensor Infinitesmal Green-Lagrange Incremental Strain Tensor JN Reddy u ) j x i Continuum Elements 7

8 Point forces Distributed force F F f STRESS VECTOR F Plane 3 F 4 Plane Continuous rigid support Roller support F f F f a nˆ t (nˆ ) nˆ Plane Stress vector t ( nˆ ) f lim a a ( nˆ ) ( nˆ ) t t JN Reddy Continuum Formulations 8

9 CAUCHY STRESS TENSOR x 3 x A tn ˆ B nˆ s t eˆ x 3 ê 3 s t ê ê tn ˆ eˆ nˆ s x t x x t 3 x 3 t x 3 xz 3 zx t σ nˆ or t n ; t eˆ, σ t eˆ eˆ eˆ ( nˆ ) ( nˆ ) i ij j i ji j j j ij i j x 33 zz xx eˆ 3 t 3 s 3 3 zy 3 yz yy yx xy

10 FIRST AND SECOND PIOLA-KIRCHHOFF STRESS TENSORS df TdAPNˆ dapda df tda nˆ da da Nˆ ( X) nˆ df PdAda da da danˆ da, danˆ da Reference configuration Current configuration JN Reddy d J d a F A df df d SdA F f F P A S A d d d d F PS, SJ F σf T P First Piola-Kirchhoff stress tensor S Second Piola-Kirchhoff stress tensor Continuum Formulations

11 CONSERVATION PRICIPLES Conservation of Mass rdv = r dv rjdv = r dv ò ò ò ò o r = rj Conservation of Linear Momentum d r vdv = r fdv + tda ò ò ò dt dv dt T r = s +r f Continuum Formulations

12 Stress Measures (continued) Cauchy Stress Tensor Configuration in which it occurs σ ij = σ ij, σ ij = σ ij Configuration in which it is measured First Piola-Kirchhoff Stress Tensor t= σ ˆn, T= P ˆn d f= td A= T d A ( σ ˆn)d A= ( P ˆn)d A

13 Stress Measures (continued) Second Piola-Kirchhoff Stress Tensor ( ˆn S)d A=d f d f= J d f J = ( x x F, ) x J= det x S = F ( P) S J F s F T F ( x) x, Deformation gradient tensor ( x) X 3

14 DESCRIPTION OF MOTION C C x 3 x 3 x 3 Q C d s P u u Q d s P u Q d s P x x x x x x Motion: x= x(x,x,x 3,t) Notation: coordinates ofa point: x, x, x volum es: V, V, V areas: A, A, A density: ρ, ρ, totaldisplacem ents ofa point: u, ρ u 4

15 Various Strain Components Displacements: u= x x or u i = x i x i u= x x or u i = x i x i u= u u or u i = u i u i Green-Lagrange Strain Tensor ( ds) =d x d x= d x i d x j i ( ds) =d x d x= d x i d x i ( ds) =d x d x= d x i d x i E ij d x i d x j =( ds) ( ds) E ij d x i d x j =( ds) ( ds) d x i = x i x j d x j, d x i = x i x j d x j JN Reddy Continuum Formulations 5

16 Strain Measures (continued) Green-Lagrange Strain Tensor E ij = E ij = ( u i + u j x j + u k ) u k x i x i x j ( u i + u j x j + u k ) u k x i x i x j Green-Lagrange Incremental Strain Tensor Linear Green-Lagrange Incremental Strain Tensor e ij = ε ij d x i d x j =( ds) ( ds) ( ui x j + = ( E ij E ij ) d x i d x j ( e ij + η ij )d x i d x j u j x i + u k x i u k x j + u k ) u k x i x j JN Reddy Continuum Formulations 6

17 Nonlinear Green-Lagrange Incremental StrainTensor η ij = u k x i u k x j Infinitesimal Green-Lagrange Incremental Strain Tensor e ij = ( ui x j + u ) j x i Updated Green-Lagrange Strain (increment) Tensor ( ε ij )d x i d x j = ( ds ) ( ds ) ) x k η ij = x i δ ij x j = ( ui + u j x j + u ) k u k x i x i x j ε ij = ( x k Strain Measures (continued) e ij + η ij e ij = ( uk u k x i x j ( ui + x j ) u ) j x i 7

18 CONSERVATION OF MASS AND CONSTITUTIVE RELATIONS Conservation of Mass ρ= ρ J, ρ= ρ J Constitutive Relations S ij = C ijk l ε kl, S ij = C ijk l ε kl C ijk l = ρ ρ x i x p x j x q x k x r x l x s C pqrs C ijk l = ρ ρ x i x ρ x j x q x k x r x l x s C pqrs JN Reddy Continuum Formulations 8

19 Virtual Work Statements General Virtual Work Statement (in configuration to be determined) δw σ:δ( e) d V δ R= V = σ ij δ( e ij )d V δ R= V δ R= f δu d V+ t δu d S V S = f i δu i d V+ t i δu i d S V Infinitesimal Green-Lagrange incremental strain tensor components e S u u i ij xj j x i Continuum Shell Element 9

20 JN Reddy Virtual Work Statement for the Total Lagrangian Formulation Write various integrals over configuration as equivalent integrals over the reference configuration σ ij δ( e ij )d V= V S ij δ( E ij )d V V f i δu i d V= V f i δu i d V V t i δu i d S= S t i δu i d S S S ij δ( E ij )d V δ( R)= V δ( R)= f i δu i d V+ t i δu i d S V V S S ij δ( E ij )d V δ( R)= Continuum Formulations

21 Total Lagrangian Formulation. Virtual Work Statement: E ij = δ( R)= V f i δu i d V+. Incremental Decompositions: e ij = ( u i x j + u j x i + u k S ij = S t i δu i d S S ij = S ij + S ij S ij = C ijkl E kl, S ij = C ijkl ε kl ( ) ( ui x j + E ij = E ij + ε ij, ε ij = e ij + η ij E ij = V ) u k x i, x j, u j + u k x i x i u k S ij δ( E ij )d V δ( R)= u k x i u k x j + x j ( u i + u j x j + u k u k x i x i x j ( ) ( ) ( ) x i J x m ), η ij = ) σ mn ( x j x n ) u k x i u k x j

22 Total Lagrangian Formulation (continued) 3. Virtual Work Statement with Incremental Decompositions: d( Eij ) d( Eij ) d( eij ) d( eij ), d( eij ) d( eij ) d( hij ) S ij δ( ε ij )d V+ S ij δ( η ij )d V V V =δ( R) S ij δ( e ij )d V V 4. Linearized Virtual Work Statement with Incremental Decompositions: S ij = C ijk l ε kl C ijk l e kl, δ( ε ij ) δ( e ij ) C ijkl e kl δ( e ij )d V+ V S ij δ( η ij )d V V =δ( R) S ij δ( e ij )d V V

23 x =x, x =y, u = u, u = v, u = ū, u = v FINITE ELEMENT MODEL OF THE TOTAL LAGRANGIAN FORMULATION C ijkl e kl δ( e ij )d V= V e ij = ( ui + x j e xx { e} = e yy e xy = ū v ū + v V u j x i + u k x i + u {δ e} T [ C]{ e} d V u k x j + ū + v u k ) u k x i x j u ū + v v u ū + v v v + ū u + v v Total Lagrangian Element 3

24 FINITE ELEMENT MODEL (continued) = ū v ū + v + u ū + v u ū + v v u ū + v v v + ū u + v v [ D] { ū v } u u u + u [ ] D u v v v + v { ū v } e xx { e} = e yy = ([D]+[D u ]){ū} e xy Total Lagrangian Element 4

25 FINITE ELEMENT MODEL (continued) V [D ]= [ D ]= {δ e} = {δ e} T [ C]{ e} d V V V = {δū} T ([D]+ [D u ]) T [ C]([D]+ [D u ]){ū} d V V, [D u ]=, δ e xx δ e yy δ e xy [ C]= u u u + u v C C C C C 66 =([D]+ [D u ]){δū} v v + v Total Lagrangian Element 5

26 V = V = = FINITE ELEMENT MODEL (continued) S ij δ( η ij )d V= {δ η} T { S} d V [ V ( δū ū S xx + δ v ( δū + ū S xy + δ v T V V δū δū δ v δ v V {δ η} T { S} d V ) v + S yy ( δū v + ū δū + v S xx S xy S xy S yy S xx S xy S xy S yy η ij = ū + δ v )] δ v d V ū ū v v ) v d V {δū} T [ D] T [ S][ D]{ū} d V (9.6.) u k x i u k x j Total Lagrangian Element 6

27 FINITE ELEMENT MODEL (continued) [ S]= {u} = {ū} = [Ψ]= S xx S xy S xy S yy S xx S xy S xy S yy { u v} = { n j= u } jψ j (x) n j= v jψ j (x) { n j= ūjψ } j (x) n j= v jψ j (x) =[Ψ]{ } { } ū = =[Ψ]{ } v [ ] ψ ψ ψ n ψ ψ ψ n { } T ={u,v,u,v,,u n,v n } { } T = {ū, v,ū, v,,ū n, v n } Total Lagrangian Element 7

28 FINITE ELEMENT MODEL (continued) V V {δ e} T [ C]{ e} d V = {δū} T ([D]+ [D u ]) T [ C]([D]+ [D u ]){ū} d V V = {δ } T {([D]+ [D u ])[Ψ]} T [ C]([D]+ [D u ])[Ψ]{ }d V V = {δ } T [B L ] T [ C][B L ]{ }d V (9.6.5) V {δ η} T { S} d V = {δū} T [ D ] T [ S][ D ]{ū} d V V = {δ } T [B NL ] T [ S][B NL]{ }d V V Total Lagrangian Element 8

29 FINITE ELEMENT MODEL (continued) δ( R)= = = δ( R)= = V V V V V S ij δ( e ij )d V {δ e} T { S} d V {δ } T [B L ] T { S} d V f i δu i d V+ t i δu i d S S {δ } T [Ψ] T { f} d V+ {δ } T [Ψ] T { t} d ( S Total Lagrangian Element 9

30 FINITE ELEMENT MODEL (continued) [K L ]= [K NL ]= { F}= { f} = [D ]= V V V { f x f y ([K L ]+ [K NL ]){ }= { F} { F} [B L ] T [ C][B L ]d V, { F}= [B L ] T { S} d V V [B NL ] T [ S][B NL ]d V [Ψ] T { f} d V+ [Ψ] T { t} d S }, { t} =, [D u ]= S { t x t y } u u u + u v v v + v Total Lagrangian Element 3

31 FINITE ELEMENT MODEL (continued) [ D ]= [ S]=, [ C]= S xx S xy S xy S yy S xx S xy S xy S yy C C C C C 66, { S}= S xx S yy S xy [B L ]= [B L ]+ [B u L ]+ [B v L ] [B L ]= ψ ψ ψ ψ ψ ψ ψ n ψ ψ ψ n ψ n ψ n Total Lagrangian Element 3

32 FINITE ELEMENT MODEL (continued) [B u L ]= [B v L ]= [B NL ]= u u ψ u ψ u ψ n u ψ u ψ u ψ n ψ + u ψ u ψ + u ψ u ψ n + u ψ n v v v ψ (9.6.7b) ψ v ψ v ψ n ψ v ψ v ψ n + v ψ v ψ + v ψ v ψ n + v ψ ψ ψ ψ ψ ψ ψ n ψ n ψ ψ ψ n ψ n (9.6.7c) (9.6.8) ψ n Total Lagrangian Element 3

33 FINITE ELEMENT MODEL (continued) [ ]{ } [K L ]+ [K N ] [K L ] {ū} [K L ] [K L ]+ [K N ] { v} { ( K ij L =h e C + u ) ( ) ψ i ψ j u Ω + ψ i ψ j C e ( + C + u ) ( u ψi ψ j + ψ ) i ψ j [( + C 66 + u ) ψi + u ] ψ i [( + u ) ψj + u ]} ψ j dx dy { ( K ij L =h e C + u ) ( v ψ i ψ j Ω + C + v ) u ψ i ψ j e [( + C + u )( + v ) ψi ψ j + u ] v ψ i ψ j [( + C 66 + u ) ψi + u ] ψ i [( + v ) ψj + v ]} ψ j dx dy = K ji L 33 = { } { F } { F } { F } { F }

34 FINITE ELEMENT MODEL (continued) K L ij K N ij =h e =h e Ω e Ω e { ( ) v ψ i C + C ( + v + C 66 [( + v [( + v ) ψj + v [ ψ i ψ j S xx + S xy + S yy ψ i ψ j Fi =h e f x ψ i dx dy + h e Ω e Fi =h e f y ψ i dx dy + h e Ω e {( Fi =h e + u ) ψi Ω e + [( + u ( ψ j + C + v ) ( v ψi ψ j + ψ i ) ψi + v ] ψ i ]} ψ j dx dy ( ψi ψ j ] dx dy = K N ij Γ e Γ e t x ψ i ds t y ψ i ds S xx + u ) ψi + u ψ i + ψ i ψ i ] S yy S xy } dx dy ) ψ i ) ψ j ) ψ j ψ j (9.6.3a) 34

35 6. NUMERICAL RESULTS Transverse deflection versus load (cantilever beam under uniformly distributed load).. Without iteration q Load, q O With iteration a = in. b = in. h = in. E = ksi. =. q =.5 b h v y a x, u Transverse deflection, v Total Lagrangian Element 35

36 NUMERICAL RESULTS Load, q Normal stresses vs. load for a cantilever beam under uniform load (updated). Cauchy stress Cauch stress (five Q8 elements) a = in. b = in. h = in. E = ksi. =. q =.5 Second Piola-Kirchhoff stress (five Q8 elements) (updated Lagrangian formulation) Axial stress, sigma-xx (Sxx) 36

37 SHELL FINITE ELEMENTS Degenerate Continuum Shell Elements It is obtained by a degeneration process from the 3D continuum equations. Less expensive than the solid element but more expensive than the shell-theory element. Also exhibit locking problems. Shell Theory Elements Based on a curvilinear description of the continuum using an specific shell theory. Analytical integration of energy terms over the thickness. Also exhibit locking problems. Continuum Shell Element 37

38 CONTINUUM SHELL ELEMENTS The degenerate continuum shell element approach was first developed by Ahmad et al. from a threedimensional solid element by a process which the 3D elasticity equations are expressed in terms of midsurface nodal variables. It uses displacements and rotational DOFs. Continuum Shell Element 38

39 CONTINUUM SHELL ELEMENTS Assumptions are:. Fibers remain straight. The stress normal to the midsurface vanishes (the plane stress condition) Discretization of the reference surface by using Cartesian coordinates where n k k Xi k(, ) ( Xi ) TOP ( Xi ) k BOTTOM (, ): k (,, ): Lagrangian interpolation functions Natural coordinates Continuum Shell Element 39

40 CONTINUUM SHELL ELEMENTS n k k xi ψk(, ) ( xi ) TOP ( xi ) k BOTTOM Discretization of the final configuration of the reference surface Equivalent to the discretization of the displacement of the reference surface as u u h n k k i k (, ) i k e3i k Continuum Shell Element 4

41 Continuum Shell Element (continued) Virtual Work Statement for the Dynamic Case ρ t+ t ü i δ( t+ t u i ) dv + C ijrs l rs δ( l ij ) dv V V + t S ij δ( η ij ) dv = t+ t R t S ij δ( l ij ) dv V Finite Element Model x i = n k= ψ k x k i, t x i = t u i = n k= n k= ψ k t u k i, u i = V ψ k t x k i, t+ t x i = n k= n k= ψ k u k i i=,,3 ψ k t+ t k k, t [M ] t+ t { l }+ ( t [K L ]+ t [K NL ]){ l }= t+ t {R} t {F} Continuum Shell Element 4

42 Numerical Examples Continuum Shell Element 4

43 Numerical Examples Continuum Shell Element 43

44 Numerical Examples 44

45 Numerical Examples Continuum Shell Element 45

46 POSTBUCKLING AND FAILURE ANALYSIS Nonlinear FE analysis Comparison with experimental results of Starnes and Rouse Progressive failure analysis a = 5.8 cm ( in.), b = 7.8 cm (7 in.), h k =.4 mm (.55 in.) 4-ply laminate: [45/-45/ /45/- 45/ /45/-45//9] s E = 3 Gpa (9, ksi), E = 3 Gpa (,89 ksi), u constant w w x u w w x u x P b w a y G = 6.4 Gpa (93 ksi), ν =.38 (graphite-epoxy) Continuum Shell Element46

47 Experimental Setup and Failure Region (Starnes & Rouse, NASA Langley) (a) Typical panel with test fixture (b) A transverse shear failure mode Continuum Shell Element 47

48 POST-BUCKLING OF A COMPOSITE PANEL UNDER IN-PLANE LOADING Finite Element Models Mesh has six elements per buckling mode half wave in each direction 7. in Elements used: () Four-node C -based flat shell element (STAGS) () Nine-node continuum shell element (Chao & Reddy) (3) For-node and nine-node ANS shell elements (Park & Stanley). in All meshes have 7 elements (9 nodes for the meshes with four-node elements and 35 nodes for meshes with nine-node elements). Continuum Shell Element48

49 Comparison of the Experimental (Moire) and Analytical Out-of-Plane Deflection Patterns Experimental Theoretical (FEM) Continuum Shell Element49

50 Comparison of the Experimental and Analytical Solutions for End Shortening Continuum Shell Element 5

51 Postbuckling of a Composite Panel Analysis Test End shortening, u /u cr.5..5 Analysis Test Maximum deflection, w /h Continuum Shell Element 5

52 Postbuckling of a Composite Panel (Stress Distributions) P/P cr =. (MPa) Axial Stress, σ xx (ksi) P/P cr =.5 P/P cr =. P a a y x y/b b -5 Continuum Shell Element 5

53 Postbuckling of a Composite Panel (Stress Distributions) Shear Stress, σ zx (ksi) from constitutive relations from equilibrium relations P/P cr =. P/P cr =.5 P/P cr = y/b (MPa) Continuum Shell Element 53

54 Summary Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell element Post-buckling of composite panels Continuum Shell Element 54

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