The Element-Free Galerkin Method applied on Polymeric Foams Author: - PowerPoint PPT Presentation

Federal University of Santa Catarina Grupo de Mecânica Aplicada e Computacional The Element-Free Galerkin Method applied on Polymeric Foams Author: Guilherme da Costa Machado Advisor: Marcelo Krajnc Alves, Ph.D. Co-Advisor: Rodrigo Rossi, Dr.Eng. Florianópolis - Brazil September, 2006

Overview The Element-Free Celular Solids   Galerkin Method Introduction  Moved List Square Characteristics   Approximation Mechanical Properties  Weight Function definition  Models examples  Imposition of Essential  Methodology  Boundary Conditions Unilateral Contact and Finite Strain Elastoplastic   Friction Formulation formulation Problem definition Kinematics of deformation   Imposition of the Contact and The concept of conjugated pairs   Friction terms The Elastoplastic model  General Algorithm  Strong, Weak and Incremental  Examples (EFG) formulations  Solution Procedure  Examples (FEM)  Conclusions  2

Cellular solids Introduction  Typically used in energy absorption structures.  Applications areas:  • Automotive/Transport industry; • Aerospace industry; • Packing industry; • Construction industry. Mechanical X low density.  3

Cellular solids 4

Cellular solids Relative density   ∗   /  m Very low = 0,001;  Conventional = 0,05 to 0,20;  Transition value 0,3 treated as a solid with isolated pores.  Types  Polymeric  Metallic (aluminum, cooper, nickel, titanium and zinc)  Ceramic (carbon)  Natural (wood, cork and coral structures)  5

Mechanical Properties  GIBSON & ASHBY(1997) Sólidos celulares 6

Cellular solids Elastoplastic foam behaviour under compression Densification process Stress lim Mechanical Properties    1 GIBSON & ASHBY(1997) Cellular walls/struts buckling plateau Linear elastic flexion Strain  7

Cellular solids  Modeling Approaches  Periodical Models (GIBSON & ASHBY) • Dependence of the geometrical idealization; • Considers the local cell characteristics; • Limited applications; • Isotropy and cellular interaction, in the majority of the cases, are not periodic phenomenon.  Random models (ROBERTS,GARBOCZIA e BRYDON) • Voronoi tessellation and Gaussian random field; • Problems to define the RVE and the coordinate number; • Problems at the micro tomography correlation; • Border effects; • Self-contact densification process involves a huge computational cost; • Explicit solver. 8

Cellular solids Elastoplastic model for polymeric foams  Methodology  Finite Stains Algorithm;  Total Lagrange Description;  Rotated Kirchoff stress ;  Hencky Logarithmic strain measure;  Volumetric hardening Law;  Finite Element Method - FEM;  Element Free Galerkin Method - EFG;  Contact formulation ( Signorini hypothesis);  Friction formulation (regularized Coulomb’s law);  9

Finite Strain Elastoplastic Formulation Required formulations  Stress/Strain elastic relation – an elastic law;  Yield function - indicating the stress level to start the plastic flow;  Stress/Strain plastic relation – material hardening/softening law;  10

Finite Strain Elastoplastic Formulation Movement, strain gradient and the  Kinematics of deformation multiplicative decomposition      j = = + ( , ) X t x X u   ¶ x =  j = F ( , ) X t   X ¶ X = e p F F F Polar decomposition, Cauchy-Green  tensor and the log strain measure p = p p e = e e F R U F R U e e = U C T e = e e C F F ( ) e = e E ln U 11

Finite Strain Elastoplastic Formulation Hill’s Principle: work rate invariability  1 1 1 1 1     = s ⋅ = t ⋅ = ⋅ = ⋅ = t ⋅ e D D P F S C E ,  r r r r r 2 o o o o Spectral decomposition  T e e e = C F F   3 å e = l Ä C ( l l ) i i i = i 1   3 å e = l Ä U ( l l ) i i i i = 1   3 1 å e = l Ä E ln( )( l l ) i i i 2 = i 1 12

Finite Strain Elastoplastic Formulation GIBSON & ASHBY(1997) Kirchoff rotated stress  e T e t t t s = ( R ) ( R ) = det( ) F where Hyperelastic Hencky’s model   t = ( r * ) e E æ ö 2 ÷ ç * * * * ( r ) = m r ( ) + ( r )- m r ( ) ( Ä ) 2 K I I ÷ D I ç ÷ ç è ø 3 * * r = [ ] r det F o 1 ( ) = d d + d d I ijkl ik jl il jk 2 g ( ) ( ) * * r = r E c E ( Ä ) = d d I I M ij kl ijkl 13

2 æ ö é ù é ù - o + o p p p p ÷ 2 ç 2 2 = + b ( , ) q p q p = 2 + a 2 - ê ú - a ê ú £   ( , ) q p q p c t ÷ c t 0 ç ÷ ç ÷ n ê ú ê ú ( ) è ø 2 2 ë û ë û p Yield Function and the Plastic Flow Potential ( ) a e p e p , v a p = V p ( ) e p H t c p v ö æ p L ÷ ç ( ) ( ) t e p = t o + e p e p = - ÷ ( ) ( ) H ln = o + e p e p = - p ç p p H ln J ÷ ç ÷ y a y a a ç c c p v v è L ø o  14

Return Mapping Algorithm Elastic Prediction p =  F 0 æ ö ¶ G ÷ ç p = D l ÷ p End F exp F ç ÷ ç ÷ n + 1 ç n ¶ t è ø teste p = p F F + n 1 + n 1 n ìé ù ï - 1 teste ( ) e = p F F F 2 ïê kb D l ú + - teste 1 p p ïê n + 1 n + 1 n ) ( ú ïë n + 1 n + 1 p , q G û ï é ù + + n 1 n 1 * r 0 ï ê ú * ïé r = o ù ï ê ú [ ] ïê n + 1 m D l 3 det F ú + - teste = ê ú 1 q q 0 íê n + 1 ) ( ú ï n + 1 n + 1 ê ú p , q G ë û ï + + n 1 n 1 ( ) T ê ú ï 0 teste teste teste e = e e C F F ï ê ú ⋅ = ⋅ ) ( teste ë û ( ) ( ) ï n + 1 n + 1 n + 1 D l q , p ,  n + 1 n + 1 ï n + 1 n + 1 ï 1 ln ï ( ) ï teste teste î e e = E C n + 1 n + 1 2 teste teste D l in terms of p q yes + + n 1 n 1 teste * e t = ( r ) E D + + n 1 n 1 D l > where 0 é ù teste teste * e p = - K ( r ) tr E ê ú teste teste e £ ë û ( q , p , ) 0 + + +  p teste n 1 n 1 n 1 + + n 1 n 1 v + n 1 3 Plastic Corrector teste teste teste = t D ⋅ t D q n + 1 n + 1 n + 1 2 no 15

Finite Strain Elastoplastic Formulation The global problem value  Strong formulation: reference configuration    • For each , determine that it is solution of u X t ( , ) t ∈  t o , t f      - r = W div ( , ) X t ( X b X t ) ( , ) 0 em P o o      = G t P ( , ) X t N X t ( , ) t X t ( , ) em o     = G u u X t ( , ) u X ( ) em o 16

Finite Strain Elastoplastic Formulation The global problem value  Weak formulation: reference configuration   u + Î • Find such that  n 1    ( ) d = " d Î u 1 ; u 0 u  F + n where         ( ) ( ) ò ò ò d = ⋅  d W - r ⋅ d W - ⋅ d u ; u P u u d b u d t u dA F  n + 1 n + 1 o o n + 1 o n + 1 o X t W W G o o o    { } = Î 1 ( ) W = G u u u W , u u em  i p o   { } = d d Î 1 ( ) W d = G u u u W , u 0 em  i p o 17

Finite Strain Elastoplastic Formulation Local Linearization ( Newton’s Method)  ( ) ⋅ ⋅ , • Considers as being enough regular F       ( ) ( ) + k 1 d = k + D k d = " d Î u ; u u u ; u 0 u  F F n + 1 n + 1 n + 1  k u + and expanding by Taylor in terms of , we obtain n 1         ( ) ( ) ( ) é ù k + D k d k d + k d D k u u ; u  u ; u D u ; u u F F F ë û + + + + + n 1 n 1 n 1 n 1 n 1       ( ) ( ) ( ) é ù ò  k d D k = k ⋅  D k ⋅  d W D u ; u u u u u d F   ë û n + 1 n + 1 n + 1 n + 1 o X X W o ¶ ¶ t P  é ( ) ù  ij ip - - - k = = 1 - t 1 1 u F F F ê ú ë û n + 1 jp ip jk lp ¶ ¶ F F ijkl  kl k kl u + n 1 18