Finite Element Analysis of Composite Laminates Juho Konno

Finite Element Analysis ofComposite Laminates20th Nordic Seminar on Computational Mechanics

Juho Könnö

[email protected]

Helsinki University of Technology

Institute of Mathematics

Outline of the talk

Mathematical model using the Reissner-Mindlinkinematic assumptions

20th Nordic Seminar on Computational Mechanics – p.1

Outline of the talk


Finite element formulation


Outline of the talk



The mixed interpolation technique for theReissner-Mindlin model


Outline of the talk



The mixed interpolation technique for theReissner-Mindlin model

The paper cockling problem


A typical configuration

z

zk+1

zk−1

g f

Ω

z

y

x

zk


Mathematical model

Basic idea:

Use Reissner-Mindlin kinematic assumptions

Take into account the in-plane displacement


Mathematical model

Basic idea:

Use Reissner-Mindlin kinematic assumptions

Take into account the in-plane displacement

Result:

Coupled problem consisting ofplane elasticity problemReissner-Mindlin plate problem


Constitutive relations

Use classical lamination theory




From the constitutive tensors of individual layers wederive:

A - the plane elasticity tensorD - the bending deformation tensorB - the coupling tensorA∗ - the shear deformation tensor




From the constitutive tensors of individual layers wederive:

A - the plane elasticity tensorD - the bending deformation tensorB - the coupling tensorA∗ - the shear deformation tensor

Each tensor is a weighted sum of the originalconstitutive tensors ⇒ easy to compute


Total energy of the plate

Π(u, w,β) =1

2

∫

Ω

ε(u) : A : ε(u)dΩ

︸︷︷︸

plane elasticity energy

+

∫

Ω

ε(u) : B : ε(β)dΩ

︸︷︷︸

coupling energy



Π(u, w,β) =1

2

∫

Ω


︸︷︷︸


+

∫

Ω


︸︷︷︸

coupling energy

+1

2

∫

Ω

ε(β) : D : ε(β)dΩ +1

2t2

∫

Ω

(∇w − β)·A∗· (∇w − β)dΩ

︸︷︷︸

plate energy



Π(u, w,β) =1

2

∫

Ω


︸︷︷︸


+

∫

Ω


︸︷︷︸

coupling energy

+1

2

∫

Ω

ε(β) : D : ε(β)dΩ +1

2t2

∫

Ω

(∇w − β)·A∗· (∇w − β)dΩ

︸︷︷︸

plate energy

−

∫

Ω

f ·udΩ −

∫

Ω

gwdΩ −

∫

Ω

G·βdΩ

︸︷︷︸

loading: planar, transverse and moment


Weak formulation

Minimize the energy with respect to the variablesu, w,β


Weak formulation


Two coupled equations: plate problem and planeelasticity problem


Weak formulation


Two coupled equations: plate problem and planeelasticity problem

Coupling is of elliptic nature


Displacement formulation

In displacement formulation the weak problem isProblem 1. Find (u, w,β) ∈ U × W × V such that ∀(v, ν,η) ∈ U × W × V

it holds

(A : ε(u), ε(v)) + (B : ε(v), ε(β)) = (f , v),

(B : ε(u), ε(η)) + (D : ε(β), ε(η))

+t−2(A∗· (∇w − β), (∇ν − η)) = (g, ν) + (G, η)


Displacement formulation

In displacement formulation the weak problem isProblem 1. Find (u, w,β) ∈ U × W × V such that ∀(v, ν,η) ∈ U × W × V

it holds

(A : ε(u), ε(v)) + (B : ε(v), ε(β)) = (f , v),

(B : ε(u), ε(η)) + (D : ε(β), ε(η))

+t−2(A∗· (∇w − β), (∇ν − η)) = (g, ν) + (G, η)

We can also treat the shear force q = t−2A∗· (∇w − β)

as an independent unknown


Mixed formulation

The mixed formulation with the shear force as anindependent unknown isProblem 2. Find (u, w,β, q) ∈ U × W × V × Γ such that

∀(v, ν,η, s) ∈ U × W × V × Γ it holds

(A : ε(u), ε(v)) + (B : ε(v), ε(β)) = (f,v),

(B : ε(u), ε(η)) + (D : ε(β), ε(η)) + (q,∇ν − η) = (g, ν) + (G, η),

(t2A∗−1q, s) + ((∇w − β), s) = 0,

where U × W × V × Γ ⊂ [H1(Ω)]2 × H1(Ω) × [H1(Ω)]2 × [L2(Ω)]2.


Mixed-interpolated finite elements

Solving Problem 1 or 2 with ”normal” basis functionsleads to locking




Solution: introduce a reduction operator on thediscrete spaces Rh : Vh → Γh and replace(∇w − β) → Rh(∇w − β)





Does not completely solve the locking problem, wefurther need






mesh-dependent stabilization






mesh-dependent stabilizationunequal interpolation


Main idea of the error analysis

Error analysis with the mixed formulation as astarting point




Problems to deal with




Problems to deal withStability issues




Problems to deal withStability issuesConsistency error





Restriction to a quasiuniform mesh needed





Restriction to a quasiuniform mesh needed

Adding the plane elasticity problem to the estimate ⇒use ellipticity


Main a priori result

We get the following error estimate

‖β − βh‖1 + ‖w − wh‖1 + t‖q − qh‖0 + ‖q − qh‖−1 + ‖u − uh‖1 ≤

Chki (‖g‖s−2,Ωi

+t‖g‖s−1,Ωi+‖F ‖s−1,Ωi

)+hb(‖g‖−1+t‖g‖0+‖F ‖0),

where F is defined as

‖F ‖2

s = ‖f‖2

s + ‖G‖2

s.


Main a priori result

We get the following error estimate

‖β − βh‖1 + ‖w − wh‖1 + t‖q − qh‖0 + ‖q − qh‖−1 + ‖u − uh‖1 ≤

Chki (‖g‖s−2,Ωi

+t‖g‖s−1,Ωi+‖F ‖s−1,Ωi

)+hb(‖g‖−1+t‖g‖0+‖F ‖0),

where F is defined as

‖F ‖2

s = ‖f‖2

s + ‖G‖2

s.

So, what’s the application then?



Cockling deals with small-scale deformations, mostlydue to moisture changes




In general, paper is a difficult material to model




In general, paper is a difficult material to modelArbritrary fiber orientation




In general, paper is a difficult material to modelArbritrary fiber orientationHeterogeneous





Simplification: divide the sheet into layers





Simplification: divide the sheet into layersFiber orientation varies ⇒ form constitutive tensorin each data block





Simplification: divide the sheet into layersFiber orientation varies ⇒ form constitutive tensorin each data blockUse classical lamination theory blockwise


Material model

Only global parameters are measurable


Material model


Local structure defined by


Material model


Local structure defined byOrientation angle


Material model


Local structure defined byOrientation angleLevel of anisotropy


Material model



Combining the above two we obtain the localconstitutive relation


Material model



Combining the above two we obtain the localconstitutive relation

Empirical models used for the moisture dependenceof global parameters


Problems in modelling

Choice of proper boundary conditions




Computational time




Computational time

Need for large deformations theory?


Effect of the boundary conditions

Figure 1: Simply supported, free plane Figure 2: Simply supported, fixed plane


Thank you for your attention


Documents

Finite Element Analysis of Composite Laminates Juho Konno