wavelet spectral finite element modeling of guided wave ... · wavelet spectral finite element...

[email protected] Materials Laboratory

Introduction - 1© The Aerospace Corporation 2015

Wavelet spectral finite element modeling of guided wave propagation in lap joints for bondline assessment

Dulip Samaratunga, PhD

The Aerospace Corporation

[email protected]

Project supervisor:

Ratan Jha, PhD

Mississippi State University

14th International Symposium on Nondestructive Characterization of Materials (NDCM 2015)June 22 – 26, 2015, Marina Del Rey, California, USA

14th Int. Symp. on Nondestructive Characterization of Materials (NDCM 2015) - www.ndt.net/app.NDCM2015


Outline

Motivation

Objectives

Governing equations derivation for

bonded joints

Spectral finite element formulation

Model validation with conventional finite

elements

Conclusions


Motivation

Section of an aircraft fuselage

Bonded joint

Certification of bonded composite primary

structures is challenging

Mechanical fasteners along with adhesives

has been the standard practice (e.g.,

Boeing 787)

Full cost and weight savings of composites

not possible with fasteners

Repeatable and reliable NDE technique for evaluating strength of joints is

an alternative for certification

Ultrasonic guided wave based NDE/ SHM techniques shown to be

promising

Modeling is an effective way to minimize development costs

Spectral finite element method is an efficient technique for solving ultrasonic wave propagation problems


Forward Fourier transform

ˆ ( ) ( ) j tF f t e dt

Forward continuous wavelet transform

f(t)

(EOM)SFE solution: u(ω),

v(ω), etc.

Forward integral

transformation

(Ex. Fourier/

wavelet transform)

Inverse integral

transformation

(Ex. Inverse

Fourier/ wavelet

transform)

u(t),

v(t), etc.

ˆ ( )F

ˆ ( , ) ( )t b

F a b f t dta

a, b – scaling and translations of

mother wavelet

Efficient approach for transient dynamics and wave propagation analysis

Exact solution obtained in the transformed domain

Computer implementation - similar to conventional FEM

Spectral finite element (SFE) method


Spectral finite element method: Progression

Doyle (1997) - isotropic beams and frames using Fourier based SFE (FSFE) for 1-D and 2-D structures

Gopalakrishnan et al - FSFE to model composite beams (2003) and plates (2005, 2006) for healthy and damaged (delamination, transverse crack) cases

Mitra and Gopalakrishnan - wavelet based spectral finite element (WSFE) to model 1-D, 2-D composite structures (2005, 2006, 2008) based on CLPT

Samaratunga, Jha and Gopalakrishnan - Shear flexible 2-D WSFE, healthy and transverse crack (2012, 2014)

No work is reported on modeling bonded joints using spectral finite element technique


Research Objectives

Development of computationally efficient numerical models

for guided wave propagation in bonded composite joints

using Wavelet Spectral Finite Element method

Study wave propagation behavior in bonded joints for

potential applications in Nondestructive evaluation


Governing equations derivation for bonded single lap joint


Displacement field: first order shear deformation theory (FSDT)

No transverse shear in adhesive layer assumed

Equations of motion (EOM) derived using Hamilton’s principle

Bonded single lap joint Overlapped regionOutside of

overlapped region

( , , ) ( , , ) ( , , )

( , , ) ( , , ) ( , , )

( , , ) ( , , )

c x

c y

c

u x y t u x y t z x y t

v x y t v x y t z x y t

w x y t w x y t

Eqn of Motion (EOM) of bonded single lap joint


Hamilton’s principle

Integration by parts and simplifications used for arriving final equations

K U V Kinetic energy, Total potential energy 2

1

0t

t

K U V dt 0

0

V

V

V

V

ˆ

hc x c x

h

y yc c c c

h

ij ij xx xx xy xy yy yy xz xz yz yz

h

u uK d z z

t t t t t t

v v w wz z dzdA

t t t t t t

U d dzdA

V

u u

0 0

x c x x c y ct dS s u h s v h p w dA

u

Equations of motion derivation


EOM for top adherand

Boundary conditions

Wavelet spectral finite element formulation is followed for solution

2 2

0 12 2

22

0 12 2

2

0 2

2 2

1 22 2

22

1 22

: 0

: 0

: 0

: 0

:

xyxx c xc x

xy yy ycc y

yx cc

xyxx c xx x x

xy yy ycy y y

NN uu s I I

x y t t

N N vv s I I

x y t t

QQ ww p I

x y t

MM uQ hs I I

x y t t

M M vQ hs I I

x y t

20

t

0

1

22

, ,

1

,

xx xx xx xxh h hx xz

yy yy yy yyy yzh h h

xy xy xy xy

h

h

N MQ

N dz M z dz dzQ

N M

I

I z dz

I z

Stress resultants and inertial terms

, ,

,nn xx x xy y ns xy x yy y n x x y y

nn xx x xy y ns xy x yy y

N N n N n N N n N n Q Q n Q n

M M n M n M M n M n

Final equations of motion (EOM)


Modeling of adhesive layer

Sx, Sy and P are interaction

forces between adhesive layer and plates

Two parameter elastic foundation approach used for interaction forces derivation

2 2 2 1 1 1

1 2

,

,

ax y s c c s

a

at c c t

a

Gs s k u h u h k

t

Ep k v v k

t

Interaction forces

Overlapped region of SLJ

Ea – adhesive Young’s modulus, Ga – shear modulus ta – adhesive layer thickness


EOM for both plates expressed in displacement terms (field variables)

Temporal and spatial components of field variables approximated using Daubechies compactly supported scaling functions

e.g.

Resulting ODEs are solved exactly in frequency domain

Relationship of nodal forces and displacements obtained ˆ ˆK ;e eF u K - Dynamic stiff.

matrix

Nodal representation of

overlapped region of lap joint

Similar procedure can be followed for

modeling greater no. of bonded

layers (e.g. double lap joint)

, , , , , ,kk

u x y t u x y u x y k k Z

Spectral element formulation of bonded plates


Model validation


Lap joint with beam adherand used for longitudinal wave propagation comparison

Abaqus® model has 5700 plane stress elements (CPS4R)

WSFE response matches very well with Abaqus FE

Computation times: WSFE ~4s, Abaqus

explicit ~82s (with 8 parallel processors)

F(t)

1 m

0.5m

0.5 m

0.01 m

X

Y

ZTip long. wave response

Long. Wave response at mark

Single lap joint (layup [0]10) with axial input load

Single lap joint: FE comparisons


Tip transverse wave response

Lap joint with beam adherand used for longitudinal wave propagation comparison

Mode conversion occurs at the overlap area boundary

Transpose response is present despite adherands are symmetric balanced

laminates

F(t)

1 m

0.5m

0.5 m

0.01 m

X

Y

Z

Single lap joint (layup [0]10) with axial input load



Flexural wave propagation features

well captured by WSFE and

compares very well with Abaqus

results

Tip transverse wave response Transverse wave response at mark

Single lap joint (layup [0]10) with transverse input load

0.25 m

0.25m

0.1 m0.01 m



Transverse excitation causes

longitudinal wave reflection due to

coupling behavior of the bonded joint

Tip longitudinal wave response

Single lap joint (layup [0]10) with transverse input load

0.25 m

0.25m

0.1 m0.01 m



Spectrum relations

Frequency characteristics derived directly from WSFE model

Fundamental axial and flexural modes propagate at any non-zero frequency

Fundamental shear mode and higher order modes have a cutoff frequency

Dispersion relations

Single lap joint: Freq. domain results


Bondline thickness varied to understand wave behavior

Fundamental modes insensitive to bondline property variations

Higher order modes reacts to the changes in bondline

May have potential applications in NDE of bonded joints

0 100 200 300 4000

2

4

6

8

10

12

Frequency (kHz)

Gro

up v

eloc

ity (k

m/s

)

baseline10% reduction20% reduction

higher order

modes

Flexural

Axial

Shear

Group velocity disp. Curves for

varied bondline thicknesses

Effect of bondline


Axial input: only an axial mode propagates

Transverse input: both axial and transverse modes present

Adhesive bonding causes coupling behavior

Axial response Transverse response Bonded double beam excited with

250 kHz toneburst

F

2 m

F

2 m

Coupled wave propagation in bonded

beams


Asymmetric loading leads to coupled wave propagation irrespective of direction

Axial response Transverse response Bonded double beam excited with

250 kHz toneburst

F

2 m

F

2 m

Coupled wave propagation in bonded

beams


Conclusions

Wavelet spectral finite element model was

developed for accurate and efficient wave

propagation analysis in bonded joints

Results were validated against conventional FEM

using a single lap joint as an example

Studied effects of bondline parameters on wave

propagation for implications in damage detection

Coupled wave propagation behavior was

observed due to the presence of adhesive layer


For additional details

Samaratunga, Dulip, Ratneshwar Jha, and S. Gopalakrishnan. "Wave

propagation analysis in adhesively bonded composite joints using the

wavelet spectral finite element method." Composite Structures 122

(2015): 271-283.

wavelet spectral finite element modeling of guided wave ... · wavelet spectral finite element...

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