View
3
Download
0
Category
Preview:
Citation preview
Shant@aero.orgSpace 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
dulip@aero.org
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
2dulip@aero.orgSpace Materials Laboratory
Outline
Motivation
Objectives
Governing equations derivation for
bonded joints
Spectral finite element formulation
Model validation with conventional finite
elements
Conclusions
3dulip@aero.orgSpace Materials Laboratory
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
4dulip@aero.orgSpace Materials Laboratory
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
5dulip@aero.orgSpace Materials Laboratory
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
6dulip@aero.orgSpace Materials Laboratory
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
7dulip@aero.orgSpace Materials Laboratory
Governing equations derivation for bonded single lap joint
8dulip@aero.orgSpace Materials Laboratory
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
9dulip@aero.orgSpace Materials Laboratory
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
10dulip@aero.orgSpace Materials Laboratory
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)
11dulip@aero.orgSpace Materials Laboratory
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
12dulip@aero.orgSpace Materials Laboratory
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
13dulip@aero.orgSpace Materials Laboratory
Model validation
14dulip@aero.orgSpace Materials Laboratory
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
15dulip@aero.orgSpace Materials Laboratory
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
Single lap joint: FE comparisons
16dulip@aero.orgSpace Materials Laboratory
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
Single lap joint: FE comparisons
17dulip@aero.orgSpace Materials Laboratory
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
Single lap joint: FE comparisons
18dulip@aero.orgSpace Materials Laboratory
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
19dulip@aero.orgSpace Materials Laboratory
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
20dulip@aero.orgSpace Materials Laboratory
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
21dulip@aero.orgSpace Materials Laboratory
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
22dulip@aero.orgSpace Materials Laboratory
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
23dulip@aero.orgSpace Materials Laboratory
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.
Recommended