Chia, Julian Yan Hon (2002) A micromechanics-based continuum damage mechanics approach to ...

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Chia, Julian Yan Hon (2002) A micromechanics-based continuum
damage mechanics approach to the mechanical behaviour of brittle
matrix composites. PhD thesis.

http://theses.gla.ac.uk/2856/

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Continuum Damage Mechanics
A Micromechanics-Based
Approach to the Mechanical Behaviour Brittle Matrix
of
Composites.
by
Chia Yan Hon, Julian
Submitted to the University Glasgow
of
for degree
the
of
Doctor Philosophy
of
in
Mechanical Engineering
© Chia Yan Hon, Julian, Sept 2002
I
Abstract
The describes development
thesis the damage
of a new continuum mechanics
(hereafter, CDM) for the deformation failure brittle
model and of matrix composites
fibres. The CDM is
large
reinforced with continuous model valid over sizes scales
compared to the spacing of the fibres and the dimensions of the damage. The composite
is damage in form
to the delamination,
allowed sustain of matrix micro-cracking, shear
tensile delamination fibre failure. The developed by
and constitutive equations are
decomposing the composite compliance into terms attributable to the fibre and matrix,
the failure by intersecting failure based
and modelling competing modes surfaces on
maximum stress theory. The fibres are treated as being weakly bonded to the matrix so
that the fibres transmit loads, fail in tension. The is
only axial and matrix modelled as
isotropic linear is treated transversely-isotropic damage has initiated.
elastic and as after
The the determined from
effect of multiple matrix cracking on stiffness was
data, failure by decay in load bearing
the
experimental while was modelled a rapid
Although the is largely to loading,
capacity. model motivated proportional matrix
damage has been by damage During
unloading and closure modelled elasticity.
is identical
the to the the
compression, matrix stiffness undamaged state with exception
that the fibres to transmit loads. The
are assumed not compressive model was
implemented FORTRAN interfaced
through the
computationally a subroutine with
ABAQUS/Standard finite
element solver.
The CDM by
model was validated comparing experimental and computational
balanced 0°-90° fibres
test
results of specimens with unidirectional and woven of a
brittle fabricated from fibres in This
matrix composite, polyester a polyester matrix.
low between fibres has
composite system exhibits elastic mismatch and matrix, and
SiC/SiC for
to
similar non-dimensionalised stress-strain response a composite proposed
diffuser Rolls-Royce EJ200 Test
the the
exhaust unit of aero-engine. specimens
fibres have been
tensioned to the
reinforced with aligned and misaligned uniaxially
damage failure To
tensile to
axis produce a range of mechanisms and processes.
demonstrate
the the to
ability of model analyse engineering structures, a range of
idealised from diffuser Rolls-Royce EJ200
the the
parts exhaust unit of aero-engine
in bending. The included
tested
were and analysed sub-structural specimens a simple
bar, bar T-shaped These
thickened
rectangular a with a cross-section, and a component.
ii
full damage
the
sub-structures showed range of mechanisms, which often occurs
in from brittle
simultaneously engineering structures made matrix composites.
The behaviour the fibre-matrix interface is to the behaviour fibre
of central of
It is that full interfacial be
reinforced composites. argued a range of properties can
by treating the interface imperfect be formalised infinite
modelled as such can as an
Interfacial developed
to the
periodic array of cracks. elements were model properties of
imperfect interface. The the to the fibres
an result support simplification used represent
load bearing in the CDM insight into the
as simple axial components model, and gave
behaviour imperfectly bonded interfaces.
of
iii
Acknowledgements
I to formally to the following
wish express my sincere gratitude persons and
for help
their to this thesis.
organisations and support complete
My deepest to Professor John W. Hancock,
gratitude goes
my academic
You have invaluable
taught
supervisor. me skills concerning research and personal
I indebted for
to the
management. am especially you encouragement, guidance, support
in
this thesis to to the Committee
and grace making come past, recommendation of
Vice-Chancellors Principles (CVCP) the Universities the United Kingdom to
and of of
the Overseas Research Student (ORS) Award, to the
receive recommendation
University Glasgow to the Mechanical Engineering Departmental
of receive
Scholarship, in department.
the
and provision of a pleasant work environment
I like Professor Mike Cowling for
to thank
would acknowledge and supporting
for the ORS Departmental Scholarship; Dr. David V. Philips for
my application and
finite Dr. Ronald D. Thomson for
techniques;
advice on element modelling advice on
Dr Bill Broughton for Dr. Neil L.
testing;
continuum mechanics; advice on composite
McCartney for Mr Ian Peden for
technical
correspondence on composite modelling;
detection Dr. Ian Watson for
the technical
work on acoustic emission system; resources
for detection Mr. Jamie Cunningham for
the
acoustic emission system; advice on
Mr Alexander Tory for
thermodynamics; the test
making mechanical rigs and
fabrication jigs for
the
experimental work on composites and sharing an office with me;
Mr Alan Birkbeck, Mr George Falconor, Mr John Davidson Mr Brian Robb for
and
Panagiotis Tsouchnikas, Christos Kastritseas,
the
shaping composite specimens;
Lubrano Andrea, Niall MacLeod for in the fabrication the
assisting and analysis of
Mr Kenny Stevenson, Ms Yassamine Mather, Walter Robinson
composite specimens;
for Ms Fiona Downie, Ms Careen Fraser, Ms Pauline Kyriacou, Ms
computing support;
Elaine McNamara, Ms Jane Livingstone for in the department;
administrative support
Ms Marion Richardson for in the faculty; Bostjan
administrative support engineering
Bezensek, Anuradha Banerjee, Kirsteen Lowe Moshiur Rahman for being
and
in fracture Dr Paul Molly, Ms Margaret
the
supportive colleagues mechanics group;
Smith, Dr Maureen Joan Perry Dr Muhammad Shahid for being
and supportive
in department; Dr Seat Han Cheng, Dr Wu Jia Jang, Dr Yeo Chiew Beng
the
colleagues
iv
Allen, Dr Chong Boon Keat Dr Mark Müller for being
and supportive colleagues and
friends faculty.
the
within and outwith engineering
I like to thank brothers in Glasgow Chinese Christian
would also my and sisters
Church (GCCC), St. Silas Church, Faith Assembly God Church (Singapore) for
and of
Thanks to GCCC for the
their encouragement and support. also go accommodation and
during in Glasgow the in Refiner's Fire to be
sustenance my stay and allowing youths a
life. Also, I like to thank Matty Chan Yiu Tung for being
special part of my would my
flatmate Deacon Paul Watson for his Scottish friendship.
and
My heartfelt to dad, David, Irene,
special and appreciation goes my and mum,
for their unfailing love to see me complete the thesis. Also to my brother Michael,
Stella, Angela Caroline, John, Grace,
sister-in-law sisters and nephew niece grandma
Cheong Foong Yin, Chia Weng Kok & family for their during
encouragement my
uncle
I like to late Chin Koon Seng late
research. would also remember my grandpa and uncle
Chia Weng Fatt for their kind influence life. Finally, I like to
on my would specially
Saviour Lord, Jesus Christ, for hope, faith love during this
thank grace, and
my and
thesis.
V
Nomenclature
Stresses, Strains Elasticity
and
C Stiffness
tensor
of a material
A.,
Unit
three
el, e2, e3 vectors of mutally perpendicular co-ordinate axis x1, x2, and
x3.
Strain
tensor
sl,
Resultant
strain vector
Normal
cn strain vector
Shear
ss strain vector
Mean
strain
sSý Hydrostratic tensor
strain
Volumetric dilatational
or strain
-C(cid:30)
Deviatoric tensor
eu strain
E Young's
modulus
FI Body force
G Shear
modulus
G Transformation for the tensor
matrix stress and strain
H Transformation for
matrix engineering strain
Engieering
y shear strain
Principal
71 73 engineering shear strain
172 '
SZ Complementary denisty the
energy or complementary energy per unit
volume
ll; Direction between the 'and
cosine co-ordinate axes x, xj
Ii 13 Invariants tensoro
of stress
I]' ', 13 ' Invariants tensors
of strain
-b
Jl, J2, J3 Invariants the deviatoric tensor
of stress s1;
J J2 'I J3' Invariants the deviatoric tensor
of strain e
in body.
Unit
n normal of a plane a continuous
between
Direction
cosines n and el, e2, e3-
ni, n;
(1)
Principal directions
nn(2), n(3)
Poisson's
v ratio
Cauchy's tensor.
stress
6f
Normal
stress vector
ßn
Shear
stress vector
6S
Principal
stress
6
& Mean
stress
6(5; Hydrostratic tensor
stress
j
Deviatoric tensor
stress
Sy
vi
Principal deviatoric
tensor
s stress
S,, Compliance tensor
of material
Jk/
93 Principal
shear strain
191 091
,
T, T; Cauchy's (or traction)
stress vector.
Displacement
Ui vector
W Strain density the
energy or strain energy per unit volume
bW Rate the density W
of change of strain energy
Damage Mechanics
Tf Kachanov (1958) damage
scaler variable
Robotnov (1968) damage
CO scaler variable
0 Murakami & Ohno (1981) damage tensor
second-order
D Cordebois & Sidoroff (1981) damage tensor
second-order
Dyk, Lemaitre & Chaboche (1978) fourth-order damage tensor
1
Void density in
wo area a plane
A Net
area
A/z Apparent
area
6 Net
stress or effective stress
Entropy density
s
S Entropy
(where
Internal
state variables a1..., n)
Internal density
energy
u
Density
p
Body heating
and radiation
r
Heat flow
q vector
T Absolute temperature
F Internal
entropy production
Helmholtz free denisty
energy
yr
Gibb's free density
energy
77
Internal
entropy production per unit mass
y
Composites Terminology
E,,, V(cid:30) EfVf
a ratio of over
failure
Emu ultimate strain of matrix
failure fibres
£ ultimate strain of
y applied shear strain
fracture to form
work a matrix crack surface
debond fibres bridging
to a unit area of matrix crack
7db energy used
vii
fibre bundle
6b stress
failure fibre bundle
a-bu ultimate stress of
o, composite stress
failure
6Cu ultimate stress of composite
fibre
6f stress
failure fibres
6fu ultimate stress of
failure fibres
brr mean ultimate stress of
fibre
0f stress when matrix cracks
fibre bundle
6b stress
failure fibre bundle
6bu ultimate stress of
Um matrix stress
initiation
6mc matrix crack stress
6mc(sat) matrix crack saturation stress
6MS matrix softening stress
failure
6(cid:30)ZU ultimate stress of matrix
in to fibre
007M radial stress matrix with respect axis
delamination
6Sd shear stress of composite
delamination
tensile stress of composite
a-td
far field
stress
a_
interfacial
shear stress
in fibre
dUf change strain energy per unit area
in
AU7z change matrix strain energy per unit area
done
dW work to create a steady state matrix crack per unit area
dl displacement fibres during
Aw or additional of matrix cracking
length
a crack
length
transient
crack
ao
E, Young's
modulus of composite
fibres
Ef Young's
modulus of
Em Young's
modulus of matrix
by Voigt
Evoigt Young's
modulus of a composite
by Ruess
ERuess Young's
modulus of a composite
G, Shear
modulus of composite
fibres
Gf Shear
modulus of
Shear
G(cid:30)Z
modulus of matrix
by Voigt
Gvoigt Shear
modulus of a composite
by Ruess
GRuess Shear
modulus of a composite
G1 fracture energy release rate of matrix per unit area
fibre-matrix interface
G11 debond energy release rate of per unit area
fibre length
L
intensity factor
K' of composite
effective stress
viii
K intensity factor
effective stress of matrix
intensity factor
KID
critical stress of composite
K; ' intensity factor
critical stress of matrix
Weibull
m modulus
density failure function
Pf probability
failure
Pf
probability of
PS
probability of survival
fibre
r0 radius
p closure pressure at matrix crack surface
distance fibres
R
centre-to centre separation of
by bridging fibres
T traction
closure
displacement
at crack surface
u
U
matrix crack surface energy per unit area
US dissipate due to frictional
energy sliding per unit area
Vf fraction fibres
volume of
fibre fraction forms
Vf
critical volume at which multiple matrix crack
crit
fraction
V.
volume of matrix
load distance
transfer
x' critical
load distance
x transfer
mean
distance
X
normalised at crack surface
Computational Model
Subscripts
stress
6 -
strain c composite
£ - -
f fibre
S
compliance
-
-
C
stiffness m matrix
-
-
E Young's
modulus me matrix cracking
-
-
G
shear modulus mc(sat) matrix crack saturation
-
-
Poisson's
ratio ms matrix softening
-
v -
delamination
T transformation td tensile
-
-
delamination
d increment
an of sd shear
-
-
failure
V fraction
volume u ultimate
-
-
A fraction
area
-
{}
Supers
cripts
vector
-3x1
increment
matrix n current
-
-3x3
increment
magnitude of n-1 previous
-
-
c- composite position system
f- fibre
position system
m- matrix principal stress position system