ABAQUS IMPLEMENTATION OF CREEP FAILURE
IN POLYMER MATRIX COMPOSITES WITH TRANSVERSE ISOTROPY
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Fengxia Ouyang
December, 2005
ii
ABAQUS IMPLEMENTATION OF CREEP FAILURE
IN POLYMER MATRIX COMPOSITES WITH TRANSVERSE ISOTROPY
Fengxia Ouyang
Thesis
Approved: Accepted ________________________________ _______________________________ Advisor Dean of the College Wieslaw Binienda George K. Haritos ________________________________ ______________________________ Faculty Reader Dean of the Graduate School Pizhong Qiao George R. Newkome ________________________________ ______________________________ Department of Chair Date Wieslaw Binienda
iii
ABSTRACT
Polymer Matrix Composites (PMC) are increasingly favored in structural
applications for their light weight and durability. However, the numerical modeling of
these materials poses several challenges. This is primarily due to the highly anisotropic
nature of the creep exhibited by these materials above the glass transition temperature.
Also, the damage and failure of the material is of particular interest to designers using
the PMCs. Recently, a sustained effort has been to provide the designer with large
computer codes containing comprehensive constitutive equations, often equipped with
large number of internal variables and with the most general mathematical forms, for
use in structural design analysis. Although elegant and useful, such constitutive laws are
often expensive in implementation. Specially for early stages of the design, a quicker
way of estimating complicated PMC behavior is needed. In this work, the constitutive
material law by Robinson and Binienda (2001) [1,2] is utilized for such an approach.
The model is successful in describing polymer matrix composite (PMC) materials
having long or continuous reinforcement fibers embedded in a polymer matrix.
Although the material law includes a single scalar parameter to describe the damage, it
retains the essential material behavior.
The material law is implemented computationally as a user defined subroutine
(UMAT) in a commercially available FEA code (ABAQUS). The material parameters
iv
are obtained from experiments of thin-walled tubular specimens reinforced with
unidirectional, helical fibers at an angle θ = 90o , 60o and 45o under tensile and shear
loading . The model correctly predicts the relation between logarithmic creep rate and
logarithmic stress. The user subroutine has robust convergence properties. The creep
strain rate and the effect of damage on the creep strain rate are presented for the
benchmark problem of a square plate with a circular hole at the center and pressure
vessel. The effect of fiber orientation on the durability of the square plate and pressure
vessel under damaging loads, is studied.
.
v
TABLE OF CONTENTS
Page
LIST OF TABLES…………………………………………………………….…...…..vii
LIST OF FIGURES........................................................................................................viii
CHAPTER
I. INTRODUCTION…………………………………………………….……...…1
II. THE ANISOTROPIC VISCOELASTIC MODEL………………………..…....3
III. THE ISOTROPIC POWER LAW MODEL…………………………….……....9
IV. USER SUBROUTIN UMAT………………………………..……………...….11
4.1 Introduction of UMAT …………………………………………..…11
4.2 Implementation of UMAT………………………………….……....13
4.2.1 Jacobian matrix for plane stress element and shell element……………………….……..17
4.2.2 Newton Ralphson Method………………………...…...20
V. PLANE STRESS ELEMENT IMPLEMENTATION ……………………..…23
5.1 Single element test……………………………….…..………….…23
5.2 Square plate ……………………………………..…………….......34
5.3 Plate with a hole……………………………………………………37
vi
VI. SHELL ELEMENT IMPLEMENTATION……………………………….…..49
6.1 Theory of shell element……..……………………………………...49
6.2 Shell element in ABAQUS……………………..………………….50
6.3 Implementation of shell element………..……………..…………..51
VII. CONCLUSION……………………………………………………...….……..62
REFERENCES………………………………………………………….………….….64
APPENDICES…………………………………………………………………….…...66
APPENDIX A UMAT………………………………………….………......67
APPENDIX B INPUT FILE FOR PRESSURE VESSEL………….…..….77
vii
LIST OF TABLES Table Page 1 Fiber orientation 90 deg under tension ………………………………….………..28 2 Fiber orientation 45 deg under tension.……………….…...…………….………..28
viii
LIST OF FIGURES Figure Page 2.1 Illustration of the isochronous damage function in the normal stress (y axis). P=1 is the linear form……………………………………….……5 2.2 Thin wall tube under tension and torsion loading. (a) thin wall tube
with fiber orientation. (b) typical experimental creep data under axial and shear loading……………………………………………………………..7 2.3 Non-dimensional log creep rate vs. log stress for the complete exploratory data set. Tensile and shear data are shown as indicated. Fiber angles o60=θ and o45 and shear data for o90=θ shifted to form a master curve…………………………….……………………………...…..8 . 4.1 UMAT Loop ………………………………………………………………….….16 4.2 Plane Stress element…………………………………………………………...…17 4.3 Plane stress transformation from local coordinate (x-y) to global coordinate (1-2)………………………………………………………….……......18 5.1 One element model problem set up with fiber orientation
(a) creep loading and (b) constant displacement loading…...……………………24 5.2 Comparison between power law and Robinson creep model for isotropic case under a constant load of 45 MPa when the scalar damage variable is not included in the Robinson creep model. (a) Time evolution of creep strain in Y direction. (b) Time evolution of creep strain in Y direction. (c) Time evolution of creep strain in XY direction……………………….….…..29 5.3 Time evolution of creep strain in the direction of loading (without damage) for orientations of the fiber 90 deg at 45MPa load…………….…….....30
ix
5.4 Time evolution of creep strain in the direction of loading (without damage). (a) for different orientations of the fiber (0, 45, 90 deg) at 45MPa load. (b) for 45 deg fiber orientation at varying loading 20, 45 and 100PMa…………………………………………………….….….…..31 5.5 Time evolution of creep strain with damage effect.(a) comparison when damage evolution is included under tensile load 60 MPa. (b) comparison between tensile loads 70, 75, 80 MPa with fiber orientation 45 deg. (c) comparison between fiber orientation 0, 45, 90 deg under shear loads 40 Mpa………………………………………….32 5.6 Time evolution of stress relaxation in the direction of
Loading (with damage)………………………………………………………..….33 5.7 Geometry and boundary condition of square plate problem…………………..….35 5.8 Time evolution of creep strain with different orientations of the fiber (0, 45, 90 deg) at 46MPa load in the direction of loading (without damage). (a) time evolution of strain at 1 direction (b) time evolution of shear strain ……………………………………………….36 5.9 Left, geometry and right, 2D quarter symmetry model for the square plate with a hole problem…………………………………………….…...37 5.10 Contour plot of (a) stress distribution of the power law for isotropic material after elastic step, (b) stress distribution of the Robinsons’ model law for isotropic material after elastic step, (c) stress distribution of the power law for isotropic material after creep step (5 hours), (d) stress distribution of the Robinsons’ model law for isotropic material after creep step (5 hours)………………………..…...…..40 5.11 Stress distribution along the hole (quarter circle) going in a counter clockwise direction for the proposed Creep Damage Model as and for isotropic Power Law Creep model. (a) stress distribution after elastic step (b) stress distribution after 5 hours…….............…………………..…...….…….41 5.12 Contour plot of Creep strain distribution after 5 hours creep response. (a) Robinson Damage Model and (b) Isotropic Power Law……..………....…..42 5.13 Creep strain along the hole (quarter circle) going in a counter clockwise direction for the Robinson Damage Model and for isotropic Power Law Creep model (after 5 hours)……….………………….…..43
x
5.14 Comparison between Analytic solution and Robinson model (FEA) in elastic step. (a) R=20mm and R/L=13%. (b) R=50mm and R/L=33%…........44 5.15 Creep / time response along the hole (quarter circle) going in a counter clockwise with damage effect under axial stress 10MPa with R=20mm, 50mm, 80mm, L=152.4mm, fiber orientation 45 deg……….....45 5.16 Creep / time response along the hole (quarter circle) going in a counter clockwise with damage effect under axial stress 10MPa with R=50mm, L=152.4mm, fiber orientation 45 deg and 90 deg………….…..46 5.17 Stress concentration zone with damage effect under axial stress 10Mpa fiber orientation 45 deg. (a) stress concentration zone with R/L=0.1. (b) stress concentration zone with R/L=0.3. (c) stress concentration zone with R/L=0.5…………………………..……...….47 5.18 Stress compression zone with damage effect under axial stress 10Mpa fiber orientation 45 deg. (a) stress compression zone with R/L=0.1. (b) stress compression zone with R/L=0.3. (c) stress compression zone with R/L=0.5……………………………...…….....48 6.1 Thin wall tube under tension and torsion with fiber orientation,…………....……53 6.2 Time evolution of creep strain under a constant tensile load 45 MPa of thin-walled tube for different fiber orientations (0, 45, 90 deg), (a) time evolution of maximum principle strain. (b) time evolution of shear strain……………………………………………........54 6.3 Pressure vessel, geometry and mesh and path……………………………….…...57 6.4 Path plot of time evolution of Maximum Strain along path1 of the vessel under inside pressure 0.5Mpa with damage evolution (a) fiber orientation 45 deg. (b) fiber orientation 60 deg. (c) fiber orientation 90 deg ………………………………………………….…...58 6.5 Maximum Strain along path1 of the vessel with fiber orientation 0, 45, 90 deg under inside pressure 0.5Mpa with damage evolution after 10 hours…………………………………………………………………......59 6.6 Time evolution of Maximum Strain along path2 of the vessel with fiber orientation 0, 45, 90 deg under inside pressure 0.5Mpa with damage evolution………………………………….…..…..60
xi
6.7 Time evolution of Maximum Principle Strain of the pressure vessel with fiber orientation 60, 90 deg under pressure 1Mpa with damage evolution. (a) fiber orientation 60 deg, (b) fiber orientation 90 deg……………....61 -
1
CHAPTER I
INTRODUCTION
The present work details the development of a computational material model for
polymer matrix composites. These materials are increasingly found in structural
applications for their light weight and durability. However, the numerical modeling of
these materials poses several challenges. This is primarily due to the highly anisotropic
nature of the creep exhibited by these materials. Also, the damage and failure of the
material is of particular interest to designers using the polymer matrix composites. Over
recent decades, a sustained effort has been to provide the designer with large computer
codes containing comprehensive constitutive equations, sometimes embodying several
state variables and the most general mathematical forms, for use in structural analysis in
support of design. Although this approach may be appropriate in the final stages of
design and where complex histories of stress and temperature are involved, it can be too
complicated for many design applications, particularly in the early stages of design, and
is seldom the path taken by the designer.
The constitutive material law based on a Norton /Bailey type of creep law by
Robinson and Binienda (2001) [1,2] is utilized for this computational model. The
model is successful in describing polymer matrix composite (PMC) materials having
long or continuous reinforcement fibers embedded in a polymer matrix. The objective
2
in choosing a numerical model based on this type of material law is to allow simplicity
and utility (on behalf of the designer), while still retaining the essence of the actual
material behavior.
When strongly reinforced PMC materials and structures operate in the creep range
of their polymer matrix (at or near Tg) they undergo time-dependent deformation and
eventually fail. Substantial resistance to creep deformation and damage is achieved in
materials where long or continuous fibers are embedded in the polymer matrix. By
design, much of the load in such materials is carried by the strong fibers that creep very
little, if at all. Evidently, design engineers need quantitative tools for predicting the
response of PMC materials and structures as a basis of achieving optimal structural
designs, e.g., optimal fiber configurations. Although it is often observed that highly
anisotropic creeping materials exhibit inelastic compressibility, cf., Robinson and
Binienda (2001) [1], creep deformation models do not commonly include dependence
on hydrostatic stress; this dependence is a principal feature of the material law that is
implemented in this work. The model incorporates a dissipation potential function that
is taken to depend on appropriate invariants of stress and material orientation consistent
with transverse isotropy. Failure and damage are introduced in the model with a
Monkman / Grant type of relationship [2,3,4,5].
The particular viscoelastic model used here is as described by following chapter.
3
CHAPTER II
THE ANISTROTROPIC VISCOELASTICITY MODEL
The transversely isotropic viscoelasticity model of concern derives from the more
general anisotropic deformation /damage model proposed in Robinson et al.(1992). The
present model is an extension of that earlier work in the sense that hydrostatic stress is
taken into account in the deformation response. The extension follows Robinson et al.
(1994) and Robinson and Binienda (2001)b. The viscoelasticity model has the form
ijeijije ε+ε= &&& (2.1)
n
o
ij1n
o
ij 123
ψσ
ΓΦ=
ε
ε −
&
& (2.2)
&( )
ψψ
ν= −+11
1m t o
m∆ (2.3)
where ε ije denotes the components of elastic strain, ije& and ijε& denote components of the
total and creep (viscous) deformation rates, respectively. σ ij are the components of
Cauchy stress; σo is a reference stress; oo t,m,n,ε& and ν are material parameters; Φ is a
dissipation potential function; ∆ is an isochronous damage function and the scalar ψ is
the Kachanov continuity. ψ = 1 corresponds to an undamaged material element; ψ = 0
indicates its total loss of load carrying capacity.
4
For transverse isotropy, the functions Φ and ∆ are taken to depend on the
following invariants of stress σ ij , deviatoric stress sij and a material orientation tensor
Dij .
J s sij ji212
= J D so ij ji= I ii= σ I Do ij ji= σ J D s sij jk ki= (2.4)
The dissipation potential function is
Φ = − − − − + −1 3 1
942
2 2 2
σξ ζ η ζ η
oo oJ J J J I[ ( ) ( ) ( ) ] (2.5)
and
( ) ijijijo
ijokjikkjikijij
ij
IDJ
DJDssDs
∂−+−−
−−+−=∂Φ∂=Γ
ηςδηζ
ξσ
492)
31()(2
)2( (2.6)
Φ is a positive, homogeneous function of degree unity in stress assuring that the
representation (1)-(6) is dissipative. The anisotropy parameters ξ η, and ζ are subject to
various inequalities based on physical limitations that are specified in Robinson and
Binienda (2001) [2]. Also the isochronous damage function is specified as:
∆ ∆ ∆= =( , ) ( , )σ ij ijD N S (2.7)
p/1
p
o
p
o
SN)S,N(
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛σα
+⎟⎟⎠
⎞⎜⎜⎝
⎛
σ=∆ (2.8)
in which, the invariant N specifies the maximum tensile stress normal to the local fiber
direction, and the invariant S denotes the local maximum longitudinal shear stress as:
N I I J J Jo o= − + + −12
142
2( ) (2.9)
2oJJS −= (2.10)
5
The angular brackets in (9) are the MacCauley brackets
Fig 2.1 illustrates the isochronous damage function in N- S space for a general
power law form as well as the special case when the function is linear in N and S (P=1).
Fig 2.1 Illustration of the isochronous damage function in the normal stress (y axis). P=1 is the linear form.
6
The material parameters are obtained from experiments of thin-walled tubular
specimens reinforced with unidirectional, helical fibers at an angle θ = 90o , 60o and 45o
under tensile and shear loading (Fig 2.2). The complete exploratory data set is plotted
in Fig 2.3. The correlation of the theoretical model and experimental data (solid lines)
and of creep response under two of the untested natural stress states (TS)-(dotted line)
and (LN)-(dashed line) validates the physics behind the proposed constitutive equation.
As a definitive measure of the strength of anisotropy, the creep rate under tensile stress
along the fiber direction (LN) is predicted as being less than one thirtieth (1/30) of that
under transverse stress (TN).
7
1
3
θ
2
FF
T
T τ
τ
σσ
time (hr)
0 1 2 3 4 5 6 7
axia
l and
she
ar c
reep
stra
in (%
)
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
axial strain-shear strain
(a)
(b) Fig 2.2 Thin wall tube under tension and torsion loading. (a) thin wall tube with fiber orientation. (b) typical experimental creep data under axial and shear loading.
8
Fig 2.3 Non-dimensional log creep rate vs. log stress for the complete exploratory data set. Tensile and shear data are shown as indicated. Fiber angles o60=θ and o45 and shear data for o90=θ shifted to form a master curve.
log(σ/σ0) or log( τ/σ0)
-0.4 -0.2 0.0 0.2 0.4
log(
ε. /ε. ΤΝ
) or
log(
γ. / V_ 3
ε. TN )
-3
-2
-1
0
1
90-ten60-ten45-ten90-shear
Correlations
(LN) - prediction(TS) - prediction
3
9
CHAPTER III
THE ISOTROPIC POWER LAW MODEL
The power-law model can be used to do the creep calculation for isotropic material.
The constitutive equation of isotropic power law is as following:
mcr tAσε =& (3.1)
where
crε& is the uniaxial equivalent creep strain rate,
σ is the uniaxial equivalent deviatoric stress,
t is the total time, and
A, n, and m are defined by the user as functions of temperature. σ is Mises equivalent
stress or Hill's anisotropic equivalent deviatoric stress according to whether isotropic or
anisotropic creep behavior is defined (discussed below). For physically reasonable
behavior and n must be positive and . Since total time is used in the
expression, such reasonable behavior also typically requires that small step times
compared to the creep time be used for any steps for which creep is not active in an
analysis; this is necessary to avoid changes in hardening behavior in subsequent steps.
Based on Robinson creep model we can derive A, n, m. The Robinson model to
calculate creep rate in loading direction is:
10
21
0
0 )1()(+
−=n
n ξσσεε && (3.2)
Because it is for isotropy we define
0=== ηζξ
Now we obtained
05.6151011.3 tσε −×=&
151011.3 −×=A , 5.6=n , 0=m
Depending on the choice of units for either form of the power law, the value of A
may be very small for typical creep strain rates. If A is less than 1710− , numerical
difficulties can cause errors in the material calculations; therefore, use another system
of units to avoid such difficulties in the calculation of creep strain increments.
11
CHAPTER IV
USER SUBROUTINE UMAT
The material law is implemented computationally as a user defined subroutine
(UMAT) in a commercially available FEA code (ABAQUS). User subroutines provide
an extremely powerful and flexible tool for analysis. This chapter defines the interfaces
for the user subroutines that are available in ABAQUS.
4.1 Introduction of User subroutine (UMAT)
User subroutine UMAT can be used to define the mechanical constitutive behavior
of a material; it must update the stresses and solution-dependent state variables to their
values at the end of the increment for which it is called it must provide the material
Jacobian matrix, for the mechanical constitutive model; it can be used in conjunction
with user subroutine USDFLD to redefine any field variables before they are passed in.
It is sometimes desirable to set up the FORTRAN environment and manage
interactions with external data files that are used in conjunction with user subroutines.
For example, there may be history-dependent quantities to be computed externally, once
per increment, for use during the analysis; or output quantities that are accumulated
over multiple elements in COMMON block variables within user subroutines may need
12
to be written to external files at the end of a converged increment for postprocessing.
Such operations can be performed with user subroutine UEXTERNALDB. This user
interface can potentially be used to exchange data with another code, allowing for
“stagger” between ABAQUS and another code.
User subroutines should be written with great care. To ensure their successful
implementation, the rules and guidelines below should be followed.
Every user subroutine must include the statement. As the first statement after the
argument list. The file ABA_PARAM.INC is installed on the system by the ABAQUS
installation procedure. It specifies either IMPLICIT REAL*8 (A-H, O-Z) for double
precision machines or IMPLICIT REAL (A-H,O-Z) for single precision machines. The
ABAQUS execution procedure, which compiles and links the user subroutine with the
rest of ABAQUS, will include the ABA_PARAM.INC file automatically. It is not
necessary to find this file and copy it to any particular directory; ABAQUS will know
where to find it.
1) User subroutines must perform their intended function without overwriting other
parts of ABAQUS. In particular, the user should redefine only those variables
identified in this chapter as “variables to be defined.” Redefining “variables
passed in for information” will have unpredictable effects.
2) When developing user subroutines, test them thoroughly on smaller examples in
which the user subroutine is the only complicated aspect of the model before
attempting to use them in production analysis work. If needed, debug output can
be written to FORTRAN unit 7 to appear in the message (.msg) file or to
FORTRAN unit 6 to appear in the data (.dat) file; these units should not be
13
opened by the user's routines since they are already opened by ABAQUS.
FORTRAN units 15 through 18 or units greater than 100 can be used to read or
write other user-specified information. The use of other FORTRAN units may
interfere with ABAQUS file operations. These FORTRAN units must be opened
by the user; and because of the use of scratch directories, the full pathname for
the file must be used in the OPEN statement.
3) Solution-dependent state variables are values that can be defined to evolve with
the solution of an analysis.
4.2 Implementation of UMAT
The equation of motion together with the constitutive law form system consisting
of an initial – boundary problem and an ordinary differential equation. The equation of
motion is solved with the help of a finite-element package (ABAQUS), and the
constitutive law by a solver for ordinary differential equations. The relevant constitutive
information is passed to ABAQUS by a subroutine UMAT which has to be supplied by
the user. Starting from an equilibrium at time ,nt ABAQUS performs an (incremental)
loading as well as with the time increment ,t∆ and an initial guess ,nε∆ for the strain
increment. The user subroutine UMAT has to supply ABAQUS with new Cauchy stress
tensor ),( ttn ∆+σ updated according to the constitutive law as well as with the
derivative of stress with respect to the strain increment. With this information, a new
guess for the strain increment is calculated and the whole procedure is iterated until
convergence. The precise information on the Jacobian is essential to achieve fast
14
convergence in Newton-type iteration performed by ABAQUS. The following is the
detail introduction of definition of Jacobian Matrix in UMAT and the Newton Ralphson
method incorporated in ABAQUS.
The following diagram (Fig 4.3) indicates a certain step of UMAT working with
ABAQUS.
STEP 1: At an equilibrium time ,nt ABAQUS will supply time increment ,t∆ and
total strain increment )( ntotal tε∆ and total strain )( n
total tε for the strain increment. And
also
)( ntσ is calculated by previous increment. All these four values will pass to UMAT to
calculate new Cauchy stress tensor )( ttn ∆+σ .
STEP 2: Stress Update.
According to Robinson creep damage Model, creep strain increment
tn ∆Γ
Φ=∆ −
0
1
23
σε
Also, the relationship between total strain, creep strain and elastic strain is
)()()( nne
ntotal ttt εεε ∆+∆=∆
so
)()()( nntotal
ne ttt εεε ∆−∆=∆
In this way we can update Cauchy Stress tensor. The stress increment is
)()( ne
n tJt εσ ∆⋅=∆
The new Cauchy Stress tensor is
)()()( nnn tttt σσσ ∆+=∆+
15
STEP 3: Strain Update
ABAQUS will update strain tensor
)()()( ntotal
ntotal
ntotal tttt εεε ∆+=∆+
STEP 4: ABAQUS will generate new total strain increment )( 1+∆ ntotal tε
STEP 5: ABAQUS equilibrium iterations at new time 1+nt , the maximum iteration
number is set to 9 in ABAQUS and error tolerance is set to TOL 5e-3. If less than 9
times iterations the error is less than TOL, it calls convergence, n=n+1 and move to
another increment. If the error is larger than TOL, ABAQUS will reduce the t∆ go to
step 1 until convegnece.
Fig 4.1 briefly illustrates one increment in UMAT calculation.
16
ABAQUS supply information at timeTime incrementTotal strain increment Total strain
t∆nt
)( ntotal tε∆
)( ntε
t∆ )( ntotal tε∆ nσ
UMAT
Dr. Robinson’s creep law to
ijnij ∆
ΓΦ=∆ −
σε 1
23
)()()( nne
ntotal ttt εεε ∆+∆=∆
)()()( nijntotalijn
eij ttt εεε ∆−∆=∆
)()( ne
n tJt εσ ∆⋅=∆
)()()(1 nnnn tttt σσσσ ∆+=∆+=+
totalnε
step n
Stress update
Fig 4.1 UMAT Loop
step3
ABAQUS update
step4
ABAQUS generate new)( 1+∆ n
total tε
step5
ABAQUS equilibriumiterations at time
)()()(1 ntotal
ntotal
ntotaltotal
n tttt εεεε ∆+=∆+=+
1+nt
convergence Not convergence
n=n+1, go to step 1 Reduce t∆
Strain update
step2
step1
17
4.2.1 Jacobian Matrix for plane stress element and shell element
A class of common engineering problems involving stresses in a thin plate or on
the free surface of a structural element, such as the surfaces of thin-walled pressure
vessels under external or internal pressure, the free surfaces of shafts in torsion and
beams under transverse load has one principal stress that is much smaller than the other
two. By assuming that this small principal stress is zero, the three-dimensional stress
state can be reduced to two dimensions. Since the remaining two principal stresses lie in
a plane, these simplified 2D problems are called plane stress problems.
Fig 4.2 Plane Stress element
Assume that the negligible principal stress is oriented in the z-direction. To reduce
the 3D stress matrix to the 2D plane stress matrix, remove all components with z
subscripts to get,
18
⎥⎦
⎤⎢⎣
⎡
yyx
xyx
σττσ
where yxxy ττ = for static equilibrium. The sign convention for positive stress
components in plane stress is illustrated in the above Fig 4.1 on the 2D element.
Jacobian matrix of the constitutive model for plane stress and shell element,
εσ ∆∂∆∂ / , where σ∆ are the stress increments and ε∆ are the strain increments can
be defined in following steps.
STEP 1 .Definition of Transformation Matrix [T]
Fig 4.3 Plane Stress Transformation from local coordinate (x-y) to global coordinate (1-2)
19
According to the Fig 4.3 we can calculate the resultant force in 1-2 coordinate:
0cossincoscossincoscossin11 =−−−−=∑ θθτθθσθθτθθσσ AAAAAF sxsy (4.1)
0coscoscossincossinsincos122 =−++−=∑ θθτθθσθθτθθστ AAAAAF sxsy (4.2)
We can obtain stress components in 1-2 coordinate:
syx mnnm τσσσ 222
1 ++= (4.3)
syx nmmnmn τσσττ )( 22
126 −++−== (4.4)
where
θcos=m θsin=n (4.5)
We conclude that:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
s
y
x
nmmnmnmnmn
mnnm
τσσ
τσσ
)(2
2
22
22
22
6
2
1 [ ] [ ][ ] yxT ,2,1 σσ =→ (4.6)
[ ]
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
s
y
x
T
γ
εε
γ
εε
21
21
6
2
1 [ ] [ ][ ] yxT ,2,1 εε =→ (4.7)
STEP 2. Transformation of the reduced stiffness matrix
According to following procedure we can calculate the transformed reduced stiffness
matrix
20
In this way we can obtain each component in Transformed Stiffness Matrix
6622
1222
224
114 42 QnmQnmQnQmQxx +++= (4.8)
Eq. (4.8) is the formula to calculate the components in Jacobian Matrix for plane stress
element with different orientation.
4.2.2 Newton-Ralphson Method in ABAQUS
Newton's method, also called the Newton-Ralphson method, is a root-finding
algorithm that uses the first few terms of the Taylor series of a function f(x) in the
vicinity of a suspected root. Newton's method is sometimes also known as Newton's
iteration, although in this work the latter term is reserved to the application of Newton's
method for computing square roots. For f(x) a polynomial, Newton's method is
essentially the same as Horner's method. The Taylor series of f(x) about the point
ε+= 0xx is given by
⇒
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
6
2
1
66
2212
1211
6
2
1
21200
00
γ
εε
τσσ
QQQQQ
[ ] [ ] ⇒
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ss
TQ
QQQQ
T
γ
εε
τσσ
21200
00
2
1
66
2212
1211
2
1
[ ] [ ] ⇒
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
s
y
x
s
y
x
TQ
QQQQ
T
γ
εε
τσσ
21200
00
66
2212
12111
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
s
y
x
sssysx
ysyyyx
xsxyxx
s
y
x
QQQQQQQQQ
γ
εε
τσσ
212
22
21
...)("21)(')()( 2
0000 +++=+ εεε xfxfxfxf (4.9)
Keeping terms only to first order,
εε )(')()( 000 xfxfxf +=+ (4.10)
This expression can be used to estimate the amount of offset ε needed to land closer to
the root starting from an initial guess 0x . Setting 0)( 0 =+ εxf and solving (4.10) for
0εε = gives
)(')(
0
00 xf
xf−=ε (4.11)
which is the first-order adjustment to the root's position. By letting 001 ε+= xx ,
calculating a new 1ε , and so on, the process can be repeated until it converges to a root
using
)(')(
0n
n
xfxf
−=ε (4.12)
Unfortunately, this procedure can be unstable near a horizontal asymptote or a local
extremum. However, with a good initial choice of the root's position, the algorithm can
by applied iteratively to obtain
)(')(
1n
nnn xf
xfxx −=+ (4.13)
For n=1, 2, 3, .... An initial point that provides safe convergence of Newton's method
is called an approximate zero.
The error 1+nε after the (n+1) st iteration is given by
1+nε = )( 1 nnn xx −+ +ε (4.14)
= )(')(
n
nn xf
xf−ε (4.15)
22
But
)( 0xf = ...)("21)(')( 2
111 +++ −−− nnnnn xfxfxf εε (4.16)
= ...)("21)(' 2
11 ++ −− nnnn xfxf εε (4.17)
)(' nxf = ...)(")(' 1 ++− nn xfxf ε (4.18)
so
)(')(
n
n
xfxf =
...)(")('
...)("21)('
11
211
++
++
−−
−−
nnn
nnnn
xfxf
xfxf
ε
εε (4.19)
≈ )('
.)("21)('
1
211
−
−− +
n
nnnn
xf
xfxf εε (4.20)
= 2
1
1
)('2)("
nn
nn xf
xf εε−
−+ (4.21)
and (4.14) becomes
1+nε = ])('2)("[ 2
1
1n
n
nnn xf
xf εεε−
−+− (4.22)
ABAQUS sets the error tolerance TOL 5e-3 and maximum iteration times 9.
If after less than 9 times iteration
TOLn <+1ε (4.23)
we can call convergence.
23
CHAPTER V
PLANE STRESS ELEMENT IMPLEMENTATION
In the previous chapter we introduce the theory of user subroutine UMAT and how
to implement user material law through UMAT. When developing user subroutines, test
them thoroughly on smaller examples in which the user subroutine is the only
complicated aspect of the model before attempting to use them in production analysis
work. First we do single element test.
5.1 Single element test
We first investigate the one-element tension test, given in Fig 5.1. The dimension
of this element is 10mm x10mm.
24
σ
Y
θ
Y
θ
u
(a) (b) Fig 5.1 One element model problem set up with fiber orientation. (a) creep loading and (b) constant displacement loading.
Fig 5.1 shows the schematic with boundary conditions for a single element model.
This particular test was done for the reduced integration plane stress elements with
linear and quadratic interpolation schemes ( ABAQUS element type CPS4R and
CPS8R). The elements were tested for response under constant load (creep) as well as
constant displacement (relaxation) boundary conditions. The material parameters used
25
for this purpose are the ones obtained in the thin cylinder experiment described above
and are as follows:
35.0=α , 6.10=ν , 125.7=m ,
5.6=n , psi74.66710 =σ , 57.0=ξ , 1.0=η , 64.0=ζ , perhour%01.00 =ε& ,
hourst 0.120 =
This particular set of parameters assumes that the isochronous damage function
described in equation (3.8) is of the order 1 (P =1).
Fig 5.2 shows comparison between power law and Robinson creep model for
isotropic case under a constant load of 45 MPa when the scalar damage variable is not
included in the Robinson creep model. The Fig 5.2.(a) indicates that at creep strain
evolution after 10 hours at 1 direction calculated by power law model and Robinson
creep model are agree with each other, the same to 2 direction (Fig. 5.2.(b)). But the
creep strains evolution after 10 hours for 1-2 direction are different for each material
model (Fig.5.2.(c)) Because Robinson Model is anisotropic constitutive equation while
Power Law is isotropic constitutive equation.
The Power Law equation to calculate creep strain rate:
21
00122211 )1()(
+
−===n
n ξσσεγεε &&&&
The Robinson Model to calculate creep strain rate:
21
0011 )1()(
+
−=n
n ξσσεε &&
21
0022 )1()(
+
−=n
n ησσεε &&
26
21
0012 )1()3(3/
+
−=n
n ξσ
τεγ &&
According to isotropy we define
0=== ηζξ
In this case the Power Law equation reduced to
n)(0
02211 σσεγεε &&&& ===
Robinson Model reduced to
n)(0
02211 σσεεε &&& ==
n)3(3/0
012 στεγ && =
Fig 5.3 shows the response of a single element creep loading test under a constant
load of 46 MPa with fiber orientation 90 deg, the scalar damage variable is not included.
After compare FEA result with experimental date we observed that both experiment and
FEA calculation is with creep rate relatively constant at 0.013%/hr. We also compare
UMAT results with experiment data under different tensile stress with fiber orientation
45 and 90 deg. The UMAT results well coincide with experiment data (Table 1 and
Table 2).
Fig 5.4 shows the response of a single element creep loading test. The Fig 5.4.a
shows the time evolution of creep strain under a constant load of 45 MPa for different
fiber orientations (0 deg, 45 deg, 90 deg) when the scalar damage variable is not
included in the material model. It is observed that when the creep loading is along fiber
directions, the PMC has the strongest behavior. The creep strains are progressively
27
higher when the creep loading is at progressively higher angle to the fiber orientation.
For the 90 deg orientation, failure can be achieved in roughly 10 hours. Similarly, for a
given orientation, the creep strains increase progressively, as the creep load is increased
(Fig. 5.4.(b)).
Fig 5.5.(a) shows the time evolution of creep with fiber orientations 45 deg under
loads equal to 60MPa when the scalar damage variable is included. We find that when
include this damage factor the plot is getting curved and the value of the creep strain is
higher than that without this damage factor. Fig 5.5.(b) shows the time evolution of
creep with fiber orientations 45 deg under different loads (when the scalar da70Mpa,
75Mpa, 80Mpa) with damage variable is included. For creep load less than 70 MPa, the
evolution of creep strain with time is linear. But above a critical load, the increased
damage results in a larger creep strain. We also plot the time evolution of shear strain
with different fiber orientation 0 deg, 45 deg and 90 deg under shear loading 40 MPa
(Fig 5.5.(c)). It can be observed that 0 deg and 90 deg have the same strain evolution
behavior and 45 deg has more strain evolution than 0 deg and 90 deg.
28
Table 1 fiber orientation 90 deg under tension
0/σσ Experiment data TNεε && / UMAT Results TNεε && /
0.94 0.64 0.67 1 1 1 1.04 1.46 1.34
Table 2 fiber orientation 45 deg under tension
0/σσ Experiment data TNεε && /
UMAT Results
TNεε && / Experiment data TNεγ && /
UMAT Results
TNεγ && / 0.54 0.005 0.0044 0.6 0.0092 0.0094 -0.0034 -0.0021 0.65 0.012 0.014
29
(a) (b) (c) Fig 5.2 Comparison between power law and Robinson creep model for isotropic case under a constant load of 45 MPa when the scalar damage variable is not included in the Robinson creep model. (a) Time evolution of creep strain in Y direction. (b) Time evolution of creep strain in Y direction. (c) Time evolution of creep strain in XY direction
Time(hr)
0 2 4 6 8 10 12
ε 11
0.0000
.0005
.0010
.0015
.0020
.0025
Robinson model power law
Time(hr)
0 2 4 6 8 10 12
ε 11
-.0012
-.0010
-.0008
-.0006
-.0004
-.0002
0.0000 Robinson modelpower law
Time(hr)
0 2 4 6 8 10 12
ε 11
0
1e-17
2e-17
3e-17
4e-17
Robinson modelpower law
30
Fig 5.3 Time evolution of creep strain in the direction of loading (without damage) for fiber orientation of 90 deg at 45MPa load.
time (hr)
0 2 4 6 8
axia
l stra
in (%
)
0.00
0.02
0.04
0.06
0.08
0.10
UMATExperiment
31
(a)
(b) Fig5.4 Time evolution of creep strain in the direction of loading (without damage). (a) for different orientations of the fiber (0, 45, 90 deg) at 45MPa load. (b) for 45 deg fiber orientation at varying loading 20, 45 and 100PMa.
Time(hr)
0 1 2 3 4 5 6
ε 11
0.0000
.0002
.0004
.0006
.0008
.0010
00
452
900
Time(hr)
0 1 2 3 4 5 6
ε 11
0.0000
.0002
.0004
.0006
.0008
.0010
20Mpa46Mpa60Mpa
32
(a)
(b)
Time (hr)
0 1 2 3 4 5 6
ε 12
0
1e-4
2e-4
3e-4
4e-4
5e-4
6e-4
0 deg45 deg90 deg
(c)
Fig 5.5 Time evolution of creep strain with damage effect. (a) comparison when damage evolution is included under tensile load 60 MPa. (b) comparison between tensile loads 70, 75, 80 MPa with fiber orientation 45 deg. (c) comparison between fiber orientation 0, 45, 90 deg under shear loads 40 Mpa.
Time(hr)
0 2 4 6 8 10 12
ε 11
0.0000
.0005
.0010
.0015
.0020
creep with damagecreep without damage
Time(hr)
0.0 .5 1.0 1.5 2.0 2.5
ε 11
0.000
.001
.002
.003
.004
.005
70Mpa75MPa80MPa
33
We also apply relaxation test on single element. We apply fixed displacement at
the right boundary (u=0.1%). The dimension is still 10mm x 10mm. We compare stress
calculated by Robinson’s creep law with that by power law in isotropic case. We find
out that after 10 hours test the stress along loading direction calculated by Robinson
Creep model is lower than the result calculated by Power law mode, which is due to the
damage effect of Robinson creep model.
Fig 5.6: Time evolution of stress relaxation in the direction of loading (with damage).
Power Law
Robinson Model
34
Results for both the creep and relaxation tests were able to match the results from
an isotropic Power Law Creep Model when damage was not included. However, the
proposed model deviates from the Power Law model when creep anisotropy due to
orientation and damage is included. These element tests are achieved for a wide range
of loads and orientations, demonstrating the robustness of the numerical scheme. Also,
the tests show the usefulness of the constitutive model despite the use of a single scalar
parameter for damage variable.
5.2 Square plate
One element tests are demonstrating the robustness of the numerical scheme of the
UMAT. In this part we extend our model to relatively complex model – a square plate
with multiple elements. The dimension of the plate is 20mm x 20 mm. This particular
test was done for the reduced integration plane stress element with linear and quadratic
interpolation schemes (ABAQUS element type CPS4R and CPS8R). Geometry and
boundary condition of the plate under tension is shown on Fig 5.7. We apply tensile
stress 46 Mpa stress along 1 direction with different fiber orientation (0 deg, 45 deg and
90 deg). Fig 5.8 shows the time evolution of creep strain with fiber orientations 0, 45
and 90 deg under tensile loads equal to 46MPa. Fig 5.8.(a) shows the 1 direction strain
distribution and Fig 5.8.(b) shows the shear strain distribution. All these results coincide
with the results obtained in one element tests.
35
Fig 5.7 Geometry and boundary condition of square plate problem.
20mm
σσ
36
Time (hr)
0 1 2 3 4 5 6
ε 11
0.0000
.0002
.0004
.0006
.0008
00
450
900
(a)
Time (hr)
0 1 2 3 4 5 6
ε 12
-1e-5
0
1e-5
2e-5
3e-5
4e-5
00
450
900
(b) Fig5.8 Time evolution of creep strain with different orientations of the fiber (0, 45, 90 deg) at 46MPa load in the direction of loading (without damage). (a) time evolution of strain at 1 direction. (b) time evolution of shear strain.
37
L=152.4mm
R=80mm
5.3 Plate with a hole in the middle
This particular test was done for the reduced integration plane stress element with
linear and quadratic interpolation schemes (ABAQUS element type CPS4R and
CPS8R). Geometry and boundary condition of the plate under tension and torsion is
shown on Fig 5.8. We apply tensile stress 10 MPa along 1 direction with different fiber
orientation (0 deg, 45 deg and 90 deg).
Fig 5.9 Left, geometry and Right, 2D quarter symmetry model for the square plate with a hole problem.
σ
38
First of the study we choose isotropic material to compare Robinson creep Model
with Power law creep model. The Power law model is one of the creep model
ABAQUS use to do creep calculation for isotropic material. Fig 5.10 shows the stress
distribution of the Robinsons’ model law for isotropic material and comparative stress
distribution for a Power Law type model. It can be seen that stress distribution after
instantaneous elastic response for the proposed Creep Damage model (Fig 5.10.(b)) is
identical to a Power Law model (Fig 5.10.(a)). This is because in elastic step both
power law and Robinson model has the same constitutive equation. But due to the
damage effect we take into consideration in Robinson model we can observe some
difference existing after 5 hours creep response in Fig 5.10.(c) and Fig 5.10.(d).
To further compare these two models we plot stress distribution along the hole
(counter clockwise) because this area has the most complex behavior of the whole plate.
Fig 5.11 shows the distribution of Stress distribution along the hole, after elastic step (a)
and after 5 hours of creep load (b). Fig 5.119.(a) shows the plots along the hole for both
creep law are identical to each other after elastic step. Fig 5.119.(b) shows lower
stresses in the Robinson Model at the second half of the plot is a result of higher stress
relaxation from higher stress concentration area due to the damage evolution in
Robinson Model. Fig 5.12 shows the creep strain distribution for Power law and
Robinson creep Model after 5 hours creep response. Robinson Model has more creep
strain existing in the high stress concentration zone. Fig 5.13 shows the creep strain
distribution along the hole (counter clockwise), after 5 hours of creep load. We can
observe that at the first part of the path Robinson model is agree with Power law model.
At the second half of the path Robinson model undergoes more creep strain than power
39
law that is due to the damage factor we include in the calculation of creep strain in
Robinson Model. We also find the highest value located in different area in these two
models. The highest value for Power law located in the very end of the path while for
Robinson model highest value of stress redistribution is in the area close to the end of
the path which is due to the anisotropic constitutive equations of Robinson Model even
for isotropic material.
To verify the accuracy of the numerical scheme we compare the numerical results
with analytical solutions. We choose the radius of the plate 20mm and 50mm with fiber
orientation 90 deg (Fig 5.14). We compare each case with analytical solution. Both two
plots indicated the accuracy of the numerical result in positive value area. We can find
that there is big difference in negative area. Analytic solution has much lower value
than numerical solution in negative zone. The discrepancy is due to the reason that
analytic solution is for infinite plate and free boundary calculation.
Fig 5.15 shows the stress distribution along the hole (counter clockwise), after 5
hours of creep load 10 Mpa with R=20mm, 50mm, 80mm, L=152.4mm and fiber
orientation 45 deg. Fig 5.16 shows the stress distribution along the hole (counter
clockwise), after 5 hours of creep load 10 Mpa with R=50mm, L=152.4mm.Fiber
orientation 45deg and 90 deg. Fig 5.17 shows Stress concentration zone with damage
effect under axial stress 10Mpa fiber orientation 45 deg (a)R/L=0.1 (b)R/L=0.3 (c)
R/L=0.5. Fig 5.18 shows Stress compression zone with damage effect under axial stress
10Mpa fiber orientation 45 deg (a) R/L=0.1 (b) R/L=0.3 (c) R/L=0.5. All these results
indicates that the computational routine (UMAT) successfully describes the rather
complex creep/damage phenomenon observed in PMCs.
40
(a) (b)
(c) (d)
Fig 5.10 Contour plot of (a) stress distribution of the power law for isotropic material after elastic step, (b) stress distribution of the Robinsons’ model law for isotropic material after elastic step, (c) stress distribution of the power law for isotropic material after creep step (5 hours), (d) stress distribution of the Robinsons’ model law for isotropic material after creep step (5 hours).
38MPa
20MPa
78MPa
50MPa
38MPa
20MPa
78MPa
50MPa
38MP
20MPa
40MPa
20MPa
41
(a) (b) Fig 5.11 Stress distribution along the hole (quarter circle) going in a counter clockwise direction for the proposed Creep Damage Model as and for isotropic Power Law Creep model. (a) stress distribution after elastic step. (b) stress distribution after 5 hours.
True distance along the hole (counter clockwise)
0 20 40 60 80 100 120 140
σ 11(M
Pa)
-20
0
20
40
60
80
100
Robinson modelpowerlaw
True distance along the hole (counter clockwise)
0 20 40 60 80 100 120 140
σ 11(M
Pa)
-10
0
10
20
30
40
50
Robinson modelpower law
42
(a)
(b) Fig 5.12 FEA plot of Creep strain distribution after 5 hours creep response. (a) Robinson Damage Model and (b) Isotropic Power Law.
0.073%
0.051%
0.033%
0.014%
0.003%
0.073%
0.051%
0.033%
0.014%
0.003%
43
Fig 5.13 Creep strain along the hole (quarter circle) going in a counter clockwise direction for the Robinson Damage Model and for isotropic Power Law Creep model (after 5 hours).
True distance along the hole (counter clockwise)
0 20 40 60 80 100 120 140
ε 11
-.0002
0.0000
.0002
.0004
.0006
.0008
.0010
.0012
Robinson modelpower law
44
(a)
(b) Fig 5.14 Comparison between Analytic solution and Robinson model (FEA) in elastic step. (a) R=20mm and R/L=13%. (b) R=50mm and R/L=33%.
θ
θ
0 20 40 60 80 100
σ 11(M
Pa)
-30
-20
-10
0
10
20
30
40
Robinson modelAnalytic solution
θ
0 20 40 60 80 100
σ 11(M
Pa)
-30
-20
-10
0
10
20
30
40
Robinson modelAnalytic solution
45
Fig 5.15 Creep / time response along the hole (quarter circle) going in a counter clockwise with damage effect under axial stress 10MPa with R=20mm,50mm,80mm, L=152.4mm, fiber orientate 45 deg.
True distance along the hole (counter clockwise)
0 20 40 60 80 100 120 140
σ 11(Μ
Pa)
-10
0
10
20
30
40
50
60
R=80R=50R=20
46
Fig 5.16 Creep / time response along the hole (quarter circle) going in a counter clockwise with damage effect under axial stress 10MPa with R=50mm, L=152.4mm, fiber orientate 45 deg and 90 deg.
θ (deg)
0 20 40 60 80 100
σ 11(M
Pa)
-10
0
10
20
30
40
450
900
47
(a)
(b)
(c) Fig 5.17 Stress concentration zone with damage effect under axial stress 10Mpa fiber orientation 45 deg. (a) stress concentration zone with R/L=0.1. (b) stress concentration zone with R/L=0.3. (c) stress concentration zone with R/L=0.5.
20MPa
40Mpa
56MPa
45Mpa
40Mpa
30Mpa
25 MPa
20MPa
20MPa
25 MPa
48
(a)
(b)
(c) Fig 5.18 Stress compression zone with damage effect under axial stress 10Mpa fiber orientation 45 deg. (a) stress compression zone with R/L=0.1. (b) stress compression zone with R/L=0.3. (c) stress compression zone with R/L=0.5.
-0.5MPa
-2Mpa
-6MPa
-3Mpa
-4Mpa
-2Mpa
-0.5MPa
-0.5MPa
49
CHAPTER VI
SHELL ELEMENT IMPLEMENTATION 6.1 theory of shell element
Shell elements are surface representations of structures that are much thinner in
one direction than the other two (thin-walled structures). The elements are
geometrically defined by three or four sided surfaces, and are located in space at the
mid-plane of the solid they are representing. The user specifies the thickness of the
elements as an input to the software. These elements are used in modeling all types of
thin-walled structures, such as airplane and automotive bodies, pressure vessels, sheet
metal, and many plastic molded parts.
Shell elements have six active degrees of freedom per node, much like beam
elements. Because of commonality in degrees of freedom, beam and shell elements are
often joined together in mixed models.
50
Shell elements are good for modeling structures that are thin. These elements are
usually formulated under the assumptions governing thin plate theory. If a structure is
too thick, the behavior of thin plates is no longer seen (shear stresses become large, etc.),
and shell elements should not be used. This limit is usually seen at a thickness to width
or length (whichever is smaller) of 1/10 and larger.
Shell elements generally have a lower limit on this ratio as well. At thickness-to-
width ratios between 1/100 and 1/1000, thin plates begin to behave like membranes,
with no bending stiffness (like a string in tension, subjected to transverse load). Because
of this shell elements cannot be used to model very thin, flexible structures such as
fabric or thin membranes.
6.2 Shell element in ABAQUS
ABAQUS includes general-purpose shell elements as well as elements that are
valid for thick and thin shell problems. See below for a discussion of what constitutes a
“thick” or “thin” shell problem. This concept is relevant only for elements with
displacement degrees of freedom. The general-purpose shell elements provide robust
and accurate solutions to most applications and will be used for most applications.
However, in certain cases, for specific applications, enhanced performance may be
obtained with the thin or thick shell elements; for example, if only small strains occur
and five degrees of freedom per node are desired.
Element type S4R is general-purpose shell. These elements allow transverse shear
deformation. They use thick shell theory as the shell thickness increases and become
51
discrete Kirchhoff thin shell elements as the thickness decreases; the transverse shear
deformation becomes very small as the shell thickness decreases.
Element type S8R should be used only in thick shell problems. Thick shells are
needed in cases where transverse shear flexibility is important and second-order
interpolation is desired. When a shell is made of the same material throughout its
thickness, this occurs when the thickness is more than about 1/15 of a characteristic
length on the surface of the shell, such as the distance between supports for a static case
or the wavelength of a significant natural mode in dynamic analysis.
6.3 Implementation of shell element
First, the computational model was applied to the thin-walled tube (7.5 in dia)
under a tensile stress of 0.5 MPa. This particular test was done for the reduced
integration shell elements with linear and quadratic interpolation schemes (ABAQUS
element type S4R and S8R). Geometry and boundary condition of thin wall tube under
tension and torsion is shown on Fig 6.1. We apply tensile stress along 1 direction with
different fiber orientation (0 deg, 45 deg and 90 deg). Because the stress and strain
distributions are evenly along the tube, we take one element to analyze the strain
development. Fig 6.2 shows the time evolution of creep strain under a constant tensile
load of 45 MPa for different fiber orientations (0 deg, 45 deg, 90 deg) when the scalar
damage variable is included in the material model. It is observed that when the creep
loading is along fiber directions, the PMC has the strongest behavior. The creep strains
are progressively higher when the creep loading is at progressively higher angle to the
52
fiber orientation. For the 90 deg orientation, failure can be achieved in roughly 6 hours.
Fig 6.3 shows the time evolution of creep strain under a constant shear load of 45 MPa
for different fiber orientations (0 deg, 45 deg, 90 deg) when the scalar damage variable
is included in the material model.
53
1
3
θ
2
FF
T
T
Fig 6.1 Thin wall tube under tension and torsion with fiber orientation
54
(a)
Time (hr)
0 2 4 6 8 10 12
ε 12
.0032
.0034
.0036
.0038
.0040
.0042
.0044
.0046
.0048
.0050
.0052
0 deg45 deg90 deg
(b)
Fig 6.2 Time evolution of creep strain under a constant tensile load (F) 45 MPa of thin-walled tube for different fiber orientations (0 deg, 45 deg, 90 deg). (a) time evolution of maximum principle strain. (b) time evolution of shear strain.
55
Further the computational model was applied to the pressure vessel (20 cm dia)
under a pressure of 0.5 MPa. This particular test was done for the reduced integration
shell elements with linear and quadratic interpolation schemes (ABAQUS element type
S4R and S8R). Fig 6.3 shows the geometry, mesh and path of the pressure vessel, the
thickness is 0.4.
First we plot the path along path 1 (Fig 6.3) with different fiber orientation. 0 deg
is the direction along the path 1, 90 deg is the direction perpendicular to the path1. It
can be observed that 0 deg has the smallest strain deformation along the path1, the
higher angle of fiber, the higher strain deformation occurs. We can also find the
movement of the strain distribution along path 1 with fiber orientation of 45 deg (Fig.
6.4) after 10 hours evolution. This phenomena is also obvious in other fiber orientation.
Fig 6.4 shows path plot of time evolution of Maximum Strain along path1 of the vessel
with fiber orientation 45 deg under inside pressure 0.5Mpa with damage evolution. Fig
6.5 shows Maximum Strain along path1 of the vessel with fiber orientation 0, 45, 90
deg under inside pressure 0.5Mpa with damage evolution after 10 hours.
Fig 6.6 shows path plot of time evolution of Maximum Strain along path2 of the
vessel with fiber orientation 45, 60, 90 deg under inside pressure 0.5Mpa with damage
evolution. It can be observed that the obvious difference in maximum principle strain
between 45, 60 and 90 deg is at the center part of the pressure vessel and the 90 deg has
the highest strain development.
Fig 6.7 shows the time evolution of Maximum Principle Strain of the pressure
vessel with fiber orientation 60, 90 deg under pressure 1Mpa with damage evolution.
And we set the failure maximum strain is 1%. We can observe that the 60 deg vessel
56
reached failure at 4 hours and 90 deg vessel reaches failure about 3 hours. The strain
distributions at the dome area for both vessels are close to each other. The obvious
discrepancy occurs in the straight part of the vessel and in this area the fiber orientation
is important for the strain evolution and 90 deg has the weakest behavior. So it takes
less time for 90 deg vessel to first get failure.
57
Fig 6.3 Pressure vessel, geometry and mesh and path
Path 1
Path 2
58
(a) (b)
(c)
Fig 6.4 Path plot of time evolution of Maximum Strain along path1 of the vessel under inside pressure 0.5Mpa with damage evolution (a) fiber orientation 45 deg. (b) fiber orientation 60 deg. (c) fiber orientation 90 deg.
Path 1
0 20 40 60 80
Max
imun
Prin
cipl
e S
train
.0027
.0028
.0029
.0030
.0031
.0032
45 deg - 10 hours45 deg - elastic step
Path 1
0 20 40 60 80
Max
imum
Prin
cipl
e S
train
.0029
.0030
.0031
.0032
.0033
.0034
.0035
.0036
60 deg - 10 hours60 deg - elastic step
Path 1
0 20 40 60 80
Max
imum
Prin
cipl
e St
rain
.0031
.0032
.0033
.0034
.0035
.0036
.0037
90 deg - 10 hours90 deg - elastic step
59
Fig 6.5 Maximum Strain along path1 of the vessel with fiber orientation 0, 45, 90 deg under inside pressure 0.5Mpa with damage evolution after 10 hours
Path 1
0 10 20 30 40 50 60 70
Max
imum
Prin
cipl
e st
rain
.0026
.0028
.0030
.0032
.0034
.0036
.0038
45 deg60 deg90 deg
60
Fig 6.6 Time evolution of Maximum Principle Strain along path2 of the pressure vessel with fiber orientation 45, 60, 90 deg under pressure 0.5Mpa with damage evolution
Path 2
0 10 20 30 40 50 60
Max
imum
Prin
cipl
e St
rain
.0015
.0020
.0025
.0030
.0035
.0040
45 deg - 10 hours45 deg - elastic step60 deg - 10 hours 60 deg - elastic step90 deg - 10 hours 90 deg - elastic step
61
Fig 6.7 Time evolution of Maximum Principle Strain of the pressure vessel with fiber orientation 60, 90 deg under pressure 1Mpa with damage evolution. (a) fiber orientation 60 deg. (b) fiber orientation 90 deg.
1%
0.6%
0.4%
0.3%
Failure area
0.8%
Failure area Failure area
1%
0.6%
0.4%
0.2%
0.8%
62
CHAPTER VII
CONCLUSION
This paper demonstrates the computational utility of the anisotropic, creep damage
model presented in a paper by Robinson, Binienda and Ruggles (2002). The references
1 and 2 describe supporting exploratory experiments are conducted on thin-walled
tubular specimens fabricated from a model PMC. Thin-walled tubes are used not for
their interest as structural components but because they are convenient specimens for
generating multiaxial stress and deformation. The computational routine (UMAT)
successfully describes the rather complex creep/damage phenomenon observed in
PMCs. The routine is used with a commercially available code. The present work
demonstrates the utility of the creep damage law in describing the essential physics
behind creep damage using a single scalar parameter. The model is successfully applied
to a benchmark problem (circular hole a square plate).
A primary assumption in the damage model, cf., Robinson et al. (2002), is that the
stress dependence of damage evolution is on the transverse tensile and longitudinal
shear traction acting at the fiber/matrix interface. Accordingly, the isochronous damage
function is taken to depend on the appropriate invariants N and S, i.e., )S,N(∆ .
Exploratory data are generated to partially define the isochronous damage curve
∆( , )N S = 1 for the model PMC; evidently, a more extensive data base is required to fully
63
define ∆( , )N S = 1 and to verify that this stress dependence correlates directly with creep
failure. This maybe of general interest and the present code needs to be extended to
describe the general power law form of the damage curve.
Also, the code may easily be extended to calculate creep rupture life based on the
deformation rate and the damage variable calculations. These are left for future work.
64
REFERENCES
1. ROBINSON, D.N., BINIENDA, W.K. AND RUGGLES, M.B. (2002). “CREEP OF POLYMER MATRIX COMPOSITES: PART 1- A NORTON/BAILEY CREEP LAW FOR TRANSVERSE ISOTROPY.” JOURNAL OF ENGINEERING MECHANICS, Vol. 129, No. 3, March 2003, pp. 310-317
2. BINIENDA, W.K., ROBINSON, D.N. AND RUGGLES, M.B. (2002). “CREEP
FAILURE OF POLYMER MATRIX COMPOSITES (PMC): A MONKMAN-GRANT RELATIONSHIP FOR TRANSVERSE ISOTROPY”. JOURNAL OF ENGINEERING MECHANICS, Vol. 129, No. 3, March 2003, pp. 318-323
3. LECKIE, F.A. (1986). “THE MICRO- AND MACRO-MECHANICS OF
CREEP RUPTURE.” ENGRG. FRACTURE MECH., 25,5, 505-521.
4. LECKIE, F.A., AND HAYHURST, D.R. (1974)“CREEP RUPTURE IN STRUCTURES.” PROC. ROYAL SOCIETY OF LONDON, A340, 323-347.
5. MONKMAN, F.C. AND GRANT, N.J., (1956). “AN EMPIRICAL
RELATIONSHIP BETWEEN RUPTURE LIFE AND MINIMUM CREEP RATE IN CREEP-RUPTURE TESTS.” ASTM, 56, 593 – 620.
6. LISSENDEN, C.J., LERCH, B.A., ELLIS, J.R. AND ROBINSON, D.N. (1997).
“EXPERIMENTAL DETERMINATION OF YIELD AND FLOW SURFACES UNDER AXIAL-TORSIONAL LOADING.” STP 1280, A.S.T.M, 92-112.
7. ROBINSON, D.N. AND DUFFY, S.F. (1990). “CONTINUUM
DEFORMATION THEORY FOR HIGH TEMPERATURE METALLIC COMPOSITES.” J. ENGRG. MECH., ASCE, 116(4), 832-844.
8. ROBINSON, D.N., BINIENDA, W.K. AND MITI-KAVUMA, M. (1992).
“CREEP AND CREEP RUPTURE OF METALLIC COMPOSITES.” J. ENGRG. MECH.,. ASCE, 118(8), 1646-1660.
9. ROBINSON, D.N. AND PASTOR, M.S. (1993). “LIMIT PRESSURE OF A
CIRCUMFERENTIALLY REINFORCED SIC/TI RING.” COMPOSITES ENGINEERING, 2, (4), 229-238.
65
10. ROBINSON, D.N., TAO, Q. AND VERRILLI, M.J. (1994). “A
HYDROSTATIC STRESS-DEPENDENT ANISOTROPIC MODEL OF VISCOPLASTICITY.” NASA TM 106525.
11. ROBINSON, D.N. AND WEI, WEI (1996). “FIBER ORIENTATION IN
COMPOSITE STRUCTURES FOR OPTIMAL RESISTANCE TO CREEP FAILURE.” J. ENGRG. MECH., ASCE, 122,(9), 855-860.
12. ROBINSON, D.N. AND BINIENDA, W.K., (2001)A. “OPTIMAL FIBER
ORIENTATION IN CREEPING COMPOSITE STRUCTURES.” J. APPL. MECHANICS, 68,(2), 213-217.
13. ROBINSON, D.N. AND BINIENDA, W.K., (2001)B. “MODEL OF
VISCOPLASTICITY FOR TRANSVERSELY ISOTROPIC INELASTICALLY COMPRESSIBLE SOLIDS.” J. ENGRG. MECH., ASCE, 127,(6).
14. ROBINSON, D.N., KIM, K.J AND WHITE, J.L. (2002). “CONSTITUTIVE
MODEL OF A TRANSVERSELY ISOTROPIC BINGHAM FLUID.” J. APPL. MECHANICS, 69,(1), 1-8.
15. ABAQUS THEORY AND VERIFICATION MANUALS, VERSION 6.2, HKS
INC.
66
APPENDICES
67
APPENDIX A
UMAT FILE
SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,
* RPL,DDSDDT,DRPLDE,DRPLDT,STRAN,DSTRAN,
* TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,MATERL,NDI,NSHR,NTENS,
* NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,CELENT,
* DFGRD0,DFGRD1,NOEL,NPT,KSLAY,KSPT,KSTEP,KINC)
C
INCLUDE 'ABA_PARAM.INC'
C
CHARACTER*80 MATERL
DIMENSION STRESS(NTENS),STATEV(NSTATV),
* DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),
* STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),
* PROPS(NPROPS),COORDS(3),DROT(3,3),
* DFGRD0(3,3),DFGRD1(3,3)
DOUBLE PRECISION J0,J,J2,KSI,NN,N,V,M
C ELASTIC PROPERTIES
EMOD1=PROPS(1)
68
EMOD2=PROPS(2)
ENU=PROPS(3)
EG=PROPS(4)
THETA=PROPS(5)
KSI=PROPS(6)
ESI=PROPS(7)
ENTA=PROPS(8)
N=PROPS(9)
E0=PROPS(10)
SIG0=PROPS(11)
V=PROPS(12)
M=PROPS(13)
T0=PROPS(14)
EBULK=1/(1.0-ENU**2*EMOD2/EMOD1)
C
C ELASTIC STIFFNESS
C
IF (KSTEP.EQ.1) THEN
DO K1=1,NTENS
DO K2=1,NTENS
DDSDDE(K2,K1)=0
END DO
ENDDO
69
C
Q11=EMOD1*EBULK
Q12=ENU*EMOD2*EBULK
Q21=ENU*EMOD2*EBULK
Q22=EMOD2*EBULK
Q66=EG
DDSDDE(1,1)=Q11*COS(THETA)**4.0
* +2.0*(Q12+2*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q22*SIN(THETA)**4.0
DDSDDE(1,2)=(Q11+Q22-4*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q12*(SIN(THETA)**4.0+COS(THETA)**4.0)
DDSDDE(2,1)=DDSDDE(1,2)
DDSDDE(2,2)=Q11*SIN(THETA)**4.0
* +2.0*(Q12+2*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q22*COS(THETA)**4.0
DDSDDE(NTENS,1)=(Q11-Q12-2*Q66)*SIN(THETA)*COS(THETA)**3.0+
* (Q12-Q22+2*Q66)*SIN(THETA)**3.0*COS(THETA)
DDSDDE(1,NTENS)=DDSDDE(NTENS,1)
DDSDDE(NTENS,2)=(Q11-Q12-2*Q66)*SIN(THETA)**3.0*COS(THETA)+
* (Q12-Q22+2*Q66)*SIN(THETA)*COS(THETA)**3.0
DDSDDE(2,NTENS)=DDSDDE(NTENS,2)
70
DDSDDE(NTENS,NTENS)=(Q11+Q22-2*Q12-2*Q66)*SIN(THETA)**2
* *COS(THETA)**2.0+
* Q66*(SIN(THETA)**4+COS(THETA)**4.0)
C
C1=DDSDDE(1,1)
C2=DDSDDE(1,2)
C3=DDSDDE(1,NTENS)
C4=DDSDDE(2,1)
C5=DDSDDE(2,2)
C6=DDSDDE(2,NTENS)
C7=DDSDDE(NTENS,1)
C8=DDSDDE(NTENS,2)
C9=DDSDDE(NTENS,NTENS)
C
C CALCULATE STRESS FROM ELASTIC STRAINS
C
DO K1=1,NTENS
DO K2=1,NTENS
STRESS(K2)=STRESS(K2)+DDSDDE(K2,K1)*DSTRAN(K1)
END DO
END DO
C
STATEV(1)=STATEV(1)+DSTRAN(1)
71
STATEV(2)=STATEV(2)+DSTRAN(2)
STATEV(3)=STATEV(3)+DSTRAN(3)
A1=STATEV(1)
A2=STATEV(2)
A3=STATEV(3)
D1=C1
D2=C5
D3=C2
C
END IF
C
C CALCULATE CREEP STRESS
IF (KSTEP.EQ.2) THEN
S11=STRESS(1)-(STRESS(1)+STRESS(2)+STRESS(3))/3.D0
S22=STRESS(2)-(STRESS(1)+STRESS(2)+STRESS(3))/3.D0
S33=STRESS(3)-(STRESS(1)+STRESS(2)+STRESS(3))/3.D0
S12=STRESS(NTENS)
D11=COS(THETA)**2.D0
D22=SIN(THETA)**2.D0
D12=COS(THETA)*SIN(THETA)
J0=D11*S11+D22*S22+2.D0*D12*S12
J=D11*(S11**2.D0+S12**2.D0)+D22*(S12**2.D0+S22**2.D0)
* +D12*S12*(S11+S22)
72
J2=.5*(S11**2.D0+S22**2.D0+S33**2.D0)+S12**2
Q1=J-J0**2
Q2=J0**2
TEMP1=(ESI-4.D0*ENTA)*(STRESS(1)+STRESS(2))**2/9.D0
TEMP2=(KSI)*Q1
TEMP3=(ESI-ENTA)*Q2
TEMP4=3.0*(J2-TEMP2-TEMP3+TEMP1)
PHI=SQRT(TEMP4)/(E0)
TA11=2.D0*S11*D11+2.D0*D12*S12
TA22=2.D0*S22*D22+2.D0*D12*S12
TA12=S12*(D11+D22)+D12*(S11+S22)
TB11=2.D0*J0*D11
TB22=2.D0*J0*D22
TB12=2.D0*J0*D12
TONE11=TA11-TB11
TONE22=TA22-TB22
TONE12=TA12-TB12
TTWO11=2.D0*J0*(D11-1.D0/3.D0)
TTWO22=2.D0*J0*(D22-1.D0/3.D0)
TTWO12=2.D0*J0*D12
TAO11=S11-KSI*TONE11-(ESI-ENTA)*TTWO11+
* 2.D0*(ESI-4.D0*ENTA)*(STRESS(1)+STRESS(2))/9.D0
TAO22=S22-KSI*TONE22-(ESI-ENTA)*TTWO22+
73
* 2.D0*(ESI-4.D0*ENTA)*(STRESS(1)+STRESS(2))/9.D0
TAO12=S12-KSI*TONE12-(ESI-ENTA)*TTWO12
RSTRAN11=3.D0*PHI**(N-1.D0)*TAO11/(2.D0*E0*SIG0*100)
RSTRAN22=3.D0*PHI**(N-1.D0)*TAO22/(2.D0*E0*SIG0*100)
RSTRAN12=3.D0*PHI**(N-1.D0)*TAO12/(2.D0*E0*SIG0*100)
DCRSTRAN11=RSTRAN11*DTIME
DCRSTRAN22=RSTRAN22*DTIME
DCRSTRAN12=RSTRAN12*DTIME
STATEV(4)=DCRSTRAN11+STATEV(4)
STATEV(5)=DCRSTRAN22+STATEV(5)
STATEV(6)=DCRSTRAN12+STATEV(6)
c
c DAMAGE calculation
c
I=STRESS(1)+STRESS(2)+STRESS(3)
I0=D11*STRESS(1)+2*D12*STRESS(NTENS)+D22*STRESS(2)
RE1=J2+.25*J0**2.0-J
IF (RE1.LE.0) THEN
RE1=0-RE1
END IF
NN=.5*(I-I0)+SQRT(RE1)
IF (NN.LE.0) THEN
NN=0.0
74
END IF
IF (Q1.LE.0) THEN
Q1=J0**2-J
END IF
S=SQRT(Q1)
DELTA=(NN+0.35*S)/(E0)
STATEV(11)=(1-DELTA**V*TIME(2)/T0)**(1/(1+M))
RDASTRAN11=3.0*PHI**(N-1.0)*TAO11/(2.0*E0*SIG0*100)
* /(STATEV(11)**N)
RDASTRAN22=3.0*PHI**(N-1.0)*TAO22/(2.0*E0*SIG0*100)
* /(STATEV(11)**N)
RDASTRAN12=3.0*PHI**(N-1.0)*TAO12/(2.0*E0*SIG0*100)
* /(STATEV(11)**N)
DDAMSTRAN11=RDASTRAN11*DTIME
DDAMSTRAN22=RDASTRAN22*DTIME
DDAMSTRAN12=RDASTRAN12*DTIME
STATEV(12)=DDAMSTRAN11+STATEV(12)
STATEV(13)=DDAMSTRAN22+STATEV(13)
STATEV(14)=DDAMSTRAN12+STATEV(14)
c
C CALCUALTE UPDATED STRESS
C
DS1=C1*(DSTRAN(1)-DCRSTRAN11)
75
DSTRESS11=C1*(DSTRAN(1)-DCRSTRAN11)+C2*(DSTRAN(2)-
DCRSTRAN22)+
* C3*(DSTRAN(NTENS)-DCRSTRAN12)
DSTRESS22=C4*(DSTRAN(1)-DCRSTRAN11)+C5*(DSTRAN(2)-
DCRSTRAN22)+
* C6*(DSTRAN(NTENS)-DCRSTRAN12)
DSTRESS12=C7*(DSTRAN(1)-DCRSTRAN11)+C8*(DSTRAN(2)-
DCRSTRAN22)+
* C9*(DSTRAN(NTENS)-DCRSTRAN12)
STRESS(1)=DSTRESS11+STRESS(1)
STRESS(2)=DSTRESS22+STRESS(2)
STRESS(NTENS)=DSTRESS12+STRESS(NTENS)
C
C CALCULATE UPDATED JACOBIAN
DO K1=1,NTENS
DO K2=1,NTENS
DDSDDE(K2,K1)=0
END DO
ENDDO
Q11=EMOD1*EBULK
Q12=ENU*EMOD2*EBULK
Q21=ENU*EMOD2*EBULK
Q22=EMOD2*EBULK
76
Q66=EG
DDSDDE(1,1)=Q11*COS(THETA)**4.0
* +2.0*(Q12+2*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q22*SIN(THETA)**4.0
DDSDDE(1,2)=(Q11+Q22-4*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q12*(SIN(THETA)**4.0+COS(THETA)**4.0)
DDSDDE(2,1)=DDSDDE(1,2)
DDSDDE(2,2)=Q11*SIN(THETA)**4.0
* +2.0*(Q12+2*Q66)*SIN(THETA)**2.0*COS(THETA)**2.0
* +Q22*COS(THETA)**4.0
DDSDDE(NTENS,1)=(Q11-Q12-2*Q66)*SIN(THETA)*COS(THETA)**3.0+
* (Q12-Q22+2*Q66)*SIN(THETA)**3.0*COS(THETA)
DDSDDE(1,NTENS)=DDSDDE(NTENS,1)
DDSDDE(NTENS,2)=(Q11-Q12-2*Q66)*SIN(THETA)**3.0*COS(THETA)
* (Q12-Q22+2*Q66)*SIN(THETA)*COS(THETA)**3.0
DDSDDE(2,NTENS)=DDSDDE(NTENS,2)
DDSDDE(NTENS,NTENS)=(Q11+Q22-2*Q12-2*Q66)*SIN(THETA)**2
* *COS(THETA)**2.0+
* Q66*(SIN(THETA)**4+COS(THETA)**4.0)
RETURN
END
77
APPENDIX B
INPUT FILE FOR PRESSURE VESSEL
*Heading
** Job name: Job-1 Model name: Model-1
*Node
1, 0., 20., 0.
……
12445, 19.9938431, -20., 0.496222973
*Element, type=S8R,ELSET=CREEP
1, 1, 291, 399, 4, 4171, 4172, 4173, 4174
……
4106, 4170, 164, 3, 163, 12425, 12445, 9401, 12444
*Nset, nset=B1
3, 164, ……., 12445
*Surface, type=ELEMENT, name=vessel
creep, SNEG
*shell SECTION,ELSET=CREEP,MATERIAL=com1
0.4
*TRANSVERSE SHEAR STIFFNESS
78
2230, 2230, 0
** MATERIALS-1
*MATERIAL,NAME=COM1
*USER MATERIAL,CONSTANTS=16,TYPE=MECHANICAL
44e4, 7.3e3, .284, 2.7e3,1.047, 0.57, 0.64, 0.1,
6.5, 46., 50., 10.6, 7.125, 12,2.7e3,2.7e3
*DEPVAR
14
*INITIAL CONDITIONS,TYPE=SOLUTION
CREEP,0.0,0.0,0.0,0.0,0.0,0.0,0.0
0.0,0.0,0.0,0.0,0.0,0.0,0.0
*Boundary
B1, 2, 2
*STEP,INC=30
PRESCRIBED TENSILE STRESS
*STATIC
1.E-7,1.E-6
*Dsload
vessel, P, 0.5
*EL PRINT,FREQ=1
S,
SDV1,
SDV4
79
*END STEP
*STEP,INC=100000
CREEP STEP
*VISCO,CETOL=1E-4
1.E-6,10
** OUTPUT REQUESTS
*PRINT,FREQ=1,RESIDUAL=YES
*EL PRINT,FREQ=1
SDV
*OUTPUT,FIELD,FREQ =1
*ELEMENT OUTPUT
S,E,CE,
SDV
*NODE OUTPUT
U
*END STEP
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