Abaqus-umat Program Example

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


0 20 40 60 80 100 120 140

σ 11(M

Pa)

-10

0

10

20

30

40

50


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).


0 20 40 60 80 100 120 140

ε 11

-.0002

0.0000

.0002

.0004

.0006

.0008

.0010

.0012


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.


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


Path 1

0 20 40 60 80

Max

imum

Prin

cipl

e St

rain

.0031

.0032

.0033

.0034

.0035

.0036

.0037


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


* +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


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



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



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)



DCRSTRAN11=RSTRAN11*DTIME



STATEV(4)=DCRSTRAN11+STATEV(4)



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)


* /(STATEV(11)**N)


* /(STATEV(11)**N)

DDAMSTRAN11=RDASTRAN11*DTIME



STATEV(12)=DDAMSTRAN11+STATEV(12)



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)


DCRSTRAN22)+



DCRSTRAN22)+


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


* +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


* +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(NTENS,2)=(Q11-Q12-2*Q66)*SIN(THETA)**3.0*COS(THETA)

* (Q12-Q22+2*Q66)*SIN(THETA)*COS(THETA)**3.0


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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Abaqus-umat Program Example