NUMERICAL SIMULATION OF CRACK GROWTH BASED ON VOID VOLUME FRACTION
F. R. Biglari1, K. Nikbin2 and N. P. O’Dowd3 1
Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran. Mechanical Engineering Department, Imperial College, University of London, UK.
[email protected]
Abstract Numerical simulation of crack growth is presented in this paper based on a local failure mechanism. The local failure mechanism in a metal containing a ductile concentration of voids is expressed in terms of void volume fraction. In the numerical simulation of crack growth, the crack length is extended when void volume fraction reaches a critical value. The main purpose of this paper is to demonstrate a new way of extending the crack length when the critical value has been reached. The results presented in this paper are the successful preliminary achievements of ongoing research work.
Keywords: Finite element method-crack propagation-void volume fraction
1. Introduction
analytical models were mainly focused on stationary
In the past, stationary analysis of failure in the
solutions, which could provide helpful information
materials has been the main research interest in
about the defect area [3]. In FE models, the region
fracture mechanics [1].
Moreover, failure of
with higher stress concentration is ideally divided
materials, nucleation of voids and flaws and
into smaller elements compared to the remote area
extension of crack length under different loading
with more uniform stress distribution.
situations has been studied for static or pseudo-static
elements are usually generated in the crack region in
conditions [2].
In contrast, in recent years,
order to predict solutions very local to the crack tip.
numerical simulation of crack growth has revealed a
In numerical analyses, the crack growth can be
new frontier to the dynamic aspects of fracture
simulated using two methods.
mechanics.
However, the discrete nature of
extend the crack length either by setting the stiffness
numerical solution is still a major challenge to
of an element at the tip of the crack equal to a near
provide more accurate results. The choice of method
zero value or by removing the element.
and the criterion for crack extension in the numerical
second method a duplicate node or new boundary
simulation will influence the crack growth rate.
conditions can be introduced to release the crack tip
Denser
First method can
In the
node when a critical value, for crack extension, at At present, finite element is the most commonly
that region is reached. In this paper, the application
used numerical method in fracture mechanics.
of the latter method will be presented. The void
Earliest applications of numerical analysis as well as
volume fraction failure criterion [4] is used to make
_________________________________ 1-Assistant Professor, PhD 2-Principal Research Fellow, PhD 3-Senior Lecturer, PhD
the decision for releasing a node at the crack tip.
2. Local Failure Mechanism
the material is considered to be fully voided, and
The void nucleation at the tip of a crack determines
will have no stress carrying capacity. Therefore, the
the amount and direction of crack extension in the
material softening is expected when it is under
material. The porous metal plasticity model can be
tensile load due to growth and nucleation of the
used to model the voided metals, which are still
voids.
considered
result in a material hardening behaviour due to
to
be
plastically
incompressible.
Therefore the pressure-dependent behaviour of a
In contrast, the compression loading will
closing of the voids and reduction in f parameter.
plastically deformed metal is related to the density Gurson’s model [4] describes a yield
The void volume fraction f can be expressed in
condition based on upper bound solution of a rigid-
terms of the current value of f and the plastic strain
of voids.
plastic material with a ductile void concentration as follows σ Φ= σ y
rate tensor εD p , which is directly related to the requirement of plastic incompressibility of the
2
+ 2q1 f cosh − q 2 3σ h 2σ y
− 1 + q3 f
(
2
)= 0
(1)
matrix metal as f = ∫ (1 − f ) εD p : I dt
where f is the volume fraction of the voids in the
(5)
material, q1 , q 2 and q3 are the material constants [5]
where εD p , is the plastic strain rate tensor which is
and may be considered as fitting parameters directly
derived from the yield potential equation and the
related to physical mechanism of void growth. They
presence of the first invariant of the stress tensor in
explicitly depend on the elastic-plastic behaviour of
the yield condition will result in non-deviatoric
the bulk material [6], σ h is the hydrostatic stress,
plastic strains as ∂Φ εD p = λD ∂σ
σ y is the yield stress of void-free material, σ is the
Von-Mises stress which is defined as σ =
3~ ~ σ :σ 2
The deviatoric stress σ~ can be expressed as ~ =σ I+σ σ h
(2)
(6)
where λD is the non-negative plastic flow multiplier and ∂Φ 3 ∂Φ ~ 1 ∂Φ σ− = I ∂σ 2σ ∂σ 3 ∂σ h
(3)
(7)
where I is second order unit tensor and σ is the macroscopic Cauchy stress tensor at the local crack region.
The hydrostatic stress σ h is therefore
3. FE Numerical Analysis The three point loading condition shown in Figure 1 is used to simulate a highly stressed region with a
defined as 1 σh = − σ:I 3
high
stress
concentration.
The
geometrical
(4)
characteristics of the test specimen are described by
The minimum value of f = 0 for volume fraction of
the crack depth, a, plate width W and the overall
the voids in the material implies that the material is
height of the specimen, L. In this FE analysis a deep
fully dense and the standard Mises yield surface can
crack a/W = 0.5 has been analysed with L/W = 1.7
be obtained. If 0 < f < 1 , the yield surface depends
and a crack length of 30 mm were used. The FE
on the hydrostatic stress σ h . However, when f = 1 ,
analysis was carried out using ABAQUS finite
element code [7]. The FE model consisted total
Due to symmetry of crack one side of the crack
number of 1883 elements and 1982 nodes.
boundary can be controlled during the analysis.
a
Loading
2L
W Figure 1: Schematic description of a three point loading condition containing a crack of length a.
3.1 Bulk Material Property The stress-strain relationship in the plastic regime for the material is described by a power law relation of the form, ε σ = ε y σ y
Figure 2:Finite element mesh used in the analysis.
n
(8)
where ε and σ are the equivalent Mises strain and
When a node is needed to be released the boundary condition of the FE model is modified so that the
stress, respectively, σ y the yield strength, n the strain
node at the crack tip can freely move in
hardening exponent and ε y = σ y / E the yield strain,
perpendicular direction to the crack line. This is
where E is the Young’s modulus. The values of E, n
performed by specifying a new set of boundary
and σ y respectively are 267 MPa, 10 and 348 MPa.
conditions automatically using an external Fortran
3.2 Automatic Crack Growth Procedure Since
the
material
is
homogenous
and
the
mechanical property is the same for the whole
program. The new boundary condition can either be specified by introducing a duplicated node or by changing the boundary condition properties of the node.
Both methods can be used here since the
crack shape has symmetry axis.
Unsymmetrical
model, the crack is expected to grow in straight line.
cases can arise from non-homogenous material, and
The geometry of the model is symmetric. Therefore,
unsymmetrical geometries.
in the FE model, half of the geometry was meshed.
In symmetrical cases,
the crack face is modelled on the boundary. In this
are released. As crack length increases the specimen
work, the change in boundary condition property of
weakens and the load carrying cross-section reduces.
the node at the crack tip was chosen. A C-Shell
This will increase the maximum value of the stress
program is always present to drive both FE and
distribution at the crack tip. Figure 4 illustrates the
Fortran
new
stress concentration at the crack tip over the period
boundary conditions are specified by the Fortran
of crack extension. Each point ahead of crack tip
program, when the FE analysis has passed the stress
experiences an increase and afterward a decrease in
analysis increment.
stress magnitude as crack tip geometrically moves
In Figure 3, it can be seen that the crack extension to
forward (see Figure 4a-d).
programs
simultaneously.
The
any length can be achieved when subsequent nodes
(a): ∆a=0.225 mm
(c): ∆a=0.75 mm
(b): ∆a=0.375 mm
(d): ∆a=1.35 mm
Figure 3: Simulation of crack growth using void volume fraction criterion.
(a): ∆a=0.225 mm
(b): ∆a=0.375 mm
(c): ∆a=0.75 mm
(d): ∆a=1.35 mm
Figure 4: History of Stress distribution during crack length extension.
4. Conclusion Numerical simulation of crack growth, using FE method, was demonstrated in this paper. A local failure mechanism based on void volume fraction was used to determine the time of crack extension, which has been modelled as modification, in the boundary conditions.
A new set of boundary
conditions, at the crack tip, are specified when a critical value for void volume parameter is reached. It was demonstrated that the crack extension to any
[2] Shih, C. F., Relationships between the J–integral and the crack opening displacement for stationary and extending crack, J. Mech. Phys. Solids 1981 29 4 305–326. [3] Lei, Y. and Ainsworth, R. A., A J estimation method for cracks in welds with mismatched mechanical properties, Int. J. Pres. Ves. & Piping 1997, 237–246. [4] Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I– Yield criteria and flow rules for porous ductile media, J. Engng. Materials and Technology 99 1997 2–15.
length is achieved when subsequent nodes at the crack tip are released simultaneously.
The results
presented in this paper are the successful preliminary achievements of ongoing research work.
References [1] Needleman, A. and Tvergaard, V. J., An analysis of ductile rupture in notched bars, Mech. Phys. Solids 32 1984 461–490.
[5] Tevergraad, V., Influence of Voids on Shear Band Instabilities Under Plane Strain Condition, Int. J. Fracture Mechanics, Vol. 17, 1981, 389-407. [6] Faleskog, J., Gao, X. and Shih, C. F., Cell model for non-linear fracture analysis—I: Micromechanics calibration, (1998) Int. J. Fracture 89 4, 355–373. [7] ABAQUS, Version 5.8, (1998). HKS Inc, Pawtucket, RI 02860, USA.
