Measurement and Identification Techniques
for Cracks: Application to Cyclic Fatigue
Julien Réthoré, Stéphane Roux and François Hild
LMT-Cachan, ENS Cachan / CNRS-UMR 8535 / Université Paris 6
61 avenue du Président Wilson, F-94235 Cachan Cedex, France
{rethore,stephane.roux,hild}@lmt.ens-cachan.fr
Introduction
One of the commonly encountered mechanical failures is fracture dominated. It usually occurs before large scale yielding.
However, in many materials, small scale yielding arises in the vicinity of the crack tip. Therefore, the classical elastic solutions
are only first order idealizations of the practical situation. Elastic-Plastic Fracture Mechanics (EPFM) aims at analyzing more
complex situations for which plasticity needs to be accounted for with global quantities. Different quantities are introduced to
describe cracks in a small scale yielding regime. The crack opening displacement was proposed as a fracture parameter to
analyze propagation under quasi-static and cyclic loading conditions. However, critical values of crack opening displacements
are not always easy to evaluate. Rice considered the potential energy changes induced by crack growth in a non-linear elastic
material, and proposed to use a path-independent contour integral, the J-integral, to characterize it. It can be used to predict
the inception of crack propagation. The so-called HRR fields were also introduced to describe more locally the stress and
strain fields in non-linear elastic materials. With the development of full-field measurement techniques, some of the above
discussed aspects can be assessed experimentally without having to use numerical simulations. Under these circumstances,
the measurable quantities are usually displacement fields. Among various techniques, Digital Image Correlation (DIC) allows
one to estimate full displacement fields based on a series of digital images of the surface of a specimen subjected to a specific
loading history. Recent advances have been achieved through a novel formulation that enables one to decompose the
searched displacement field onto a suited library of such fields. The latters are for instance finite element shape functions,
which open the way to a subsequent identification step. Furthermore, in the same spirit as performed in the framework of
numerical simulations of cracked bodies with XFEM, an extended interpolation scheme is proposed herein. It includes
discontinuous functions as well as singular functions in the vicinity of the crack tip.
A multiscale approach has been proposed to avoid spurious local minima, and to provide a robust and accurate displacement
measurement. The uncertainty that is reached lies typically in the range 10-2 to 10-3 pixel size for displacements. The interest
of this experimental tool is that it provides full kinematic fields. For cracked samples or structures, it thus gives access to a
wealth of data that are exploited to estimate mechanical properties. Two routes are followed. First, by using a standard least
squares technique, stress intensity factors and crack tip locations are determined by using a known displacement basis.
Second, an interaction integral formulation gives also access to stress intensity factors. By choosing suitable test functions,
the minimization of a scalar product with respect to measurement noise yields the optimal basis to extract stress intensity
factors.
A cracked steel sample (CCT geometry) subjected to a fatigue test is analyzed in the sequel. A far field microscope provides
pictures such that the physical size of one pixel is about two micrometers. Consequently, displacement fluctuations on the
order of tens of nanometers are measurable. The value of stress intensity factors is analyzed and the two identification
techniques are compared.
Extended finite element discretization
To account for the presence of a crack, the standard finite element shape functions basis is enhanced by using discontinuous
and singular functions. As initially proposed by Moës et al. [1], the partition of unity property of finite element shape functions
is exploited. This allows one to introduce local “patches” with extended degrees of freedom associated with extended
functions supported by a partition of unity. Then, the following expression holds for the discrete approximation of the
displacement field
u=
∑ N i (x )ui +
i∈N
4
∑ N i (x )H i (x )d i + ∑
i∈N cut
∑ N i (x )F j (x )bij
j=1i∈N tip
where Ni are standard finite element shape functions covering the set of nodes N, ui their associated degrees of freedom, Hi
discontinuous functions, di additional degrees of freedom associated with those discontinuous functions, Fj branch functions
and bij their associated degrees of freedom (Fig. 1). Ntip is the set of nodes holding a singular enrichment; i.e. those of the
element containing the crack tip. Ncut contains the nodes of elements that are cut by the crack without containing the crack tip.
Figure 1. Enrichment strategy: squares, respectively circles, denote nodes that hold singular, respectively discontinuous,
degrees of freedom.
The discontinuous functions are derived from the Heaviside step function which is 0 below the crack surface and 1 above
( )
H i ( x ) = H ( x ) − H xi
The branch functions describe the square root behaviour of the displacement field in the vicinity of the crack tip
{F }= ⎧⎨
⎫
⎛θ ⎞
⎛θ ⎞
⎛θ ⎞
⎛θ ⎞
r sin⎜ ⎟ , r cos⎜ ⎟ , r sin⎜ ⎟sin(θ ), r cos⎜ ⎟sin(θ )⎬
⎝2⎠
⎝2⎠
⎝2⎠
⎝2⎠
⎩
⎭
Digital image correlation principle
The principle of digital image correlation is to relate two digital images described by their gray level functions g and f. These
images correspond to a deformed and initial state of the solid of interest. The displacement field between these two states has
to be determined as accurately as possible. The local optical flow conservation reads
g (x ) = f (x + u ( x ))
Because a discretization scheme is used over the entire region of interest, an integral form of this basic principle is searched
for. The aim is now to minimize
η2
defined as follows
η 2 = ∫∫ [g ( x ) − f ( x + u ( x ))]2 dS
Because of the highly non-linear nature of this objective function, a Newton iterative procedure is adopted assuming f is
differentiable. Inserting the above proposed discretization scheme in the minimization algorithm, a linear system is to be
solved for each iteration i [2]
MUi = B
where U
i
contains the nodal unknowns at iteration i
[
(
)][
(
)]
[
(
)]
M mn = ∫∫ ψ m .∇f x + u i −1 ( x ) ψ n .∇f x + u i −1 (x ) dS , Bm = ∫∫ ψ m .∇f x + u i −1 ( x ) [g (x ) − f ( x )]dS
and
ψn
are the vector form of the discretization functions. This iterative determination of the displacement field is inserted
into a multi-grid resolution strategy in order to avoid local minima. Because of the very irregular texture of the images, Gauss
quadrature is not well suited for the integration of the M matrix. A pixelwise integration is preferred. Within the elements that
are cut by the crack, the displacement field is discontinuous and the gradient of the initial image is computed over two distinct
areas.
A priori performance
Using an artificially deformed image, the performance of the proposed extended technique is demonstrated. An optimum
element size of 32 pixels is obtained (Fig. 2). The enriched degrees of freedom are supported by bilinear shape function, and
then the interpolation of the displacement jump is linear within the elements cut by the crack. Consequently, even though the
uncertainty should decrease with the element size, a limit due to the discretization error made within the elements that hold
only the discontinuous functions is reached. The typical uncertainty is about 0.01 pixel, which allows one to have a good
estimate of the displacement field.
Figure 2. Uncertainty analysis and estimated displacement jump along the artificial crack.
Analysis of a fatigue experiment
The studied sample has a CCT geometry (Fig. 3-c) and is subjected to cyclic tension with a load ratio R = 0.4 [4]. In the
present analysis, only the stage corresponding to the maximum load level is considered after about 300,000 cycles for which
the crack size 2a = 14.5 mm. Pictures are taken by using a long distance microscope and a CCD camera (resolution: 1024 x
1280 pixels, dynamic range: 12 bits) so that the physical size of one pixel is 2.08 µm. At this magnification, the raw surface is
observed (Fig. 3). The studied material is an XC48 (or C45) steel. The Young's modulus is 190 MPa and the Poisson's ratio
0.3.
Figure 3. Initial, deformed images and geometry of the CCT specimen.
The displacement analysis is carried out first by using standard bilinear shape functions with 32-pixel elements over the entire
image (Fig. 4-a). The kinematics close to the crack is not well described. Second, the analysis is carried out by using the
proposed extended digital image correlation technique (Fig. 4-b).
Figure 4. Displacement normal to the crack for the CCT specimen expressed in pixels:
(a) by using standard bilinear shape functions, (b) by using extended digital image correlation.
Stress intensity factors measurement
From the displacement field obtained by the previous analysis, it is possible to determine the stress intensity factors KI and KII.
The first route is a least square minimization between the measured displacement field and its analytic asymptotic
development around the crack tip. Rigid body contributions are included in this minimization as well as mode I and mode II
fields. Also included are the term due to the T-stress and sub-singular terms. A value of 23 MPa.m1/2 for KI and 0.5 MPa.m1/2
for KII. The uncertainty of this estimation is of the order of 1.5 MPa.m1/2.
Second, an interaction integral is used to obtain the stress intensity factors from the displacement field. Here, it is proposed to
improve the approach proposed by Réthoré et al. [3]. Let us start with the domain independent interaction integral from
I = −2
(1 − ν )
E
∫∫ [σ ml
aux
(
)]
um,l δij − σ ijaux ui,k + σ ij ui,aux
k qk, j dS
where the superscript aux refers to an auxiliary field. The latter can be chosen to extract the singular contributions from the
displacement field u. The usual choice is the asymptotic crack tip field in mode I and mode II. Then, the following “interaction”
Irwin relationship holds
aux
I = K Iaux K I + K II
K II
aux
aux
aux
aux
= 1, K II = 0 , or K I = 0, K II = 1 , and the interaction integral
One may choose the auxiliary fields such that K I
yields KI, respectively KII. The other special feature of this interaction integral is the virtual crack extension field q. The latter
enables one to write the interaction integral as a domain integral. It is constrained by the following propositions
|| q ||=1 at the crack tip, q is tangent to the crack faces, || q ||=0 outside the integration domain
The variation of q between the crack tip and the boundary of the integration domain are not constrained. In the proposed
approach, this freedom is exploited to minimize the influence of a measurement uncertainty on the estimation of the stress
intensity factors. By adopting two schemes (for the displacement and the virtual crack extension field), the interaction integral
is rewritten as
I =UT L Q
th
where L collects the contribution to the interaction integral of the i shape function of the displacement discretization scheme
th
and j shape function of the virtual crack extension discretization scheme. Then the perturbation of a generic stress intensity
factor K due to a measurement uncertainty
δu
reads
δK = δU T L Q
Because of the linear dependence, the mean of this perturbation is proportional to the mean of
δu .
Figure 5. Optimal virtual crack extension field for mode I and mode II.
Illustration of the noise sensitivity reduction as a function of the domain size.
Therefore, for a white noise, the mean perturbation is vanishing. Furthermore, the variance of this perturbation reads
δK 2 = Q T LT L Q δu 2
Since the variance of the stress intensity factor perturbation is a quadratic form of the virtual crack extension field, a
minimization leads to an optimal virtual crack extension field. That is to say that q can be optimized to minimize the influence
of a measurement uncertainty on the estimation of stress intensity factors. Moreover, as L depends upon the desired mode, q
is specialized for mode I and mode II. The improvement is illustrated in Fig. 5. One observes that the uncertainty sensitivity
-3
for the optimal field varies as r , this result can also be deduced from a limit analysis. A classical conical field, being singular
-2
at the crack tip, displays a slower decrease as r . As a consequence, the improvement of the uncertainty sensitivity varies as
-1
r . For example, compared to the results obtained with a conical virtual crack extension field, the variance of δK is decreased
by a factor of 100 for a domain size of 500 pixels. The values of the stress intensity factors obtained with this method are
presented in Fig. 6. One can check the domain independence of the estimation, which leads to a value of 23 MPa.m1/2 for KI
and 0 MPa.m1/2 for KII. This result is in good agreement with those obtained by using the least squares method. However, the
uncertainty of this estimation is about 1.0 MPa.m1/2.
Figure 6. Stress intensity factors obtained with the optimal virtual crack extension field.
Summary
This contribution presents the development of an extended digital image correlation technique for cracked bodies.
Discontinuous as well as singular enrichments are added to a finite element approximation exploiting their partition of unity
property. This enhanced discretization scheme is used to solve the minimization problem associated with the correlation of
two gray level images by the “passive” advection of their texture due to a displacement field. The performance of this
approach is illustrated through an artificial test case and an experimental analysis. After measuring the displacement field, two
routes are followed to estimate stress intensity factors. First, a least squares minimization between measured and analytical
fields is proposed. Second, an improvement of the interaction integral is developed. It aims at exploiting the not so
constrained virtual crack extension field in order to reduce the sensitivity of the stress intensity factors estimation to
measurement uncertainties.
Acknowledgments
This research was funded by the CETIM Foundation through a grant entitled PROPAVANFIS: “Advanced methods for the
experimental and numerical analysis of crack propagations under complex loadings.”
References
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Journal for Numerical Methods in Engineering, 46, 133-150 (1999).
2. Besnard, G., Hild, F. and Roux, S., “Finite-element displacement fields analysis from digital images: Application to PortevinLe Châtelier bands,” Experimental Mechanics, 46, 789-803 (2006).
3. Réthoré, J., Gravouil, A., Morestin, F. and Combescure, A., “Estimation of mixed-mode stress intensity factors using digital
image correlation and an interaction integral,” International Journal of Fracture, 132, 65-79 (2005).
4. Hamam, R., Hild, F. and Roux, S., “Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue,”
Strain, accepted.