ABSTRACT:
This paper presents parameter-estimation methods developed to determine crack-tip field parameters in anisotropic elastic solids from full-field experimental data. The crack-tip field parameters of interest include not only stress intensity factors but also effective crack-tip positions. Two approaches that are based on least-squares method and conservation integrals were presented. In the approach based on the least-squares method, the Stroh representation of anisotropic elasticity was used to determine the coefficients in the asymptotic expansion of crack-tip fields in anisotropic solids. On the other hand, conservation integrals in fracture mechanics, such as J-integral, M-integral, Interaction integrals, were also used to extract the parameters including the effective crack-tip position. Both approaches were employed to analyze crack-tip displacement fields obtained by using Digital Image Correlation technique during translaminar fracture test of a fiber- reinforced polymer matrix composite. The applicability of homogeneous isotropic plane-elasticity models to translaminar fracture processes in laminated composite materials is also discussed.
INTRODUCTION
Fiber-reinforced composite materials have been widely used in aerospace, automotive and other industries owing to their high strength-to-weight ratio. However, it is well known that heterogeneous microstructures in composite materials introduce a variety of failure mechanisms and damage modes, which makes it difficult to develop reliable theoretical models to describe and predict such failure processes. Therefore, research efforts to understand the complexity of fracture processes in composites heavily relies on experimental observations and measurements. Up until recently, most of the experimental investigations of composite fracture are based on measuring far field loads and overall deformations during mechanical tests. Then, the crack-tip field parameters, such as stress intensity factors, are inferred through analytical formula or numerical calibrations in order to characterize fracture behaviors of composites.
On the other hand, owing to the advances in full-field optical measurement techniques, the crack-tip field quantities, such as stress intensity factors, can be directly determined from measured near-tip deformation fields. Thus, the applications of optical techniques to investigate fracture behavior of engineering materials have become routine tasks. Nevertheless, a relatively fewer number of investigations based on optical techniques were conducted to investigate fracture processes of fiber reinforced composite materials. This is because the extraction and interpretation of the crack-tip field parameters in composite require special considerations on material heterogeneity and anisotropic behavior, when compared to homogeneous isotropic materials.
In this study, we integrated the digital image correlation technique into conventional fracture tests to provide better insight to the progression of fracture processes in composite materials during the tests. Firstly, the description of asymptotic crack-tip fields in anisotropic elastic solids was presented following Stroh representation [1, 2] of general anisotropic elasticity. Based on the asymptotic expansion of crack-tip fields, two approaches to determine the crack-tip field parameters from near-tip deformation fields were presented, namely the least-squares method [3, 4] and the conservation integrals [5-7]. Special considerations were made to determine not only the stress intensity factors but also the effective position of a propagating crack-tip, so that crack growth resistance-curve (R-curve) can be constructed solely from the optical measurements. Both approaches were employed to characterize translaminar fracture behavior of a fiber-reinforced polymer matrix composite.
ESTIMATION OF THE CRACK-TIP FIELD PARAMETERS IN ANISOTRPIC ELASTIC SOLIDS
Asymptotic expansion of crack-tip fields in anisotropic solids
Consider an in-plane elastic field of a general anisotropic solid. Then, the in-plane displacement components uk (x,y) can be represented by using the Stroh formalism of anisotropic elasticity as follows,
where the functions
are analytic functions of complex variables,
and the 2 x 2 matrix, .
depends only on the elastic stiffness tensor,
. Each column of the matrix Aka and the characteristic complex root pa are the eigenvector and the eigenvalue of the following eigenvalue equation, respectively
Then, the stress components cr 2 can be represented as
where, the matrix Lka is given by
A general form of crack-tip fields in a semi-infinite crack problem with the origin at the crack tip can be expressed using the Stroh formalism. From the traction-free boundary condition at the crack plane located in the negative real axis, the eigenfunction expansion of the elastic fields can be obtained from Hilbert arc problem as
and both fn and gn are column vectors with two real-number elements. In this
and both fn and gn are column vectors with two real-number elements. In this expression, the stress intensity vector defined as
is related to the f0 as follows,
where mode-I and mode-II stress intensity factors are K} and Kn , respectively. In this representation, the displacement and stress fields including the higher order terms are represented as eigenfunction expansions similar to Williams-type expansion for the isotropic case.
Least squares method
Least squares method (linear and non-linear) is the most commonly-used data-fitting method. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model.
In the present study, the full-field displacements around the crack tip are obtained by DIC. The analytical eigenfunction expansions for displacement components around the crack tip are known in terms of Stroh representation. If we also consider the rigid body translation in the results obtained by DIC, the expansions for the displacements around the crack tip can be rewritten in more general forms as
or in the matrix notation
Here x and y are the coordinates in the coordinate system with origin at the crack tip, N represents the number of terms in the expansion used in calculation (theoretically
), and hn are constants related to Fn and Gn in Stroh representation, which need to be determined in the least-squares.
Then the sum of squared residuals of the displacements for points can be presented as
Here ui and vi are the experimental results of data i obtained by DIC. The minimum value of E occurs when the gradient is zero, i.e.
Then the following linear equations are obtained as:
It is known that u and v are linear functions with respect to the coefficients hn and non-linear functions with respect to the coordinates x and y . If the crack tip position is known (so are the coordinates x and y ), linear least-squares method can be used to get the unknowns by solving the above equations directly [3].
However, in the present study of composite, a damage zone usually occurs in front of the crack tip. The crack tip position is not easy to be determined visually. Thus we use the non-linear least-squares which treat the crack tip positions as unknowns as well [4]. However, the non-linear least squares method has one problem that sometimes it can not converge to reasonable values especially when the higher order terms of the eigenfunction expansions are included in the calculation. A possible solution is to combine the linear and non-linear least-squares. The method is also based on an iteration procedure. The visible crack tip position is firstly used as the initial guess. Then the other unknowns are obtained by linear least-squares method. Secondly the obtained unknowns are back substituted into the nonlinear least-squares part to get the crack tip position. Then this obtained crack tip position is used as the guess for the next iteration step. The procedure keeps moving until it finds a convergent result. In the present study, the linear least squares method is combined with the nonlinear least-squares method called Levenberg-Marquardt algorithm. Another thing which is worth mentioning here is that it has been reported that the number of the terms can influence the accuracy of the calculation [3, 4]. Generally, it is found from our analysis to the artificially constructed analytical displacement field that more number of the terms should be included in the calculation to get a convergent result with smaller error.
Conservation integrals
There are various conservation integrals (path independent integral) in Fracture Mechanics such as J and M integrals [5-8]. J integral is the most famous conservation integral because it represents the strain energy release rate and is related to the stress intensity factors. The explicit expressions of J and M are:
Here w is the strain energy density,
in linear elasticity. Ti are the components of the traction vector,
are the components of the unit vector normal to r . r is a path connecting any two points on the opposite sides of the crack surface and enclosing the crack tip. ds is an element of arc length along r . ui are the displacement components.
It has been derived by Suo [2] that J is related to the stress intensity factors in anisotropic solids in terms of the following relations:
Here
A and L are both 2 x 2 matrices defined in Stroh representation. H is a 2 x 2 symmetric matrix.
Another important conservation integral is the J-based interactional integral [8], which is firstly introduced to separate mode I and Mode II stress intensity factors in the mixed mode problems. Suppose two independent elastic states "A" and "B" for the plane problems, which represent the unknown and auxiliary field respectively. Now consider a superposed state "A+B", the J integral of this state is
Here
is defined as the J-based interactional integral, which has the explicit expression as
Here
Based on the relation of
the relation between the interactional integral
and the stress intensity factors for the two states k A and k B can be obtained as
To get
for the unknown field, two auxiliary field ‘B1" and ‘B2" with
and
interactional integrals are constructed using Stroh representation with the same material properties. Then the corresponding
A Then k can be obtained by the following equations:
For a semi-infinite crack, if the coordinate of the crack tip in y direction is assumed to be zero, then the coordinate of the crack tip in x direction is x0 = M / J with respect to an arbitrary origin.
