Properties of the crack resistance of layered composite and simulation of a crack quasi-static growth

The outcomes of standard test of a carbon/epoxy layered composite using DCB specimen for measuring of the I-mode interlaminar fracture toughness (IFT) were analysed. Using the test results and earlier developed the model of quasi-static growth of crack (delamination) in layered composite, the investigation of variable properties of IFT was continued. It is shown that the use of stationary continuous functions for mathematical description of IFT does not allow to completely satisfy to the data of test. The simplest type of stationary discontinuous function is selected for analysis and simulation: the periodic piecewise-constant function. Using the model of quasi-static growth of crack and the mentioned type of variable IFT function the process of a crack propagation in DCB sample at quasi-static loading with control of displacement is simulated. Comparison of IFT function and the crack resistance (R-curve) together with other outcomes of simulation are the base of proposed procedure of variable IFT estimation from DCB Standard test.


Introduction
The quasi-static growth of a crack is the continuous process of an infinitely slow transition from one equilibrium state to another during which the parameters of any interstate satisfy the equation of energy balance.In a case of a crack growth the main internal parameters of an elastic body change as results of external loading and the increment of crack surface.
In contrast to the classic Griffith theory [1] of brittle fracture and its developments [2,3] the process of quasi-static growth assumes the continuous crack propagation induced by the external load increasing.
There were numerous attempts of analytical description of the quasi-static growth of a crack under increasing load.Critical analysis of the most popular of them was done by [4].Apparently, the first semi-empirical attempt was done by [5].The most perspective are the models based on the energy balance equation.In the approximate analysis [6] the plastic energy dissipation rate is presented as some sum of two parts.One of them is defined by load increase and other by crack size growth.This introduction of the plastic energy dissipation rate allows to describe the stable growth of a crack.Using the Dugdale's model [7] of plastic zone and corresponding estimate of the plastic energy dissipation rate [8] obtained a nonlinear differential equation for load/crack length function.Similar results were obtained also by [9][10][11][12][13][14][15].Comparison of some mentioned energy-based models is done by [16].The new concept is developed in [17,18].
A simplified uniparameter description of quasi-static crack growth is given by [19] at the assumption of the small-scale Dugdale-type cohesive zone.The improved version of a model is presented in [20].The main advantage of this model is invariance of the governing differential equation in respect of geometrical configuration and the means of loading.In the paper [20] there is considered some application of mentioned model for damage reconstruction in the double-cantilever beam (DCB) sample of a layered composite.Important conclusion of this research is on the non-homogeneity of the interlaminar fracture toughness (IFT) of this material.Note that here and anywhere below IFT means the I-mode interlaminar fracture toughness.
This problem was investigated more detail in assumption of the continuous stationary change of the IFT for two shapes of this function [21,22].
The presented paper is focused to the piecewise-constant option of IFT function.

Material, sample, setup and technology of testing, measurement
To obtain experimental information on the interlayer resistance of the layered composite, special test was performed in accordance to the requirements of the standard [23].The DCB-samples of the carbon/epoxy layered composite cute of special industrial panel of 21 prepreg layers (10 Hex ply M21/40%/285T2 + 11 Hex ply M21/40%/285T6) and structure 0°/3 ±45°/3+90°/1.Nominal dimensions of a sample: total length is equal to 160 mm, the width of cross-section 20 mm, total thickness 6 mm.The forces are applied via two loading blocks.Six samples were tested for obtaining statistical data.
The DCB-sample is the most popular type for measuring interlaminar fracture toughness of composite.The standard testing of the unidirectional carbon/epoxy layered composite was done for measurement of the interlaminar crack resistance properties of this material.The quasi-static tests of DCB sample (figure 1, a), according to the standard, were carried out on an Instron 8800 hydraulic testing machine with controlled extension at a constant rate of 5 mm/min.To increase the accuracy of measurement of small values of the load, an S2M meter of small loads (HBM Test and Measurement), with the upper measurement limit of 1 kN, was connected in series in the loading circuit as the basic force transducer (figure 1, b).After each jump-like increment of delamination, the loading was interrupted, the crack (delamination) length was visually registered (Fig. 3b) using low-sensitive microscope (x10).

Testing results and their pre-processing
A typical original record of the load/extension process (model number 2) is presented in figure 2. It includes 15 loading stages at a speed of 5 mm / s.Each stage ended with a load interruption for visual The same data in the detailed presentation are shown in figure 3. The main record is represented by a dot line with circular markers.In portions of stable growth of the crack, the markers form a continuous bold line, and in the portions of the crack jump only a dot line is visible.Dashed straight lines approximately represent the lines of complete unloading, after which residual extensions are formed.The maximum value residual extension at test end is equal 1 mm (see figure 2).The mentioned above features of process of DCB sample destruction can be considered more details using the 'crack length/force' and the 'crack length/extension' functions (figure 4 and figure 5).Note that crack length was basically defined by compliance method with correction using results of direct measurement and correction item of Standard [23] for accounting of effect of rotation of cross-sections at delamination front.For here presented sample this correction is equal to 4.394 mm.

Several main outcomes of test.
Full report on the test outcomes can be found in [22].Below are presented only three main results important for that paper.
First, there is defined that only 2.73-8.23% of total crack surface increment in DCB sample Standard test occurs at the crack stable growth, but more than 92% of total surface of destruction corresponds to unstable non-controlled stages.Second, for any unstable stage of crack growth there were obtained boundary values of crack resistance:1) the initial IFT that corresponds to start of unstable growth   (maximum) and 2) the ending value of IFT at crack arrest   (minimum).Generalized information of this outcome for all tests and samples is presented by figure 4.There is registered approximate stationarity for both parameters and a relatively large scattering of statistical estimates.
Third, the estimates of mean rate of crack unstable growth were obtained.Existence of the close correlation between mean rate of crack unstable growth and difference ∆ = (  −   ) was determined.
The statistic of mean rate of crack unstable growth is important result of this investigation.Particularly, it is determined that maximal rate of crack unstable growth is not more than 180 mm/s.

On the aim of study and the variable IFT option
Here there is continued the heuristic analysis of variable properties of IFT which was started earlier.If to assume that the crack resistance is some variable function of crack length with the reference parameter  0 , then the crack resistance may be conveniently represented by equation (1).
=  0 (1) where  is a dimensionless function of crack .If the average crack resistance is a constant, then it is usually declared as the IFT of material.So,  can be called as the normalized crack resistance function.
In [21] the version of a smooth IFT function was considered as trigonometrical sinus.The piecelinear periodic function with variable amplitude is selected for description of variable IFT in [22].Both those options partially satisfied the experimental data.In both cases the stable part of crack growing can be satisfactory described by equation of the separate integral curve of theory of quasi-static crack growth.But in both cases description of crack growth is incorrect about critical force of each separate step of loading.
In general, the  is random function, at least, with two parameters (position and scale).For isotropic material the normalized crack resistance  must be periodic or steady random function and satisfied with the condition of isotropy of structural properties.Here the analysis is focused on the periodic piecewiseconstant option of IFT function which the mean value is equal to 1.One period of function by length  contains  intervals of length   those satisfy to condition: and in interval   the function deviation from mean value is equal to   .
More detail if crack length is in interval below:   ≤  <  +1 where and  is currant number of period, then Here the set of horizontal solid segments indicates the piecewiseconstant IFT normalized function.The IFT function mean value is constant and is equal to 1.The normalized crack length is represented here as the ratio to initial crack length  0 .The same length unit is used for normalized representation of period of IFT function and each containing interval of this period.
In general, the IFT all parameters (,   ,   , ) are random variables, but for aims of this research there is sufficiently to assume that the numerical characteristics of those random variables are independent from crack length (condition of stationarity of IFT function).Estimation of main parameters of IFT can be obtained from Standard test and this problem briefly is discussed below.

Modified invariant equation and solution for DCB sample.
In this case the invariant equation of theory of a crack quasi-static growth [22] transforms to follow: where  = /  ,  = / 0 ,   =    0 ⁄ and  =    ⁄ .The  and   are current and critical forces,  and  0 are current and initial crack length,   is critical size of cohesive zone of a crack.
Note that here in equation ( 5)  is the strain energy release rate which is defined by equation (6).
where  is a current force,  is corresponding extension,  is the width of the cross-section of a sample, and  is the cylindrical stiffness of cross-section.The crack length  is given as a sum of actual crack length and correction member in accordance with the Modified Beam Theory method of standard [23].More detailed description of equation ( 5), resolving method, properties, and examples of solution for some configurations of a cracked body can find in [18,19].
For DCB sample the normalized strain energy release rate as the function of normalized force and crack length  =  2  2 (7) For this case the equation ( 5) can be converted to follow:  Below the piecewise-constant periodic IFT function with variable amplitude is used with the input data of function () is given above.In figure 6 DCB sample loading diagram with displacement control is represented.The general quasi-static solution of equation ( 8) is shown by the dote-type curve which on the intervals of the stable crack growth is covered by solid curves.Each of these curves is a part of separate integral curve of solution of equation ( 8).Here is shown also the analytical continuation (to level of zero force) of the full stable part of the integral curve.The vertical dash-type segments correspond to unstable increment of a crack and transition to the stable part of next integral curve.
The same system of designations is used for presentation of other parameters of problem.In figure 7 there is shown the function between a force and crack length at considered kind of loading.The variable piecewise-constant IFT causes the spasmatic crack growth: stable stages alternate with jump-type crack increments.The general trend of force decreasing is observed.Figure 8 illustrates the predicted process of crack growth at quasi-static loading with control of displacement.The basic characteristics of crack resistance R-curve is interrupted (figure 9) Figure 9 shows that the R-curve as the general characteristic of crack resistance of material cannot be obtained directly from Standard test [23] if this material IFT is variable.It is seen that more than 64% of the new surface of the crack occurs in dynamic regime at the unstable growth of a crack.It means that information on the crack resistance in this part of surface will be lost.

Discussion and conclusion. Estimation of IFT from DCB test.
The presented paper focuses on the continuation of the study of the non-homogeneity of the first mode of interlaminar fracture toughness (IFT) of a layered composite.As is known, the motive for this study is caused by the inadequate spasmodic growth of delamination (interlaminar crack) in some types of unidirectional layered composite during the standard DCB-sample test to determine the IFT.This nature of the crack growth can be satisfactorily explained if to assume that IFT is a local characteristic of crack resistance, which varies along the surface of the crack and only its mean value is a constant of the material.
The model of quasi-static crack growth can be used for investigation of the general regularities of the interlaminar crack resistance of a layered composite of mode 1.The heuristic approach to the analysis of this problem involves the free choice of the type of crack resistance function, which seems promising and satisfies the requirement of stationarity.At each stage of stable crack growth, the parameters of the integral curve of solution of the invariant differential equation of the theory of quasistatic crack growth are determined from the condition of best compliance with the experimental data obtained in standard tests of the DCB sample.Simplified simulation of this procedure was performed for two types of periodic continuous crack resistance functions [21,22].In the paper [21] the option of sinusoidal shape there is examined for description of IFT function.Other option of IFT function there is used for the same purpose in [22].There the continuous periodic piece-linear function (saw-type) simulates the hypothetic IFT in layered material.In both cases the satisfied prediction of crack stabile growth there is obtained everywhere excluding surround of the point of the maximal force.In this point the prediction gives: But in tests this derivative is usually more than zero (it can be seen in figure 2).Note it seems that condition ( 9) is common for all sets of continuous IFT functions.Therefore, in this paper the piecewise-constant periodic function accepted as the simplest description of discontinuous variable IFT.The presented analysis shows that this description provides the satisfaction of all requirements and allows to considerate inverse problem: to define the IFT function from DCB test using theoretical relationship between the piecewise-constant IFT function corresponding R-curve.In figure 10 is shown IFT function and corresponding Rcurve graphs.As also above (in figure 9) the R-curve stable parts are shown by solid lines, and unstable parts by dash-type lines.The IFT function (like figure 5) is presented here by the set of dot-type horizontal segments.It is seen that the value   of IFT is upper boundary for corresponding stable part of Rcurve.The bottom boundary is precise limit of the quasi-static solution of R-curve.For any part of stable growth of crack the procedure of estimation of upper boundary of IFT is caused by invariant equation (8).From this equation the relative value of IFT function () is as follow: where the set of parameters , ,   ⁄ can be selected for any point of considered part of stable growth of a crack, and theoretically the result of () calculation must be the same.It seems that in this case the minimal error will correspond to the upper point of considered part of stable growth of a crack.But the integral or least square methods can give more reliable estimates.
In the figure 10 is shown that the unstable crack propagation partly continues at upper limit of IFT.If   is total length of unstable part, then it equal to sum of two intervals   =   +   .At the interval   crack resistance is defined by bottom boundary of IFT, and the crack growing will accelerate.In It is seen in figure 10 that the IFT value of the part of unstable growth of a crack is the bottom boundary of R-curve and coincides with minimum of quasi-static solution of R-curve of considered part.At the interval   the quasi-static solution corresponds its descending branch () with parameters  −1 ,  −1 of end point of the part  −1 and bottom boundary of IFT   .In generalized form this equation can be presented as follows: 1 (, ,   ,  −1 ,  −1 ) = 0 (11) Here   is the bottom relative IFT which should be defined.The interval   defined by the ascending branch of integral curve () in part   of quasi-static solution and initial conditions in point of crack arrest  () ,  () .
In generalized form this equation can be presented as follows:  2 (, ,   ,  () ,  () ) = 0 (12) Here   is the upper relative IFT which definition is described above by equation (9).From DCB test must be obtained additional data on the crack unstable growth length   .The system of two equations (11) and (12) defines the relation between the bottom boundary of IFT   and the crack length, and the value of   can be estimated using information on the mean rate of crack unstable growth.

Figure 3 .
Figure 3. Basic record in more detailed presentation

Figure 4 .
Figure 4. Statistics of crack resistance

Figure 5 .
Figure 5. Example of the periodic piecewise-constant IFT function

Figure 10 .
Figure 10.The IFT function and corresponding Rcurve