Microdamage Modeling in Laminates

Abstract

During service life structures made of laminated composites are subjected to complex combinations of thermo-mechanical and environmental loads. The final macroscopic failure of composite laminate is preceded by initiation and evolution of several microdamage modes in layers. This is because the transverse tensile strain to failure of unidirectional composites is lower than other failure strain components. Therefore transverse cracking of layers with off-axis orientation with respect to the main load direction, caused by combined action of transverse tensile stress and shear stress, is usually the first mode of damage (Parvizi and Bailey, J. Mater. Sci. 13:2131–2136, 1978 [1]; Jamison et al., ASTM STP 836:21–55, 1984 [2]).

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

During service life structures made of laminated composites are subjected to complex combinations of thermo-mechanical and environmental loads. The final macroscopic failure of composite laminate is preceded by initiation and evolution of several microdamage modes in layers. This is because the transverse tensile strain to failure of unidirectional composites is lower than other failure strain components. Therefore transverse cracking of layers with off-axis orientation with respect to the main load direction, caused by combined action of transverse tensile stress and shear stress, is usually the first mode of damage [1, 2].

The crack, see Fig. 6.1a, is usually well defined, it runs parallel to fibers in the layer and the crack plane is transverse to the laminate middle-plane. Often they cover the whole thickness of the layer and propagate over the whole width of the tensile test specimen (may be except for laminates with very thin layers and/or in low stress cyclic (fatigue) loading). These cracks which in this chapter we call intralaminar cracks are called also matrix cracks, tunneling cracks, transverse cracks or inclined cracks (in off-axis layers with different orientation than 90°).

Fig. 6.1

Intralaminar cracks in cross-ply laminate: a schematic showing of laminate with cracks; b multiple cracks with rather uniform distribution; c the crack tip region at the 0/90 interface with local interlayer delamination and fiber breaks in the 0-layer

Intralaminar cracks do not usually cause the final failure of a laminate, but may significantly impair the effective properties of the laminate [3] and serve as a source for other damage modes initiation, such as delamination [4, 5] (Fig. 6.1b), “stitch cracks” [6] and fiber breaks (Fig. 6.1c) in the adjacent plies.

With increasing load or with the number of cycles in fatigue loading the number of cracks increases. Initiation, evolution and effect of these cracks on laminate stiffness has been discussed in many papers, see for example review papers [7, 8]. The extent of cracking in a layer is quantified with an average measure called crack density: number of cracks in a layer over certain distance measured transverse to the crack plane. So the crack density in kth layer is $\rho_{k}$ (cracks/mm). Number inverse to the crack density is called average crack spacing $2l_{k} = 1/\rho_{k}$. Slightly deeper analysis based on features of an elastic stress solution reveals that the distance between cracks measured in, for example, millimeters is what we can measure but it is not the most informative characteristic of the damage state. For example, spacing 1 mm between cracks does not give any information how close to each other they are in the sense of interaction of corresponding stress perturbations. Instead, if we know that the crack dimension in vertical direction (which is equal to the cracked ply thickness $t_{k}$) is 1.0 mm we know that the distance between two cracks in the previous example is equal to the crack size as it approximately is in Fig. 6.1b. This is a very high crack density, close to the maximum possible reached in tests (often called saturation region or the region with strong crack interaction). If the spacing between cracks is larger than $4t_{k}$ cracks may be considered as non-interactive (as a consequence of St. Venant’s principle the stress perturbations caused by them do not overlap). Therefore, a proper measure of distance between cracks is so-called normalized crack spacing $2l_{kn} = 2l_{k} /t_{k}$ with corresponding normalized crack density $\rho_{kn} = \rho_{k} t_{k}$.

Each cracking event in a layer creates two new traction free surfaces. This means that the transverse stress and the in-plane shear stress on the crack surface is equal to zero and new cracks close to existing crack cannot be expected. With increasing distance from the crack the transverse and in-plane shear stresses start to recover and when the distance is large they may asymptotically approach the value as it was in the layer before cracking (called far-field stress). The stress transfer mechanism from the undamaged layer to the damaged is through high intra-laminar shear stresses at the layer interface in the vicinity of the crack. The efficiency of the stress transfer (the distance needed to recover the far-field stress state) depends on the interface quality. The distance is much shorter in case the interface is not damaged and can be very large in case of delaminations starting from the intralaminar crack. The described stress transfer mechanism resulting in stress recovery in the damaged layer allows for creation of many cracks in the same layer (multiple cracking).

A relevant question following this description is: if the distance from the crack to recover the far-field stresses in a laminate with ply thickness 0.25 mm is only a couple of mm why we do not have at least 50 cracks created simultaneously at the same load in a specimen with gauge length 100 mm? Experiments show that each new crack requires an increase of the applied load. The reason is that the transverse and shear failure properties are not the same along the transverse direction of the layer: there are some weaker positions where the first cracks occur and more strong positions requiring larger macroscopic load. As it will be shown in Sect. 6.3 the transverse failure properties have statistical distribution, for example, it can be Weibull distribution for strength [9–11]. In result only a few cracks are created at relatively low load because there are just a few weak locations. Then the cracking rate increases with increasing load because we are reaching loads where the material has the highest probability density of failure and many positions have almost the same failure properties. After that the multiple cracking slows down because there are only a few positions with high value of failure properties; the probability density curve approaches to zero.

There is another mechanism slowing down the rate of cracking in the high crack density region. The stress distribution between two cracks depends on the distance between them (normalized spacing). When the normalized crack density is very high, there is no enough distance for stress recovery and even the maximum values of the in-plane stresses between two cracks become significantly lower than the far-field value. The creation of a new crack between two existing requires significantly higher macroscopic load being applied to the laminate. This in addition to fact that the remaining positions for cracking are very strong slows down and eventually stops the intralaminar cracking.

Due to progressing microcracking, the macroscopic thermo-mechanical properties of the laminate are degraded. We will illustrate the degradation mechanism on symmetric $\left[ {0/90} \right]_{S}$ cross-ply laminate as an example. Assume that we apply to this laminate macroscopic average tensile stress $\sigma_{x}^{LAM}$ and compare its axial deformation before cracking and in the presence of intralaminar cracks in the 90-layer. It is important to realize that in both cases the macroscopic deformation of the 0-layer is equal to the macroscopic deformation of the laminate (this is why strain gauges and extensometers are located on the specimen surface) and therefore the 0-layer strain can be used as a measure of the average axial strain of the laminate $\varepsilon_{x}^{LAM}$ and from there the axial modulus of the laminate $E_{x}^{LAM}$ is calculated. In undamaged state the stress distribution in layers does not depend on coordinate and the Classical Laminate Theory (CLT), which is based on iso-strain assumption can be used to calculate the strain $\varepsilon_{x0}^{LAM}$ and stresses in layers, for example $\sigma_{x0}^{90^\circ }$ (index 0 is added to specify the case with zero damage) and $\sigma_{x0}^{0^\circ }$. If the 90-layer has a crack the $\sigma_{x}^{90^\circ }$ at the crack face is zero. Due to stress transfer over layer interface the stress increases with the distance from the crack and somewhere far from the crack it could reach $\sigma_{x0}^{90^\circ }$. So,

$$\sigma_{x}^{90^\circ } \le \sigma_{x0}^{90^\circ }$$

(6.1)

in any point of the damaged 90-layer. Due to axial force balance the axial force has to be the same in any cross-section of the laminate. This means that because of (6.1) stress $\sigma_{x}^{0^\circ }$ in any position of the 0-layer is larger than in the undamaged laminate,

$$\sigma_{x}^{0^\circ } \ge \sigma_{x0}^{0^\circ }$$

(6.2)

Larger axial stress results in larger local axial strains in the 0-layer, leading to larger macroscopic deformation of the layer. Obviously, the result is larger laminate strain, $\varepsilon_{x}^{LAM} \ge \varepsilon_{x0}^{LAM}$ meaning that the damaged laminate axial modulus is lower, $E_{x}^{LAM} \le E_{x0}^{LAM}$. Quantitative estimation of the change requires knowledge of the stress distribution between cracks.

The quantification simplifies realizing that the average values of in-plane stresses are governing the stiffness degradation and details of the stress distribution are not important. It can be shown using divergence theorem [12] that the average stress applied to the laminate has a rule of mixtures (RoM) relationship to average stresses in layers. Since the average stress between two cracks is always lower than $\sigma_{x0}^{90^\circ }$, from RoM and the force equilibrium follows that the average stress in the 0-layer is higher than $\sigma_{x0}^{0^\circ }$ with the same consequences as described in the previous paragraph.

The simplest way of accounting for average stress reduction in a damaged ply in a model is by replacing the cracked layer with “effective layer” which has “effective” = reduced thermo-elastic properties. How much the properties have to be reduced is an open question. An extreme case of this approach is the well know ply-discount model, commonly used together CLT. Physically this approach is not correct: thermo-mechanical constants of the material in the damaged layer have not changed. Nevertheless, the reduction of elastic constants is a simple way to incorporate the effect of reduced average stress in the layer, still keeping the concept of iso-strain which in non-bending case builds the basis of CLT. However, the common assumption in this approach that transverse and shear properties of a ply with cracks are zero is very conservative and does not reflect the real situation where the number of cracks is increasing in a stable manner during the service life. The ply-discount assumption corresponds to case with an infinite number of cracks when the in-plane stresses between cracks approach to zero. Therefore, requirement that laminate stiffness with increasing crack density approaches the ply-discount model prediction must be satisfied in all predictions based on stress distribution models.

The basic approach, called micromechanics modeling, (see review for example in [13]), is based on perturbation stress analysis. Most of the models are focused on an approximate analytical description of the local stress distribution in the repeating element between two cracks. The simplest calculation schemes used are based on shear lag assumption or variational principles [13–16]. Most of the analytical solutions are applicable to cross-ply type of laminates with cracks in 90-layers only. The most accurate numerical routines based on Reissner’s variational principle are presented in [17]. “Equivalent constraint model” was introduced in [18] to determine the effective properties of the damaged layer. In Zhang et al. [18] his approach was used together with shear lag model. With improved stress model it has the potential of accounting for interaction of cracks belonging to different layers of laminates.

Using the calculated stress distributions between cracks one could find the average value of the stress change in the cracked layer to be used to predict laminate stiffness degradation. In this work we will use a different method to account for the average stress change. The change of in-plane average stresses due to cracking is proportional to the average values of the crack face opening (COD) and sliding displacements (CSD). Certainly the change depends also on crack density in the layer as well as elastic and geometrical constants. So, dependent on the suitability of the used model the stiffness change can be expressed in terms of average stress change or in terms of average COD and CSD. The former method is more suitable when analytical stress distributions are available, whereas the latter is preferable when crack face displacements have been calculated (for example, using FEM). Therefore the damaged laminate stiffness can be expressed also in terms of density of cracks and two parameters: average COD a CSD as done in the GLOB-LOC model [19, 20]. These two rather robust parameters depend on the normalized crack spacing (crack density). In Sect. 6.4 stiffness expressions based on COD and CSD approach are given for a general symmetric laminate with cracks in all layers.

The relationship between the average stress change between cracks and the average value of COD and CSD can be easy explained. If we imagine that somehow the corresponding points on both crack faces are kept together (it would require application of tractions to points on crack surfaces), not allowing the crack to open or the faces to slide (COD = CSD = 0), the stress between cracks would be the same as in undamaged laminate and the laminate thermo-elastic properties would not change. However, we know that under in-plane tension cracks open; under shear their surfaces slide and under compression they are closed. The latter case with closed cracks is very interesting because, as just described, the stress state is as in undamaged laminate and, hence, the stiffness of the damaged laminate in compression equals to the undamaged laminate stiffness.

In the tensile case, as soon as we allow for separation of points on crack faces (opening or sliding), the stress between cracks is reduced. The larger the COD and CSD, the larger is the average stress reduction. The most extreme case is fully delaminated unit between two cracks. Then there is no stress transfer between layers and in-plane stresses in the cracked layer are zero. This corresponds to the maximum possible COD and CSD which can be easy estimated knowing that the 90-layer material in this case is not deformed and the whole load is carried by the rest of layers.

Most of the existing stiffness models use assumption that cracks are uniformly distributed in the layer, with equal spacing between them. It simplifies analysis and is expected to give sufficient accuracy. However, the crack distribution in the layer may be highly non-uniform. This is more typical in the beginning of the cracking process when the average crack density is relatively low. The reason is the random distribution of transverse failure properties along the transverse direction of the layer. At low crack density the stress distribution between two existing cracks has a large plateau region with constant high stress and any position there is a site of possible failure. At high crack density there is a distinct maximum in the stress distribution between cracks and a new crack most likely will be created in the middle between existing cracks.

The possible inaccuracy introduced in laminate stiffness prediction by using assumption of uniform spacing between cracks in a layer has been addressed in [21, 22]. In [21] so-called “double-periodic” approach was suggested to calculate the COD of a crack in a non-uniform case: the COD is found as average from two solutions for periodic crack systems representing the two different distances to neighbouring cracks. Very good agreement of this approach was found with direct FEM solution for non-uniform cracks. It was shown that at fixed crack density the elastic modulus reduction is highest if the cracks are uniformly distributed and in this sense periodic crack distribution models give lower bond to modulus.

2 Experimental Methods for Damage State Characterization

Crack density in a layer enters all expressions for stiffness reduction and it is the main output of damage evolution modeling. Therefore quantification of the damage state is of primary importance for accurate predictions. A short overview of most common experimental methods (optical microscopy, X-ray images, acoustic emission) is given in this section discussing their accuracy, complexity of measurements and other drawbacks.

The effect of each individual crack on stiffness depends on its geometrical features: cracks with large, severe damaged zone at the crack tip and delaminations between layers starting from the crack tip, see Fig. 6.1, lead to more stiffness reduction than “ideal” straight cracks because dependent on these features the crack opening (and the average stress between cracks) may be very different. These details of the damage used in models have to be studied experimentally and below we briefly describe the use of Electronic Speckle Pattern Interferometry (ESPI) for COD measurement and Raman spectroscopy for stress concentration analysis. Stress concentrations caused by intralaminar cracks are triggering new damage modes as interlayer delamination and fiber breaks in adjacent layers and the initial shape of stress concentrations is changing.

Optical observations belong to the simplest group of methods. Specimen edge inspection under microscope is preferable for Carbon fiber (CF) reinforced composite laminates: thermal tensile stresses in layers are high due to high manufacturing temperature and large mismatch in thermal expansion coefficients between layers. In result, intralaminar cracks in CF composites are opened, the polished fiber cross-sections are bright and shiny and even without any applied mechanical load cracks are well seen in a microscope as dark belts, see Fig. 6.1b.

For CF laminates the most accurate is to remove the specimen from the testing machine after loading it to certain level of strain for inspection in a microscope. An example of the crack density dependence on the applied strain level in CF/EP IM8/8552 cross-ply laminate obtained in this way is shown in Fig. 6.2.

Fig. 6.2

Crack density increase with axial strain in 90-layer of CF/EP cross-ply laminate. Different symbols correspond to edge data for different specimens of the same plate. Measurement length was 80 mm along the specimen edge

For GF laminates this method is not applicable because we would not see any cracks in unloaded specimen (thermal stresses are lower and due to worth reflective properties the contrast is lower). If a small tensile testing machine is linked to the microscope, one can see that applying even small strains we start to see cracks, see Fig. 6.1c. Since this option is not always available, counting can be made on the loaded specimen edge or on the surface without unloading and taking it out from the testing machine. Good accuracy can be achieved adjusting the position and orientation of a light source. Cracks can be counted on edges in reflected light as well as on the specimen surface as dark lines in transmitted light. The surface observations are not decisive if several damaged layers of the same orientation are in the laminate: some cracks are too close to each other and it is not possible to distinguish which crack belongs to which of these layers.

Since the described procedure is rather time consuming, edge replicas are often used: instead of microscopy observation of the specimen edge a replica (“print” of the crack on plastic film) taken from the edge of loaded specimen is analyzed in a microscope after the mechanical test. The advantage is that replicas are taken in loaded state when the crack is most open. Since the quality of replicas is never better than the quality of the polished surface, a thorough calibration of the data obtained from replicas with respect to the direct microscopy has to be performed for every used laminate lay-up and material. Generally speaking in replicas we can lose some cracks and count as cracks some artifacts like polishing scratches. Observing the specimen in microscope and slightly changing the focus we can get more “in-depth” information than observing replicas.

The accuracy of these techniques decreases with decrease of layer thickness, because the crack size is limited by the layer thickness and small cracks at the same applied strain open proportionally less and are difficult to distinguish.

Observation of local interlayer delaminations using optical methods described here is very difficult because the delamination cracks are usually not opened. One can see them in Fig. 6.1b but the measured delamination length would be very uncertain.

Yet, the most critical in evaluating this method is the question how representative are the observations done on the specimen edge for the bulk of the specimen. First is to check whether the crack density values obtained from both specimen edges are the same. Then the specimen has to be cut longitudinally to introduce two new edges which after careful polishing are inspected and compared with data from edges. Only after this validation the edge data can be used as representative for the material. The differences can be particularly large for laminates with thin layers and especially in fatigue loading: more damage is in the edge region. The differences between edge and the interior can be very large for secondary damage like local delaminations: often they are present on specimen edges only.

X-ray images Penetrant liquid has to fill the crack to obtain X-ray image of the crack and actually what we see is the penetrant. Images, for example, in [23], show cracks in both layers of a cross-ply laminate subjected to tensile fatigue loading. Cracks in 0-layers are due to mismatch in Poisson’s ratios between layers and often they are initiated from cracks in the 90-layer. Local delaminations at cracks and especially in their crossing regions can be identified. Similarly as with transmitted light technique, using X-ray technique it is difficult to distinguish between cracks in the top and in the bottom 0-layer of the laminate. A limiting factor is the condition that penetrant has to enter the crack in order it to be seen. This means that the crack to be seen has to be in the surface layer or connected with the specimen edge. Cracks inside the material, not connected with edges, other cracks or surfaces are invisible. The penetrant enters more easily inside of large opened cracks. It can be much more difficult if the size of the tunneling crack in laminate thickness direction is small (thin cracked layer).

Acoustic emission Often the energy released during intralaminar crack propagation is larger than necessary to create the new crack surface. Part of the excess of the energy goes in acoustic signal in the frequency region 10 Hz–100 kHz. Using a sensor on the specimen surface these signals can be recorded. Recording simultaneously the applied strain dependence of time we can determine the number of cracks corresponding to certain level of strain. An example with data recorded for GF/EP cross-ply laminate [23] is shown in Fig. 6.3. The cracking in the 90-layer of this laminate was unstable (after initiation the crack instantaneously propagated across the whole width of the specimen) and the acoustic signals shown as peaks in Fig. 6.3 are very distinct and the number of cracks is easy to count. As follows from Fig. 6.3 the applied strain was increasing linearly until value 0.6 % was reached. After that the strain was reduced. No new cracks (acoustic events) were observed during the unloading.

Fig. 6.3

Acoustic signals caused by multiple cracking in 90-layer of GF/EP [02/902] cross-ply laminate during increasing applied strain [23]

If a more sophisticated equipment and software is available, two acoustic sensors may be placed on the specimen in two different positions. Then from the time delay in both signals the exact position of the created crack can be identified. This helps to separate signals from cracks in the gauge length region from cracks created outside it (close to the clamping region, under tabs).

The described technique has problems to identify and register cracks at high crack density. The signals become much less distinguishable than in the initial stage of cracking shown in Fig. 6.3. Part of the reason is that the stress state between two existing cracks where the new crack is created is very complex and the new cracks (tunnels) are not so well defined: they are not straight cracks covering the whole thickness and width of the layer. Instead we have curved cracks close to existing ones; so-called “delta cracks” and growing interlayer delamination. The acoustic emission method in the way as it is described here cannot distinguish between these modes of damage or to quantify them.

Another not resolved problem is detection and quantification of stable crack growth or growth in small increments (jumps) as it is typical for cracks in very thin layers and especially during cyclic loading. In these cases the acoustical emission signals are not distinctive: there is an emission but it is not in form of countable peaks with each peak corresponding to one damage event. An alternative in this case would be to obtain the total emitted energy as a function of time and try to correlate it with the damage state. Even in this case there is a problem of separating the acoustic energy corresponding to different simultaneous modes of damage.

Delaminations at the crack tip and deviations from ideal straight crack geometry (branched cracks, delta cracks) affect the opening (COD) and the sliding (CSD) of crack surfaces and in this way the amount of stiffness reduction. Therefore experimental information regarding COD and CSD is required to compare it with values from idealized models. For example, the COD of an ideal straight crack with no damage at the interface would be much smaller than in a case with delamination.

The first measurements of the CODs were reported in [24, 25]. The distance between crack faces as a function of layer thickness coordinate was measured from micrographs and also using image analysis. In spite to measurement errors due to uncertainties in focusing the microscope to the exact crack surface position and due to filtering when image analysis was used, it was found that the average value of COD depends on stiffness ratio of the cracked and the supporting layer and on their thickness ratio. These findings were later confirmed with FEM calculations.

Recently, more accurate and reliable COD and CSD measurements were performed using one of the methods of full-field strain measurements, the Electronic Speckle Pattern Interferometry (ESPI) [26]. The ESPI is based on interference of two coherent laser beams: reference beam and observation beam that form the same angle with respect to the studied surface, so the corresponding speckle patterns also interfere [27]. From recording the phase maps for the initial and the final stress states the relative displacement at each point on the specimen surface is calculated. The discontinuities correspond to the crack locations and are seen as displacement jumps. COD from model with ideal crack geometry was validated for cracks in inside layers [26]. The technique is very time consuming and, whereas it renders very unique information for the research community, is not suitable for industrial application. Since this technique is optical it can be used on the edge and surfaces of the laminate only.

More details regarding the damage state at the intralaminar crack tip (local strain distribution) can be obtained using Raman spectroscopy [27]. The basis of this technique is the fact that, for example, in Kevlar fibers the Raman wavelength depends on the applied stress. This property is used to obtain the strain distribution along the fiber. In order to use the fiber as a local strain gauge, the fiber is first calibrated to obtain the relationship between the applied stress and the Raman wavenumber. The fiber is embedded in the composite with a special function to serve as sensor. This technique has very high resolution, and the strain values are obtained directly from the fiber. Unfortunately, as for all optical techniques, the measurements are on edges and outer surfaces, or the matrix has to be transparent. The method is relatively slow (the data collection time for a single measurement in a fixed position along the fiber takes several seconds).

In [29] this technique was used to measure the local stress concentration distribution in the 0-layer of a cross-ply laminate close to the tip of a intralaminar crack, plotting the stress as a function of the distance from crack tip. It was found that the shape of the stress concentration changes after higher load application: the normalized maximum becomes lower and the concentration zone is wider. This behavior was explained in [29] by fiber breaks introduced in the 0-layer close to the crack tip, see also Fig. 6.1c, which are “softening” part of the 0-layer adjacent to the layer interface.

3 Damage Initiation and Growth

3.1 Initiation Stress and Propagation Stress

Two phases in development of each individual intralaminar crack can be identified: initiation and propagation (growth). The stress state based analysis is similar in quasi-static and in cyclic (fatigue) loading. We assume that a necessary condition for intralaminar crack initiation and propagation in the layer under consideration is that the transverse stress, which consists of thermal and mechanical part, is tensile or at least non-negative. Existence of large in-plane shear stresses in the layer contributes to cracking. Many experiments show that the tensile transverse stress has a major role for crack initiation and also for the crack growth: the crack opening mode (Mode I) is the dominant mode of crack propagation. Often in modeling the in-plane shear stresses contribution to the cracking process is even neglected. This simplification of the analysis reflects the fact that the resistance to crack growth in shear (Mode II) is usually much higher than in Mode I. Ignoring the effect of shear stresses in this section is not critical for the simulation methodology described and mixed mode criteria can be easy implemented if available.

First we will give a rather “diffuse” definition of the initiation and propagation terms. Initiation is a process on fiber/resin scale (microscale) which leads to development of a mesoscale damage entity (defect, flaw, crack…?) which further development (propagation) can be analyzed ignoring the microstructure and considering the layer in the laminate as a homogeneous material.

Detailed analysis of initiation is very complex and it is outside the scope of this book. Generally speaking the sequence of micro-events is known: it is a combination of failure of interface leading to prolonged debonds and the resin failure following by coalescence of these small damage entities into large damage entity which starts to grow unstable in the layer thickness direction, see Fig. 6.4. The unstable growth in thickness direction is arrested when the crack approaches the interface with the neighboring layer. This process has been analyzed using Linear Elastic Fracture Mechanics (LEFM) in homogenized layer analytically [30] as well as numerically with FEM [31]. The most probable region for initiation is at specimen edges where the transverse in-plane stress is slightly higher than in the interior of the specimen. Another important parameter is geometrical, it is the local variation of fiber volume fraction: in several locations fibers are very close to each other or even touching. There: (a) the stress concentrations are higher than in a unit cell with average fiber content; (b) due to very limited space between fibers impregnation with resin could be of lower quality than in average, leading to reduced interface and resin mechanical properties. These locations are randomly distributed in the specimen.

Fig. 6.4

Intralaminar crack formed in result of coalescence of fiber/matrix interface debonds

Since suitable and reliable micromechanics analysis for evaluating the necessary applied strain or stress level for initiation does not exist, we will employ here a pragmatic approach stating that there is a material system dependent stress level $\sigma_{in}$ required to initiate a defect large enough to grow unstably in thickness direction and then propagate along fibers to become intralaminar crack. Notation $\sigma_{in}$ can be considered as a symbolic notation for simultaneously acting transverse and shear stress in the layer, most probably expressed through stress invariants. It is convenient to assume that the initiation stress level does not depend on the thickness of the layer. Certainly, a weak reduction with increasing volume (layer thickness) is still possible because of the increasing probability to find locations with very unfortunate combinations of geometrical and failure parameters. For example, changing the thickness several times has a small effect in the Weibull strength distribution [9]. The initiation stress $\sigma_{in}$ is expected to be higher than the transverse tensile strength of the same unidirectional (UD) composite, $\sigma_{T}^{ + }$. The reason is that comparing with free standing UD composite the severity of large defects at the layer surface inside the laminate is reduced due to bond with the neighboring layer. Simple analysis in [32] shows that suppressing surface defect growth the initiation stress can be by 50 % higher than the transverse tensile strength

$$\sigma_{in} = 1.12\sqrt 2 \sigma_{T}^{ + } .$$

(6.3)

LEFM can be applied for propagation of the initiated crack: the available potential energy has to be equal or larger than the work required creating a new crack surface. In terms of LEFM for Mode I growth: the Mode I energy release rate has to be equal or larger than its critical value $G_{IC}$ which is material property (energy needed to create a unit of new crack surface). In the presence of in-plane shear stresses mixed mode propagation is possible and a criterion which involves Mode I and Mode II energy release rate as well the corresponding critical values may be required.

The energy release rate, G is proportional to the square of the in-plane stress in the layer in the location of cracking. G is a linear function of the intralaminar crack size in thickness direction which is equal to the cracked layer thickness. From here and the propagation criterion the stress level $\sigma_{prop}$ for crack growth can be calculated. From the above description we conclude that $\sigma_{prop}$ depends on the cracked layer thickness $t_{k}$.

Two different scenarios can describe what follows after crack initiation. If the stress level which was necessary to initiate the crack is high and the layer thickness is large

$$\sigma_{in} > \sigma_{prop}$$

(6.4)

In this case the crack will propagate in unstable manner as soon as it is initiated. In terms of energies it means that at the initiation stress level the available strain energy release rate is much higher than the critical value $G_{c}$. In this case the initiation stress governs the multiple cracking process and fracture mechanics is not an applicable tool for analysis.

On the other hand, if the stress for the initiation, $\sigma_{in}$ is rather low and/or the ply is very thin,

$$\sigma_{in} < \sigma_{prop}$$

(6.5)

the crack after initiation will not propagate. In other words at the initiation stress level the available strain energy release rate for crack propagation is lower than $G_{c}$. In this case nothing will happen directly after the initiation and higher stress has to be applied to get the crack propagating. This stress level for propagation can be calculated using fracture mechanics. In this case cracking is propagation governed.

The phenomenon that the laminate strain for first ply failure in thin layers becomes ply thickness dependent was experimentally observed in [34, 35] and called “in-situ strength”. Using critical strain energy release rate based failure criterion the stress for crack propagation $\sigma_{prop}$ has $1/\sqrt {t_{k} }$ dependence on ply thickness $t_{k}$ and therefore stress level for crack propagation in thin layers is higher. So, according to our description in thin layer crack is initiated at roughly the same stress as in thick layers but its propagation is delayed. Unfortunately, the thickness of the layer at which transition takes place from the initiation stress $\sigma_{in}$ controlled cracking behavior to ply thickness dependent propagation stress $\sigma_{prop}$ (fracture mechanics) controlled behavior is different for different composite systems and has to be determined in tests. Comparison of crack density curves for several ply thicknesses is the most correct way to find the transition point.

3.2 Statistical Nature of Initiation Stress Distribution

We will explain the statistical effects considering transverse stresses only. The ideas and methodology are applicable for general in-plane loading and expressions are easy to generalize.

Statistical distribution of the transverse cracking initiation stress $\sigma_{in}$ is assumed in simulations. In its transverse direction each layer, where cracks may be expected, is considered as consisting of many small elements. Each element has its individual intralaminar cracking initiation stress. Weibull strength distribution is assumed to describe the variation in $\sigma_{in}$ between elements but the geometrical position of element with a given $\sigma_{in}$ is random.

A proper test to obtain the Weibull distribution parameters is tensile loading of cross-ply specimens with counting intralaminar cracks after reaching different loading levels (far-field mechanical + thermal stress in the local axes of the layer with cracks). Two to three specimens usually give enough data for statistical analysis. To obtain initiation stress $\sigma_{in}$ distribution, test has to be performed on laminates with relatively thick 90-layer (0.5 mm and more). This is necessary to ensure that cracking is initiation governed: the initiated crack propagates at once through the whole specimen and, hence, the crack counting on the specimen edge is validated. The above is valid assuming that the initiation stress does not depend on the layer thickness.

In our earlier work [36] we found values around $m_{\sigma } \approx 18$ for the Weibull shape parameter using different composites. In analysis the length of the element in the transverse direction is related to the maximum possible crack density and is selected so that each element may crack not more than once. As a rough estimate, one can assume that in internal layers at highest crack density the average distance between cracks is equal or slightly larger than the kth cracked layer thickness

$$\rho_{{k,{ \hbox{max} } }} = \frac{1}{{t_{k} }}$$

(6.6)

So, the element size has to be taken equal or smaller than the ply thickness. In surface layer the spacing at highest crack density (saturation) is approximately two times larger.

$$\rho_{{k,{ \hbox{max} } }} = \frac{1}{2{t_{k} }}$$

(6.7)

These values are suggested based on observations with many composite systems and are applicable as long as the straight crack assumption is applicable: after crack saturation new damage modes occur with increasing loading (delta cracks, curved cracks, local delaminations) but they are difficult to quantify and to include in models. The effective stiffness of the layer at this stage is already significantly reduced and the laminate stiffness is approaching the ply discount model prediction.

The analysis presented in this section is not a Monte Carlo type of simulation where failure of each individual element is considered explicitly based on local stress distribution and its individual failure properties. In Monte-Carlo simulations each element has to be significantly smaller. Here we express the probability of cracking through crack density without specifying which particular element has failed.

The Weibull distribution for crack initiation stress $\sigma_{in}$ is in form

$$P_{in} = 1 - { \exp }\left[ { - \frac{V}{{V_{0} }}\left( {\frac{{\sigma_{T} }}{{\sigma_{in0} }}} \right)^{{m_{\sigma } }} } \right]$$

(6.8)

In (6.8) $P_{in}$ is the probability that crack is initiated when the transverse tensile stress in the element is $\sigma_{T}$; $m_{\sigma }$ and $\sigma_{in0}$ are the shape and the scale parameters obtained in tests with reference specimens of element volume $V_{0}$; V is the volume of the considered element which has to be included if the identified shape parameters are used for cracking probability calculation in layers of different thickness.

Sometimes $m_{\sigma }$ is estimated from standard deviation of transverse tensile strength data for UD composite specimens. Then the $\sigma_{in0}$ value may be estimated from the average tensile transverse strength $\sigma_{T}^{ + }$ of the UD composite. Due to suppression of surface defect growth when the layer is in a laminate we obtain increase of initiation stress described by relationship (6.3). It is reasonable to assume that the same relationship is valid for scale parameters in Weibull distributions

$$\sigma_{in0} = 1.12\sqrt 2 \sigma_{0}$$

(6.9)

where $\sigma_{0}$ is the scale parameter we would have for UD composite transverse strength distribution for specimens with volume $V_{0}$. It is related to the average transverse strength $\sigma_{T}^{ + }$ of large US specimens (with volume $V_{UD}$) [37] by

$$\sigma_{T}^{ + } = \sigma_{0} \left( {\frac{{V_{0} }}{{V_{UD} }}} \right)^{{1/m_{\sigma } }}\Gamma \left( {1 + \frac{1}{{m_{\sigma } }}} \right)$$

(6.10)

In (6.10) $\Gamma$ is gamma function. From (6.9) and (6.10)

$$\sigma_{in0} = \frac{{\sigma_{T}^{ + } 1.12\sqrt 2 }}{{\Gamma \left( {1 + \frac{1}{{m_{\sigma } }}} \right)}}\left( {\frac{{V_{UD} }}{{V_{0} }}} \right)^{{1/m_{\sigma } }}$$

(6.11)

Even if (6.11) is mathematically correct, it should be used with caution. The volume ratio $V_{UD} /V_{0}$ of UD specimen and an element in the cracked layer is very large and the strength recalculation to smaller volumes may be inaccurate, especially if we remember that $m_{\sigma }$ value is obtained using data on a limited number of broken UD specimens.

More advisable is to use multiple cracking evolution data (crack density) for 90-layers in cross-ply laminates to determine parameters in (6.8). The probability of initiation $P_{in}$ can be defined as the number of elements with initiated cracks $M_{in}$ versus the total number of elements M in the layer. This definition is valid for low crack densities where cracks are not interacting. The number of initiated cracks in the considered thick layer case is equal to the number of fully developed cracks, $M_{in} = M_{cr}$ and therefore

$$P_{in} = \frac{{M_{cr} }}{M}$$

(6.12)

According to (6.6) the number of elements in interior layer with index k is

$$M = \frac{L}{{t_{k} }} = L \cdot \rho_{{k,{ \hbox{max} }}}$$

(6.13)

In surface layer it is approximately two times different. Similarly, the number of cracks when the stress in the layer is $\sigma_{T0}^{(k)}$ is related to average spacing between them

$$M_{cr} = \frac{L}{{2l_{k} }} = L \cdot \rho_{k} \left( {\sigma_{T0}^{(k)} } \right)$$

(6.14)

Substituting (6.13), (6.14) in (6.12) we obtain relationship linking the probability of crack initiation $P_{in}$ with the crack density at certain stress level in a layer with index k

$$P_{in}^{(k)} = \frac{{\rho_{k} \left( {\sigma_{T0}^{(k)} } \right)}}{{\rho_{{k,{ \hbox{max} }}} }}$$

(6.15)

Using crack density data for the 90-layer in cross-ply laminate in (6.15) we obtain $P_{in}^{90}$ dependence on transverse stress $\sigma_{T0}^{90}$. The stress is calculated using CLT from the applied load (or strain) and from the temperature difference $\Delta T$ between manufacturing and testing temperature

$$\sigma_{T0}^{90} = \sigma_{T0}^{90mech} + \sigma_{T0}^{90thermal}$$

(6.16)

Index “0” is used for stresses according to CLT i.e. laminate without damage. Then, expecting that (6.8) is valid, standard procedure is applied to the obtained experimental relationship: double logarithm of $P_{in}^{90}$ we plot against $ln\left( {\sigma_{T0}^{90} } \right)$. If the data really follow Weibull distribution, the obtained relationship is linear. Using linear fit to these data (trend line in EXCEL, for example) we obtain parameters $m_{\sigma }$ and $\sigma_{in0}$ from fitting function.

More accurate values of Weibull parameters can be found comparing Monte-Carlo simulations with experiment [37] or using the probabilistic approach developed in [38]. Both approaches require analytical models for stress distribution between two cracks.

Now we will discuss how the Weibull distribution for crack initiation stress (6.8) with known parameters $m_{\sigma }$ and $\sigma_{in0}$ can be used to predict the number (density) of initiated cracks in an arbitrary kth layer of a multidirectional laminate subjected to increasing general thermo-mechanical loading. The proposed approach is simple and has no ambition for high accuracy. The application routine differs dependent on the level of the transverse stress $\sigma_{prop}$ for propagation with respect to the initiation stress region given by (6.8), (6.15).

In thin layers the propagation stress may be very high and most of the cracks in the layer may be initiated before the transverse stress becomes equal to $\sigma_{prop}$. In this situation simulations of growing density of initiated cracks can be performed combining (6.8) and (6.15) (this time for initiated cracks that are not propagating)

$$\rho_{k}^{in} = \rho_{{k,{ \hbox{max} }}} \left\{ {1 - { \exp }\left[ { - \frac{V}{{V_{0} }}\left( {\frac{{\sigma_{T0}^{(k)} }}{{\sigma_{in0} }}} \right)^{{m_{\sigma } }} } \right]} \right\}$$

(6.17)

We still assume that only one crack can initiate in each element and therefore in (6.17) $\rho_{{k,{ \hbox{max} }}}$ is the highest possible crack density in the kth layer discussed above. All initiated cracks are relatively short in the fiber direction and in different elements they can be in different positions along fibers. Because of that interaction between initiated cracks may be neglected and the CLT transverse stress in the layer, $\sigma_{T0}^{(k)}$ can be used in (6.17). Since we use assumption that crack initiation is only weakly dependent on ply thickness, this expression can be used for all layers independent on their thickness.

Another extreme case is when the propagation stress is low and each initiated crack immediately propagates leading to “fully developed” cracks. The transverse stress $\sigma_{T}^{(k)}$ in any point between two cracks (including the maximum point which is in the middle) is lower than the stress in undamaged layer at the same applied load

$$\sigma_{T}^{(k)} \le \sigma_{T0}^{(k)}$$

(6.18)

The average value of the stress is also lower

$$\sigma_{T(av)}^{(k)} = k_{\sigma } \sigma_{T0}^{(k)} \quad k_{\sigma } < 1$$

(6.19)

Coefficient $k_{\sigma }$ depends on the density of fully developed cracks. Expressions (6.8) and (6.15) still are applicable, but the question is what should we use for $\sigma_{T}^{(k)}$. One obvious option is to use the maximum value of the stress between two cracks but one could argue that even if the stress reaches maximum in one point the strength is randomly distributed. It could be very high in that particular point and the next crack may come where stress is lower but the strength is even lower. Another alternative is to use the average stress between cracks when estimating the probability of occurrence of a new crack. Independent on the choice the expression for crack density in this case is

$$\rho_{k}^{in} = \rho_{k} = \rho_{{k,{ \hbox{max} }}} \left\{ {1 - { \exp }\left[ { - \frac{V}{{V_{0} }}\left( {\frac{{\sigma_{T0}^{(k)} k_{\sigma } }}{{\sigma_{in0} }}} \right)^{{m_{\sigma } }} } \right]} \right\}$$

(6.20)

The most complex is the case when the propagation stress $\sigma_{prop}$ is somewhere in the middle of the initiation stress distribution region. In this case as long as $\sigma_{T0}^{(k)} < \sigma_{prop}$ initiation follows (6.21)

$$\rho_{k}^{in} = \rho_{{k,{ \hbox{max} }}} \left\{ {1 - { \exp }\left[ { - \frac{V}{{V_{0} }}\left( {\frac{{\sigma_{T0}^{(k)} }}{{\sigma_{in0} }}} \right)^{{m_{\sigma } }} } \right]} \right\},\quad \sigma_{T0}^{(k)} < \sigma_{prop}$$

(6.21)

Just before $\sigma_{T0}^{(k)} = \sigma_{prop}$ we have initiated crack density $\rho_{k0}^{in}$. As soon as we reach equality all initiated cracks propagate and instantly become fully developed with crack density $\rho_{k} = \rho_{k0}^{in}$. Since some scatter in fracture toughness for crack propagation is inevitable in reality the crack density jump will not be instant. With increasing load new cracks initiate but they will immediately grow in fully developed cracks according the rule

$$\rho_{k}^{in} = \rho_{k} = \rho_{{k,{ \hbox{max} }}} \left\{ {1 - { \exp }\left[ { - \frac{V}{{V_{0} }}\left( {\frac{{\sigma_{T0}^{(k)} k_{\sigma } }}{{\sigma_{in0} }}} \right)^{{m_{\sigma } }} } \right]} \right\},\quad \sigma_{T0}^{(k)} k_{\sigma } > \sigma_{prop}$$

(6.22)

In this simulation approach interaction is accounted only for cracks in the same layer. Interaction between cracks in different layers during damage evolution is neglected because at present this problem is not studied sufficiently.

We can summarize the sequence of calculation steps used to construct crack density curves, stress-strain curves using this approach.

• (a) For each layer estimate $\rho_{{k,{ \hbox{max} }}}$ according to (6.6) or (6.7) and select an arbitrary set of crack density $\rho_{k}$ values in the region [0; $\rho_{{k,{ \hbox{max} }}}$].

• (b) Using (6.17)–(6.22) find the corresponding $\sigma_{T0}^{(k)}$ set

• (c) Subtract thermal stresses found using CTL from the $\sigma_{T0}^{(k)}$ set to find the set of pure mechanical stresses in the layer $\sigma_{T0}^{(k)mech}$ responsible for the selected crack densities.

• (d) Use CLT to calculate the set of corresponding strains applied to the laminate $\varepsilon_{x0}^{LAM}$. Since it is linearly related to $\sigma_{T0}^{(k)mech}$, calculation for linear elastic material is simple. More complex macroscopic mechanical loading cases are analyzed similarly

• (e) Plot the crack density in each layer versus $\varepsilon_{x0}^{LAM}$ and fit the curves with monotonously increasing functions. From these functions we can find the damage state in each layer at arbitrary selected $\varepsilon_{x0}^{LAM}$ to be used in following simulations.

• (f) Use thermo-elastic properties reduction expressions in Sect. 5.4 to find the degraded laminate thermo-elastic properties for these damage states ($E_{x}^{LAM} ,\;\nu_{xy}^{LAM} ,\;\alpha_{x}^{LAM}$ etc.).

• (g) Calculate damaged laminate strains $\varepsilon_{x}^{LAM}$ using the values of $\varepsilon_{x0}^{LAM}$ and the degraded elastic constants.

The last step on this list requires more detailed explanation. When stress $\sigma_{x0}^{LAM}$ is applied to the laminate the strain in the damaged laminate, $\varepsilon_{x}^{LAM}$ will be larger than the strain in the undamaged laminate, $\varepsilon_{x0}^{LAM}$. It is because the elastic modulus $E_{x}^{LAM}$ is lower than $E_{x0}^{LAM}$. The stress-strain relationship for the damaged and undamaged laminate are

$$\sigma_{x0}^{LAM} = E_{x0}^{LAM} \varepsilon_{x0}^{LAM} \;,\quad \sigma_{x0}^{LAM} = E_{x}^{LAM} \varepsilon_{x}^{LAM} \;,\quad \varepsilon_{x}^{LAM} > 0$$

(6.23)

From here we obtain expression

$$\varepsilon_{x}^{LAM} = \frac{{E_{x0}^{LAM} }}{{E_{x}^{LAM} }}\varepsilon_{x0}^{LAM}$$

(6.24)

The crack density curves obtained in step (e) as functions of $\varepsilon_{x0}^{LAM}$ now can be plotted versus the real laminate strains $\varepsilon_{x}^{LAM}$ using (6.24).

3.3 Energy Release Rate Based Analysis of Intralaminar Crack Propagation

Intralaminar crack propagation along the fiber direction in a layer of laminate is illustrated in Fig. 6.5 (L-is the fiber direction). Except for the specimen edge region where the stress state is 3-dimensional and except the very beginning of the intralaminar crack propagation when the crack is relatively short, the intralaminar growth is in a self-similar manner (so-called steady state growth). Condition for that is that the crack front is far away from both specimen edges.

Fig. 6.5

Schematic showing of intralaminar (tunneling) crack with length a propagating parallel to fibers in the layer

The meaning of a self-similar propagation is explained in Fig. 6.6 where a section of the laminate parallel to the crack plane in the position of the crack is shown. In Fig. 6.6a the crack length is a. The shape of the crack front and the stress state there are very complex and not known. However, the stress state at the crack front $x_{1} = a + da$ in Fig. 6.6b is the same as in Fig. 6.6a at $x_{1} = a$, the stress state is just shifted in $x_{1}$ direction by da. The stress state at specimen edges is also complex but it is not changing due to crack propagation by da. Using the made assumption that the crack front is far away from specimen edges we conclude that between specimen edges and the crack front we have rather large regions where the stress distribution is not dependent on $x_{1}$ (plateau region). When the crack propagates by da the plateau region to the left of the crack front increases by da whereas the plateau region to the right decreases by. This is the only change and the corresponding change of potential energy can be easy calculated.

Fig. 6.6

Intralaminar crack propagation along $x_{1}$ coordinate: a crack length; b crack length $a + da$

The new created region of length da in $x_{1}$ direction on the left of the crack front is shown in Fig. 6.7b. The middle layer there contains a fully developed crack. The “lost” part of the plateau region to the right of the crack front of the same length da is shown in Fig. 6.7a. It does not contain crack and the stress distribution there can be calculated using CLT.

Fig. 6.7

Schematic showing of the undamaged region (a) replaced with region containing crack (b)

Obviously the potential energy change in the system, which is needed in fracture mechanics analysis, is equal to the energy change when an element without crack in Fig. 6.7a is replaced with a cracked element in Fig. 6.7b. The change could be determined by first calculating the stress states (for example, using CLT for element in Fig. 6.7a and FEM for element in Fig. 6.7b) and then find strain energy. However, there is a simpler way: the potential energy change is equal to the work which has to be performed on crack surfaces to close it, thus bringing the element back to the undamaged state. We will demonstrate the procedure in details for crack opening case and for a crack which does not interact with other cracks in the same layer (low crack density), see Fig. 6.8.

Fig. 6.8

Schematic showing of a force applied to the element on crack surface and b the work to move this element to position when the crack is closed

We consider an element with area dzda on the deformed (opened) crack face and apply to it increasing force $dF_{2}$ in the $x_{2}$ direction until it is closed. To close it we have to apply displacement $u_{2}^{(k)} \left( z \right)$ which is equal to 1/2 of the opening of the crack at this z-coordinate. The transverse stress when it is closed is $\sigma_{T0}^{(k)}$, the same as before cracking. In the definition used in this chapter COD is equal to $u_{2}^{(k)} \left( z \right)$, i.e. it is 1/2 of the distance between corresponding points on both crack surfaces.

The force in the closing instant is $dF_{2} = \sigma_{T0}^{(k)} dzda$ and the performed work by closing

$$dW_{2} = \frac{1}{2}u_{2}^{(k)} \left( z \right)dF_{2} = \frac{1}{2}\sigma_{T0}^{(k)} u_{2}^{(k)} \left( z \right)dzda .$$

(6.25)

The work to close both crack surfaces (two surfaces is the reason for coefficient 2 in (6.26)) is

$$W_{2} = 2 \cdot \frac{da}{2}\sigma_{T0}^{(k)} \mathop \smallint \limits_{{ - \frac{{t_{k} }}{2}}}^{{ + \frac{{t_{k} }}{2}}} u_{2}^{(k)} \left( z \right)dz .$$

(6.26)

The energy release rate $G_{I}$ is defined as the work to close the crack divided with area $da \cdot t_{k}$ and therefore

$$G_{I} = \sigma_{T0}^{(k)} \cdot \frac{1}{{t_{k} }}\mathop \smallint \limits_{{ - \frac{{t_{k} }}{2}}}^{{ + \frac{{t_{k} }}{2}}} u_{2}^{(k)} \left( z \right)dz .$$

(6.27)

Average value of any function $\phi (z)$ in segment $\left[ { - \frac{{t_{k} }}{2}, + \frac{{t_{k} }}{2}} \right]$ is defined as

$$\phi_{a} = \frac{1}{{t_{k} }}\mathop \smallint \limits_{{ - \frac{{t_{k} }}{2}}}^{{ + \frac{{t_{k} }}{2}}} \phi \left( z \right)dz .$$

(6.28)

Hence, the integral in (6.27) divided by $t_{k}$ is the average value of the COD over the crack surface (here assumed independent on $X_{1}$)

$$G_{I} = \sigma_{T0}^{(k)} u_{2a}^{(k)} .$$

(6.29)

In (6.29) we see the importance of the average crack opening displacement $u_{2a}^{(k)}$ for the energy release rate in Mode I, $G_{I}$. In linear elastic solution all stresses strains and displacements are proportional. Therefore the average COD, $u_{2a}^{(k)}$ is proportional to the far-field transverse stress in the layer, $\sigma_{T0}^{(k)}$ and to the dimensions of the model which can be represented by the transverse size of the crack which is equal to the ply thickness $t_{k}$. It motivates to use normalized average COD defined as follows

$$u_{2an}^{(k)} = \frac{{u_{2a}^{(k)} }}{{\sigma_{T0}^{(k)} t_{k} }}E_{T} = \frac{{E_{T} }}{{\sigma_{T0}^{(k)} t_{k}^{2} }}\mathop \smallint \limits_{{ - \frac{{t_{k} }}{2}}}^{{ + \frac{{t_{k} }}{2}}} u_{2}^{(k)} \left( z \right)dz .$$

(6.30)

In (6.30) factor $E_{T}$ is introduced to have $u_{2na}^{(k)}$ as a dimensionless parameter. In terms of normalized average COD the strain energy release rate given by (6.29) can be rewritten as

$$G_{I} = \left[ {\sigma_{T0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot u_{2an}^{(k)}$$

(6.31)

Due to its importance here and also in stiffness reduction predictions, see Sect. 6.4, the normalized average COD, $u_{2an}^{(k)}$ and its dependence on elastic and geometrical parameters of layers and the laminate lay-up has been studied extensively using numerical methods [19, 20, 39–41]. It was found that $u_{2an}^{(k)}$ is a robust parameter slightly dependent on cracked layer and neighboring layer stiffness ratio in direction transverse to the crack and on the thickness ratio of these layers. Simple fitting expressions for $u_{2an}^{(k)}$ obtained in FEM parametric analysis are presented in Appendix.

Similar analysis can be performed for crack face sliding and the strain energy release rate in Mode II cracking. The crack face sliding displacement, $u_{1}^{(k)} \left( z \right)$ is introduced as half of the distance in fiber direction between corresponding points on both crack surfaces. The normalized average CSD is introduced as

$$u_{1an}^{(k)} = \frac{{u_{1a}^{(k)} }}{{\sigma_{LT0}^{(k)} t_{k} }}G_{LT} = \frac{{G_{LT} }}{{\sigma_{LT0}^{(k)} t_{k}^{2} }}\mathop \smallint \limits_{{ - \frac{{t_{k} }}{2}}}^{{ + \frac{{t_{k} }}{2}}} u_{1}^{(k)} \left( z \right)dz$$

(6.32)

and the energy release rate in Mode II is

$$G_{II} = \left[ {\sigma_{Lt0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{G_{LT} }} \cdot u_{1an}^{(k)}$$

(6.33)

Expressions (6.31) and (6.33) for energy release rates are applicable only for so-called “non-interactive” cracks when after the crack closing the stress in the crack position is equal to the far-field value which comes from CLT. At higher crack density it is not correct because the stress at the closed crack is lower because of neighboring cracks.

At higher crack density we can assume having 2 cracks with distance between them $2l_{k}$ as shown in Fig. 6.9a. For simplicity we assume that the considered element between the two vertical lines is a repeating element. First we consider case when the next crack propagates in the middle between these two existing cracks as shown in Fig. 6.9b focusing on Mode I. We have to calculate the strain energy release rate when the new crack is created.

Fig. 6.9

A “new” crack propagating between two “old” cracks: a damage state with two old cracks; b “new” crack in the middle between two “old” cracks; c “new” crack in an arbitrary position between two old cracks

We can use the approach used above and calculate the work to close the crack but the obtained expression would contain the stress $\sigma_{T}^{(k)}$ in the middle between these two “old” cracks which in an unknown way depends on the “old” crack density (spacing $2l_{k}$). Instead we will find the work to close the “new” crack in a different way: we can close ALL cracks by first closing the “new” one and then the “old” ones. The work to close all cracks in the repeating element at once is denoted $W_{all \to 0}$, the work to close the “old” cracks when the “new” crack is already closed is $W_{old \to 0}$, the unknown work to close the “new” crack in the presence of both “old” cracks is $W_{all \to old}$. Obviously we can first close the “new” crack and then the “old” cracks and summary work is

$$W_{all \to 0} = W_{all \to old} + W_{old \to 0}$$

(6.34)

or

$$W_{all \to old} = W_{all \to 0} - W_{old \to 0} .$$

(6.35)

The work to close all cracks at once can be calculated as in the first part of this section, because after closing the stress in the layer is the far-field stress. We have one “new” crack in the middle and two halves of the “old” cracks to close and the crack spacing is $l_{k}$.

The expression is

$$W_{all \to 0} = 2\left[ {\sigma_{T0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot u_{2an}^{\left( k \right)} (l_{k} ) \cdot t_{k} da .$$

(6.36)

Notation $u_{{2an}}^{\left( k \right)} (l_{k} )$ is used to emphasize that the normalized average COD is calculated using (6.30) for crack spacing $l_{k}$ corresponding to crack density $\rho_{k} = 1/l_{k}$. The work to close the “old” cracks (two halves of them belong to the analyzed element) when the “new” is already closed is

$$W_{old \to 0} = \left[ {\sigma_{T0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot u_{2an}^{\left( k \right)} (2l_{k} ) \cdot t_{k} da .$$

(6.37)

In (6.37) $u_{2an}^{\left( k \right)} (2l_{k} )$ is the normalized average COD for crack spacing $2l_{k}$. Expressions (6.36) and (6.37) are substituted in (6.35) obtaining the work to close the “new” crack

$$W_{all \to old} = \left[ {\sigma_{T0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot \left[ {2u_{2an}^{\left( k \right)} \left( {l_{k} } \right) - u_{2an}^{\left( k \right)} (2l_{k} )} \right]t_{k} da .$$

(6.38)

Dividing by the new created surface $t_{k} da$ we obtain

$$G_{I} = \left[ {\sigma_{T0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot \left[ {2u_{2an}^{\left( k \right)} \left( {l_{k} } \right) - u_{2an}^{\left( k \right)} (2l_{k} )} \right] .$$

(6.39)

Similar expression for Mode II cracks propagation reads

$$G_{II} = \left[ {\sigma_{LT0}^{(k)} } \right]^{2} \frac{{t_{k} }}{{G_{LT} }} \cdot \left[ {2u_{1an}^{\left( k \right)} \left( {l_{k} } \right) - u_{1an}^{\left( k \right)} (2l_{k} )} \right] .$$

(6.40)

Expression for work to close the “new” crack is valid if the crack is in the middle between two “old” cracks. This may not be correct and the distance to the closest crack to the left and right for the “new” crack may be $l_{k}^{left}$ and $l_{k}^{right}$ respectively. The COD on the left face of the crack is different than on the right face.

The energy release rate when a crack is not in the middle was analyzed in [21]

$$\begin{aligned} G_{{nonuniform}} &= \frac{1}{2}\left[ {\sigma _{{T0}}^{{(k)}} } \right]^{2} \frac{{t_{k} }}{{E_{T} }} \cdot \left[ {2u_{{2an}}^{{\left( k \right)}} \left( {l_{k}^{{right}} } \right) - u_{{2an}}^{{\left( k \right)}} \left( {2l_{k}^{} } \right)} \right. \hfill \\ &\quad+ \left. { 2u_{{2an}}^{{\left( k \right)}} \left( {l_{k}^{{left}} } \right) - u_{{2an}}^{{\left( k \right)}} (2l_{k}^{} )} \right] \hfill \\ \end{aligned}$$

(6.41)

To use the derived expressions for energy release rates criteria for crack propagation have to be formulated. In a simplest case they are $G_{I} = G_{Ic}$ and $G_{II} = G_{IIc}$. A large variety of mixed mode criteria are available, but at present there is rather limited knowledge about interaction between different fracture modes in composites.

Finally, it has to be reminded that the fracture toughness $G_{Ic}$ and $G_{IIc}$ may have different values in different locations. It is not clear whether the location with the lowest initiation stress has also the lowest fracture toughness. Actually it is not relevant, relevant is a question what is the fracture toughness in the material surrounding the initiation location. It is not difficult to accept the thought about variation of fracture toughness due to non-uniform fiber distribution. However, it is not clear whether the variation can be described by Weibull distribution as it was assumed in [42].

4 Stiffness of Damaged Laminate

4.1 Calculation Expressions

We consider symmetric N layer laminate the upper part of which is shown in Fig. 6.10. The kth layer of the laminate has thickness $t_{k}$ and fiber orientation angle $\theta_{k}$. Direction 1 is fiber direction and direction 2 is transverse to fibers. The total thickness of the laminate, $h = \mathop \sum \nolimits_{k = 1}^{N} t_{k}$. In result of loading layer may contain intralaminar cracks. The damage state is characterized by crack density in the kth layer $\rho_{k}$ defined as the number of cracks per mm length measured transverse to the crack plane. The crack density in a layer is $\rho_{k} = 1/(2l_{k}^{edge} \left| {{ \sin }\theta_{k} } \right|)$. The average half distance between cracks in the layer measured on the specimen edge is $l_{k}^{edge}$. Dimensionless crack density $\rho_{kn}$ in layer is introduced as $\rho_{kn} = t_{k} \rho_{k}$. “Vector” and matrix objects are denoted by $\left\{ {} \right\}$ and $\left[ {} \right]$ respectively. The macroscopic stiffness matrix of the damaged laminate is $\left[ Q \right]^{LAM}$ and the stiffness of the undamaged laminate is $\left[ Q \right]_{0}^{LAM}$. The laminate compliance matrix $\left[ S \right]_{0}^{LAM} = \left( {\left[ Q \right]_{0}^{LAM} } \right)^{ - 1}$. The thermal expansion coefficient vectors of the undamaged and damaged laminate are $\left\{ \alpha \right\}_{0}^{LAM}$ and $\left\{ \alpha \right\}^{LAM}$. A bar above the matrix and vector entities in following text indicates layer characteristics in the global coordinate system x, y, z.

Fig. 6.10

Schematic showing of the upper part of symmetric laminate with cracks in different layers

In this section exact expressions derived in [19, 20] for macroscale thermo-elastic constants of the damaged laminate RVE are presented. They were obtained using so-called “GLOB-LOC” approach establishing link between global thermo-elastic properties of damaged laminate and the local stress state close to the crack. Expressions for general symmetric laminate with intralaminar cracks in plies were obtained. The effect of the stress perturbation caused by crack is expressed in terms of crack face opening and sliding displacements whereas the number of cracks is presented by normalized crack density. The expressions are as follow

$$\left[ Q \right]^{LAM} = \left( {\left[ I \right] + \mathop \sum \limits_{k = 1}^{N} \rho_{kn} \frac{{t_{k} }}{h}\left[ K \right]_{k} \left[ S \right]_{0}^{LAM} } \right)^{ - 1} \left[ Q \right]_{0}^{LAM}$$

(6.42)

$$\left\{ \alpha \right\}^{LAM} = \left( {\left[ I \right] + \mathop \sum \limits_{k = 1}^{N} \rho_{kn} \frac{{t_{k} }}{h}\left[ S \right]_{0}^{LAM} \left[ K \right]_{k} } \right)\left\{ \alpha \right\}_{0}^{LAM} - \mathop \sum \limits_{k = 1}^{N} \rho_{kn} \frac{{t_{k} }}{h}\left[ S \right]_{0}^{LAM} \left[ K \right]_{k} \left\{ {\bar{\alpha }} \right\}_{k}$$

(6.43)

The $\left[ K \right]_{k}$ matrix-function for a layer with index k is defined as

$$\left[ K \right]_{k} = \frac{2}{{E_{2} }}\left[ {\bar{Q}} \right]_{k} \left[ T \right]_{k}^{T} \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & {u_{2an}^{k} } & 0 \\ 0 & 0 & {\frac{{E_{2} }}{{G_{12} }}u_{1an}^{k} } \\ \end{array} } \right]\left[ T \right]_{k} \left[ {\bar{Q}} \right]_{k}$$

(6.44)

The stiffness matrix of the damaged laminate, $\left[ Q \right]^{LAM}$ is symmetric as requested by thermodynamics considerations. These matrix expressions for thermo-elastic properties contain elastic ply properties, details of laminate lay-up and dimensionless density of cracks in each layer. The influence of each damage entity is represented by the 3 × 3 displacement matrix in (6.44) which contains the normalized average COD, $u_{2an}^{k}$ and normalized average CSD, $u_{1an}^{k}$ of the crack surfaces in kth layer. It is assumed that all cracks in the same layer are equal: they have the same crack face displacements and the crack distribution is uniform. In (6.42) and (6.43) $\left[ I \right]$ is the identity matrix.

When the distance between cracks in a layer is much larger than the crack size, the stress perturbations of two neighboring cracks do not overlap and cracks in this region are called non-interactive. The normalized average COD and CSD in this crack density region are independent on the value of the crack density. Superscript 0 is used to indicate values in this region, $u_{1an}^{0k}$, $u_{2an}^{0k}$.

Parametric analysis of $u_{1an}^{0k}$ and $u_{2an}^{0k}$ using FE was performed in [19] to identify the most significant geometrical and material constants affecting crack face displacements. In result simple and relatively accurate fitting expressions were obtained to calculate $u_{1an}^{0k}$ and $u_{2an}^{0k}$ as a function of neighboring layer properties. These expressions, see Appendix, are considered to be sufficiently general to be used for cracks in any laminate. Hence, there is no need to use FEM in any of simulations presented in this paper.

4.2 Examples of Calculation and Experiments

The model described in Sect. 6.4.1. has been validated comparing with experimental data and FEM calculations in [19, 20, 40–42]. In this section we will illustrate some results for quasi-isotropic laminate and the agreement with experimental data for cross-ply type of laminates. The material is Glass fiber/epoxy unidirectional layer with longitudinal modulus 44.73 GPa, transverse modulus 12.76 GPA, in-plane shear modulus 3.5 GPa, in-plane Poisson’s ratio 0.30 and ply thickness 0.138 mm.

We start with simulations of elastic constants for [0/+ 60/−60]s laminates with intralaminar cracks in layers shown in Fig. 6.11.

Fig. 6.11

Elastic properties reduction in [0/60/−60]s GF/EP laminate with increasing intralaminar crack density in layers: solid lines represent case when the same crack density is in all layers, whereas symbols represent results for a case when off-axis layers only have cracks with the same crack density

Two damage states are compared. In the first damage state all layers have the same crack density. It may be the case when cracks are introduced by thermal loading: subjecting the quasi-isotropic specimen to very low temperature transverse tensile thermal stresses in all layers would be the same. We can assume that it would introduce approximately the same number of cracks in all layers. Certainly it is just an assumption used in elastic properties simulation. In reality differences in crack density may be caused by different layer thickness (−60 layer is two times thicker) and different location of layers (surface layers have different constraint conditions). In the second damage state the laminate has cracks only in off-axis layers. It could be a case when laminate is subjected to axial tensile loading. The crack density is assumed the same. This is also an approximation which may be justified if (1) the ply thickness is sufficiently large and damage state is governed by initiation stress and (2) the crack density is not high because the saturation crack density would be lower for cracks in thick layer.

Disregarding the above discussion about how realistic the two damage states are, we can use Fig. 6.11 to (a) to see how different elastic constants of the laminate are affected by cracking; (b) to evaluate the effect of cracks in the 0-layer on stiffness reduction in this particular laminate. First we see that the axial modulus is not at all affected by cracks in the 0-layer and results in both simulation cases almost coincide. Similar result was earlier obtained analyzing cross-ply laminates with cracks in all layers using FEM [19]. This is because cracks in 0-layer are oriented in x-direction which is the loading direction to define $E_{x}$. Elastic modulus $E_{y}$ is reduced much more in presence of 0-layer cracks because in this case these cracks are perpendicular to the loading direction in the defining test. The shear modulus reduction in case of cracks in 0-layer is much more severe an also Poisson’s ratio is reduced more than in absence of these cracks. In general, we can see 10–20 % reduction of all elastic constants which in most practical cases would not be acceptable.

Comparison with experimental data has to be the final validation. Some test results are presented in Fig. 6.12 together with simulations. In Fig. 6.12 the axial modulus $E_{x}^{{}}$ and the Poisson’s ratio $\nu_{xy}^{{}}$ (both normalized with respect to initial values) of GF/EP cross-ply laminate with layer elastic constants given above are shown as dependent on intralaminar crack density in the 90-layer. The agreement with test data is excellent.

Fig. 6.12

Reduction of normalized modulus and Poisson’s ratio of [02,904]s GF/EP laminate with increasing crack density in the 90-layer

The results presented in this section demonstrate the generality of the model to deal with any symmetric laminates with arbitrary intralaminar damage state in layers and the accuracy of the obtained laminate stiffness predictions in cases when the damage state is known from experiments.

5 Conclusions

The methodology presented in this chapter allows predicting the thermo-elastic properties of laminates with intralaminar cracks in layers. The agreement with test data is good and the methodology can be considered as reliable. The prediction of the damage initiation and growth is still in the development stage. In this chapter strength based approach to initiation of intralaminar cracks was suggested and experimental procedure for parameter determination in the model was described. The damage propagation is expected to follow rules of fracture mechanics with the corresponding methodology described in the chapter. The approach can be used for damage prediction in quasi-static as well as tension-tension fatigue loading.

Appendix: Expressions for COD and CSD

The COD, $u_{2an}^{0}$ of non - interactive crack is considered in a coordinate system where the cracked layer has 90-orientation with respect to x-axis. In other words x-direction is direction 2 for the layer with crack. Index k denoting the layer is omitted in expressions below. A distinction has to be made between cracks in surface layers and cracks in inside layers. Obviously the normalized average COD of surface cracks is larger because the cracked layer is supported only from one side. The fitting expressions are presented for symmetric case where the bottom support layer has equal properties, orientation and geometry as the top support layer. The expression for $u_{2an}^{0}$ is

$$u_{2an}^{0} = A + B\left( {\frac{{E_{2} }}{{E_{x}^{S} }}} \right)^{n}$$

(6.45)

In (6.45) $E_{x}^{S}$ is the Young’s modulus of the support layer measured in the x-direction which is the transverse direction for the cracked layer. For a crack in internal layer

$$\begin{aligned} A & = 0.52\quad B = 0.3075 + 0.1652\left( {\frac{{t_{90} }}{{2t_{s} }} - 1} \right) \\ n & = 0.030667\left( {\frac{{t_{90} }}{{2t_{s} }}} \right)^{2} - 0.0626\frac{{t_{90} }}{{2t_{s} }} + 0.7037 \\ \end{aligned}$$

(6.46)

In (6.46) $t_{s}$ is thickness of the adjacent support layer and $t_{90}$ is thickness of the cracked layer.

For a crack in surface layer

$$\begin{aligned} A & = 1.2\quad B = 0.5942 + 0.1901\left( {2\frac{{t_{90} }}{{t_{s} }} - 1} \right) \\ n & = - 0.52292\left( {\frac{{t_{90} }}{{t_{s} }}} \right)^{2} + 0.8874\frac{{t_{90} }}{{t_{s} }} + 0.2576 \\ \end{aligned}$$

(6.47)

Suggestions for calculations in more realistic cases when the support layers from different sides are different are given in [42].

Crack face sliding displacements (CSD), $u_{1an}^{0}$, see [20] for details, also follows a power law

$$u_{1an}^{0} = A + B\left( {\frac{{G_{12} }}{{G_{xy}^{S} }}} \right)^{n}$$

(6.48)

In (6.48) $G_{xy}^{S}$ is the in-plane shear modulus of the support layer.

For cracks in internal layer

$$A = 0.3\quad B = 0.066 + 0.054\frac{{t_{90} }}{{2t_{s} }}\quad n = 0.82$$

(6.49)

For cracks in surface layer

$$A = 0.6 \quad B = 0.134 + 0.105\frac{{t_{90} }}{{t_{s} /2}}\quad n = 0.82$$

(6.50)

Expressions (6.45)–(6.47) and (6.48)–(6.50) show that the normalized average COD and CSD are larger for less stiff surrounding layers and approach to certain asymptotic value with increasing support layer and cracked layer stiffness ratio. For thicker support layers the COD and CSD is smaller. This effect of neighbouring layers on the crack face displacements is called “constraint effect”.

Due to nonlinear shear stress-shear strain response the secant shear modulus of the layer will change with increasing laminate strain and will affect the value of $u_{1an}^{0}$ calculated according to (6.48).

When the distance between cracks decreases (high dimensionless crack density) the stress perturbation regions of individual cracks overlap and the normalized average COD and CSD start to decrease. The $u_{2an}^{k}$ has been related to COD of non-interactive cracks, $u_{2an}^{0k}$ by relationship [40]

$$u_{2an}^{k} = \lambda_{k} \left( {\rho_{kn} } \right)u_{2an}^{0k}$$

(6.51)

The crack interaction function $\lambda$ is a function of the crack density in the layer and generally speaking it depends on material and geometrical parameters of the cracked layer and surrounding layers. For non-interactive cracks $\lambda = 1$.

Detailed analysis of the effect of different parameters on interaction function was performed in [40] using FEM. Weak interaction (2–5 %) is observable at normalized spacing $2l_{90} /t_{90} = 2.5$. Further decrease of spacing leads to dramatic drop of the values of the interaction function to 0.3. The interaction of cracks in Glass fiber/epoxy laminates is stronger than in Carbon fiber/epoxy laminates. In the latter at high stiffness ratio the interaction function is not sensitive to layer thickness ratio. In the former with lower layer stiffness ratio the interaction is stronger if the support layer is thicker.

The calculated values of the interaction function were fitted by an empirical relationship with an origin in a simple shear lag model. The interaction function according to the shear lag model is

$$\lambda_{k} = { \tan }h\left( {\frac{k}{{\rho_{kn} }}} \right)$$

(6.52)

The shape function (6.52) was used to obtain the k value from the best fit. The best fit with this function to data corresponding to CF laminates (k _CF  = 1.12) and for GF laminate (k _GF  = 0.84). The interaction effect on $u_{2an}^{k}$ for cracks in surface layer was analyzed in [41] where also more accurate interaction functions for internal cracks are presented. The effect of nonuniform crack distribution on COD was analyzed in [21].