Literature review
2.1
Bolted joints on composites
The introduction of bolts in a composite structure leads to a complicated three-dimensional stress field near the bolt hole, as stated in [8]; however, they are still used in aerospace structures, despite the many advantages of adhesive bonding, because of the ease of assembly/disassembly and airworthiness certification. In general bolted joints can fail in many different modes, as described in [9], including: net-section, bearing, shear-out and bolt failure (Figure 2.1).
• Net-section failure (Figure 2.1a) is the complete breakage of the specimen along a line transverse to the direction of the load. Failure initiates in proximity of the hole edge and occurs, for a joint subjected to uniaxial loading, when the ratio of by-pass load to bearing load is high or when the ratio of hole diameter to plate width (d/w ) is high.
• Bearing failure (Figure 2.1b), occurs in the material immediately adjacent to the contact area between fastener and laminate and is caused primarily by compressive stresses acting on the hole surface. It is likely to occur when the ratio d/w is low or when the ratio of by-pass load to bearing load is low.
Bearing failure is strongly affected by the lateral constraint (clamping force), since it prevents the delamination of plies and buckling of fibers.
• Shear-out failure (Figure 2.1c), is caused by shear stresses, and occurs along shear-out planes on the hole boundary in the principal fastener load direction. Shear-out failure occurs when the hole is too close to the end-edge of the spec-imen. Shear-out failure also occurs in highly orthotropic laminates, regardless of the end distance.
• Bolt failure, Figure 2.1(d), is caused by shear stresses in combination with bending stresses in the fastener.
Normally, all possible failure modes should be considered in the design of a joint but, in specific conditions, some failure modes are more important than others. In practice, a good design helps to avoid most of them: shear-out failure can be avoided by imposing restrictions on allowable laminate configurations and distance of the hole from the edge of the plate, net-section failure can be avoided changing the ratio between bolt diameter and width of the plate and bolt failure is probably not a primary failure mode but rather a secondary failure following bearing. Results presented in literature indicate that in practical aircraft applications (wing struc-tures for example) bearing and net-section failure are the most important failure modes. Bearing, in particular, being impossible to avoid just using a good design requires special attention.
The problem of bearing failure of pin loaded joints has a standard testing pro-cedure, described in the ASTM document D5961, [10]. The procedure has many variations, but in general it consists in connecting a composite specimen to a rigid fixture using one or more bolts. The load (or displacement) is applied by a testing machine on the fixture and is transferred to the specimen through the bolt, while its far edge is held by a grip. The resulting displacement (or reaction force) is recorded during the test until failure. The specimen, shown in Figure 2.2, is designed to promote bearing avoiding the other previously described failure modes.
In particular the distance e of the hole from the edge and the ratio D/w between diameter of the hole and width were chosen with this criterion. The length L was chosen big enough to avoid the negative influence of the grip on the stress state near the hole location.
Such tests can be performed with two methods: single-lap and double-lap. In a single-lap test the specimen is connected to the fixture only on one side, while in a double-lap test the specimen faces the fixture on both sides. The ASTM standard bearing test single and double-lap fixtures are shown respectively in Figure 2.3 and 2.5. Both testing methods are useful to get information about the strength of a joint
Figure 2.2: ASTM standard bearing failure test specimen
and are commonly used in practice, [11]. Single-lap test reproduces more accurately the conditions in which bolts usually work on real structures, e.g. in aerospace appli-cations the connection between skin and frames or between multiple skin panels. In this situation, when the two connected parts are pulled, the entire joint is subjected to bending that tends to nullify the momentum caused by the eccentricity of the pulling forces. This greatly increases the severity of the loading condition on the bolt, introducing localized stress concentrations as graphically shown in Figure 2.4. Double-lap test is instead a more controllable testing method thanks to the absence of bending and the related pure-shear loading condition and also represents a desir-able condition that should be always applied on real structures when possible. With this testing method it is possible to better control the influence of single parameters like stacking sequence, material or design on the overall strength of the joint.
Figure 2.3: ASTM standard single-lap bearing failure test fixture, [10]
Bearing failure is well documented in literature in the case of straight holes. Var-ious authors, [12], have observed that shear cracks formed by the accumulation of in-plane compression failure in each individual ply are the primary failure mode in pin-bearing failure. These cracks eventually will propagate and reach the free surface or will merge, initiating unstable delamination growth and bringing to catastrophic failure. Results appear to be dependent on the laminate thickness, the presence of a washer and the clamping force, but independent of ply orientation. The clamp-ing force in particular seems to help containclamp-ing and delayclamp-ing failure due to friction. In [13], the authors draw the same conclusions about the process of damage accu-mulation of shear cracks, but also state that the presence of delamination plays a primary role on the resistance of the joint. Delamination is the cause of stiffness loss and local instabilities that lead to catastrophic failure and must be included in the analysis of mechanically fastened joints in composite laminates.
Only a few authors analyzed the problem in the presence of countersunk holes, that greatly complicate the problem adding stress concentrations and asymmetry. Most articles, [14], present only experimental results about the strength of bolted joints with countersunk hole, but some, [15], present interesting investigations on the differences in strength and damage progression between countersunk and straight holes. The effects of geometry, bolt torque, clearance and other parameters are also considered. It is observed that the introduction of the countersink increases the tendency of creating delamination, that is primarily located at the corner between straight and countersunk region.
This result, together with the conclusions of [13] about the role of delamination in bearing failure, confirms the importance of an accurate investigation on the effects of the presence of this kind of damage in proximity of a bolted joint, regardless of its source.
Historically delamination has been studied as a crack in the material, that in this case is typically brittle and can be effectively modeled with the Linear Elastic
Fracture Mechanics (LEFM). This theory considers three ways of applying the force to propagate a crack: Mode I or opening mode, Mode II or in-plane shear mode and Mode III or out-of-plane shear mode (Figure 2.6). For each of this modes at first Griffith, [16], and later Irwin, [17], have used an energetic approach to avoid the problem of the stress singularity at the crack tip of the linear elasticity theory. The energetic problem is solved by computing the energy required to create new free surfaces during the propagation of a crack, taken from the potential and elastic energy stored in the specimen during loading. The energy required to create a unit fracture surface is called Strain Energy Release Rate and is a property of the material, related to Fracture Toughness.
2.2
Analytical and numerical modeling
Many attempts have been made to predict the stress state around the hole or the strength of a bolted joint using analytical models. Ref. [18], for example, presents an approach to evaluate the stress distribution around the hole starting from dis-placement expressions in the form of trigonometric series that satisfy the boundary conditions of the problem. Such results can be used to validate a numerical solution or, in combination with a failure criteria, to predict the strength of a bolted joint like attempted with good agreement in [19]. Other authors, [20], implemented the same approach into a Finite Element model to allow the introduction of more com-plex damage models or failure criteria and therefore obtain better results using the computational speed and efficiency of modern calculators.
In material science failure criteria are generally composed by a set of equations that are function of the stress components and the ultimate strength values of the material or its constituents, whose solution will indicate if the material is damaged or not. In the last decades several failure criteria have been proposed in the attempt to predict damage and failure in composite materials. A very simple one is, for example, the Maximum Stress criterion. It simply states that, for a single lamina, failure occurs when one among the critical strength values S1T, S1C, S2T S2C, or S12,
referred to the principal axis, is reached by the correspondent stress component in tension or in compression. This model does not take into account the interaction between failure modes nor can distinguish them. It has however its utility and is still used in practice.
A more refined criterion is the one proposed by S. Tsai and E. Wu, [21]. In this case all material strength values and stress components are combined in a single equation that takes into account the interaction between different failure modes but is not able to distinguish between, for example, matrix or fiber failure yet.
distinguishes between 4 failure modes: fibers breakage under tensile loads, fibers buckling under compressive loads and matrix cracking both in tension and in com-pression. Each failure mode is represented by its own equation for a total of 4 and the interaction between different modes is also considered. The criterion is shown in Eq. 2.11, respectively for fiber in tension, fiber in compression, matrix in tension and
matrix in compression. If one of these equations reaches a value equal to or greater than 1 the correspondent damage mode is present at the considered location. More refined models exist, for example the one proposed by A. Puck, [23] that considers several additional failure modes, but Hashin’s model is implemented in most FE software like Abaqus (even if only for Plane Stress shell elements) and it has proven to be accurate and reliable.
(
σ1 S1T)
2+ α(
τ12 S12)
2≥ 1
if σ
1≥ 1
(
σ1 S1C)
2≥ 1
if σ
1≤ 1
(
σ2 S2T)
2+ (
τ12 S12)
2≥ 1
if σ
2≥ 1
(
σ2 2S23)
2+ [(
S2C 2S23)
2− 1]
σ2 S2C+ (
τ12 S12)
2≥ 1
if σ
2≤ 1
(2.1)
Many books and manuals, [7], [24], carefully describe the methodology and problems encountered when trying to properly simulate the behavior of composite materials using commercial Finite Element software. Their complex non-linear, orthotropic,
trained users. Obtaining a proper FE model to predict the stress state in the partic-ular case of bolted joint with countersunk hole, [25], represents already a challenge, but many attempts are present in literature, [26], [15], [27], to also implement a dam-age model and predict strength and failure modes of the joint. This is usually done by implementing user-defined subroutines, based on a failure criterion, that check the presence of damage after each step of the non-linear solution. Two-dimensional (2D) FE stress and failure analyses have been conducted in the past because of the limited performances of computer processors, in which plane stress approxima-tion and equivalent orthotropic laminate properties were assumed. Such models suffer from many significant drawbacks, one of which is the homogenization of the laminate which neglects the interface of consecutive layers and stacking sequence which can lead up to 20% error in predicting the joint failure strength, [28]. In ad-diction, the plane stress approach assumes that the through-thickness normal and shear components of the stress tensor (τ13, τ23, σ3) are negligible in comparison with
their in-plane counterparts (σ1, σ2, τ12). In practice however the onset and growth
of delamination is governed by such inter-laminar stress components and therefore cannot be properly simulated using 2D elements. Recent advances in computational processors and improved FE formulations have allowed the development of more comprehensive 3D models to be solved with improved computational capabilities. Widely used commercial FE codes offer layered structural solid elements which can be used to model laminated composites, however particular attention is required in this case because using a single element in the thickness yields relatively inaccurate inter-laminar stress predictions. In order to achieve more accurate results, one ele-ment per layer has been used by some researchers, [8], [29], but depending on the problem the accuracy may not be sufficient, requiring the use of more elements per single ply thickness, [7].
Many attempts have been also made to simulate onset and growth of delam-ination damage in composites using the Cohesive behavior, [30], available in the
software Abaqus. This method can be effectively used to simulate Mode I crack opening delamination, [31], but also supports Mode II and Mode III: a very inter-esting publication on this topic is ref. [32] in which a model to simulate onset and growth of delamination in the case of pin-loaded joint is described.
The Cohesive behavior is a particular model initially developed to simulate ad-hesive bonds. Its particular formulation however allows the simulation of bonds with a thickness value of zero, representing a good way to model those between the plies of a composite laminate. This model is based on a bi-linear formulation. The initial response is elastic linear and follow a Traction Vs Separation law, an elas-tic constitutive relation that relates the “separation” between two initially bonded (eventually coincident in case of zero-thickness) nodes to the “traction”, the result-ing force that keeps them together. The constituent matrix considers the separation in all the three opening modes of cracks. The response is linear up to a critical user-defined strength value, followed by initiation and evolution of damage. The evolution may follow different laws but the most simple is the linear degradation, as shown in Figure 2.7.
linearly, with a different slope, until it reaches zero and the complete breakage of the bond between the nodes in exam. The slope of the second line is defined with an energy-based approach: the grey area under the two lines, indicated as GC in
Figure 2.7, is the energy dissipated due to failure or Strain Energy Release Rate for the particular opening mode in exam. The user has to define the energy value for all the three opening modes.
2.3
Damage characterization
Observation and characterization of damage in composites is not an easy task: a composite layup may present many different damage modes that need to be recog-nized to properly evaluate their effects on the overall properties and the resistance of the part. Most non-destructive inspection (NDI) techniques developed for metals can be also used on composites and the choice depends on the application and the information required. Radiography, [33], in combination with a dye-penetrant liquid opaque to X-rays, such as zinc iodide, can be used to locate matrix cracks and de-lamination, but only those close to an external surface where the ink can effectively penetrate. Of more practical utility is the Ultrasonic inspection, [34], where the returning echo of a sound pulse is used to locate many different types of damage and defects (Figure 2.8).
This method is available in different forms and allows the detection of matrix cracks and delamination and can return information about the severity of damage or its position along the thickness. It can also be used to detect manufacturing imperfections like resin-rich regions or excessive porosity which produces a local change in the density of the part causing a detectable difference in the speed of the sound pulse. Both X-ray and Ultrasonic inspection however are not able to easily detect fiber breakage. To detect this important type of damage the best method
Figure 2.8: Ultrasonic inspection visual explanation
remains a destructive visual inspection such as microscopy of cross sections, [35]. In this method the specimen has to be cut along a cross section of interest, allowing to visual inspect the zone with a microscope. A very important parameter is the preparation of the cross section’s surface: to be able to properly observe damage a very good polishing is required. Good results can be obtained using progressively finer sanding paper and a polishing cloth soaked with abrasive particles suspension. If the surface is properly treated, the image can return very detailed information at single fiber level of detail, allowing the direct observation of many types of defects and damage modes, as shown in Figure 2.9
However, if used alone, this method can return only a bi-dimensional view of the material’s state, allowing to see only damage located on the observed surface. It is therefore important to combine different inspection methods in order to obtain a good understanding of the damage state. A good combination is, for example, ultrasonic inspection and microscopy: the first one can be used to locate critical areas that will be later sectioned and observed.
Figure 2.9: Example of microscopy on a damaged composite specimen. damage induces changes in many other parameters, like impedance or capacity that can be revealed.
Another idea is to use the sound emitted by the formation of a crack, [37]. Using a set of multiple piezo-electric sensors positioned around a location of interest it is possible not only to reveal the presence of cracks but also, by triangulating the position basing on the time of arrival of the signal, to locate its position. Researches on this topic are currently active at San Diego State University.
