Fracture envelope

For fibres made from the same polymer but with different degrees of chain orientation the end points of the tensile curves, a5, are approximately located on a hyperbola. Typical examples of this fracture envelope are shown in Figs. 8... 

Fig. 8 Tensile curves of cellulose II fibres measured at an RH of 65% (1) Fibre B, (2) Cor-denka EHM yarn, (3) Cordenka 700 tyre yarn, (4) Cordenka 660 tyre yarn and (5) Enka viscose textile yarn [26]. The solid circles represent the strength corrected for the reduced cross section at fracture. The dotted curve is the hyperbola fitted to the end points of the tensile curves 1,3 and 5. The dashed curve is the fracture envelope calculated with Eqs. 9,23 and 24 using a critical shear stress rb=0.22 GPa... 

The presented explanation for the existence of the fracture envelope will be used in formulating a fracture criterion for polymer fibres. Let us suppose a hypothetical polymer fibre with chains having a single orientation angle in the unloaded state. The shape of the fracture envelope is now calculated by taking into account the shear deformation of the chains only. For this case the work per unit volume up to fracture is given by... 

The relation between the end points of the tensile curve, ab and eh (= b), can be calculated with Eqs. 9,23 and 24. This relation is now by definition taken as the fracture envelope. Note that these equations only hold for elastic deformation. In order to account for some viscoelastic and plastic deformation, a value gv is used, which is somewhat smaller than the value for elastic deformation g. The dashed curves in Figs. 8 and 9 are the calculated fracture envelopes (neglecting the chain extension) for the cellulose II and the POK fibres, respectively. These figures show a good agreement between the observed and calculated fracture points. 

As shown in Sect. 2, the fracture envelope of polymer fibres can be explained not only by assuming a critical shear stress as a failure criterion, but also by a critical shear strain. In this section, a simple model for the creep failure is presented that is based on the logarithmic creep curve and on a critical shear strain as the failure criterion. In order to investigate the temperature dependence of the strength, a kinetic model for the formation and rupture of secondary bonds during the extension of the fibre is proposed. This so-called Eyring reduced time (ERT) model yields a relationship between the strength and the load rate as well as an improved lifetime equation. 

The dashed line connects all failure points, in ductile as well as in brittle failure this line is called the failure envelope or fracture envelope... 

FIG. 13.72 Shear stress-strain curves for PMMA at 22 °C under different pressures at a strain rate of approximately 4 x 10-4 s. The filled circles connect all fracture points in a fracture envelope. 

A simple explanation for the shape of the fracture envelope starts with the assumption that the tensile curve is linear with modulus E1. The work of fracture or the strain energy per unit volume up to the fracture point is given by... 

This paper presents results from a study of assemblies composed of glass fibre reinforced epoxy composites. First, tests performed to produce mixed mode fracture envelopes are presented. Then results from tests on lap shear and L-stiffener specimens are given. These enabled failure mechanisms to be examined in more detail using an image analysis technique to quantify local strain fields. Finally the application of a fracture-mechanics-based analysis to predict the failure loads of top-hat stiffeners with and without implanted bond-line defects is described. Correlation between test results and predictions is reasonable, but special attention is needed to account for size effects and micro-structural variations induced by the assembly process. 

Keywords. Fracture envelope. Stiffener, Top-hat, Debonding, Image analysis. 

The paper is presented in three parts. First, the tests employed to determine the mixed mode fracture envelope of a glass fibre reinforced epoxy composite adhesively bonded with either a brittle or a ductile adhesive are briefly described. These include mode I (DCB), and mixed mode (MMB) with various mixed mode (I/II) ratios. In the second part of the paper different structural joints will be discussed. These include single and double lap shear and L-specimens. In a recent European thematic network lap shear and double lap shear composite joints were tested, and predictions of failure load were made by different academic and industrial partners [9,10]. It was apparent that considerable differences existed between different analytical predictions and FE analyses, and correlation with tests proved complex. In particular, the progressive damage development in assemblies bonded with a ductile adhesive was not treated adequately. A more detailed study of damage mechanisms was therefore undertaken, using image analysis combined with microscopy to examine the crack tip strain fields and measure adherend displacements. This is described below and correlation is made between predicted displacements and failure loads, based on the mixed mode envelope determined previously, and measured values. 

Fracture mechanics characterisation tests have been performed to determine the mixed mode fracture envelope of an epoxy bonded glass/epoxy composite. Analysis of lap shear, and L-stiffener geometries has shown that for this relatively brittle adhesive reasonable first estimations of failure loads can be obtained for both cracked and uncracked specimens. An image analysis technique has been developed which enables failure mechanisms to be... 

The retention capacity of individual closures, according to the conditions necessary for hydraulic seal failure, can be predicted from Fig. 12 by the following simple method, (i) Select the aquifer trend most relevant to the particular prospect (based on depth and location), (ii) Extrapolate a hydrocarbon fluid gradient upwards from the closure elevation. There is a risk of hydraulic failure if the fluid gradient intersects with the crestal elevation within the fracture envelope. 

Fracture Envelope Measured Aquifer Gradient (rft) Estimated Aquifer Gradient Crestal Reservoir Pressure... 

By associating aquifer gradients (Fig. 12) with first-order spatial pressure domains and depth of burial, aquifer pressures for individual prospects can be predicted. Retention capacity as dictated by the pressure difference between the reservoir aquifer pressure and seal pore-pressure or fracture envelope can then estimated. The critical stage in this method is the selection of the correct aquifer pressure. Of the other variables required, the crestal elevation of the prospect is usually known with a reasonable degree of confidence, and seal pore-pressures are coincident with the fracture gradient which in turn is confirmed by measured (LOT/FIT) data. Application of this method within the GEA suggests that pre-Cretaceous seals retain hydrocarbon columns within the range from 200 to over 750 m. 

The goal of this research was to study the fracture properties of adhesively bonded composite structures in mode I, II, and mixed-mode I/II loading to create fracture envelopes for these modes to be used in design. The main emphasis was dynamic loading of these structures, in order... 

Figure 11.17 Shear stress-strain curves for PMMA showing fracture envelope. (Reproduced with permission of Rabinowitz, Ward and Parry, J. Mater. Set, 5, 29 (1970))...

The general procedure to prediet fracture in mixed-mode problems is similar to traditional linear elastie fracture meehanics (LEFM) procedures that is, the applied stress intensity, K[, is compared to the critical stress intensity at crack propagation ATjc (a material property). The crack propagates when Ki = K q. The difference with mixed-mode problems is that the critical stress intensity or strain energy release rate is a function of the mode mix, so that in order to characterize the strength of the interface, a fracture envelope that describes the dependence of the total critical stress intensity, Kq, on the mode mix must be constructed experimentally ... 

An example of such a fracture envelope, in this case developed for the case of mixed-mode cracking of an adhesive joint system consisting of 7075-T6 aluminium adherends bonded with a 0.4 mm thick structural epoxy appears in... 

Fig. 5. Fracture envelope, Gc versus phase angle, f, for FPL-etched 7075-T6 aluminum ad-herends bonded with 0.4 mm Cybond 4523GB epoxy adhesive cured at 150 C for 45 min...

Once the particular coating system is characterized, any geometry utilizing this system can be described by the fracture envelope. Of course, the applied stress intensity or strain energy release rate must be found in terms of the mode 1 and II components, for comparison with Eq. 21 i.e. the crack propagates when... 

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