ABSTRACT
Layered structures are used in protection systems such as personal and heavy armor, windshields and also in thermal barriers. Such materials have mismatch in the properties, both elastic and fracture, from layer to layer. The focus of this study is to understand the behavior of cracks in such systems, especially when the crack orientation is such that there are property changes along the crack front. Plates comprising of layers of epoxy and PMMA were prepared by bonding together sheets of 6 mm nominal thickness with an epoxy adhesive. Single edge notched (SEN) specimens were loaded in bending. The thickness averaged stress intensity factor (SIF) was obtained through photoelasticity. Subsequently the behavior of crack-propagation in these materials was also investigated by loading SEN specimen dynamically. The crack tip fields were recorded using high-speed imaging coupled with dynamic photoelasticity, from which the thickness averaged fracture parameters are extracted.
Introduction
Layered structures are used in many applications including, protection systems, thermal barriers, windshields and heavy armor. A layered architecture offers the scope for choosing the layer material and properties in order to optimize the overall performance of the structure. However this advantage comes with added complexities in terms of material characterization and analysis. Particularly one can expect the fracture oriented failure of such structures to be sensitive to the layer architecture, change of elastic and fracture properties from one layer to the other and also to the type of loading experienced and crack orientation. Without a thorough understanding on these aspects, one cannot successfully garner the full potential of layered architecture for critical applications.
Fracture of layered materials has received considerable attention over the years. There have been several investigations attempting to explore the fracture of bi-material systems. A bi-material system consists of two material layers joined together along an interface across which there are property jumps. The behavior of stationary cracks both oriented along the interface and oriented normal to the interface have been investigated by several researchers [1-5]. The behavior of propagating cracks along the interface of a bi-material has also received extensive attention [6-10]. Crack propagation cross the interface in a multilayer system has also been studied [11-12]. In all these investigations, the crack orientation is such that there is no property change across the crack front and therefore a two dimensional approach is applicable.
A through thickness edge crack in a layered plate can have both elastic properties and the fracture toughness varying along the crack front. In this situation it is logical to anticipate that the stress intensity factor (SIF) will vary along the crack front and this variation will be sensitive to the variation of the elastic properties along the crack front. Recent studies [13, 14] indicate that if the elastic property variation along the thickness (crack front) in a cracked plate is continuous, then under in plane bending, the SIF variation is coterminous with the elastic modulus. This implies that the stiffer layer will have a higher SIF compared to the compliant layer. Therefore, the critical condition for a crack to become unstable may not be reached simultaneously at all points across the thickness (crack front). This can either lead to an overall increase of the critical load or its decrease depending on the relative variation of the SIF and the fracture toughness across the thickness. The purpose of this study is to bring out the fracture behavior of layered plates having cracks with property gradients along the crack front. To this extent, thin plates made of two different polymers, Poly Methyl Methacrylate (PMMA) and Epoxy (LY556) having both elastic mismatch and fracture toughness mismatch were bonded together using an Epoxy adhesive. An edge crack in this plate was loaded in quasi-static and dynamic three point bending. The thickness averaged SIF is obtained through photoelasticity.
Experimental details
Specimen preparation and characterization
PMMA sheet used in this study was a commercial grade sheet of nominal thickness 5.5 mm. The epoxy sheets were cast in house and were of nominal thickness 5.8 mm. The elastic modulus and Poisson’s ratio of the materials were determined though tensile tests performed as per ASTM D638 using a 20kN UTM. The longitudinal and lateral strains were measured using a pair of strain gages. Fracture tests were also conducted using the three point bend specimen geometry following ASTM D5045-99. The fracture specimens had a natural crack which was initiated from a machined notch by tapping with a sharp razor blade. Five samples were tested for obtaining the fracture toughness. The elastic and fracture properties of the sheets are listed in table 1. One can easily observe from table 1 the variation of elastic modulus and fracture toughness from one layer to the other. The fringe constant of the materials was also determined by loading circular discs of the materials under diametrical compression and recording the fringes through a circular polariscope equipped with Tardy compensation. The two materials have different sensitivities, with the Epoxy almost 13 times more sensitive than the PMMA.
Table 1. Properties of materials used
|
Material |
Elastic Modulus (GPa) |
Poisson’s ratio |
Fringe constant (MPa-m/fr) |
Fracture toughness (MPaVm) |
|
Epoxy |
3.44 |
0.34 |
0.018 |
0.53 ± 0.04 |
|
PMMA |
2.67 |
0.34 |
0.239 |
0.95 ± 0.11 |
The specimen for fracture testing was prepared by bonding together a PMMA sheet to an Epoxy sheet using the two part epoxy adhesive, Araldite®, which has similar characteristics to the Epoxy sheet. The bonding surfaces were first abraded with fine grit paper and then cleaned with methanol. Then a thin layer of the premixed adhesive was applied with a serrated tool. The two sheets were then assembled and placed in a fixture under slight pressure for overnight curing. After curing the sheets were sized to a length of 200 mm and width of 50 mm. From the measured thickness of the specimen after bonding and the thickness of the individual sheets, it was estimated that the adhesive layer had an average thickness of around 170 micrometers. A notch of the required length was made in the specimen using a band saw and a natural crack was subsequently extended from the notch tip using a sharp razor blade.
Static testing
Single edged notch (SEN) specimens were subjected to both four-point bending and three-point bending in a UTM. The specimen was placed in a light field circular polariscope during loading and the isochromatic fringes were recorded using a CCD camera for further analysis.
Dynamic testing
SEN specimens were subjected to dynamic three-point loading using a Hopkinson bar. A hollow polymeric bar of 3 meter length was used for this purpose. Tests were performed on individual Epoxy and PMMA sheets as well as the combined PMMA-Epoxy specimens also. A SIM02-16 ultra high speed camera coupled with a circular polarizer was used to capture the isochromatic fringes during the fracture process. Sixteen images were captured at framing rates in the range of 100,000 to 150,000 frames per second. Figure 1 shows the schematic of the experimental setup. A make trigger circuit attached on the impact face of the bar was used to trigger the camera.
Fig. 1 Schematic of the experimental setup for dynamic loading
The flash lamps and the strain gage data acquisition system was triggered by the camera itself. The captured images were analyzed to determine the crack speed and also the SIF as explained in the next section.
Analysis of isochromatics
The analysis of the crack-tip isochromatics to extract the fracture parameters (SIF) is a well established procedure for a homogeneous plate [15]. However, in the present study, the sample consists of two materials having different elastic and optical properties. Because of this, the stress field may not be constant through the sample thickness. The optical retardation is a function of the principal stress difference (a1-a2) and the fringe constant of the materials; both vary along the optical path (z-axis) in this case. The net relative retardation, A, can be written as
where, fa1 and fa2 denote respectively the optical fringe constant of material 1 and material 2 and h1 and h2 the respective thicknesses. For a cracked plate with elastic modulus varying along the crack front subjected to in plane bending, it has been shown that the variation in the stresses along the plate thickness is same as the elastic modulus variation, implying more or less an iso-strain type of situation [14]. Using this assumption, we can write the above equation as
In equation 2, Ee is an equivalent elastic modulus and
is the principal stress difference in a homogeneous plate having an elastic modulus of Ee, where,
is the total thickness. In the present case, the elastic modulus and the fringe constant do not vary within a layer, hence we can define an equivalent fringe constant, fae, for the whole plate as follows.
The optical fringe constant calculated using the above equation was used for analyzing the isochromatic fringes in this study, however, it should be pointed out that the usage is valid only for an iso-strain situation. A three-dimensional finite element analysis was carried out to investigate the variation of the stresses along the thickness. The contours of (a1-a2)IE for a homogeneous specimen and a layered specimen are shown in figure 2. The crack occupies the negative y axis. Figure 2(a) shows the variation of (ara2)/E at four locations across the thickness, two near surface (s) and two internal planes (i) for a homogeneous material having elastic modulus same as Ee. One can see that there is some variation of the stresses across the thickness even in a homogeneous plate. Figure 2(b) shows the contours of (ara2)/E for an Epoxy-PMMA layered plate. Interestingly, the variation of (a1-a2)IE across the thickness in the PMMA-Epoxy plate is almost identical to that in a homogeneous material indicating that once the stresses are normalized with the local elastic modulus, their variation is identical to that of a homogeneous plate. The method for extracting fracture parameters, particularly SIF from isochromatics is detailed in [15]. The procedure essentially involves fitting the near-tip stress field expressions to the experimentally observed fringe order using the stress-optic law. Recent investigations report that, continuous variation of elastic properties along the crack front, do not affect the structure of the first three terms, corresponding to r(1/2\ r0) and rj/2) in the stress field [13]. In the PMMA-Epoxy plate, the elastic properties are constant within each layer and vary only across the interface. Therefore we will assume the near-tip stress field to be identical to that in a homogeneous material. The SIF was calculated from the isochromatics following the over deterministic non-linear least square method using the equivalent optical fringe constant, fae, in the stress optic law. The SIF thus obtained will be the thickness averaged SIF, Ke, and the SIF in the individual layers can be calculated as [14]
where, subscripts P and E refer respectively to PMMA and Epoxy.
Fig. 2 Contours of (aI-a2)/E at four different planes along the thickness for (a) homogeneous plate and (b) PMMA-Epoxy plate with an edge crack subjected to in plane bending. (s-close to surface, i-internal)
Results and discussion
Static tests
The results of the three-point and four-point bend tests will be discussed in this section. As mentioned earlier SENB specimens having a span of 200 mm and width of 50 mm were loaded in four-point bending and three-point bending. The thickness averaged SIF, Ke as a function of the applied moment in the four-point test is shown in figure 3 for two crack lengths. The solid line in the figure is the SIF calculated using the analytical solution for an edge crack in a homogeneous plate subjected to four point bending. It can be observed that Ke values in figure 3 are in good agreement with the theoretical values validating the use offae in the stress optic law. Edge cracks were also subjected to quasi-static three point bend test until failure of the specimen. Figure 4 shows the load-displacement curve till failure of the sample. The load increased and close to a load of 300 N the crack in the epoxy layer jumped with a small drop in the load. The load further increased with stable crack growth in the epoxy layer. Simultaneously the crack in the PMMA layer also grew stably; however, the crack tip in PMMA lagged the crack tip in epoxy leading to crack tunneling. Figure 5 shows the SIF as a function of applied load. In figure the solid symbols represent the equivalent SIF, Ke, before crack jump and the open symbols are the SIF calculated from the fringes assuming that the fringes are caused by only the epoxy layer crack. The solid line is the SIF calculated using the theoretical expressions available for an edge crack subjected to three-point bending in a homogeneous plate. Once again close agreement between the theoretical SIF and the experimentally determined SIF can be observed. The isochromatic fringes just before first crack jump, at the end of first crack jump and just before the final failure are shown in figure 6. One can see two crack tips in figure 6(c), one in the epoxy layer and one in the PMMA layer, suggesting crack tunneling. The equivalent SIF at the onset of first crack jump is 0.61 MPa-m1/2 which is about 15% larger than the fracture toughness of epoxy (see table 1). This indicates that the presence of the tougher PMMA layer helps in increasing the fracture resistance of the pate and also delays the onset of final unstable failure.
Fig. 3 Equivalent stress intensity factor, Ke, as a function of applied moment
There was no sign of delimitation in the specimen.
Fig. 4 Load displacement record for three point bend test
Fig. 5 Stress intensity factor as a function of load in three point bending
Fig. 6 Stable crack growth in a PMMA-Epoxy plate under three point bending (a) just before first crack jump (load 310 N) (b) after first crack jump (load 300 N) and (c) just before final unstable failure (load 388 N)
Dynamic loading
SEN specimens were subjected to dynamic three-point bending as explained in section 2.3. The isochromatics recorded during the experiment were analyzed to determine the crack speed and also the value of the equivalent SIF at the point of crack propagation. Figure 7 shows the isochromatics in a PMMA-Epoxy specimen loaded dynamically. The pictures are separated in time by 10 microseconds. The first three pictures show the development of the opening mode stress field around the crack tip. In the fourth picture the crack has already started moving. From the seventh picture onwards, another butterfly shaped fringe (indicated by arrow in the picture) can be seen behind the crack tip (indicated by the vertical line). This fringe continues to follow the crack tip. From this observation, we can deduce that the crack propagation started only in the epoxy layer in picture four and the crack tip in PMMA starts propagating only in the seventh picture. The crack-tip location history obtained from the photographs of two nominally identical experiments is shown in figure 8. The initiation of the crack in epoxy layer could be recorded only in one case (test 02). In both experiments, the crack propagated with a constant velocity of 240 m/sec. From the plot of the crack-tip location versus time (test 02 in figure 8), the exact crack initiation time was determined as 26 microseconds after the first picture was taken. The crack tip in PMMA initiated about 30 microseconds after the crack tip in epoxy layer initiated. The equivalent SIF as a function of time was extracted from the isochromatic fringes and the variation of Ke with time for PMMA-Epoxy specimen is shown in figure 9. The SIF corresponding to a time of 26 microseconds was estimated as 0.6 MPa-Vm, which is close to the dynamic initiation toughness of epoxy layer.
Fig. 7 Crack propagation in a PMMA-Epoxy layered plate subjected to dynamic three-point bending. The vertical line indicates the crack tip in Epoxy layer and the arrow indicates the crack tip in PMMA. The time interval between two pictures is 10 microseconds.
Fig. 8 Crack-tip location history for dynamic fracture of PMMA-Epoxy layered plates under dynamic loading
Fig. 9 Stress intensity factor history up to crack initiation in PMMA-Epoxy layered plates under dynamic loading
Conclusions
The fracture behavior of layered plates having an edge crack subjected to quasi-static and dynamic three-point loading is investigated. The layered plate was prepared by bonding together PMMA and Epoxy sheets which have 30% mismatch in elastic modulus and 80% mismatch in the fracture toughness. Particularly the effect of sudden change in elastic modulus and fracture toughness across the crack front on the fracture behavior is investigated. The thickness averaged SIF was determined from the photoelastic fringes recorded during the fracture phenomena. The results of the study indicate that the presence of the relatively tough PMMA layer can increase the fracture resistance of the material and also delay the onset of unstable fracture under quasi-static bending. When subjected to dynamic loading, the crack tips in Epoxy and PMMA propagated at different time instants with the crack tip in the relatively brittle epoxy layer initiating earlier than that in PMMA. Further studies are in progress to understand the beneficial effects of layered structure on their fracture tolerance under dynamic loading.
