When dimensioning glass structures, the material strength must be considered as a key parameter. In simulations of glass fracture, it is an important parameter for capturing crack formations and the fragmentation as realistically as possible. However, pointing out a unique value is difficult due to the large variations observed experimentally. Several aspects are contributing to the strength definition, which in particular are the surface quality, the size of the glass element, the loading (time and intensity), and the environmental conditions. As the material is brittle and therefore incapable of redistributing stresses, it is very sensitive to surface flaws and other defects as those result in stress concentrations. Like other brittle materials, a glass element will suddenly fail without noticeable warnings when the stress intensity at a crack tip loaded in tension reaches its critical value.
In the quasi-static load regime, where the strength of glass is known to be insensitive to moderate loading rates, the strength prediction is well-described by the theory of linear elastic fracture mechanics (LEFM). For brittle materials, it is common only to consider the Mode I crack opening(tensile opening). The arising elastic stress intensity around a crack tip can then be represented by a stress intensity factor KI, which first was introduced by Irwin (1957). Once the stress intensity factor reaches a critical value, denoted as the fracture toughness KIc, a crack starts growing until failure. In a humid surrounding environment, however, a crack may also start growing slowly below that critical value when exposed to a positive crack opening stress. This is known as sub-critical crack growth and is the reason why the strength of loaded glass decreases over time(Wiederhorn 1967; Wiederhorn and Bolz 1970).
Covadis 101 Avec Crack
The same testing technique was used to study strain-rate effects on the flexural strength of borosilicate glass (BSG) in four-point bending and equibiaxial bending considering different surface conditions (Nie et al. 2009; Nie et al. 2010). Most recently, a servo-hydraulic high-speed test rig was used by Meyland et al. (2019a) to test the flexural strength of small circular soda-lime-silica glass specimens with two different surface conditions in a ring-on-ring configuration. Despite the differences in the applied testing methods, all studies confirm a strength increase with loading rate, which by some of the mentioned authors is explained by the fact that the effect of sub-critical crack growth has been reduced or even eliminated at the observed loading rates.
The finite element method (FEM) is a versatile tool for many engineering applications. For simulation of dynamic events such as blast waves from explosions or impact scenarios, the explicit FEM analysis is widely used. An extensive effort has been put into the research of the numerical simulation of crack initiation and formation in solid materials. Different approaches do exist such as the cohesive zone element (Camacho and Ortiz 1996), the extended finite element method (XFEM) (Moës et al. 1999), the meshless method (Belytschko et al. 1995) or the particle conversion method (Johnson et al. 2002).
However, many challenges are still to deal with such as the dynamic failure of thin-walled structures. Especially a certain effort needs to be directed towards the simulation of windows commonly covering large surfaces. Those will consequently require a massive amount of elements to capture the crack formation, which is a disadvantage with respect to computational time and therefore a limiting factor for the practically use.
In the following, the focus will be given to a more crude approach to crack simulation, namely the element deletion technique, which is practically applicable for the simulation of glass under low-velocity impact and blast. It comes with the great benefit that it is relatively simple to implement, and any failure criterion or damage formulation can be coupled with it. However, it is inherently mesh dependent, as a crack only can extend by deleting an element. Also, the element deletion is not an optimal solution to apply for structural applications where it is expected that created cracks close again. In most engineering problems, a more coarse mesh is used, which makes it difficult to predict the correct state at a crack tip leading to an overestimation of the fracture energy (Unosson et al. 2006). Thus, the use and interpretation of fracture models using the element deletion technique should be done with caution.
A less comprehensive approach for fracture simulation of brittle materials in the finite element method is the smeared crack model introduced by Hillerborg et al. (1976) for concrete, which uses the cohesive zone concept. The term smeared crack denotes that the model does not represent a micro-crack explicitly, but accounts for the effect of a crack by an elastic stiffness reduction, or even elimination, at the integration points of an element. A bilinear traction-separation law, as illustrated in Fig. 3, characterises the stresses over the crack for the cohesive zone model that is used for the built-in material model *Brittle Cracking in ABAQUS/Explicit.
As soon as full fracture is obtained, i.e. at a crack opening width of δcf, the element can no longer carry any tension loads in that direction. The element can still withstand compression loads when the crack closes. However, the model is to be applied with caution, as it removes elements that are fully fractured from the model and, therefore may not represent the correct structural behaviour in the case where cracks close again. For a pure Mode I failure, it may be reasonable to use the brittle cracking failure criterion to remove elements, as a crack only is loaded in tension.
As the smeared crack model initially was developed for concrete and other brittle/quasi-brittle materials in the quasi-static load regime, where the rate of loading is assumed not to affect the mechanical material properties, one has to assign different strength properties in a blast simulation depending on the considered loading rate. However, in most blast-related engineering problems, the loading rate may vary and a model that can account for such variations isto prefer. Nevertheless, this paper still includes this model in further discussions exactly because of its simplicity and the widely seen practically application.
A more straight forward approach to the failure simulation of glass with strain-rate effects included is aimed at, as the smeared crack model is considered being too elementary for the application inblast-related problems, and the more comprehensive JH-2 model requires a wide range of quasi-static to dynamic experimental data for the determination of the numerous model parameters, which not always are sufficiently available. Moreover, the latter has its great strength in simulating projectiles impacting solid ceramic blocks for the determination of for instance the penetration depth.
Moreover, no oscillations are seen after the stiffness removal. However, in case the failing element is surrounded by elements that have not reached the failure criterion, the situation may be different. It is reported by Pelfrene et al. (2016a) that surrounding elements will experience heavy oscillations at a sudden release. In larger simulations, those stress wave reflection may cause the unexpected failure of elements away from the original crack propagation (Song et al. 2008).
The fracture strength must be considered as one of the most decisive parameters for the crack formation in the numerical simulations. However, different approaches in defining the strength are used by the three fracture models discussed in the previous section. A strain-rate dependent strength model is applied to the JH-2 ceramic model and the immediate element failure model, where as a specific strength value isto be pointed out for the smeared crack model. The latter two uses strength values provided by Meyland et al. (2019a).
For *Brittle Cracking in ABAQUS, some extra parameters need to be defined in addition to those given in Table 3. From Meyland et al. (2019a) the dynamic fracture strengths are taken as σ0,a= 229 MPa and σ0,b= 297 MPa for the two considered target piston velocities, respectively. The fracture energy for soda-lime-silica glass is set to GIc=8.0J/m2in accordance with the value stated in Section 3.2. Based on that, the element length criterion is not met for any element in the applied mesh. Hence, the crack opening displacement for tensile failure, δcf, is determined from Eq. (11). The displacement for shear failure, δcs, and ultimate failure, δcu, is chosen to equal the tensile failure displacement, i.e. δcs= δcu= δcf.
Fig. 10 shows the simulated loading curves for both tested loading rates together with the corresponding experimental results. A more natural post-failure behaviour is obtained due to the steep strength reduction seen after crack initiation. However, at the lower loading rate, the load is not dropping down to zero after reaching the fracture strength. At that point, the first cracks are initiated on the bottom side of the specimen without being fully extended through the thickness, resulting in a stiffness reduction until complete failure occurs.
Similar behaviour cannot be observed for the higher loading rate, which presumably can be explained by the more rapid loading that forces the cracks to open until full failure is achieved. Both simulated loading rates initiate failure at loads comparable to the experiments. This is not surprising due to the assigned fracture strengths that are taken from the corresponding experiments as the fracture model does not account for strain-rate dependencies directly.
Usually, one would expect a larger number of fragments at a higher loading rate as the fracture strength is assumed to increase. However, this behaviour could not be captured in the simulations. Nevertheless, the smeared crack model is considered not being a suitable solution for the simulation of blast-related engineering problems due to the missing strain-rate dependency. 2ff7e9595c
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