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From a reality-based fracture mechanics viewpoint, it is to be appreciated at the outset that cracks in ice nucleate and grow on different planes, on many orientations, from un-cracked surfaces, from within, and propagate from existing crack tips. The sought-after fracture mechanics for sea ice should handle crack formation from an un-notched state, crack propagation from small to large, and the fracture characteristics will differ given different settings (such as load control or CMOD control and varying stiffness of the loading device).


In its unloaded state, the fabricated (sawn) crack may be viewed as having a well-defined crack tip with a point-sized process zone. On being loaded, this through-the-thickness crack has no well-defined traction-free crack tip (figure 2a) but rather may be viewed as a traction-free crack attached to a fracture process zone, within which the localized micro-cracking, bridging and micro-crack coalescence accumulates at the same time as (normal to the crack plane) the tensile closing stress decreases (while the separation of two adjacent points on either side of the crack plane increases). Before the application of an opening load, the effective crack length may truly be taken equal to the pre-sawn length; this is not true once the load has been applied. Loading causes micro-cracking and bridging, and the effective crack length increases. In the VFCM, the separation normal to the crack plane due to the micro-cracking activity (not including deformation due to the straining of intact ligaments) is all assumed to occur solely on the crack plane (ahead of the traction-free crack).


Fictitious crack model: (a) the physical loaded through-the-thickness crack in sea ice; (b) stress distribution in FPZ before, and (c) during, growth of the traction-free crack; (d) stress-separation curve (this figure is a duplicate of fig. 2 in [18]).


For any sea ice fracture test, the SSC as well as the viscoelastic characterization of the sea ice is not known. This information must back-calculated such that predicted results match the experimental results. The SSC is assumed to be dependent on the separation distance as well as on the rate of separation and the bulk material is considered to be linearly viscoelastic. The procedure used to carry out the match between the model and experiment is shown as a flow-chart in [15] (see fig. 5 therein). The computations were initiated by assuming a Dugdale-type of stress-separation curve followed by a linear stress-separation curve. It was found that none of these two simple approximations could predict all the experimental observations. To improve the approximations further, bilinear, trilinear, etc., shapes of the stress-separation curve were tried. Each shape tried was kept between the linear and the Dugdale shape, which had formed the bounds. The whole procedure is considered converged when the following criteria are satisfied to a reasonable accuracy: (1) the CMOD and COD as a function of time are predicted correctly; (2) the CTOD as a function of time is predicted correctly; (3) the peak load reached is same as the experimental peak load and is reached at the experimentally observed time; (4) the CTOD at macroscopic crack growth is predicted correctly and occurs at the experimentally observed time. The initial slopes of the load-CMOD and load-CTOD curves are sensitive to the initial slope of the SSC and the magnitude of σt. Once the initial slopes of the computed and experimental load versus CODs have been matched, then the remaining shape and slope of the SSC was influenced more by the consideration that the point of instability must occur at the experimentally obtained CTOD.


At a given ice temperature and salinity, the tensile fracture characteristics of a single crack in first-year sea ice will vary with scale (test size) L, initial crack length A (another scale), test methodology (load control or displacement control) [20], until the test sizes are of the order of several metres, and the cracks are of the order of a metre in size. With the all-important availability of the SSC, the fracture behaviour can be predicted by the VFCM (as demonstrated in [23]).


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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).


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.


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. 350c69d7ab


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