Experimental strength tests are performed on two series of nominally equal plate specimens of annealed soda-lime glass subjected to either ring-on-ring or ball-on-ring bending. The Weibull effective area which represents a fictitious surface area exposed to uniform tension is calculated using closed-form solutions. Finite-size weakest-link systems are implemented numerically in a computationally intensive procedure for random sampling of plates extracted from a virtual jumbo pane whose surface area contains a set of stochastic Griffith flaws. A non-linear finite element analysis is conducted to compute the bending stresses. The glass surface condition is represented in different flaw-size concepts that depend on a truncated exponentially decaying flaw-size distribution. Stress corrosion effects are modelled by implementation of subcritical crack growth. The effective ball contacting radius is determined in a numerical computation. The results show that surface size effects in glass are not only a matter of strength-scaling, as also the shape of the distribution changes. While the lowest strength value, as per the major in-plane principal stress at the recorded fracture origin, in the respective data sets is very similar, the strongest specimen observed in ball-on-ring testing is over 70% stronger than the correspondingly strongest specimen observed in ring-on-ring bending. The Shift function is used to make visual comparisons of the difference in quantiles in the observed data sets. Use of an ordinary Weibull distribution leads to non-conservative strength predictions on smaller effective areas, and to too low strength predictions than are viable for glass design on larger areas. The numerical implementation of finite-size weakest-link systems can produce better predictions for the strength-scaling compared to a Weibull distribution, in particular when the flaw-size concept is modified to include a doubly stochastic flaw-size distribution or a random noise added to each subdivided region of the discretized surface area. The simulated ball-on-ring fracture origins exhibit greater spread from the centre point than otherwise observed in laboratory tests. It is indicated that the chosen representation of surface condition may not be accurate enough for the modelling of all fracture origins in the ball-on-ring setup even though acceptable results are obtained with the ring-on-ring model. There is a need for more insight into the surface condition of glass which can be conducive to the development of flaw-size based weakest-link modelling.
Surface flaws in glass are represented by cracks and the extension of a crack is modelled by an energy balance (Griffith 1920). Crack growth is prompted by either of three modes of deformation, viz. mode I, mode II and mode III (Irwin 1958). Mode I refers to crack opening due to displacements normal to the crack plane surfaces. Mode II and III describe in-plane and out-of-plane shearing displacement cracking (Broek 1983). As a simplification, only the impact of Mode I displacements are considered. Failure is governed by the critical release rate of elastic strain energy. The mode I stress intensity factor (SIF) for a sharp crack subjected to far-field tensile stress \(\sigma \) acting perpendicular to the crack plane is
Charles.Proxy.v4.2.Cracked.All-In-One - FSOCIETY Crack
where a is the crack size and Y is a geometrical configuration factor whose value is roughly equal to unity (Irwin 1957; Hellan 1984). For example, for a straight-fronted planar edge crack it is \(Y=1.12\), and for a half-penny shaped crack it is \(Y=0.73\) at the deepest point on the crack contour (Irwin 1958; Newman and Raju 1981). If the crack plane is inclined at an angle \(\phi \) in the coordinate system of the principal stresses \(\sigma _1\) and \(\sigma _2\) with \(\phi \) measured from the major axis, then the stress component acting perpendicular to the crack plane is calculated as
While subjected to tensile stress in an atmosphere that contains water moisture, environmentally assisted crack growth occurs in glass due to stress corrosion (Charles 1958a, b). The general shape of the logarithm of crack growth velocity as function of the magnitude of SIF is illustrated in Fig. 1. For structural glass design considerations, the influence of regions II and III on the time-to-failure can be neglected (Fischer-Cripps and Collins 1995). Subcritical crack growth in Region I is modelled using Eq. (6) in which \(v_0\) and n are stress corrosion parameters and \(K_I,\text th\) is a threshold value of SIF below which crack growth arrest occurs (Evans 1974). It is assumed that \(n=16\) (Mencik 1992).
It can be shown that combining Eqs. (3) and (6) and introducing \(a_i\) for the initial crack size and a(t) for time-dependent size after crack extension, the following equation holds (Haldimann 2006).
In subsequent analysis, failure of the single-link is evaluated assuming that a given flaw has a representation as a crack (1) with a geometrical configuration, and; (2) with an orientation of the crack plane with respect to the stress field, and; (3) which is possibly operated on by stress corrosion producing time-dependent subcritical crack growth. In Lamon (2016), this notion of the physical processes taking place inside the link is referred to as a flaw-size based approach to fracture in contrast to an elemental strength approach which is not based specifically on a representation of flaws in terms of a crack. In the latter case, only the density function of elemental strengths is required as in, e.g., the Matthews et al. (1976) failure model. The value of plate thickness used in the stochastic failure modelling is equal to the average value recorded in Table 2.
The basis for Eq. (8) is that (1) the average number of cracks, \(\mu A\), is Poisson distributed, Eq. (A.34); (2) cracks fail independently of each other, and; (3) cracks are distributed over the surface according to a uniform random distribution with Pareto distributed sizes, Eq. (A.32), where \(a_0\) and c are the Pareto scale and shape parameters, respectively. Equation (8) can be written in the form of the Weibull function (A.30) with,
Here it is assumed that only the major in-plane principal stress component contributes to failure so that with the fracture criterion, Eq. (5), crack planes are always oriented perpendicular to \(\sigma _1\). The original area, A, in Eq. (9) has been replaced in Eq. (13) by an effective area, \(A_\text eff\), according to
The surface area A of a jumbo sheet is represented by a uniform rectangular meshgrid, the nodes of which correspond to potential fracture sites. Only nodes that correspond to the surface area are considered so edge failures are neglected. Each node is located at the centre of a unit cell of some predefined size. In the present work, the cell size is 1 mm\(^2\) and the spacing between the nodes is 1 mm in the longitudinal and transversal directions, respectively. An average number of \(N_0=\mu A\) cracks are uniformly distributed over the cells; the total number on a given plate is a Poisson random number with expectation \(N_0\), Eq. (A.34). If a given cell contains more than one crack, then the largest one only is selected. An orientation is assigned to each crack with the angle uniformly distributed in \([0,\pi )\). Each crack is assigned a size and shape according to some assumed distribution and geometry. Virtual specimens are extracted from the basic plate and analyzed as follows.
In a geometrically nonlinear analysis, the in-plane stress components, and time-variant mode I SIF, Eq. (3), are calculated at each crack-containing node and at a predefined number of time frames each of length \(\Delta t\) for \(i=1,\ldots ,N_t\) with \(N_t\) being the total number of time frames. The number of time frames is chosen large enough so that the maximum stress increase anywhere on the plate is smaller than approximately 0.1 MPa. Subcritical crack growth is calculated based on Eq. (7) with modified integration limits a(t) replaced by \(a_i+\Delta a\) on the left-hand side, and t by \(\Delta t\) on the right-hand side. Also, to calculate the integral on the right-hand side of Eq. (7), the stress is linearized in each time frame. Subcritical crack growth is allowed to occur only in time frames that correspond with a SIF that exceeds the threshold limit value, \(K_I,\text th\). Subsequently, the SIF history is adjusted to reflect the influence of subcritical crack growth. An algorithm is used to search through the SIF history for the first instance of a failed crack, fracture being governed by the criterion in Eq. (5). According to the weakest-link principle, failure of the specimen is deemed when the first crack fails. It is assumed that unit cells fail independently of each other. The strength is defined as the major in-plane principal stress at the failure node. However, because the SIF history is discretized, the time step associated with failure corresponds to a SIF equal to or larger than the fracture toughness. The strength is calculated by linear interpolation of the maximum principal stress at time steps before and after failure based on the corresponding SIF values.
Flaw-size approach in Yankelevsky (2014) The straight-fronted edge crack shape is adopted by Yankelevsky (2014) with an exponentially decaying size distribution that is parametrized by the size of the largest crack, \(a_\text max\), that is present in the population, see Eq. (A.37). It is assumed that for new glass in the as-received condition, a representative value is 1 flaw per square centimetre density. It is claimed that with this model, there is no need for calibration because it is independent of test results. Only the maximum crack size that is present in the population is required as input and this number can be inferred from the standards, e.g. EN 572-2 (2012), and the fact that manufacturers inspect the glass for large defects. Subcritical crack growth is not considered and all crack planes are assumed to be oriented perpendicular to the major in-plane principal stress. 2ff7e9595c