EULER AND EXPONENTIAL ALGORITHM IN VISCOELASTIC ANALYSES OF LAMINATED GLASS

Laminated glass combines two remarkable materials: glass and a polymer ply. While glass is stiff and brittle, the polymer ply is a rate-dependent compliant material. Together, they form a material which keeps the aesthetic value of glass, and due to the polymer, no fragile collapse appears. The polymer ply exhibits time- and temperature-dependency, whereas glass suffers from brittle fracture, which makes the analysis difficult. In this article, a 2D sectional plane-stress model for the viscoelastic analysis of laminated glass is presented. This study presents one step in the development of a phase-field-based damage solver for laminated glass to select the optimal time-integration scheme for a quasistatic-analysis and later also for dynamics. The validation against experimental data is provided, and the model reduction is also discussed

Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme of $\operatorname{GL}(n,q)$. In particular we generalise a theorem of Perin on subgroups of $\operatorname{GL}(n,q)$ acting transitively on $t$-dimensional subspaces. We survey transitive subgroups of $\operatorname{GL}(n,q)$, showing that there is no subgroup of $\operatorname{GL}(n,q)$ with $1<t<n$ acting transitively on $t$-dimensional subspaces unless it contains $\operatorname{SL}(n,q)$ or is one of two exceptional groups. On the other hand, for all fixed $t$, we show that there exist nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on linearly independent $t$-tuples of $\mathbb{F}_q^n$, which also shows the existence of nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam-Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in $\operatorname{GL}(n,q)$. Many of our results can be interpreted as $q$-analogs of corresponding results for the symmetric group.Comment: 28 page

Intersection theorems for finite general linear groups

A subset $Y$ of the general linear group $\operatorname{GL}(n,q)$ is called $t$-intersecting if $\operatorname{rk}(x-y)\le n-t$ for all $x,y\in Y$, or equivalently $x$ and $y$ agree pointwise on a $t$-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if $n$ is sufficiently large compared to $t$, the size of every such $t$-intersecting set is at most that of the stabiliser of a basis of a $t$-dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of $Y$ is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\operatorname{GL}(n,q)$ that intersect not necessarily pointwise in $t$-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\operatorname{GL}(n,q)$. These results may be viewed as variants of the classical Erd\H{o}s-Ko-Rado Theorem in extremal set theory and are $q$-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\operatorname{GL}(n,q)$.Comment: 34 pages, minor change

Simulating progressive failure in laminated glass beams with a randomized phase-field solver

Laminated glass achieves improved post-critical response through the composite effect of stiff glass layers and more compliant polymer films, manifested in progressive layer failure by multiple localized cracks. As a result, laminated glass exhibits greater ductility than non-laminated glass, making structures made with it suitable for safety-critical applications while maintaining their aesthetic qualities. However, as shown in our previous studies, such post-critical response is challenging to reproduce using deterministic failure models, which mostly predict failure through a single through-thickness crack localized simultaneously in all layers. In this numerical-experimental study, we explore the extent to which progressive failure can be predicted by a simple randomized model, where layer-wise tensile strength is modeled by independent, identically distributed Weibull variables. On the numerical side, we employ a computationally efficient, dimensionally-reduced phase field formulation -- with each layer considered to be a Reissner-Mindlin beam -- to study progressive failure through combinatorial analysis and detailed Monte Carlo simulations. The reference experimental data were obtained from displacement-controlled four-point bending tests performed on multi-layer laminated glass beams. For certain combinations of the glass layer strengths, results show that the randomized model can reproduce progressive structural failure and the formation of multiple localized cracks in the glass layers. However, the predicted response was less ductile than that observed in experiments, and the model could not reproduce the most frequent glass layer failure sequence. These findings highlight the need to consider strength variability along the length of a beam and to include it in phase-field formulations.Comment: 30 pages, 17 figures, and 2 table

Managing Business Corruption: Targeting Non-Compliant Practices in Systemically Corrupt Environments

This article focuses on strategies of ‘managing business corruption’ at the firm level and offers insights for practitioners in systemically corrupt environments. Our study of 110 CEOs and owners of companies operating in Russia tested a new, ethnographic approach to managing corruption at a firm level. We conceptualize ‘managing business corruption’ as devising and implementing strategies that mitigate corruption-related risks in an effective way. We argue that such strategies have to target specific non-compliant practices, identified bottom-up, yet also amount to a pragmatic, problem-solving framework at the firm level, implemented top-down. Leadership is a key factor which defines effectiveness of corruption management in systemically corrupt environments. While the latter are generally conducive to tolerance and passive attitudes to corruption among business leaders, we identify proactive modes (preventive and controlling) and possible channels (formal hierarchy and informal networks) for leadership action. The proposed approach can also be used for leadership training

A graph based model for the detection of tidal channels using marked point processes

In this paper we propose a new method for the automatic extraction of tidal channels in digital terrain models (DTM) using a sampling approach based on marked point processes. In our model, the tidal channel system is represented by an undirected, acyclic graph. The graph is iteratively generated and fitted to the data using stochastic optimization based on a Reversible Jump Markov Chain Monte Carlo (RJMCMC) sampler and simulated annealing. The nodes of the graph represent junction points of the channel system and the edges straight line segments with a certain width in between. In each sampling step, the current configuration of nodes and edges is modified. The changes are accepted or rejected depending on the probability density function for the configuration which evaluates the conformity of the current status with a pre-defined model for tidal channels. In this model we favour high DTM gradient magnitudes at the edge borders and penalize a graph configuration consisting of non-connected components, overlapping segments and edges with atypical intersection angles. We present the method of our graph based model and show results for lidar data, which serve of a proof of concept of our approach.Ministry of Environment, Energy, and Climate ProtectionMinistry of Science and Culture of Lower Saxon

Phase-field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-static Bending

A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time/temperature-dependent material properties we derived recently for two commonly used polymer foils made of Polyvinyl Butyral or Ethylene-Vinyl Acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin-Francfort-Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. So, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks localised using the quasi-static analysis, whereas multiple cracks evolved during the experiment. It has also been observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.}Comment: 29 pages, 20 figures, and 6 table

Effect of Interlayer Mechanical Properties on Quasi-static and Free Vibration Response of Laminated Glass

Laminated glass fulfills the demands on safety and security in transparent structural elements used in architecture and other fields of engineering. It can be constructed as forced-entry, bullet, or blast resistant. The basic three-layer configuration consists of two glass panes connected with a polymer or ionomer interlayer; the advanced products contain also other layers. The foil ensures shear coupling and provides post-breakage resistance and damping. For the design of laminated glass structures and their analysis, knowledge of mechanical properties of interlayers is essential. In numerical simulations, the interlayer is most typically described by the generalized Maxwell chain ‒ a classical viscoelastic model which can capture the time/temperature-dependent response of polymers under shear. Its parameters can be found for common interlayer types in the literature. However, they differ even for the same material, because of a slightly different content of additives, different test setups, and different data processing procedures. In this contribution, the dependence of the response of a laminated glass element on the material parameters of the polymer interlayer is studied by means of numerical modelling and experiments. Two examples are shown and discussed, i.e., quasi-static analysis of a simply-supported beam and modal analysis of a free-free beam. Numerical predictions are obtained by a layer-wise model based on the finite element method. These predictions are validated against the detailed experimental data. We demonstrate that using the Maxwell model parameters from the literature determined even for the same material type but not for the concrete foil may lead to unrealistic predictions