Fracture of solar-grade anisotropic polycrystalline Silicon: A combined phase field–cohesive zone model approach

Artículo Open Access en el sitio web del editor. Pago por publicar en abierto.This work presents a novel computational framework to simulate fracture events in brittle anisotropic polycrystalline materials at the microscopical level, with application to solar-grade polycrystalline Silicon. Quasi-static failure is modeled by combining the phase field approach of brittle fracture (for transgranular fracture) with the cohesive zone model for the grain boundaries (for intergranular fracture) through the generalization of the recent FE-based technique published in [M. Paggi, J. Reinoso, Comput. Methods Appl. Mech. Engrg., 31 (2017) 145–172] to deal with anisotropic polycrystalline microstructures. The proposed model, which accounts for any anisotropic constitutive tensor for the grains depending on their preferential orientation, as well as an orientation-dependent fracture toughness, allows to simulate intergranular and transgranular crack growths in an efficient manner, with or without initial defects. One of the advantages of the current variational method is the fact that complex crack patterns in such materials are triggered without any user-intervention, being possible to account for the competition between both dissipative phenomena. In addition, further aspects with regard to the model parameters identification are discussed in reference to solar cells images obtained from transmitted light source. A series of representative numerical simulations is carried out to highlight the interplay between the different types of fracture occurring in solar-grade polycrystalline Silicon, and to assess the role of anisotropy on the crack path and on the apparent tensile strength of the material.Unión Europea FP/2007–2013/ERC 306622Ministerio de Economía y Competitividad MAT2015–71036-P y MAT2015–71309-PJunta de Andalucía P11-TEP-7093 y P12-TEP- 105

Power Law Scaling in the World Income Distribution

We show that over the period 1960-1997, the range comprised between the 30th and the 85th percentiles of the world income distribution expressed in terms of GDP per capita invariably scales down as a Pareto distribution. Furthermore, the time path of the power law exponent displays a negatively sloped trend. Our findings suggest that the cross-country average growth process appears to be scale invariant but for countries in the tails of the world income distribution, and that the relative volatility of smaller countries' growth processes have increased over time.Growth

Dynamic nonlinear crack growth at interfaces in multi-layered materials

Finite thickness interfaces, such as structural adhesives, are often simplified from the modelling point of view by introducing ideal cohesive zone models that do not take into account the finite thickness properties in the evaluation of the interface stiffness and inertia. In the present work, the nonlinear dynamic response of those layered systems is numerically investigated according to the finite element method. The weak form of the dynamic equilibrium is written by including not only the contribution of cohesive interfaces related to the virtual work exerted by the cohesive tractions for the corresponding relative displacements, but also considering the work done by the dynamic forces of the finite thickness interfaces resulting from their inertia properties. A fully implicit solution scheme both in space and in time is exploited and the numerical results for the double cantilever beam test show that the role of finite thickness properties is remarkable as far as the crack growth kinetics and the dynamic strength increase factor are concerned

On the mean/variance relationship of the firm size distribution: evidence and some theory

In this paper we make use of firm-level data for a sample of European countries to prove the existence of a positive linear relationship between the mean and the variance of firms’ size, an empirical regularity known in mathematical biology as the Taylor power law. A computerized experiment is used to show that the estimated slope of the linear relationship can be fruitfully employed to discriminate among alternative theories of firms’ growth.Taylor power law; Firm size distribution; Stochastic growth

A multi-scale numerical method for the study of size-Scale effects in ductile fracture

The use of a stress-strain constitutive relation for the undamaged material and a traction-separation cohesive crack model with softening for cracking has been demonstrated to be an effective strategy to predict and explain the size-scale effects on the mechanical response of quasi-brittle materials. In metals, where ductile fracture takes place, the situation is more complex due to the interplay between plasticity and fracture. In the present study, we propose a multi-scale numerical method where the shape of a global constitutive relation used at the macro-scale, the so-called hardening cohesive zone model, can be deduced from meso-scale numerical simulations of polycrystalline metals in tension. The shape of this constitutive relation, characterized by an almost linear initial branch followed by a plastic plateau with hardening and finally by softening, is in fact the result of the interplay between two basic forms of nonlinearities: elasto-plasticity inside the grains and classic cohesive cracking for the grain boundaries

A generalized electric model for mono and polycrystalline silicon in the presence of cracks and random defects

Damage, micro-cracks, grain boundaries and other defects in solar cells are impacting on the electric power-loss of photovoltaic modules, their actual solar conversion efficiency and also their lifetime. In the present contribution, a one-dimensional model for simulating the electric current distribution in solar cells accounting for a distributed series resistance is generalized to the presence of partially conductive cracks. The proposed model is used to perform a quantitative analysis of electroluminescence (EL) images of cracked monocrystalline silicon solar cells. A further generalization in a stochastic direction is also proposed in order to take into account randomly distributed defects typical of polycrystalline silicon

Simulated hail impacts on flexible photovoltaic laminates: testing and modelling

The problem of simulated low-velocity hail impacts on flexible photovoltaic (PV) modules resting on a substrate with variable stiffness is investigated. For this type of PV module it is shown that the prescriptions of the IEC 61215 International Standard for quality control used for rigid (glass-covered) PV modules should be augmented by taking into account their real mounting condition and the stiffness of the substrate in the simulated hail impact tests. Moreover, electroluminescence inspection of the crack pattern should be made in addition to electric power output measurements.An implicit finite element simulation of the contact problem in dynamics is also proposed, with two different degrees of accuracy, to interpret the experimentally observed extension of cracking. Results pinpoint the important role of stress wave propagation and reflection in the case of soft substrates

Dynamic formulation of phase field fracture in heterogeneous media with finite thickness cohesive interfaces

Robust numerical prediction of crack propagation in heterogeneous media has been a matter of relevant importance in many engineering applications. In this study, a modelling framework for triggering dynamic fracture events in heterogeneous media, like layered materials, with internal finite thickness cohesive interfaces is proposed through the exploitation of the combined use of the phase field approach to fracture and the interface cohesive zone model to simulate the interplay between bulk and interface cracking. The proposed formulation is constructed via a consistent variational formalism leading to a coupled system of equations, which are solved using a staggered solution scheme. Representative applications examine the robustness of the computational approach, exhibiting results consistent with experimental evidences available in the literature

A global/local approach for the prediction of the electric response of cracked solar cells in photovoltaic modules under the action of mechanical loads

AbstractA numerical approach based on the finite element method to assess the impact of cracks in Silicon solar cells on the electric response of photovoltaic modules is proposed. A global coarse-scale finite element model of the composite laminate is used for carrying out the structural analysis. The computed displacements at the edges of each solar cell are passed via a projection scheme as boundary conditions to a 3D local fine-scale finite element model of the cells which accounts for cohesive cracks. The evaluated crack opening displacements along the crack faces are finally used as input to an electric model characterizing the grid line/solar cell ensemble. The identification of the relation between the localized electric resistance due to cracks and the crack opening, to be used as a constitutive model of cracks, is finally discussed in reference to experimental tests performed in the laboratory