Multiple cracks identification for piezoelectric structures

We investigate through molecular dynamics simulations the fracture properties of boronitrene (BN), graphene, and their interfaces. Four types of interfaces between boronitrene and graphene are considered. It is found that the fracture toughness of graphene is highest among the examined models and is 3.61 and 4.24 \(\hbox {MPa}\sqrt{\hbox {m}}\) in the armchair and zigzag directions, respectively. Compared to graphene, boronitrene exhibits approximately 12 and 21% smaller values of the fracture toughness in the armchair and zigzag directions, respectively. In the armchair direction, the fracture toughness of the interface between boronitrene and graphene with B–C bonds in the interface is weakest and is about 2.49 \(\hbox {MPa}\sqrt{\hbox {m}}\), while the interfacial fracture toughness with C–N bonds in the interface is very close to that of graphene. In the zigzag direction, the interfacial fracture toughness is close to that of BN sheet. Under tension in the zigzag direction, a centered crack, which is initially perpendicular to the tensile direction, kinks at both tips in graphene and boronitrene regions. Since graphene has larger fracture toughness than that of boronitrene, an initial crack in their interface is forbidden to penetrate the graphene region; i.e., the crack can only propagate in the boronitrene region or along their interface of the hybrid BN/graphene sheets. The crack shape in the hybrid BN/graphene sheets depends on the arrangement of B–C–N atoms around the interface and the initial crack tip region.     

Fast crack propagation correlated with crack tip stress in 2D hexagonal atomic lattices

We construct strip finite element models of 2D hexagonal atomic lattices with initial cracks to simulate dynamic crack propagation under mode \(\mathrm{I}\) displacement loading, in which the atomic bonds of 2D lattices are represented by Timoshenko beam elements. Series of 2D lattices, including graphene, hexagonal boron nitride and virtual graphene-like materials, are modeled by varying the nonlinear constitutive relations of beam elements. Branching and oscillation phenomena inevitably occur in fast-propagating crack when the crack speed reaches a critical value, which is closely related to the stress field near the crack tip. Our results reveal that the size of nominal plastic zone \(r_{p}\) around crack front varies with different 2D lattices at both crack initiation and branching. The critical branching speeds \(V_{\mathrm {C}}\) change with material properties, and is correlated with the local stresses around the crack front. Further, we find that \(V_{\mathrm {C}}\) increases with the increment of conditional yield stresses of 2D lattices, but \(V_{\mathrm {C}}\) decreases with the increment of \(r_{p}\) monotonously and linearly at crack branching. Therefore, nonlinear zone, formed by redistributed singular stresses at crack tip, dominates crack kinking or branching during fast crack propagation.     

3D shear-mode fatigue crack growth in maraging steel and Ti-6Al-4V

Fatigue crack growth tests in mixed-mode II + III were performed on maraging steel and Ti-6Al-4V. The 3D evolutions of the crack fronts -measured by SEM after interrupted tests- were analyzed, taking into account the reduction in effective crack driving force by the interlocking and friction of the asperities of the crack surface. Under small-scale yielding conditions, the mixed-mode crack growth rates were found to correlate best with \({\sqrt{{\Delta {\rm K}}_{\rm II}^{{\rm eff}^{2}}+1.2\Delta {\rm K}_{\rm III}^{{\rm eff}^{2}}}}\) in maraging steel, while for Ti-6Al-4V, \({\sqrt{\Delta {\rm K}_{\rm II}^{{\rm eff}^{2}}+0.9\Delta {\rm K}_{\rm III}^{{\rm eff}^{2}}}}\) appeared suitable. For extended plasticity, a crack growth prediction method is proposed and validated for Ti-6Al-4V. This method is based on elastic-plastic F.E. computations and application, ahead of each node of the crack front, of a shear-dominated fatigue criterion.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

On the use of tensor paths to estimate the nonproportionality factor of multiaxial stress or strain histories under free-surface conditions

Nonproportional (NP) strain hardening is caused by multiaxial load histories that induce variable principal stress/strain directions, activating cross-slip bands in several directions, due to the associated rotation of the maximum shear planes. This effect increases the strain-hardening behavior observed under proportional loads, those with fixed principal directions, and must be considered in multiaxial fatigue calculations, especially for materials with low stacking fault energy, such as austenitic stainless steels. NP hardening depends on the material and on the shape of the multiaxial load history path in a stress or strain diagram as well. It can be evaluated by a nonproportionality factor \({F_{\rm NP}}\) that varies from zero, for a proportional load history, to one, for a \({90^{\circ}}\) out-of-phase tension–torsion loading with the same normal and effective shear amplitudes. Originally, \({F_{\rm NP}}\) was estimated from the aspect ratio of a convex enclosure that contains the load history path, such as an ellipse or a prismatic enclosure, but such convex enclosure estimates can lead to poor predictions of \({F_{\rm NP}}\). Another approach consists on evaluating the shape of the six-dimensional (6D) path described by the six normal and shear components of the stress tensor, where the stress path contour is interpreted as a homogeneous wire with unit mass. The moment of inertia (MOI) tensor of this hypothetical wire is then calculated and used to estimate \({F_{\rm NP}}\). The use of 6D stress paths to estimate \({F_{\rm NP}}\) is questionable, since 6D formulations implicitly include the effect of the hydrostatic stress, while NP hardening is caused by the deviatoric plastic straining, not by stresses alone or by their hydrostatic component. In this work, the NP factor \({F_{\rm NP}}\) of a multiaxial load history is estimated from the eigenvalues of the MOI tensor of the plastic strain path, which are associated with the accumulated plastic straining in the principal directions defined by the associated eigenvectors. The presented formulation assumes free-surface conditions, but allows a surface pressure, covering the conditions of most critical points, which indeed are located on free surfaces. Experimental results for 14 different tension–torsion multiaxial histories prove the effectiveness of the proposed method.     

Rate and microstructure influence on the fracture behavior of cemented carbides WC-Co and WC-Ni

Tungsten carbide has both industrial and military applications, from high strength end mill dies and geological drilling, to kinetic energy penetrators. In these extreme environments, an understanding of the dynamic fracture properties and the potential influence of grade microstructure is necessary. The present work investigates fracture behavior of cobalt and nickel cemented tungsten carbide with varying grain size and binder content. Notched hardmetal WC-Co and WC-Ni samples are impacted under mode-I (opening) fracture conditions, and the dynamic stress intensity factor is determined from digital image correlation using ultra high-speed imaging, and compared with quasi-static values. In both grain size and binder content variants examined, the dynamic fracture toughness increased from the quasi-static by a factor of 1.51–2.44. In addition, a 7% increase in cobalt binder content (while maintaining nominally identical average grain size) resulted in a 20% increase in quasi-static fracture toughness, from 8.62 to 10.38 MPa\(\sqrt{\text {m}}\); while the same binder increase resulted in a 34% decrease in critical SIF from 21.07 to 15.72 MPa\(\sqrt{\text {m}}\). The 6% nickel binder WC was found to have a 4.5% higher quasi-static fracture toughness than the 6% cobalt binder WC of the same grain size, but a statistically insignificant difference under dynamic loading. Overall, there is a 28% increase in the quasi-static fracture toughness of tungsten carbide samples with an increase of average grain size from 1 to 3 \(\upmu \)m, and under dynamic loading the larger grain WC shows a nominally identical increase in fracture toughness. These findings are discussed within the theory of classical dynamic fracture mechanics, the implications of the experimental configurations pursued, and the microstructural features are examined using fractography.     

The unsteady hydrodynamic force during the collision of two spheres in a viscous fluid

The time-dependent Stokes equations were solved in the vicinity of two spheres colliding in a viscous fluid with viscosity ν to determine the rate of change of the hydrodynamic forces during large accelerations associated with Hertzian mechanical contact of small duration \({\tau_{\rm c}}\). It was assumed that the gap clearance remains finite during contact and is approximately equal to the height σ of surface micro-asperities. The initial condition corresponds to the steady-state axisymmetric solution of Cooley and O’Neill (Mathematika 16:37–49, 1969), and the initial value problem for the time-dependent Stokes streamfunction was solved using Laplace transform methods. Assuming that σ is small compared to the sphere radius a, we used singular perturbation expansions and tangent-sphere coordinates to obtain an asymptotic solution for the viscous flow in the gap and around the moving sphere. The solution provides the dependence of the resistance, added mass and history forces on σ, the sphere velocity and acceleration, and the ratio of the sphere diameters. We found that the relative importance of viscous and mechanical forces during contact depends on a new Stokes number \({St_{\rm c}=\sigma^2/\nu \tau_{\rm c}}\). Integration of Newton’s equation for the motion of the sphere during mechanical contact showed that there is a critical \({St_{\rm c}=O(\sigma/a)}\) for which there is no rebound at the end of contact.     

Structural,magnetic and photocatalytic characterization of Bi<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>La<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>FeO<Subscript>3</Subscript> nanoparticles synthesized by thermal decomposition method

Single-phase La-substituted bismuth ferrite (Bi\(_{\boldsymbol {1-x}}\)La\(_{\boldsymbol {x}}\)FeO\(_{\mathbf {3}}\)) nanoparticles have been synthesized by thermal decomposition of a glyoxylate precursor. The crystal structure transition of BiFeO\(_{\mathbf {3}}\) from the rhombohedral (R3c) to the cubic \(\boldsymbol {Pm}\bar {\mathbf {3}}\boldsymbol {m}\) structure by La addition was confirmed by X-ray diffraction and infrared spectrometry methods. Furthermore, the Bi\(_{\boldsymbol {1-x}}\)La\(_{\boldsymbol {x}}\)FeO\(_{\mathbf {3}}\) nanoparticles showed a weak ferrimagnetism behaviour, while the magnetization increased from 0.18 to 0.48 emu g\(^{\mathbf {-1}}\) with La substitution. The Bi\(_{\boldsymbol {1-x}}\)La\(_{\boldsymbol {x}}\)FeO\(_{\mathbf {3}}\) nanoparticles exhibited strong absorption in the visible region with the optical band gap calculated from Tauc’s plot in the range of 2.19–2.15 eV. Furthermore, the effects of La substitution on the photodegradation of the methylene blue (MB) under visible light were also studied. The photodegradation of MB dye was enhanced from 64 to \(\sim \)99% with increasing La substitution from \(\boldsymbol {x =}\) 0 to 0.1 and then decreased to 8% for \(\boldsymbol {x =}\) 0.15.     

Rejoinder on: Some recent work on multivariate Gaussian Markov random fields

I thank the discussants, Miguel A. Martinez-Beneito, Fedel Greco, Carlo Trivisano, Stephan R Sain, and Reinhard Furrer, for their insightful and stimulating commentary. The rejoinder is organized in five sections: (1) the M-based models, (2) posterior sensitivity to prior choices for \({\varvec{C}}\) and \({\varvec{\varSigma }}\), (3) stationary and non-stationary (M)GMRFs, (4) various approaches to model formulation and related applications, and (5) statistical computation.     

Measurement Uncertainty Budget of the PMV Thermal Comfort Equation

Fanger’s predicted mean vote (PMV) equation is the result of the combined quantitative effects of the air temperature, mean radiant temperature, air velocity, humidity activity level and clothing thermal resistance. PMV is a mathematical model of thermal comfort which was developed by Fanger. The uncertainty budget of the PMV equation was developed according to GUM in this study. An example is given for the uncertainty model of PMV in the exemplification section of the study. Sensitivity coefficients were derived from the PMV equation. Uncertainty budgets can be seen in the tables. A mathematical model of the sensitivity coefficients of \(T_{\mathrm{a}}\), \(h_{\mathrm{c}}\), \(T_{\mathrm{mrt}}\), \(T_{\mathrm{cl}}\), and \(P_{\mathrm{a}}\) is given in this study. And the uncertainty budgets for \(h_{\mathrm{c}}\), \(T_{\mathrm{cl}}\), and \(P_{\mathrm{a}}\) are given in this study.     

Microstructure and Magnetic Properties of Sn<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript>Ni<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>2</Subscript> Thin Films Prepared by Flash Evaporation Technique

An effort was made to develop semiconductor oxide-based room temperature dilute magnetic semiconductor (DMS) thin films based on wide band gap and transparent host lattice with transition metal substitution. The Sn\(_{\mathrm {1}-x}\)Ni\(_{x}\textit {O}_{\mathrm {2}}\) (\(x\,= \mathrm {0.00, 0.03, 0.05, 0.07, 0.10, and \,0.15}\)) thin film samples were prepared on glass substrates by flash evaporation technique. All the samples were shown single phase crystalline rutile structure of host SnO\(_{\mathrm {2}}\) with dominant (110) orientation. The Ni substitution promotes reduction of average crystallite size in SnO\(_{\mathrm {2}}\) as evidenced from the reduction of crystallite size from 40 (SnO\(_{\mathrm {2}}\)) to 20 nm (Sn\(_{\mathrm {0.85}}\)Ni\(_{\mathrm {0.15}}\textit {O}_{\mathrm {2}}\)). In the energy dispersive spectra as well as X-ray photoelectron spectra of all the samples show, the chemical compositions are close to stoichiometric with noticeable oxygen deficiency. The crystalline films were formed by coalescence of oval-shaped polycrystalline particles of 100 nm size as evidenced from the electron micrographs. The energy band gap of DMS films decreases from 4 (SnO\(_{\mathrm {2}}\)) to 3.8 eV (x \(=\) 0.05) with increase of Ni content. The magnetic hysteresis loops of all the samples at room temperature show soft ferromagnetic nature except for SnO\(_{\mathrm {2}}\) film. The SnO\(_{\mathrm {2}}\) films show diamagnetic nature and it converts into ferromagnetic upon substitution of 3 % Sn\(^{\mathrm {4+}}\) by Ni\(^{\mathrm {2+}}\). The robust intrinsic ferromagnetism (saturation magnetization, 21 emu/cm\(^{\mathrm {3}}\)). Further increase of Ni content weakens ferromagnetic strength due to Ni-O antiferromagnetic interactions among the nearest neighbour Ni ions via O\(^{\mathrm {2-}}\) ions. The observed magnetic properties were best described by bound magnetic polarons model.     

Spin-orbit coupling in graphene,silicene and germanene: dependence on the configuration of full hydrogenation and fluorination

We investigate the effect of full hydrogenation and fluorination on the spin-orbit coupling (SOC) in graphene, silicene and germanene. In chair conformation, the fluorination of graphene increases the spin-orbit splitting (\(E_{\mathrm{so}})\), while the hydrogenation and fluorination of other structures reduce the \(E_{\mathrm{so}}\) at the \(\Gamma \)-point. In case of boat conformation, the hydrogenation and fluorination reduce the symmetry of honeycomb structure, which in turn remove the degeneracy of valence band maximum at the \(\Gamma \)-point. The change in band gaps due to SOC is very small in boat conformation structures as compared to that in the corresponding chair conformation structures.     

A comparative study on dielectric behaviours of Au/(Zn-doped PVA)/<Emphasis Type="Italic">n</Emphasis>-4H-SiC (MPS) structures with different interlayer thicknesses using impedance spectroscopy methods

Three different thicknesses (50, 150 and 500 nm) Zn-doped polyvinyl alcohol (PVA) was deposited on n-4H-SiC wafer as interlayer by electrospinning method and so, Au/(Zn-doped PVA)/n-4H-SiC metal–polymer–semiconductor structures were fabricated. The thickness effect of Zn-doped PVA on the dielectric constant (\(\varepsilon ^{\prime }\)), dielectric loss (\(\varepsilon ^{{\prime }{\prime }}\)), loss-tangent (tan \(\delta \)), real and imaginary parts of electric modulus (\(M^{\prime }\) and \(M^{{\prime }{\prime }})\) and ac electrical conductivity \((\sigma _{\mathrm{ac}})\) of them were analysed and compared using experimental capacitance (C) and conductance (\(G/\omega \)) data in the frequency range of 1–500 kHz at room temperature. According to these results, the values of \(\varepsilon ^{\prime }\) and \(\varepsilon ^{{\prime }{\prime }}\) decrease with increasing frequency almost exponentially, \(\sigma _{\mathrm{ac}}\) increases especially, at high frequencies. The \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) values were obtained from the \(\varepsilon ^{\prime }\) and \(\varepsilon ^{{\prime }{\prime }}\) data and the \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) vs. f plots were drawn for these structures. While the values of \(\varepsilon ^{\prime }\), \(\varepsilon ^{{\prime }{\prime }}\) and tan \(\delta \) increase with increasing interlayer thickness, the values of \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) decrease with increasing interlayer thickness. The double logarithmic \(\sigma _{\mathrm{ac}}\) vs. f plots for each structure have two distinct linear regimes with different slopes, which correspond to low and high frequencies, respectively, and it is prominent that there exist two different conduction mechanisms. Obtained results were found as a strong function of frequency and interlayer thickness.     

Static mode fatigue crack propagation and generalized stress intensity correlation for fatigue–brittle polymers

Stable fatigue crack propagation is predominantly described by the Paris power law correlation of the crack growth rate with the amplitude cyclic stress intensity. The Paris relationship works well for most ductile materials but does not capture the response for fatigue–brittle materials lacking a cyclic damage mechanism, including ceramics and many polymers. Instead, crack growth rate of fatigue–brittle materials correlates to the peak cyclic stress intensity factor, \(\hbox {K}_{\mathrm{max}}\). This work shows that \(\hbox {K}_{\mathrm{max}}\) correlation of fatigue crack growth is derived directly from static mode crack tip behavior with constant correlation coefficients, and that \(\Delta \hbox {K}\) correlations are not generally applicable for static mode crack propagation in fatigue–brittle polymers. This derivation predicts load ratio, frequency, and waveform effects, which are included in a general static mode fatigue crack propagation law. Fatigue crack propagation data of a known fatigue–brittle polymer are presented to demonstrate static mode crack propagation behavior correlation with \(\hbox {K}_{\mathrm{max}}\) with constant parameters.     

Measurement of the Water Relaxation Time of $${\upvarepsilon }$$-Polylysine Aqueous Solutions

\({\upvarepsilon }\)-Polylysine is an effective food preservative. In this paper, the \({\upbeta }\)-relaxation time of \({\upvarepsilon }\)-polylysine aqueous solutions, which represents the rotational speed of a single water molecule, was measured by broadband dielectric spectroscopy at various temperatures and concentrations. The broadband dielectric spectrum of each sample containing water ranging from 35 wt% to 75 wt% at temperatures ranging from \(0\,^{\circ }\hbox {C}\) to \(25\,^{\circ }\hbox {C}\) was measured using a co-axial semirigid cable probe. The measured dielectric spectra of the samples were composed of several Debye relaxation peaks, including a shortest single molecular rotational relaxation time of water, the \({\upbeta }\)-relaxation time, longer than that of pure water. This result represents that \({\upvarepsilon }\)-polylysine suppresses the molecular kinetics of water. It is also found that the \({\upbeta }\)-relaxation time of an \({\upvarepsilon }\)-polylysine solution that contained more than 35 wt% water showed a typical Arrhenius plot in the temperature range from \(0\,^{\circ }\hbox {C}\) to \(25\,^{\circ }\hbox {C}\). The activation energy of each sample depends on the water content ratio of the sample. As indicated by its long \({\upbeta }\)-relaxation time, \({\upvarepsilon }\)-polylysine is expected to possess high abilities of suppressing freezing and ice coarsening.     

Stability Evaluation and Calibration of Type C Thermocouples at the Pt–C Eutectic Fixed Point

Tungsten–rhenium thermocouples (type C thermocouples) are used to measure temperatures higher than 1500 \({^{\circ }}\)C under protective, inert, or vacuum conditions in a wide range of industries, such as metallurgy, power generation, and aerospace. Generally, the measurement uncertainty of a new tungsten–rhenium thermocouple is about 1 % (20 \({^{\circ }}\)C at 2000 \({^{\circ }}\)C), and a significant drift is always observed above 1200 \({^{\circ }}\)C. Recently, the National Institute of Metrology, China, has spent great efforts to calibrate tungsten–rhenium thermocouples with high-temperature fixed points of up to 2000 \({^{\circ }}\)C. In the present work, three tungsten–rhenium thermocouples made by two manufacturers were calibrated at the Pt–C eutectic fixed point (1738 \({^{\circ }}\)C) and their stability was investigated. A linear fitting and extrapolation method was developed to determine the melting and freezing temperatures of the Pt–C eutectic fixed point for avoiding the effect of thermal resistance caused by the sheath and protection tube. The results show that the repeatability of the calibration is better than 0.9 \({^{\circ }}\)C from the melting curve of the Pt–C fixed point and better than 1.2 \({^{\circ }}\)C from the freezing curve of the Pt–C fixed point, and a good agreement was obtained for the calibration with the melting and freezing temperature plateau through the linear fitting and extrapolation method. The calibration uncertainty of the thermocouples at the Pt–C eutectic fixed point was 3.1 \({^{\circ }}\)C (k \(=\) 2).     

Fracture criterion on the basis of uniformity of plastic work of polycrystalline ductile materials under various stress states

Studies on the fracture criterions of structural materials would be of great significance to the service security of engineering structures. This note proposes a novel criterion on the basis of the uniformity of plastic work under various stress states. In order to realize abundant stress states, modified Arcan fixtures were designed and integrated with a self-made in situ tensile device. Accordingly, by changing the stress ratio of tensile to shear components, true stress–strain relationships of a typical polycrystalline ductile material (Gr-4 titanium alloy) specimens on the basis of various stress states were obtained and piecewisely fitted. By, respectively, adopting linear and polynomial fitting in the elastic and plastic deformation stages, the plastic work, namely the envelope areas of true \({\sigma_{\rm t}}\)–\({\varepsilon_{\rm t}}\) curves, was quantitatively calculated. Approximate uniformity of plastic work was verified as no correlation between plastic work and stress state was observed. Moreover, the evolution behavior of microvoids inside a single grain and the equivalent average slip distance of polycrystalline ductile materials during trans-granular fracture process were also analyzed theoretically to explain the uniformity. The trans-granular slip and fracture behavior of Gr-4 titanium alloy specimen and orientations of crack propagation of extruded AZ61B magnesium alloy specimens were also examined experimentally.     

Bootstrap in semi-functional partial linear regression under dependence

This paper deals with the semi-functional partial linear regression model \(Y={{\varvec{X}}}^\mathrm{T}{\varvec{\beta }}+m({\varvec{\chi }})+\varepsilon \) under \(\alpha \)-mixing conditions. \({\varvec{\beta }} \in \mathbb {R}^{p}\) and \(m(\cdot )\) denote an unknown vector and an unknown smooth real-valued operator, respectively. The covariates \({{\varvec{X}}}\) and \({\varvec{\chi }}\) are valued in \(\mathbb {R}^{p}\) and some infinite-dimensional space, respectively, and the random error \(\varepsilon \) verifies \(\mathbb {E}(\varepsilon |{{\varvec{X}}},{\varvec{\chi }})=0\). Naïve and wild bootstrap procedures are proposed to approximate the distribution of kernel-based estimators of \({\varvec{\beta }}\) and \(m(\chi )\), and their asymptotic validities are obtained. A simulation study shows the behavior (on finite sample sizes) of the proposed bootstrap methodology when applied to construct confidence intervals, while an application to real data concerning electricity market illustrates its usefulness in practice.