Indirect measurement of $$\mu (I)$$ relation from finite granular avalanche down an inclined narrow reservoir of smooth bed

This work examines the algebraic \(\mu -I\) relation proposed for steady uniform dry granular flows via unsteady granular avalanche experiments of finite nearly identical dry glass spheres down an inclined narrow reservoir of smooth bed. Lateral high-speed digital imaging permits particle tracking velocimetry with which we can evaluate bulk local instantaneous volume fraction and velocity components to conduct a quasi-two-dimensional control volume analysis of streamwise momentum assuming an internal shear stress based on the \(\mu -I\) rheology, a hydrostatic normal stress and a Coulomb yielding condition at lateral walls. Hence, the desired \(\mu \) is a function of flow dynamics and a wall friction coefficient \(\mu _w\). Complementary sliding table experiments were conducted to estimate an upper bound of \(\mu _w=0.17\) which was used with a chosen nonzero lower bound \(\mu _w=0.05\) to extract possible range of \(\mu \) at a local instantaneous inertial number I. The so-obtained local instantaneous \(\mu -I\) data conform to the non-linear monotonically increasing trend proposed for steady inertial flows above a crossover value \(I_c\approx 0.03\). Below \(I_c\), a peculiar segment of decaying \(\mu \) with I was revealed agreeing to the rheology tests in quasi-static regime.     

Study of sedimentation of non-cohesive particles via CFD–DEM simulations

The sedimentation process of granular materials exists ubiquitously in nature and many fields which involve the solid–liquid separation. This paper employs the coupled computational fluid dynamics and discrete element method (CFD–DEM) to investigate the sedimentation process of non-cohesive particles, including the hindered settling stage and the deposition stage. Firstly, the coupled CFD–DEM model for sedimentation is validated by the hindered settling velocity at different solid volume concentrations of suspension \(\phi _{0} \), i.e., \(\phi _0 =\) 0.05–0.6. Two typical modes of sedimentation are also presented by the concentration profiles and the equal-concentration lines. Then, the comparisons between mono- and poly-dispersed particle system are detailed. In the sedimentation of the poly-dispersed particle system, the segregation phenomenon is simulated. Furthermore, this segregation effect reduces with the increase of the initial solid concentration of suspension. From the simulations, the contact force between every pair of particles can be obtained, hence we demonstrate the “effective stress principle” from the view of the particle contact force by giving the correspondence between the particle contact force and the “effective stress”, which is a critical concept of soil mechanics. Moreover, the deposition stage can be simulated by CFD–DEM method, therefore the solid concentrations of sediment bed \(\phi _{\mathrm{max}} \) on different conditions are studied. Based on the simulation results of \(\phi _{\mathrm{max}} \) and the theory of sedimentation, this paper also discusses a method to calculate the critical time when sedimentation ends of two typical modes of sedimentation.     

Study of optical properties of potassium permanganate ($$\hbox {KMnO}_{4}$$) doped poly(methylmethacrylate) (PMMA) composite films

We have developed films of pure polymethylmethacrylate (PMMA) (0.5, 1, 2 and 5%) and potassium permanganate \((\hbox {KMnO}_{4})\)-doped PMMA composite films of thickness (\(\sim 100\, \upmu \hbox {m}\)) using the solution-cast technique. To identify the possible change that happen to the PMMA films due to doping, the optical properties were investigated for different concentrations of \(\hbox {KMnO}_{4}\) by recording the absorbance (A) and transmittance (\(T\%\)) spectra of these films using UV–Vis spectrophotometer in the wavelength range of 300–1100 nm. From the data obtained from the optical parameters viz. absorption coefficient (\(\alpha \)), extinction coefficient (\(\kappa \)), finesse coefficient (F), refractive index (\(\eta \)), real and imaginary parts of dielectric constant (\(\varepsilon _{\mathrm{r}}\) and \(\varepsilon _{\mathrm{i}})\) and optical conductivity (\(\sigma \)) were calculated for the prepared films. The indirect optical band gap for the pure and the doped-PMMA films were also estimated.     

Structural,dielectric and piezoelectric study of Ca-, Zr-modified $$\hbox {BaTiO}_{3}$$ lead-free ceramics

We prepared a lead-free ceramic (\(\hbox {Ba}_{0.85}\hbox {Ca}_{0.15})(\hbox {Ti}_{1-x}\hbox {Zr}_{x})\hbox {O}_{3}\) (BCTZ) using the conventional mixed oxide technique. The samples were prepared by an ordinary mixing and sintering technique. In this study we investigated how small amounts of \(\hbox {Zr}^{4+}\) can affect the crystal structure and microstructure as well as dielectric and piezoelectric properties of \(\hbox {BaTiO}_{3}\). X-ray diffraction analysis results indicate that no secondary phase is formed in any of the BCTZ powders for \(0 \le x \le 0.1\), suggesting that \(\hbox {Zr}^{4+}\) diffuses into \(\hbox {BaTiO}_{3}\) lattices to form a solid solution. Scanning electron microscopy micrographs revealed that the average grain size gradually increased with \(\hbox {Zr}^{4+}\) content from 9.5 \(\upmu \!\hbox {m}\) for \(x = 0.02\) to 13.5 \(\upmu \!\hbox {m}\) for \(x = 0.1\); Curie temperature decreased due to the small tetragonality caused by \(\hbox {Zr}^{4+}\) addition. Owing to the polymorphic phase transition from orthorhombic to tetragonal phase around room temperature, it was found that the composition \(x = 0.09\) showed improved electrical properties and reached preferred values of \(d_{33} = 148\) pC \(\hbox {N}^{-1}\) and \(K_{\mathrm{p}} = 27\%\).     

Enhanced run-out of dam-break granular flows caused by initial fluidization and initial material expansion

We report results of the run-out of experimental dam-break flows in a horizontal channel generated from the collapse of columns of fine (75 \(\upmu \)m) particles fluidized at various degrees. We find that the flow run-out (x) made dimensionless by the initial column length (\(x_{o})\) is a power function of the initial column height-to-length ratio (r), as shown in previous works with non-fluidized flows. The run-out of flows initially fluidized at different degrees is accounted by \(x/x_{o }=\alpha r^{n}\). For initially non-fluidized flows, our values of \(\alpha \) are significantly higher than those reported earlier for flows of coarser granular material (\(>\)0.15 mm), showing that finely grained flows have longer run-outs compared to their coarser counterparts. The coefficient \(\alpha \) is a function of the initial degree of fluidization, with a higher growth above 93 % of fluidization, which coincides with the onset of bed expansion, and it accounts for a flow run-out increase being up to more than twice that of non-fluidized flows. The parameter \(\alpha \) is well correlated with the amount of initial bed expansion, which undergoes a sharp transition at high degrees of fluidization that has shown to be an important mechanism for reducing flow friction. Our results are consistent with earlier findings that showed that bed expansion significantly increases pore pressure diffusion timescales in static columns, suggesting that the long run-out of initially expanded finely grained flows is due to their ability to diffuse pore pressure slowly.     

Vortex Shedding from an Object Moving in Superfluid $$^4\hbox {He}$$ at mK Temperatures and in a Bose–Einstein Condensate

Vortex shedding from a microsphere oscillating in superfluid \(^4\hbox {He}\) at mK temperatures is compared with that from a laser beam moving in a Bose–Einstein condensate as observed by other authors. In particular, in either case a linear dependence of the shedding frequency \(f_v\) on \(\varDelta v = v - v_c\) is observed, where v is the velocity amplitude of the sphere or the constant velocity of the laser beam above a critical velocity \(v_c\) for the onset of turbulent flow: \(f_v = a \,\varDelta v\), where the coefficient a is proportional to the oscillation frequency \( \omega \) above some characteristic frequency \(\omega _k\) and assumes a finite value for steady motion \(\omega \rightarrow 0\).     

New aspects for friction coefficients of finite granular avalanche down a flat narrow reservoir

This work examines the bulk internal friction coefficient, \(\mu \), and effective wall friction coefficient, \(\mu _w\), for finite number of nearly identical dry glass spheres in avalanche down a narrow inclined reservoir of smooth frictional bed using a validated discrete element scheme. Instantaneous deviatoric strain rate tensor \(\dot{\gamma }^d_{ij}\) and stress tensor \(\tau _{ij}\) are computed locally to evaluate a three-dimensional constitutive model developed based on the rheology of steady homogeneous surface flows. On one side, the algebraic \(\mu -I\) relation conforms to conventional relation for glass beads, \(\mu =0.34+0.31/(1+0.15/I)\) (Jop et al. in J. Fluid Mech. 541:167–192, 2005, Midi in Eur. Phys. J. E 14:341–365, 2004, Jop in Comptes Rendus Phys. 16:62–72, 2015), when the inertial number \(I>I_{c}=2\times 10^{-2}\). The assumption of collinear \(\tau _{ij}\) and \(\dot{\gamma }^d_{ij}\), however, does not hold and such misalignment agrees to the findings in non-uniform inhomogeneous flows (Cortet et al. in Europhys. Lett. 88(1):14001, 2009). Below \(I_c\), we observe a decaying \(\mu -I\) as found in slowly deforming rheology tests and a simplified model is developed in view of shear-induced dilatation upon yielding. Non-constant effective wall friction coefficient is measured to grow in time and with I towards the sphere-wall sliding friction coefficient in the contact model while preserving the depth-weakening feature as in confined steady surface flows (Richard et al. in Phys. Rev. Lett. 101:248002, 2008, Brodu et al. in Phys. Rev. E 87:022202, 2013). The fact that rotation at one sphere center can divert surface relative velocity across the contact area to render lower sliding friction is considered to develop a model describing how \(\mu _w\) drops with the ratio between rotation-induced velocity and sliding velocity, \(\varOmega \). The simulation data compares fairly well to the predicted monotonic decay of \(\mu _w\) with \(\varOmega \).     

Study of $$\upbeta $$-phase development in spin-coated PVDF thick films

A study was conducted to ascertain the effect of variation in spin speed and baking temperature on \(\upbeta \)-phase content in the spin-coated poly(vinylidene fluoride) (PVDF) thick films (\({\sim }4{-}25\,\upmu \hbox {m}\)). Development of \(\upbeta \)-phase is dependent on film stretching and crystallization temperature. Therefore, to study the development of \(\upbeta \)-phase in films, stretching is achieved by spinning and crystallization temperature is adjusted by means of baking. PVDF films are characterized using Fourier transform infrared spectroscopy, X-ray diffraction, differential scanning calorimetry, and scanning electron microscopy. It is observed that crystallization temperature lower than \(60^{\circ }\hbox {C}\) and increase in spin speed increases the \(\upbeta \)-phase content in PVDF films. Crystallization temperature above \(60^{\circ }\hbox {C}\) reduces \(\upbeta \)-phase content and increases \(\upalpha \)-phase content. It was also observed that viscosity of the PVDF solution affects the \(\upbeta \)-phase development in films at a particular spin speed.     

Strong laws for weighted sums of $$\psi $$-mixing random variables and applications in errors-in-variables regression models

In this paper, we establish strong laws for weighted sums of identically distributed \(\psi \)-mixing random variables without any conditions on mixing rate. The classical Kolmogorov strong law of large numbers is extended to weighted sums of \(\psi \)-mixing random variables. Two types of weights are considered for the weighted sums. These results are applied to the least-squares estimators in the simple linear errors-in-variables regression model when the errors are \(\psi \)-mixing random vectors.     

In vitro bioactivity evaluation of $$\upalpha $$-calcium sulphate hemihydrate and bioactive glass composites for their potential use in bone regeneration

To combine the self-setting property of \(\upalpha \)-calcium sulphate hemihydrate (\(\upalpha \)-CSH) with the bioactive property of bioactive glass (BG), BG was added into \(\upalpha \)-CSH to prepare \(\upalpha \)-CSH/BG composites. The in vitro bioactivity and cytocompatibility of the \(\upalpha \)-CSH/BG composites were assessed by soaking the composites in simulated body fluid (SBF) and co-culturing with the osteoblasts, respectively. Formation of a bone-like apatite layer on the composite surface was studied by X-ray diffraction (XRD), Fourier transform infrared spectrometry (FTIR), scanning electron microscopy (SEM) and energy-dispersive spectroscopy (EDS). Variations in ionic concentration and pH of the SBF solution were detected. The incorporation of BG into \(\upalpha \)-CSH, effectively compensated for the pH decrease caused by the dissolution of \(\upalpha \)-CSH and the ion exchange. Osteoblast-like cells (MG63) were cultured on the samples, and the MTT results confirmed that the composites containing BG were more favourable for the proliferation of these cells. Hence, \(\upalpha \)-CSH/BG composites might have great potential for the use as a bone regeneration material.     

Constitutive modeling of polycarbonate over a wide range of strain rates and temperatures

The mechanical behavior of polycarbonate was experimentally investigated over a wide range of strain rates (\(10^{-4}\mbox{ to }5\times 10^{3}~\mbox{s}^{-1}\)) and temperatures (293 to 353 K). Compression tests under these conditions were performed using a SHIMADZU universal testing machine and a split Hopkinson pressure bar. Falling weight impact testing was carried out on an Instron Dynatup 9200 drop tower system. The rate- and temperature-dependent deformation behavior of polycarbonate was discussed in detail. Dynamic mechanical analysis (DMA) tests were utilized to observe the glass (\(\alpha \)) transition and the secondary (\(\beta \)) transition of polycarbonate. The DMA results indicate that the \(\alpha \) and \(\beta \) transitions have a dramatic influence on the mechanical behavior of polycarbonate. The decompose/shift/reconstruct (DSR) method was utilized to decompose the storage modulus into the \(\alpha \) and \(\beta \) components and extrapolate the entire modulus, the \(\alpha\)-component modulus and the \(\beta\)-component modulus. Based on three previous models, namely, Mulliken–Boyce, G’Sell–Jonas and DSGZ, an adiabatic model is proposed to predict the mechanical behavior of polycarbonate. The model considers the contributions of both the \(\alpha \) and \(\beta \) transitions to the mechanical behavior, and it has been implemented in ABAQUS/Explicit through a user material subroutine VUMAT. The model predictions are proven to essentially coincide with the experimental results during compression testing and falling weight impact testing.     

Effect of Defect Size on Subsurface Defect Detectability and Defect Depth Estimation for Concrete Structures by Infrared Thermography

This study aims to reveal the effect and correlation of delamination size and defect shape for using infrared thermography (IRT) through FE modeling to enhance the reliability and applicability of IRT for effective structural inspections. Regarding the effect of delamination size, it is observed that the temperature difference between sound and delaminated area (\(\Delta \)T) increases as the size of delamination increases; however, \(\Delta \)T converges to a certain value when the area is 40 \(\times \) 40 cm and the thickness is 1 cm. As for the shape of delamination, it can be assumed that if the aspect ratio which is the ratio of the length of the shorter side to the longer side of the delamination is more than 25%, \(\Delta \)T of any delaminations converges to \(\Delta \)T of the same area of a square/circular-shaped delamination. Furthermore, if the aspect ratio is 25% or smaller, \(\Delta \)T becomes smaller than the \(\Delta \)T of the same area of a square/circular-shaped delamination, and it is getting smaller as the ratio becomes smaller. Furthermore, this study attempts to estimate depths of delaminations by using IRT data. Based on the correlation between the size of delamination and the depth from the concrete surface in regard to \(\Delta \)T, it was assumed that it was possible to estimate the depth of delamination by comparing \(\Delta \)T from IRT data to \(\Delta \)T at several depths obtained from FE model simulations. Through the investigation using IRT data from real bridge deck scanning, this study concluded that this estimation method worked properly to provide delamination depth information by incorporating IRT with FE modeling.     

Goodness-of-fit tests in semiparametric transformation models

Consider a semiparametric transformation model of the form \(\varLambda _{\theta }(Y)\) \(= m(X) + \varepsilon \), where \(Y\) is a univariate dependent variable, \(X\) is a \(d\)-dimensional covariate, and \(\varepsilon \) is independent of \(X\) and has mean zero. We assume that \(\{ \varLambda _{\theta } : \theta \in \varTheta \}\) is a parametric family of strictly increasing functions, while \(m\) is an unknown regression function. The goal of the paper is to develop tests for the null hypothesis that \(m(\cdot )\) belongs to a certain parametric family of regression functions. We propose a Kolmogorov–Smirnov and a Cramér–von Mises type test statistic, which measure the distance between the distribution of \(\varepsilon \) estimated under the null hypothesis and the distribution of \(\varepsilon \) without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of \(\theta \) and a local polynomial estimator of \(m(\cdot )\). The limiting distributions of these two test statistics are established under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedures, and we apply our tests to data on the scattering of sunlight in the atmosphere.     

Thermo-transport properties of Zn-substituted layered Li-nickel oxide, $$\hbox {LiNiO}_{2}$$

The layered Li-TM-\(\hbox {O}_{2}\) materials have been investigated extensively due to their application as cathodes in Li batteries. The electrical properties of these oxides can be tuned or controlled either by non-stoichiometry or substitution. Hence the thermo-transport properties of Zn-substituted \(\hbox {LiNi}_{1-x}\hbox {Zn}_{x}\hbox {O}_{2}\) for \(0 \le x \le 0.16\) have been investigated in the temperature range of 300–900 K for potential application as a high-temperature thermoelectric material. For \(x < 0.08\), the compounds were of single phase belonging to the space group R-3mH while for \(x > 0.08\) an additional minority phase, ZnO forms together with the main layered phase. All the compounds exhibit a semiconducting behaviour with electrical resistivity, varying in the range of  \(\sim 10^{-4}\) to \(10^{-2}\,\,\Omega \hbox {m}\) between 300 and 900 K. The electrical resistivity is found to increase with increasing Zn-substitution predominantly due to a decrease in the charge carrier hole mobility. The activation energy remains constant, \(\sim \)10  meV, with Zn-substitution. The Seebeck coefficient of the compounds is found to decrease with increasing temperature and increase with increasing Zn-substitution. The Seebeck coefficient decreases from \(\sim \)95 to \(35\ \upmu \hbox {V K}^{-1}\) and the corresponding power factor is \(\sim \)12\(\ \upmu \hbox {W m}^{-1}\ {\hbox {K}}^{-2}\) for the \(x = 0.16\) compound.     

Measurement of the Nonlinearity of Heat-Flux Sensors Employing a $$\hbox {CO}_2$$ laser

Heat-flux sensors are widely used in industry to test building products and designs for resistance to bushfire, to test the flammability of textiles and in numerous applications such as concentrated solar collectors. In Australia, such detectors are currently calibrated by the National Measurement Institute Australia (NMIA) at low flux levels of 20 W \(\cdot \) m\(^{-2}\). Estimates of the uncertainty arising from nonlinearity at industrial levels (e.g. 50 kW \(\cdot \) m\(^{-2}\) for bushfire testing) rely on literature information. NMIA has developed a facility to characterize the linearity response of these heat-flux sensors up to 110 kW \(\cdot \) m\(^{-2}\) using a low-power \(\hbox {CO}_2\) laser and a chopped quartz tungsten–halogen lamp. The facility was validated by comparison with the conventional flux-addition method, and used to characterize several Schmidt–Boelter-type sensors. A significant nonlinear response was found, ranging from (\(3.2 \pm 0.9\))% at 40 kW \(\cdot \) m\(^{-2}\) to more than 8 % at 100 kW \(\cdot \) m\(^{-2}\). Additional measurements confirm that this is not attributable to convection effects, but due to the temperature dependence of the sensor’s responsivity.     

Structural,elastic, optoelectronic and magnetic properties of $$\mathbf{CdHo }_\mathbf{2}{} \mathbf{S}_\mathbf{4}$$ spinel: a first-principle study

We report the results of the full-potential linearized augmented plane wave (FP-LAPW) calculations on the structural, elastic, optoelectronic and magnetic properties of \(\hbox {CdHo}_{2}\hbox {S}_{4}\) spinel. Both the generalized gradient approximation (GGA) and Trans-Blaha modified Becke-Johnson potential (TB-mBJ) are used to model the exchange-correlation effects. The computed lattice parameter, internal coordinate and bulk modulus are in good agreement with the existing experimental data. According to the calculated elastic moduli, \(\hbox {CdHo}_{2}\hbox {S}_{4}\) is mechanically stable with a ductile nature and a noticeable elastic anisotropy. The ferromagnetic phase of \(\hbox {CdHo}_{2}\hbox {S}_{4}\) is energetically favourable compared to non-magnetic one, with a high magnetic moment of about 8.15 \(\upmu _{\mathrm{B}}\). The calculated band structure demonstrates that the title compound is a direct band gap semiconductor. The TB-mBJ yields a band gap of \(\sim \)1.86 and \(\sim \)2.17 eV for the minority and majority spins, respectively. The calculated optical spectra reveal a strong response in the energy range between the visible light and the extreme UV regions.     

Effect of Pressure on Deep-Ocean Thermometers

A laboratory experiment was devised and performed to investigate the pressure dependence of Sea-Bird Electronics SBE35 and SBE3 deep-ocean thermometers. The thermometers were mounted in a massive brass comparator together with a calibrated standard platinum resistance thermometer. The measurements were performed in a pressure chamber in the pressure range 0.1 MPa to 60 MPa. The results showed that both the investigated SBE35 and SBE3 thermometers are pressure dependent, with a pressure sensitivity of +41 \(\upmu \)K\(\cdot \)MPa\(^{-1}\) and \(-77\) \(\upmu \)K\(\cdot \)MPa\(^{-1}\), respectively. Nevertheless, the results obtained in only one individual device per model (one SBE35 and one SBE3) cannot be generalized and further investigations of a larger number of devices per model are needed.     

NRC Microwave Refractive Index Gas Thermometry Implementation Between 24.5 K and 84 K

The implementation of microwave refractive index gas thermometry at the National Research Council between 24.5 K and 84 K is reported. A new gas-handling system for accurate control and measurement of experimental gas pressure has been constructed, and primary thermometry measurements have been taken using a quasi-spherical copper resonator and helium gas at temperatures corresponding to three defining fixed points of the International Temperature Scale of 1990 (ITS-90). These measurements indicate differences between the thermodynamic temperature T and ITS-90 temperature \(T_{90}\) of \(\left( T - T_{90} \right) = -0.60 \pm 0.56\) mK at \(T_{90} = 24.5561\) K, \(\left( T - T_{90} \right) = -2.0 \pm 1.3\) mK at \(T_{90} = 54.3584\) K, and \(\left( T - T_{90} \right) = -4.0 \pm 2.9\) mK at \(T_{90} = 83.8058\) K. The present results at \(T_{90} = 24.5561\) K and \(T_{90} = 83.8058\) K agree with previously reported measurements from other primary thermometry techniques of acoustic gas thermometry and dielectric constant gas thermometry, and the result at \(T_{90} = 54.3584\) K provides new information in a temperature region where there is a gap in other recent data sets.     

Characterization and optical properties of $$\hbox {Pr}_{2}\hbox {O}_{3}$$-doped molybdenum lead-borate glasses

\(\hbox {Pr}^{3+}\) doped molybdenum lead-borate glasses with the chemical composition 75PbO?[25–(x \(+\) y)\(\hbox {B}_{2}\hbox {O}_{3}]\)–\(y\hbox {MoO}_{3}\)–\(x\hbox {Pr}_{2}\hbox {O}_{3}\) (where \(x = 0.5\) and 1.0 mol% and \(y = 0\) and 5 mol%) were prepared by conventional melt-quenching technique. Thermal, optical and structural analyses are carried out using DSC, UV and FTIR spectra. The physical parameters, like glass transition \((T_{\mathrm{g}})\), stability factor \((\Delta T)\), optical energy band gap \((E_{\mathrm{gopt}})\), of these glasses have been determined as a function of dopant concentration. The \({T}_{\mathrm{g}}\) and optical energy gaps of these glasses were found to be in the range of 290–350\({^{\circ }}\hbox {C}\) and 2.45–2.7 eV, respectively. Stability of the glass doped with \(\hbox {Pr}^{3+}\) is found to be moderate (\(\sim \)40). The results are discussed using the structural model of Mo–lead-borate glass.     

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.