Parabolic Function Research Articles

A study on the compressive mechanical properties of 316L diamond lattice structures manufactured by laser powder bed fusion based on actual relative density

Lattice structure design is a novel structural lightweight method and has many advantages, but some researches ignored the effects of the factors studied on actual relative density and skipped actual relative density to discuss the effects on mechanical properties. In this paper, the effects of process parameters and geometrical parameters on the compressive mechanical properties of 316L diamond lattice structures manufactured by laser powder bed fusion were investigated on the basis of actual relative density. The results show that the actual relative density (28.0 % → 25.4 %) and the compressive strength in the building direction (55.7 ± 1.3 MPa → 42.0 ± 1.5 MPa) decrease as the hatch spacing (0.05 mm → 0.09 mm) increases, and they conform to the parabolic function relationship better than the Gibson-Ashby relationship, but the opposite in the XY direction. The actual relative densities and compressive strength inevitably conform to the Gibson-Ashby relationship regardless of the nominal relative densities and cell sizes, and the modulus (857 ± 54 MPa → 1019 ± 51 MPa) increases with the increasing of strut diameter (0.54 mm → 3.21 mm). The compressive strength could decrease slightly with the increasing of contour size (10 mm → 30 mm), while the modulus (964 MPa → 682 MPa) decreases rapidly. To sum up, the process parameters and geometrical parameters could affect the actual relative density significantly, and the actual relative density is the key to correctly analyze the effects of various parameters. The conclusions of this work are helpful to optimize the manufacturing process of lattice structures and design geometrical parameters according to the required performance.

The Hermitian symmetric space Fokas–Lenells equation: spectral analysis and long-time asymptotics

Abstract Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas–Lenells (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL equation is transformed into the solution of a Riemann–Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space FL equation with the aid of the parabolic cylinder function.

Mechanical Properties and Failure Mechanism of the Weakly Cemented Overburden in Deep Mining

With increases in the mining depth and area in the Ordos coal field, the failure law of the super thick sandstone in the Zhidan group leads to frequent disasters, such as rock bursts and mine earthquakes, which have become a significant issue, restricting large-scale continuous mining. To adequately understand the movement mechanism of the super-thick and weakly cemented overburden, and to promote the large-scale mining of the coal resources under it, this study analyzes the physical and mechanical properties, along with the microstructural characteristics, of the weakly cemented overburden of the Yingpanhao Coal Mine through mechanics tests, scanning electron microscope tests (SEM) and hydrolysis experiments. A two-dimensional discrete element model of the survey region is then built to explore the temporal and spatial evolution laws of the overburden failure. The results show that, even though poorly cemented strata such as the Cretaceous Zhidan group sandstone and the Zhiluo group sandstone are weak in lithology, their unique mineral composition and microstructural characteristics give them a greater rigidity when their thickness reaches a certain value. The surface subsidence exhibits a sudden increase, and the dynamic disaster range of the overlying strata is wide when deep multi-face mining was carried out under the super-thick and weakly cemented overburden. The temporal and spatial evolution laws of the strata subsidence and influence boundary are closely related to their depth, and their relationships evolve into the Boltzmann function and Boltzmann–parabolic function, respectively. The failure mode of the super-thick and weakly cemented overburden is ‘beam–arch shell–half arch shell’, and the failure boundary exhibits arch fractures.

Comparison and study of several methods for solving the hydraulic jump values in pumping well

This paper was envisaged to provide deeper insights of the hydraulic jump value in pumping well in the loess aquifer. Herein, we discussed the research status and problems of the value, compared the differences between the analytical solution method and the empirical formula method. Subsequently, we proposed a fitting method with improvement and the tangent method to estimate the hydraulic jump value, and quantitatively analyzed the linear variation rule of the water level curve around the well. The results show that the value calculated by the traditional empirical formula was about one tenth of the theoretical value in constant flow and the water level around the well was found to be logarithmic in the range of N1∼N3. When the range was extended, the water level curve changed to a parabolic nature and power function in turn. The hydraulic jump value obtained by h-r extension of the modified water level curve fitting equation was improved greatly, and the average error was found to be less than 1 m. The stability of the hydraulic jump value using the tangent method was poor, and the estimated horizontal distance was larger than 3 m. The results show that the reliability of the improved curve fitting method was better, while the empirical formula method and the tangent method exhibited a larger error and poor stability. When the error correction was performed, the improved curve fitting method could be used to estimate the hydraulic jump value under the same conditions, and can replace the theoretical calculation value.

Dipole modelling of temperature-dependent magnetic flux leakage

Due to the nonlinear coupling, assessing the direct effect of temperature on magnetic flux leakage (MFL) signal is a complicated task. If temperature induces inner stress, it makes the problem doubly difficult, so few models are available for predicting the MFL signal under this condition. To model the effect of temperature on MFL signal, the temperature-dependent magnetic dipole models are proposed. In the first case, where the direct thermal effect is involved only, the dipole model is improved via the modified temperature-dependent Jiles-Atherton (J-A) model. While in the second case, where the combined effects of temperature and thermal stress are considered, the magnetomechanical J-A parameters are further introduced into the dipole model. The thermal stress distribution around a cylindrical through-hole defect is solved by thermoelastic and solid mechanics theories. The magnetomechanical theory is employed to analyse the stress-dependent magnetisation distribution, the key parameter in the magnetic dipole model. The verified experiments are conducted on an M250-50A non-oriented grain (NO) silicon steel specimen with a cylindrical through-hole defect. And the MFL signals predicted by both proposed models agree with the experimental results. When the direct effect of temperature is involved only, the peak-to-peak amplitude of the MFL signal (MFLpp) presents approximately linear dependence on temperature in the range from −40 °C to 60 °C. In addition, when both temperature and thermal stress are considered, the MFLpp changes as a parabolic function of temperature, this being much more significant than the direct effect. The proposed models can act as effective tools to understand the temperature and thermal stress influences on MFL signals. They are also appropriate to solve the inverse problem of sizing the defects accurately when the temperature is involved.

An efficient and flexible window function for a memristor model and its analog circuit application

The memristor is a novel nanostructured resistive tuning two-terminal electronic device that has been widely explored in the areas of neuromorphic computing systems, memory, digital circuits, analog circuits, and many more new applications. In this article, an efficient and flexible window function is presented for a linear drift memristor model. The proposed parametric cubic parabolic window function provides a unique feature (controllable window function discontinuity at the boundaries) to a linear drift memristor model by which the distorted pinched hysteresis loop problem is resolved and the number of programming resistance states of the memristor is improved. Five control parameters are introduced in the proposed window function in order to correct the existing problems (such as boundary effect, boundary lock and inflexibility) and are able to provide asymmetric nonlinearity at the boundaries of the device, making it feasible for tracking the resistive switching dynamic of a futuristic oxide-based memristive device with different inert electrodes. The proposed model is validated with a solution-processed ZnO-based fabricated memristive device. A programmable analog gain amplifier circuit is ultimately executed to simulate the utilization of the evolved memristor model, and the effect of memristance resolution is investigated.

Analytical solution of stability and vibration problem of clamped cylindrical shells containing functionally graded layers within shear deformation theory

The aim of the study is to present a new approach to the analytical solution of the vibration and stability problem of clamped sandwich cylindrical shells (SCSs) covered by functionally graded (FG) coatings under an axial compressive load in the framework of shear deformation theory (ST). After modeling the micro and macro mechanical properties of FG coated SCSs with various configurations, the constitutive relations and basic equations are derived depending on the stress, deflection and two angles of rotation functions using modified Donnell type theory. An attempt is made to analytically solve the fundamental differential equations for SCSs covered by FG coatings using novel approximation functions satisfying the clamped boundary conditions. Three different shear stress functions (SSFs) such as parabolic shear stress function (Par-SSF), cosine-hyperbolic shear stress function (Cos-Hyp-SSF) and uniform shear stress function (U-SSF) are used in the analysis. After confirming the accuracy of the obtained results, the influences of coating profiles, volume fractions and layer arrangement variations on the critical parameters for three different transverse shear stress functions are investigated in detail.

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

In this paper, the phase space representation for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. We have found the Husimi distribution function for the stationary states of the oscillator model under consideration for both cases without and with the applied external homogeneous field. The obtained function is expressed through the double sum of the parabolic cylinder function. Different special cases and the limit relations are discussed, too.

A mesoscale tensile model for woven fabrics based on Timoshenko beam theory

The yarn weave architectures highly influence the mechanical response of woven fabrics. To provide an accurate prediction of the mechanical behaviors of woven fabrics under tension, a mesoscale model is developed. The model takes into account yarn properties, yarn weaving pattern, and interactions between yarns. The yarn properties such as tensile modulus and yarn strength of yarns are considered as random variables. Each yarn in fabric is modeled as a Timoshenko beam with a weaving shape modeled by a parabolic function. The yarn shape evolution and mechanical response of undulated yarn under tension are computed based on Castigliano’s second theorem. The damage initiation and propagation of a yarn in fabric is introduced through a damage variable. The model is demonstrated on woven jute fabrics. Analytical and numerical analyses are carried out and show good prediction on the macroscopic tensile response of woven jute fabric. The model provides a quantitative understanding between yarn geometrical parameters and the expected material response, and makes it helpful for designing woven structural composites with high performance.

Developing an evaluation model based on unascertained measurement for evaluation of tunnel squeezing

Tunnel squeezing brought great difficulties to the construction and severely threatened the safety of on-site operators. The researches regarding large deformation evaluation have been widely developed, but actual conditions of tunnels are considerably complex, producing a large variety of uncertainty information existing in the evaluation process. Therefore, we constructed an unascertained measurement model incorporating four membership functions for evaluation of tunnel squeezing based on the collected datasets. Simultaneously, information entropy was introduced to objectively calculate the index importance for each index. For the first group data (GPI), the accuracy associated with four membership functions are 100%, 83.33%, 50%, and 83.33%, respectively, while the accuracy of GPII are 70%、77.5%、67.5%, and 70%, respectively. Linear function and parabolic function show better performance on uncertainty information interpretation according to the classification results. The results revealed that the uncertainty model constructed in this study can enrich the available uncertainty evaluation system.

Feature curve-based registration for airborne LiDAR bathymetry point clouds

The airborne LiDAR bathymetry (ALB) system is widely used in the fields of sea–land measurement, including seafloor topographic feature description, 3D seafloor model construction, coral reef monitoring, and underwater archaeology. A manned ALB data registration method based on feature curves was proposed considering problems such as sparse ALB data features, low point cloud density, and difficult corresponding feature extraction. First, triangulation network interpolation was used to extract the isoline points of the seafloor point cloud. The curves describing the seafloor topographic trend were generated through cubic parabolic spline function interpolation. Second, the curve deformation energy function was constructed based on the curve features, and a similarity measurement was carried out on the registration curve by integrating the energy function with the longest common subsequence (LCSS) algorithm. The RANSAC and ICP algorithms were used for rough and fine registration, respectively. The effectiveness and robustness of the proposed method were evaluated using two sets of experimental data. The experimental results showed that the average distance between corresponding points in the coarse registration stage reached 0.128 m and 0.136 m. After fine registration, the average point distance between the point cloud coordinates and ground truth reached 0.073 m and 0.267 m, providing an effective and reliable solution for ALB data registration.

Contrasting water use characteristics of riparian trees under different water tables along a losing river

• Water use of riparian trees under various water table depth (WTD) along a losing river was determined. • Riparian trees accessed more upper soil water and had lower leaf δ 13 C under deep WTD compared to shallow WTD. • Water source contribution was a parabolic function of WTD and 2.1 m was the optimum WTD to utilize deep water. • Leaf δ 13 C had a positive linear relation with proportional contribution of deep water. Rivers losing flow into surrounding aquifers (‘losing’ rivers) are common under changing climates and groundwater overexploitation. The riparian plant-water relations under various water table dynamics along a losing river remain unclear. In this study, the water isotopes (δ 2 H and δ 18 O), leaf δ 13 C, and MixSIAR model were used combinedly for determining the root water uptake patterns and leaf water use efficiency (WUE) of Salix babylonica (L.) at three sites (A, B, and C) with different water table depths (WTDs) in the riparian zone of Jian and Chaobai River in Beijing, China. The correlations of water source contributions with WTD and WUE were quantified. The riparian S. babylonica primarily took up upper (0–80 cm) soil water (71.5%) with the lowest leaf δ 13 C (−28.8 ± 1.1 ‰) at site A under deep WTD (20.5 ± 0.5 m). In contrast, deep water below 80 cm depth including groundwater contributed 55.1% to S. babylonica at site B with fluctuated shallow WTD (1.9 ± 0.4 m), where leaf δ 13 C was highest (−27.9 ± 1.0 ‰). The S. babylonica mainly used soil water in 30–170 cm layer (56.9%) with mean leaf δ 13 C of − 28.2 ‰ ± 0.7 ‰ at site C with stable shallow WTD (1.5 ± 0.1 m). It was found that both the contributions of upper soil water in 0–80 cm and deep water below 80 cm had significantly quadratic correlations with WTD under shallow water table conditions ( p < 0.05). Leaf δ 13 C was negatively correlated with contributions of upper soil water above 80 cm depth, but it was positively related to the contributions of deep water below 80 cm in linear functions ( p < 0.001). The results indicated that 2.1 m was the optimum WTD for riparian trees, because they maximized the use of deep water sources to get the highest WUE. This study provides insights into managing groundwater, surface water resources and riparian afforestation in losing rivers.

A 3D parametric Martian magnetic pileup boundary model with the effects of solar wind density, velocity, and IMF

Using global magnetohydrodynamic simulations, we construct a 3D parametric model of the Martian magnetic pileup boundary (MPB). This model employs a modified parabola function defined by four parameters. The effects of the solar wind dynamic pressure, the solar wind densities and velocities, and the intensity and orientation of the interplanetary magnetic field (IMF) are examined using 267 simulation cases. The results from our parametric model show that (1) the MPB moves closer to Mars when the upstream solar wind dynamic pressure (Pd) increases, the subsolar standoff distance decreases and the flaring degree of the Martian MPB increases with an increasing Pd according to the power-law relations. For the same Pd, a higher solar wind velocity (a lower density) leads to a farther location of the MPB from Mars, along with a larger flaring degree, which is explained by the higher solar wind convection electric fields and a stronger magnetic pileup process under these conditions. (2) Larger Y or Z components of the IMF, BY or BZ, result in a thicker pileup region and a farther MPB location from Mars, as well as a decrease in the flaring degree. The radial IMF component, BX, has little effect on the geometry of the MPB. (3) In most of the simulations used to derive the current parametric model, the strongest Martian crustal magnetic field is located on the dayside. However, for a larger value of the southward IMF, the Martian MPB is located farther away in the northern hemisphere instead of the southern hemisphere. The north-south asymmetry of the Martian MPB with the southern hemisphere being farther away is observed for other IMF directions. We suggest that the magnetic reconnection of the southward IMF with the crustal field that occurs at middle latitudes of the southern hemisphere results in different magnetic field topologies and the closer location of the MPB under these conditions. Our model results show a relatively good agreement with the previous empirical and theoretical models.

Green and Low-Carbon Commuting Evaluation and Optimization of a Cross-Border Metropolitan Region by the Subway Network: The Case of Shenzhen and Hong Kong, China

In the context of the One Country–Two Systems policy, Shenzhen and Hong Kong have rapidly grown into a dual-core international metropolis. However, a huge barrier is the commuting cost incurred by the tariff policy, which blocks the road to the high-quality development of the Guangdong–Hong Kong–Macao Greater Bay Area (GBA). Based on traffic accessibility and kernel density analysis of the subway network and points of interest (POIs) of urban functional areas with ArcGIS, this study analyzed the commute time and convenience level of the functional areas in the two separate tariff zones. The traffic (commute) time in 75 min was taken as the turning point of the efficiency of the Shenzhen–Hong Kong connection, and it was found that both the customs clearance time and the accessible areas show the shape of a parabolic function with an upward opening. It was also found that, in Shenzhen and Hong Kong, the commute time in most of the business, commercial, residential, and industrial areas is 60–75 min. In particular, the comfortable range within 45 min was not achieved. It is necessary to further promote the effectiveness of the Shenzhen–Hong Kong customs clearance system and continuously reduce the customs clearance cost so as to promote the integration of Shenzhen and Hong Kong in order to implement the national strategy of “dual circulation” and innovation-driven development and efficiently drive the coordinated construction of the GBA.

High-temperature and dynamic mechanical characterization of closed-cell aluminum foams

Metallic foams exhibit great application prospects in some extreme working conditions such as high temperatures and/or high-velocity impact environments, but predominantly, their mechanical behavior has remained to be fully understood to date. In this study the uniaxial and biaxial tests are conducted on closed-cell aluminum foams at elevated temperatures. It is indicated that both drop stress and initial failure strength exhibit approximately linear relationship with the applied temperature, while the densification strain keeps almost at a constant value. The higher foam density or lower applied temperature, the larger the initial failure surface characterized in the von Mises - mean stress plane. However, the normalized failure surfaces nearly do not depend on foam density or temperature, and can be well fitted in terms of elliptical or parabolic function. Dynamic compressive tests under constant strain-rates reveal that the compressive strength is correlated to strain-rate positively, but the densification strain negatively. A novel rate-dependent constitutive model is proposed to describe the compressive constitutive behavior of Al foams. Finally, the crushable foam model with the tabular input of yield ratio approach can produce satisfactory numerical predictions compared to dynamic experimental results. This study provides new insights into the intrinsic mechanical properties of metallic foams under some extreme conditions.

(Digital Presentation) Calculation of the Energy Band Diagram and Estimation of Electronic Transport Parameters of Metastable Amorphous Ge2Sb2Te5

Phase change memory (PCM) is a high speed, high density non-volatile resistive memory technology that utilizes different phases (crystalline and amorphous) of phase change materials such as Ge2Sb2Te5 (GST) to store information [1]. Here, the material undergoes two types of reversible switching phenomena: (i) Ovonic Threshold Switching (OTS), which causes the amorphous phase of the material to switch from a highly resistive state to conductive state with application of high electric fields, resulting in current flow and (ii) Ovonic Memory Switching (OMS), which is due to the change of the phase of the material between amorphous and crystalline phases induced by heating [2]. Even though PCM entered high volume manufacturing, the electronic properties of the phase change materials are still not well understood [3,4]. In this work, we construct the energy band diagram of amorphous GST as a function of temperature using a temperature dependent model of effective activation energy of conduction in metastable amorphous GST [5], which we obtained from high-speed experiments on GST line-cells [6]. Assuming the bandgap energy to linearly decrease with temperature [7,8] and p-type conduction (based on positive Seebeck coefficients measured in a wide temperature range [9]) , we determine a temperature dependent Fermi energy level from 0K to melting temperature, Tmelt ~ 858 K. Liquid GST is expected to become metallic (bandgap collapsing to 0) at ~ 894K, based on the experimental results. We also estimate the carrier concentration at Tmelt utilizing the latent heat of fusion (126 kJ/kg) [10] to be pmelt = 1.47 x 1022 cm-3. Using the melt resistivity measured on GST thin film, we calculate the carrier mobility at melting point as µmelt ~0.187 cm2/V-s, close to the previously reported value of 0.15 cm2/V-s based on crystalline state mobility and a density of states calculation [2]. Assuming a weak temperature dependence of the mobility [5], we obtain the carrier concentration of ~3.37× 1017 cm-3 at room temperature which lies within the ~1017-1018 cm-3 range estimated in a former study [3]. Finally, we calculate conduction activation energy of as-deposited amorphous GST from temperature dependent Seebeck coefficient measured simultaneously with the resistance [9]. The activation energy varies as a parabolic function of temperature where it starts from 0 eV at 0 K, reaches a peak of ~0.257 eV near glass transition temperature (~400 K) with the room temperature value of ~0.24 eV and becomes 0 eV again at ~810 K. We also utilize the Seebeck coefficient measurements along with the band edges and Fermi energy level information from the energy band diagram to calculate the ratio of minority carrier concentration to the total carrier concentration as a function of temperature; this is useful to predict the temperature beyond which bipolar conduction becomes significant.Acknowledgment: Analysis is performed with the support of US National Science Foundation (NSF) award ECCS 1711626. The experimental data used for this analysis were collected with the support of US NSF DMR-1710468 on devices fabricated with the support of US Department of Energy Office of Basic Energy Sciences.

Studi Awal Sonifikasi Pada Data Gerak Parabola Berbantuan Arduino Uno R3 Atmega328

Parabolic motion is the typical phenomenon in classical mechanics. The students are able to understand the concept by reading the data in the parabolic function. However, there is limited material for visual impaired students to study the parabolic motion data. One of the solutions to overcome this problem is the parabolic motion experiment development using sonification method. Research has been carried out to analyze the accuracy of the frequency produced by the speakers in data conversion assisted by Arduino Uno. The research also analyzes the effect of elevation angle variations on the resulting frequency changes. The data collection process is carried out by analyzing the percentage of accuracy using the average technique of the data recorded by the Phyphox application. The data analysis of the Phyphox application is compared with the data obtained from the Tracker application. From the results of data analysis obtained the percentage value of the velocity for the angle 30o, 45o, and 70o were 99.92%, 99.67%, and 98.42%, respectively. Therefore, it can be said that the preliminary sonification study is feasible to be used in physics experiments on parabolic motion learning, especially for visual impaired students. The final plotting result of this research shows that the frequency versus time at each angle produces a graph in the parabolic form. While the effect of angle variations on frequency changes is that the larger the angle, the slower it will reach the peak frequency.

A quasi-contagion process modeling and characteristic analysis for real-world urban traffic network congestion patterns

The congestion propagation pattern in urban traffic networks has long been viewed as a complex question. Inspired by the fact that virus spread patterns are similar to traffic congestion propagation in some ways, this paper proposes a congestion propagation model (SIS-CP) to describe the congestion propagation patterns in large-scale traffic networks with very few parameters. In this SIS-CP model, the attribute of the real traffic network in the actual operation process is considered, a method based on Hilbert–Huang transform (HHT-based) is proposed to determine traffic states of links, and the congestion propagation patterns are analyzed. The real-world data of ‘DiDi Chuxing GAIA Open Dataset Initiative of Chengdu city from March to June is used to calibrate and validate the proposed models. The results show that the basic and limit value of congestion links’ ratio do exist in real traffic networks; the relationship between congestion links’ ratio and its derivative is conformed to the parabolic function; the HHT-based method can determine traffic states of links accurately; and the SIS-CP model can describe the congestion propagation patterns correctly.

Experimental characterization and modeling of complex anisotropic hardening in quenching and partitioning (Q&P) steel subject to biaxial non-proportional loadings

Quenching and partitioning (Q&P) steels, as one of the third-generation advanced high strength steels, exhibit complex anisotropic hardening behavior under biaxial non-proportional loadings. In this study, a wide range of strain path changes (SPCs) defined as an angle (χ) between stress deviators before and after the SPC is experimentally applied to characterize the anisotropic hardening of a Q&P steel with a strength grade of 980 MPa (QP980). The investigated SPCs include uniaxial tension followed by compression (UT-UC; χ=180°), uniaxial compression followed by tension (UC-UT; χ=180°), uniaxial tension followed by plane strain (UT-PS; χ=30°), equi-biaxial tension followed by plane strain (EBT-PS; χ=30°), followed by uniaxial tension (EBT-UT; χ=60°) and followed by simple shear (EBT-SH; χ=90°). These SPCs are achieved by the program-controlled biaxial tensile testing equipment using novel arm-reinforced cruciforms. The experimental findings in the QP980 steel sheet are summarized as follows: (1) Strong strain-dependent Bauschinger effect, transient hardening and permanent softening are observed in both UT-UC and UC-UT loadings; (2) Bauschinger effect increases as strain increases, and it exhibits obvious tension-compression asymmetry; (3) The transient hardening behaviors are path-dependent under SPCs with various χ; (4) The Bauschinger coefficient is approximately a parabola function of cosχ; (5) The most significant strength softening occurs under EBT-SH loading. In order to describe the complex hardening behavior of the QP980 steel, a distortional hardening model is proposed on the basis of non-associated flow rule. A coupled quadratic and stress-invariant-based yield function is used to describe stress anisotropy, strength differential effect, and anisotropic hardening for proportional loadings. Then, the Bauschinger effect related hardening and path-dependent permanent softening are captured by a modified homogeneous anisotropic hardening model. The calibrated model parameters and experimental validations demonstrate that the strain- and path-dependent Bauschinger effect can be well predicted by the proposed constitutive model. Especially, the model enables to accurately describe both the asymmetric anisotropic hardening and yield surface distortions of the QP980 steel at various biaxial SPCs.

Action-based embodied design for mathematics learning: A decade of variations on a theme

Embodied cognition theory emphasizes that bodily interaction with the environment is important for all forms of learning, including mathematics. This theoretical trend coincides well with developments in motion responsive technology, and has resulted in numerous embodied technologies for mathematics learning. This review aims to contribute to clarifying theoretically and empirically grounded design principles of action-based embodied designs for mathematics learning. We analyzed 79 publications between 2010 and 2019, containing 15 studies assessing 15 sensorimotor problems for five mathematical domains (proportion, angle, area, parabola, and sine function), and explicated the characteristics of the technologies, their learning sequences and elicited learning processes, and the influence of within-topic ask variations on students’ learning. We found that action-based designs pose motor control problems using continuous motion feedback to facilitate learners to discover and practice a challenging new ways of moving their hand(s) in which to ground mathematical cognition. The state of discovery of the sensorimotor solution is important, and passive and readymade designs are cautioned. The learning sequence in which these technologies are embedded, elicit mathematical knowing through necessary and sequential phases in which personal idiosyncratic experiences increasingly converge into a culturally shared mathematical discourse. In the qualitative stage, an acting step elicits students to actively establish new motor coordination-patterns through the emergence of new perceptual structures known as attentional anchors. In the subsequent reflecting step, students’ personal sensorimotor experiences and attentional anchors become the ground for referencing in (a shared) mathematical discourse through multimodal (words, gestures) collaboration with a tutor. In the quantitative stage, measuring artifacts (grids, protractors, numbers, variables) are included in students’ field of promoted action, which discretize and formalize students’ actions and subsequent reflections into culturally recognizable quantitative forms. Critically, task factors such as the type of objects students manipulate (cursors icons, bars, rectangle), and the direction these objects are moved (parallel, orthogonal), affect students’ attentional anchors and subsequent reflections in the qualitative stage, but converge to similar mathematical insights in the quantitative stage. These insights help to better use (new) motion responsive technology in eliciting child–computer interaction that can lead to mathematical cognition and beyond.