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Thermoelectric (TE) materials have a broad range of application in engineering such as cooling for thermal protection. Due to the brittle nature, TE semiconductors are prone to crack/voids under thermal shock. The investigation of the transient problem in cracked thermoelectric materials is essential for engineering applications. This paper describes a thermal-mechanical coupling method to analyze the transient response of the TE medium of finite size with an arbitrarily located inner crack. Based on the Fourier and Laplace transform, the crack problem is simplified into a system of singularity integral equations which can be solved by simple allocation. Besides, the results of this paper are verified through the finite element method (FEM). Furthermore, the effects of the crack position, crack's thermal and electrical permeability are discussed. Also, the effect of crack on the thermoelectric energy conversion efficiency is studied. Results show that the field concentrations at the crack tip become more significant when the crack is centrally located in the vertical direction. The electrical permeability of the crack has a limited influence on the energy conversion efficiency. However, as thermal permeability of the crack goes up, the efficiency falls significantly.

The deformation characteristic of an ultra-heavy plate in thickness direction is described by introducing a parameter, called deformation penetration coefficient, and the rolling force model taking this parameter into account in the broadside stage is established. In this paper, the deformation penetration coefficient is defined as the ratio of the actual deformation depth to the overall plate thickness, which is shown as the function of the initial thickness, the pass reduction and the critical shape factor. Since the plate width speard can be neglected during broadside rolling, a two-dimensional velocity field is constructed by simplifying a classical three-dimensional velocity field. For the purpose of solving the problem of functional integration originated from the nonlinear Mises specific plastic power, the simplified velocity field is analyzed by the geometric approximation (GA) yield criterion, and the internal deformation power is obtained. Moreover, the friction power and the shear power are obtained by the inner product method of the strain vector and the mean velocity intergral method, respectively. The analytical solution of rolling force is obtained by the variational method, and its value for each pass is compared with the measured data from a domestic factory. The comparison result shows that the rolling force model in this paper has a high predictive accuracy, since the maximum error is less than 8.6%. The rolling parameter analysis shows that the deformation penetration coefficient, the pass reduction and the ratio of plate thickness to diameter affect the rolling force of the ultra-heavy plate apparently.

Fluid-conveying micro/nano tubes are key tools, which have great applications in biological devices and especially smart drug delivery in order to target the cancer cells. Furthermore, exploiting the smart materials and their combination with drug delivery systems may positively affect the instability control and improve the efficiency and adaptability of design. Recently a specific size-dependent behavior for piezoelectric materials, known as flexoelectric effect, has drawn a great deal of attention. It is proven that this effect, which is resulted by coupling between the strain field and electric polarization, is of significant importance in structures with nano dimensions. This paper is carried out to investigate the vibrations and instability analysis of fluid-conveying piezoelectric nanotubes on the basis of flexoelectricity approach. The fluid-conveying nanotubes made for drug delivery targets are commonly in contact with soft tissues, which could be modeled as a Kelvin-Voigt foundation. The nonlocal strain gradient theory (NSGT) constitutive relations are employed in order to model the problem. An appropriate electric potential distribution is determined using the Maxwell's equation and Gauss's law. The Euler-Bernoulli beam theory and slip boundary conditions are exploited to derive the governing fluid-structure interaction (FSI) equation, which contains flexoelectric and surface effect terms. Galerkin's principle is hired to discretize the equation leading to an eigenvalue problem. Afterwards, the obtained characteristic equation is solved straightforwardly to gain the eigenvalues. The instability of the nanotube is investigated throughout presenting the eigenvalue diagrams. Some illustrations are employed to analyze the effect of different involved parameters on the vibrations and instability behavior of the system. The reported results in the numerical section of the paper may be helpful to achieve an efficient and accurate design of fluid-conveying nanotubes.

This paper studies the bifurcation behavior in an inflated bilayer tube of arbitrary thickness under inflation and uni-axial extension. It is assumed that both layers are composed of the Gent material with each layer having its own Jm, where Jm is a material parameter in the Gent model that signifies the maximum extensibility. First, we determine several critical parametrical regions where localized bulging disappears for a single-layer tube. Then we investigate localized bulging in an inflated bilayer tube, where one layer (layer I) of the tube cannot bulge whereas the other part (layer II) can. Surprisingly, we find that such a composite tube is still susceptible to localized bulging and localized bulging can be prevented only if the proportion of layer I exceeds a critical value, no matter whether layer I occupies the inner side or the outer side. Even for a very thin bilayer tube, the same feature holds. The cases of fixed axial force and fixed axial stretch are both studied, and the critical geometrical parameters marking the transition between bulging and no bulging are determined. Moreover, we carry out a numerical analysis by use of the finite element method to verify the applicability of an explicit bifurcation condition and the predicted bifurcation behavior. This paper offers a possible way to avoid bulging formation in a cylindrical tube while retaining moderate extensibility.

A new frictional contact model of an arbitrarily oriented substrate loaded by a rigid flat indenter is proposed in the present paper, in which a spatially varying friction coefficient is considered. With the aid of Fourier integral transformation, the governing equation is obtained and solved semi-analytically. The surface contact stress field, the moment applied to make the indenter move vertically to the substrate's top surface, as well as the nominal friction coefficient are investigated comprehensively. It is found that the contact behavior can be well improved by choosing proper friction coefficient, inhomogeneity parameter as well as gradient orientation angle. The new finding of the present paper should be helpful to access the surface wear resistance and design various inhomogeneous composites.

This paper is concerned with a computationally efficient numerical study on the dynamic response of a partially filled freight train subject to abrupt braking using the moving element method. The motion of liquid inside the freight container is modelled based on the potential flow theory. A trigonometric function is employed to represent the free surface elevation of the liquid cargo. The vehicle components are represented by interconnected rigid multi-bodies and the track is modelled by a Euler beam resting on a viscously damped Pasternak-type foundation. The accuracy of the liquid sloshing model is examined by comparison with available results in the literature. This paper also investigates the effect of the liquid cargo filling level, the initial train speed and the braking torque on the dynamic response of the coupled liquid freight train-track system.

The paper presents an efficient and numerically stable calculation model of a journal plain floating ring bearing. The computational model is based on the numerical solution of the Reynolds equation in combination with the analytical description of the resulting variables. This model is used in a virtual turbocharger assembled in multibody systems. This approach allows to effectively solve transient events, such as turbocharger run-up, considering issues of rotor dynamics, tribology of bearings, flows of lubricant in the channels and potentially also gas flow through the sealing system. The model of the bearing includes the influence of the inlet and outlet channels and the non-cylindrical shape of the bearing surfaces. Influence of lubricant and structure temperature changes caused by shear stresses in lubrication film and related changes in the lubricant properties is also considered. The paper also presents the numerical implementation of the computational models and the verification of these models using technical experiments with the turbocharger of a diesel engine.

This paper presents a new low-order electric field model for Macro-Fiber Composite devices with interdigitated electrodes. Specifically, the paper proposes a continuous electric field model, where the differential form of Gauss's law is used to obtain spatial terms of the electric field, and the integral form of Gauss's law is used to obtain temporal term of the electric field. By neglecting three-dimensional effects, the method of matched asymptotic expansion is employed to obtain the electric field between two pairs of interdigitated electrodes under quasi-electrostatic conditions. By averaging the solution along with the thickness of piezoceramic fibers and expanding it to the device's entire geometry by Fourier series, the low-order model of the electric field between the interdigitated electrodes is obtained. The matched asymptotic expansion and the low-order model agree well with finite element results. Generalized Hamilton's principle is employed to obtain the coupled electromechanical equations of motion for a beam with the Macro-Fiber Composite device. The Euler-Bernoulli assumptions are used to approximate strain distribution. Under current assumptions, equivalent capacitance and electromechanical coupling terms are obtained without using discontinuous functions (e.g., Heaviside or piecewise functions) for electric field terms. The equivalent capacitance is within 2% of the values reported by the manufacturer. Frequency response functions are obtained for the output voltage, tip displacement, and optimum working load of the energy harvester, and the current and the tip displacement for the actuator by assumed modes solution for a cantilever beam. It is shown that the actuator's current is related to the optimum working load of the energy harvester by Ohm's law. In the limited case, the proposed continuous electric field model reduces to the constant electric field model. When the constant electric field model is compared to the continuous electric field model, it is shown that the constant electric field model overpredicts the equivalent capacitance and electromechanical coupling. The relative difference between the two models for equivalent capacitance is around 41%, and for electromechanical coupling is around 22%. These relative differences are in agreement with the empirical correction factors defined in the earlier constant field models.

In this paper, we investigate the instability behavior of thin elastic circular disks subjected to two concentrated edge loads arranged along a diameter. Despite the determination of critical load intensities for disks, made of conventional, linear elastic material, under some specific boundary conditions has been published in the past, we show some new and quite interesting findings. Particularly new are the investigations of tensile buckling of completely free disks as well as the post-buckling behavior of such disks under tension and compression, respectively. Comparisons with already existing results are presented, too. As far as the buckling load is concerned, all results are expressed in terms of a non-dimensional buckling factor. It is well known that the use of auxetic materials provides some potential for improving the behavior of lightweight structures. This fact has motivated us to consider, in which way the variation of the Poisson’s ratio ν influences the stability behavior of the disks both under compression and under tension not just for the so far not investigated completely free stretched disks but also for the already published configurations. The Poisson’s ratio is varied in the full thermodynamically admissible range [ − 1.0, 0.5] and some quite peculiar results are found and explained, especially for disks made of auxetic materials, i.e. for ν 0 . Although the stress fields are independent of the value of ν , the dependency of the critical load intensities is not simply proportional to the dependency of the plate’s bending stiffness on ν , and the buckling modes show significant changes when ν is varied. The buckling factors for disks under tensile loading are by an order of magnitude larger than those for the compressed disks. Of course, also the buckling modes differ completely between compression and tension. Furthermore, the post-buckling behavior is qualitatively and quantitatively significantly different, too. The considerations and the achieved results are interesting for scientists working in the field of structural stability and for engineers in lightweight design of structures. Furthermore, there are applications in the design of sensors and actors as well as of flexible electronics. Since similar thin membrane structures appear in biological tissues, the paper might be interesting also for biologists.

Creep-fatigue life prediction is essential to plate-fin heat exchangers (PFHE) work at high temperatures and pressures in addition to thermal cycles. As the core of PFHE, the plate-fin structure is a complex pore structure and it is pretty difficult to carry out the stress analysis by conventional finite element method. Thus in this paper, a homogeneous method is proposed to do the strength design for plate-fin structure. This method treats the pore structure as an equivalent solid structure, and the key is to calculate the equivalent material properties and the equivalent stress. The equivalent mechanical properties have been obtained in our previous work [1], and in this paper the equivalent thermophysical properties including thermal conductivity, coefficient of thermal expansion, specific heat and density have been derived. This equivalent method is also verified by the three-dimensional finite element method. Besides, creep and fatigue tests of plate-fin structure are applied to calculate the stress and strain magnification factors. Basing on the equivalent material properties, a full scale stress analysis of PFHE by the equivalent finite element model has been carried out successfully to calculate the equivalent stress. After that, we multiply the equivalent stress-strain by the stress and strain magnification factors to calculate the local stress and strain. Then the creep and fatigue damages are predicted according to design curves of base metal. The result of equivalent model agrees well with that of the classical model, which concludes that this homogeneous method is effective to predict the macroscopic performance of plate-fin structure.

This paper presents a rate-dependent isotropic‒kinematic hardening model in tension‒compression of TRIP and TWIP steel sheets. The isotropic‒kinematic hardening model is widely utilized to describe the Bauschinger effect, transient behavior and permanent softening under reverse loading which are indispensable for numerical simulation of springback in sheet metal forming. The isotropic‒kinematic hardening model, however, has not yet been suggested for the strain rate effect higher than several tens per second although the high strain rate prevails in practical automotive sheet metal forming. This paper proposes a rate-dependent model based on tension‒compression tests of TRIP980 and TWIP980 steel sheets at various strain rates ranging from 0.001 s−1 to 100 s−1. A proposed rate-dependent model is extended from the rate-independent Chaboche type model based on single-surface plasticity. Among three Chaboche type models, the Zang's model is selected as the basic rate-independent model considering both a small change of the work-hardening rate in monotonic loading and a constant stress offset of permanent softening in reverse loading. With the basic rate-independent model, the material parameters are acquired at each strain rate to check their dependency on the strain rate and then formulated as linear or exponential functions of the logarithmic scale of the strain rate. Consequently, the present rate-dependent model is proposed with incorporation of the basic rate-independent model and the rate-dependent functions for the material parameters.

The frictional receding contact problem for two graded layers pressed by a rigid punch is considered in this paper. The punch is subjected to both normal and tangential loads thereby resulting in frictional contact with the upper layer. It is also assumed that the contact between the layers is frictional and the lower layer is fixed. It is further assumed that the gradation in the layers follows an exponential variation through the thickness with different profiles while Poissons ratios are taken as constants. Using standard Fourier transform, the contact problem is converted to a system of two singular integral equations in which the contact pressures and the contact widths are the unknowns. The integral equations are then solved numerically using Gauss–Jacobi integration formula. The Finite Element Method was additionally employed and both exponential and power law material gradation is used to solve the investigated problem and the obtained numerical and analytical results are in good agreement. The primary intention of this paper is to investigate the effect of material gradation, friction coefficients, layers thicknesses and material property mismatch at the interface between the layers on the contact pressures and contact widths. © 2018 Elsevier Ltd

This paper presents an analysis of the transient contact solutions of wheel-rail frictional rolling impacts calculated by an explicit finite element model of the wheel-insulated rail joint (IRJ) dynamic interaction. The ability of the model to simulate the dynamic behavior of an IRJ has been validated against a comprehensive field measurement in a recent paper (Yang et al., 2018). In addition to the measured railhead geometry and bi-linear elastoplastic material model used in Yang et al. (2018), this study adopts a nominal railhead geometry and an elastic material model for the simulations to provide an overall understanding of the transient contact behavior of wheel-IRJ impacts. Each simulation calculates the evolution of the contact patch area, stress magnitude and direction, micro-slip distribution, and railhead nodal vibration velocity in the vicinity of the joint during the wheel-IRJ impacts. The simulations apply small computational and output time steps to capture the high-frequency dynamic effects at the wheel-IRJ impact contact. Regular wave patterns that indicate wave generation, propagation and reflection are produced by the simulations; this has rarely been reported in previous research. The simulated waves reflect continuum vibrations excited by wheel-rail frictional rolling and indicate that the simulated impact contact solutions are reliable.

Rotating annular disk has many practical applications, and its inner and outer edge boundary conditions must be selected to apply the rotating disk with different purpose. There are several potential inner and outer edge boundary conditions that can be selected for the rotating disk, but the corresponding dynamics characteristics become a problem that makes the disk work stably and efficiently. In this paper, 25 full combinations of edge boundary conditions are considered for the rotating flexible annular disk. Then their natural frequency, dynamic stability, critical and limit speeds, and steady state response amplitude under initial transversal runout are discussed and compared systematically. This paper is meaningful and beneficial to select the most available and suitable scheme of edge boundary conditions in the rotating disk application.

Recent studies have shown the coupling of optical and mechanical energy in a class of liquid crystal elastomers that contain light-sensitive molecules. As shown experimentally in the literature, these materials can undergo large, reversible elastic deformation under light illumination, and therefore geometric nonlinearity effects can be significant. In this paper, we present a large deflection model for the light-induced bending vibration of a liquid crystal elastomer beam. In this regard, the von Karman's nonlinear strain–displacement relationship is used to account for the large deflection of the beam. The effect of light on the liquid crystal elastomer beam is modeled as an inhomogeneous and time-dependent light-induced contraction strain and the dynamic equations of the beam are derived using the Hamilton's principle. Finite element formulation is developed to analyze the nonlinear dynamic response of the beam under uniform light illuminations. In addition, numerical results are presented and effects of different physical and geometrical parameters, including light intensity, light source position, contraction coefficient, and the thickness of the beam on the vibration characteristics of the beam are investigated. The model developed in this paper can be used to design liquid crystal elastomer based structures such as optically sensors, photo-mechanical energy harvesters, or other reversible opto-mechanically active structures.

In many engineering applications of roll-to-roll manufacturing, the inhomogeneity in the tension profile applied at both the ends of web is apparent, which cannot be completely ignored in the modeling so as to accurately predict the dynamic behaviors of the travelling web. In this paper, transverse vibration and instability of axially travelling web subjected to parabolic tension profile are investigated. The governing equations of motion are derived by employing the Hamilton principle. The Galerkin method is used to compute the complex natural frequencies and mode shapes. The instability of travelling web versus different axial velocities is studied. This paper focuses on the influence of tension inhomogeneity on the critical velocity and mode shape. The sensitivity of aspect ratio and average tension to the tension inhomogeneity are also discussed. The critical velocities of the travelling web may be greatly decreased because of the tension inhomogeneity and a small tension inhomogeneity is enough to change the mode shapes completely.

The dynamic behavior of a Timoshenko nanobeam would be significantly different from a macro-one due to the large ratio of surface area to volume of nanomaterials. Furthermore, the shear deformation effect would be obvious for a Timoshenko nanobeam in contrast to an Eulerian one. In this paper, a recently developed elastic theory is adopted in order to predict the resonant frequency of a Timoshenko nanobeam, in which not only the surface effect but also the shear deformation effect and the rotary inertia one are considered. In contrast to the existing surface effect theories, surface effect of nanomaterials is characterized by the surface energy density in the adopted theory. The resonant frequency of both a fixed-fixed nanobeam and a cantilevered one is analyzed. It is found that the dynamic behavior of nanobeams deviates significantly from the one predicted by both the classical Timoshenko beam theory and the Euler–Bernoulli one due to the surface effect. Furthermore, the shear deformation effect and the rotary inertia effect cannot be neglected in nanobeams with a relative small aspect ratio, which cannot be precisely characterized by the Euler–Bernoulli beam theory. In addition, the influencing mechanism of surface effect on the dynamic behavior of nanobeams would depend on the boundary conditions. The resonant frequency of a fixed–fixed Timoshenko nanobeam would be improved, while that of a cantilevered one would be weakened by the surface effect in contrast to the corresponding classical solutions. The results in this paper should be useful for precise design of nano-devices and helpful for reasonable assessment of test results of nano-instruments.

Over the last decade, a novel nondestructive evaluation (NDE) method based on the application of highly nonlinear solitary waves has emerged. The method is based on the actuation and detection of solitary waves propagating along a medium made of uniform spherical particles, the last of which is in contact with the material to be assessed noninvasively. The hypothesis is that the dynamic interaction between the wave and the material/structure to be inspected is dependent upon the condition of the material/structure. The study presented in this paper aims at designing and developing the NDE framework for the detection of localized corrosion in metals. In particular, this paper presents a numerical investigation of the interaction of highly nonlinear solitary waves with plates in pristine and corroded conditions. A coupled discrete/finite element model is used to predict the time of flight and the relative amplitudes of the reflected solitary waves as localized corrosion progresses in the plate. The sensitivity of the method is quantified in terms of different parameters such as plate thickness, chain length, and particles' mechanical and geometric properties. It is found that the sensitivity, i.e. the ability to detect corrosion at earlier stages, of this novel NDE technique is inversely proportional to the plate thickness and is proportional to the diameter and the stiffness of the particles. In the future, the findings of this study can be used to optimize the design of solitary wave transducers, which are devices that can be used to trigger, sustain, and detect the propagation of the solitary waves and their interaction with the structure of interest.

A formulation for studying the vibration characteristics of the rectangular plate with general boundary conditions excited by the turbulent boundary layer is developed in this paper. The Rayleigh-Ritz method, modified Fourier spectral approach, and the classical theory of elastic plates are employed to establish the vibration model of the rectangular plate. Besides, the sound pressures are calculated by the Rayleigh integral and the turbulent boundary layer pressure field is described by the cross-spectral density expression of the Corcos model. The power spectral density of vibration characteristics can be obtained according to the stochastic theory. The results indicated that the power spectral density of responses calculated in this paper are consistent with those in the references, which validate the accuracy of the developed formulation. The innovation of this formulation lies in the application of modified Fourier spectral approach to the turbulent-flow-induced vibration for the first time. The velocity power spectral density of the plate with classical and elastic boundary conditions are studied, and the influences of the boundary conditions are discussed. The boundary conditions affect the vibration response by controlling the system stiffness of the plate. Moreover, the influences of turbulent flow speed and direction under different boundary conditions are studied. It is found that the increasing turbulent flow speed leads to the increase of the vibration level, and the growth rate is similar under different boundary conditions. The velocity power spectral densities are slightly different when a turbulent flow along the short side and long side of the plate.

This paper aims to improve the existing electromechanically coupled analytical model that was developed for defect-induced phononic crystals (PnCs) designed for piezoelectric energy harvesting (PEH). The work outlined in this paper improves the model by considering lateral movements, which are not negligible when the slenderness ratio of the rods is small. Based on the Rayleigh-Love rod theory, the proposed method uses the generalized Hamilton's principle to derive two governing equations in the mechanical and electrical domains. An explicit form of the solutions to the governing equation is then used to derive an improved electromechanically coupled transfer matrix. Based on the transfer matrix method and the S-parameter method, the proposed electromechanically coupled analytical model enables the prediction of defect bands and PEH performance. To evaluate the predictive capabilities under a small slenderness ratio of the rods, the results from the proposed Rayleigh-Love rod-theory-based analytical model are compared with those found by the existing analytical model based on classical rod theory and by a finite element model. The novelties of the research outlined in this paper are as follows: 1) the proposed method considers, for the first time, lateral inertia in analytical modeling of defect-induced PnCs for PEH purposes and 2) three coefficients are newly presented to quantify the effects of lateral motions on the electromechanically coupled transfer matrix.

High temperatures generated in cutting processes significantly affect the surface integrity of machined parts and tool wear, leading to workpiece thermal damage, tensile residual stresses in the workpiece and a reduction in tool life. In recent years, different analytical thermal models to predict cutting temperatures have been developed in literature based on 2D modeling of the cutting process and the assumption that thermal conductivities of workpiece and chip are not dependent on temperature. However, this dependence of conductivity on temperature may have a significant influence on predicted temperatures and must be taken into account. In this paper, a thermal model of the orthogonal cutting process that considers thermal conductivity of materials (chip and tool) to be dependent on temperature is developed. A linear variation of thermal conductivity with temperature is assumed for chip (workpiece) and tool materials. The model is based on application of: (1) the Kirchhoff transformation in order to convert the nonlinear heat conduction problem into a linear one, (2) the theory of moving and stationary heat sources in semi-infinite and infinite mediums in order to model primary and secondary deformation zones and (3) imaginary heat sources to meet adiabatic boundary conditions in the chip and tool. Imaginary heat sources were defined in the thermal model proposed in this paper in such a way that the effect of the tool-chip interface dimensions and of cutting tool width on the tool temperature could be taken into account. This allows the temperature on the rake face and lateral faces of the tool to be predicted. To this end, a new methodology that considers the temperature-dependent thermal conductivity of materials was developed in order to estimate heat partition ratio along the secondary heat source (tool-chip interface), which is assumed to be non-uniform. Orthogonal cutting tests were also performed in order to verify model predictions by comparing them to tool temperature distributions measured using an IR camera.

In this paper, we present the mechanical sensitivity improvement of a disk resonator gyroscope (DRG) by optimizing the thickness distribution of the nested rings. We calculate the mechanical sensitivity of the DRG with some given ring thickness distributions by using finite element analysis (FEA). The comparison results suggest that the ring thickness distribution has great influence on the mechanical sensitivity of the DRG. Then we introduce the bio-inspired particle swarm optimization (PSO) to study the ring thickness distribution. The globally optimized ring thickness distribution providing the maximal mechanical sensitivity is obtained and detailedly discussed. Based on this, we study how do basic structure parameters of the DRG affect the optimum ring thickness distribution. The results of this investigation can give detailed ring thickness design rules for designing series of DRGs providing the maximal mechanical sensitivity. Meanwhile, the investigation method of this paper could be expanded to include other objectives, constraints, or variables relevant to the DRG or other MEMS devices.

The paper aims at clarification of the role of reduction in yield locus curvature on forming limit diagrams. To this end, a cross-hardening model showing a reduction of yield surface curvature is used which accounts for dynamic and latent hardening effects associated with dislocation motion during loading. The model's three-dimensional tensorial as well as reduced plane-stress vector formulations are given. The first quadrants of forming limit diagrams are numerically produced using finite element models of the Marciniak-Kuczyński test with spatially correlated random defect distribution as localization triggering mechanism. The effect of cross hardening is investigated in detail. It is demonstrated that for plane strain loading path there occurs no difference in localization predictions of the models with and without cross hardening whereas for biaxial strain paths a delayed localization is observed in the cross hardening model as compared to the one without cross hardening effects. This is in accordance with the relative bluntness of the yield surface at the points of load path change towards localization. These results are complemented by Nakazima test simulations where similar observations are made. The paper aims at clarification of the role of reduction in yield locus curvature on forming limit diagrams. To this end, a cross-hardening model showing a reduction of yield surface curvature is used which accounts for dynamic and latent hardening effects associated with dislocation motion during loading. The model's three-dimensional tensorial as well as reduced plane-stress vector formulations are given. The first quadrants of forming limit diagrams are numerically produced using finite element models of the Marciniak-Kuczyński test with spatially correlated random defect distribution as localization triggering mechanism. The effect of cross hardening is investigated in detail. It is demonstrated that for plane strain loading path there occurs no difference in localization predictions of the models with and without cross hardening whereas for biaxial strain paths a delayed localization is observed in the cross hardening model as compared to the one without cross hardening effects. This is in accordance with the relative bluntness of the yield surface at the points of load path change towards localization. These results are complemented by Nakazima test simulations where similar observations are made.

The main goal of this paper is to develop a comprehensive beam model based on the micropolar elasticity theory which is as general, as easy to use, and as convenient as the classical beam theories. Uncomplicated torsion and bending theories for micropolar elastic beams deforming in three-dimensional space and under different types of external loading and boundary conditions are presented in this paper. Unlike the classical beam models, the developed beam model includes the effect of microinertia and contains new material parameters to capture the microstructure-dependent size effects which could be useful when dealing with micro scale beams. The presented micropolar beam model generalizes the Duleau torsion and Timoshenko bending beam models to include the microstructure effects. Hamilton's principle and a variational approach are used to derive the dynamic equations of the micropolar beam with longitudinal, torsional, and bending deformations. Then the governing dynamic equations are solved numerically by using a finite element approach and numerical results for a simply supported micropolar beam are provided. The static and dynamic behaviors of the developed micropolar beam model are studied and compared against the classical beam models. In particular, the conditions for recovery of the results of the classical beam theories, i.e. Duleau and Timoshenko theories, are addressed.

In this paper, a mathematic expression is presented to describe the mode jumping and the equilibrium path reversal characteristics of thin film secondary wrinkling based on the non-linear bifurcation buckling theory. With the usage of variable transformation and the introduction of dimensionless parameters, the non-linear Von-Karman equilibrium equation considering the two-direction loading characteristics, is transformed into a non-linear boundary value problem with zero trivial solution. Based on the bifurcation theory, the eigenvalue problem is addressed through the linearization of non-linear boundary value problem. An approach to critical load prediction of thin film secondary wrinkling is proposed by introducing a critical aspect ratio parameter, since the mechanism of thin film secondary wrinkling is double eigenvalues splitting. Furthermore, this paper presents an analysis in detail on the critical load of the rectangular thin film secondary wrinkling under shear, indicating that the critical load is linearly proportional to the initial stretching displacement, while its relationship with the size of the free edge is nonlinear. The validity of this critical load prediction approach is verified via the digital image correlation experiment. The results provide strong supports for the controlling and tuning of the wrinkles for high precision pre-tensioned thin film structures.

Photo-sensitive characteristic of the liquid crystal networks (LCNs) provides a distinct way to control the deformation of the LCN membrane. In this paper, the spontaneous photo-deformation of a simply supported square liquid crystal network membrane is investigated. Firstly, the governing equation for the LCN membrane with an arbitrary pre-designed uniaxial director (common orientation vector) pattern is established. By using the finite difference method, the deflection configurations are obtained, and four basic flexure modes for the LCN membrane are observed. The evolution of the flexure modes with respect to the director orientation is discussed in detail, and the critical division condition is given. The results given in this paper provide a theoretical guidance on the design of LCN based intelligent light driven membrane elements.

Due to the essential roles of entangled microtubule networks in complicated cellular activities such as mechanotransduction, improving the insight into their mechanical properties and dynamic load-transfer mechanisms can be inspiring to design new bio-nano-metamaterials. Damping effects arising from the polymeric nature of the microtubules as well as the surrounding cytosol in vivo and stabilizing agents in vitro can play significant roles in the dynamic behavior of the individual microtubules and their networks. Hence, this paper presents a comprehensive analysis of damping effects on elastic wave-propagation characteristics of microtubule-based metamaterials. For this purpose, microtubules are taken as the constituting elements of architected periodic structures of various topologies, and the interactions between the structures and their surrounding media are modeled by drag forces and viscous damping. The model is then used to investigate in-plane and out-of-plane motions of the microtubule-based networks. The paper starts with an explanation of the finite element method applied to derive the governing equations of undamped elastic wave propagation from obtaining the dispersion curves of the periodic structures. Also, considering the damping matrix proportional to the stiffness matrix, the equations of damped wave propagation in microtubule-based periodic structures are obtained and solved using the state-space method. The results prove that considering both of the in-plane and out-of-plane motions have significant effects on the formation of the dispersion curves and wave attenuation properties of microtubule networks. Moreover, the damping effects of the surrounding medium can lead to forming different dispersion curves and, consequently, showing different wave attenuation characteristics due to various damping-induced phenomena such as branch overtaking and branch cut-off/cut-on. The results of the current study are counted as the initial steps towards more realistic modeling of bio-nano-filters. They may soon be used as a basis to design bio-nano-metamaterials with unique wave-filtering properties and higher biocompatibility.

Based on the nonlocal elastic continuum theory, the band structures of the anti-plane transverse elastic wave propagating normally in nanoscale periodic laminates is investigated in detail in this paper. The localization factor computed by using the transfer matrix method is employed to describe the band structures. The variations of the displacement are calculated and analyzed. The influences of the size-effect of each sub-layer, the material and structural parameters on the cut-off frequency and the dense band zones are discussed in detail. The study presented in this paper provides a theoretical guide for the design and applications of the novel nanoscale wave devices.

Though several studies have tried to explain and model the effect of thickness on crack growth behavior, this remains a controversial topic. The present paper proposes an approach to model the fatigue crack growth with different thicknesses, which requires crack growth experimental data with different specimen thicknesses and out-of-plane constraint distribution relations for CT specimens. Empirical formula, for the distributions of two types of constraint factors Tz and T33, were fitted according to the FEA results. The stress state variation induced by the thickness was quantified using the formulas; then, the crack driving force was modified to establish the novel crack growth model. The predictions using this model largely agree with the experimental data. So, the newly developed model was verified. This paper proposes a new perspective on quantifying the effect of crack thickness, and provides an alternative to the traditional method of modeling the crack growth behavior.

Accurate modeling of thermoelastic damping (TED) is a significant and challenging task in the design of multilayer micro-resonator devices. Several analytical models were proposed to predict TED in bilayer and trilayer microbeams. However, previous models theoretically only considered one-dimensional (1-D) heat conduction in the thickness direction. 1-D model cannot capture the effects of length-to-thickness ratio, structural boundary conditions and flexural-mode order, which is only suitable for the long slender beam. To address these issues, this paper presents an analytical TED model based on the two-dimensional (2-D) heat conduction in the thickness and length directions in a bilayer microbeam with a rectangular cross-section. The expression for TED in the form of two infinite series is more explicit and simpler than others. And two key factors that determine the difference between the 2-D model and the 1-D model are first summarized in this paper. The simulation results of the present model match better with those of the finite element method (FEM) than the previous 1-D model. The shape of TED spectrum is affected by the ratio of the thermal diffusivity of materials. Two peaks occur in the case of ratio of thermal diffusivity higher than 100. However, two peaks will reduce to one as the ratio decreases to 1. The convergence rate of the present model is weaker than the 1-D model.

For the problem of a functionally graded thick-walled tube subjected to internal pressure, we have already presented the elasticity solution based on the Voigt method with the assumption of a uniform strain field within the representative volume element. This paper discusses the thermoelastic problem of the functionally graded thick-walled tube subjected to both axisymmetric mechanical and thermal loads, and gives the solution in terms of volume fractions of constituents. We assume that the tube consists of two linear elastic constituents and the volume fraction of one phase is a power function varied in the radial direction. The theoretical solutions of the displacement and the stresses are presented under the assumption of a uniform strain field within the representative volume element. Comparisons of the theoretical solutions and the finite element analysis demonstrate the validity of the assumption. Based on the relation of the volume average stresses of constituents and the macroscopic stresses of the composite material in micromechanics, the present method can avoid the assumption of the distribution regularities of unknown overall material parameters appeared in existing papers. Further, the present method is valid for the materials with different Poisson׳s ratios of constituents. The effects of the volume fraction, the ratio of two thermal expansion coefficients and the ratio of two thermal conductivities on the displacement and stresses are systematically studied.

This paper studies the impact and rebound behaviour of a thin-walled elastic–plastic circular ring after it impinges onto a rigid wall with initial velocity V0 by finite element method. Through the dimensional analysis and a systematic simulation with different ring geometries and material properties, three non-dimensional parameters are identified, which dominate the impact duration and rebound velocity of the ring; and they are: (1) the ratio of wall thickness to average radius, η=h/R; (2) the yield strain of the material, Y/E; and (3) the non-dimensional initial velocity of the ring, ν≡V0/VY, where VY≡Y/(Eρ)1/2 denotes the yield velocity of the material [3] . When the initial velocity is low, the impact between the ring and rigid plate remains elastic, the compression duration could be analytically obtained, and it agrees well with the numerical results, while the restitution duration is about 3/4 of the compression duration. The corresponding coefficient of restitution (COR) is found to be independent from the material property, whereas it increases from 0.75 to 0.78 when the thickness ratio changes from 1/20 to 1/40. With the increasing of initial velocity, a four-hinge crushing mode and a subsequent five-hinge mode are identified, which are different from the crushing mode of a ring under static compression. The variations of compression and restitution durations, rebound velocity and COR are discussed in detail, while some of them are compared with those of the impact of a solid ball onto a rigid wall. The effects of the ring geometry and material properties on these variables are also presented. The rebound velocity is found to reach the maximal at about a half of material׳s yield velocity when the initial velocity of the ring is about twice of the yield velocity, resulting in COR being 0.25 for all the materials and geometries adopted in this paper.

Decreasing energy consumption is a key issue for stratospheric airship to maintain its high altitude long-endurance feature. The airship envelope covered micro-grooves is considered as a feasible way to reduce resistance in order to save energy. The key to using grooves as the drag reduction technique is to evaluate its drag reduction ability under the condition of rarefied gas which is different from the original continuum atmospheric state. In this paper, the drag reduction ability of V-shape and rectangular micro-grooves under both continuum and rarefied condition is evaluated using the Lattice Boltzmann Method (LBM). This method modified for rarefied flow has been validated by the analytical solution in this paper. The local vortices are obviously observed within the grooves in the continuum and slippery flow regimes. The mechanism of drag reduction with grooves is elucidated at first and the ability of drag reduction under the rarefied condition is then evaluated by comparison of that under continuum condition. In the end, both the V-shape and rectangular grooved wall flow with different height and width are simulated. The relationships of drag reduction ability and the groove geometry are also determined and evaluated. Our results reveal that the grooves can reduce the resistance with a drag reduction rate of about 6% in the continuum. The drag reduction ability of the same grooves, however, decreases to 2.5% or less in the slippery regime. The grooves with appropriate size which make the vortices full within grooves can reduce shear force between fluid and wall. The results obtained in this paper are good references to the design of stratospheric airship envelope and the drag reduction technique.

An improved Karman equation for hot-rolled strip was deduced to generate a new rolling pressure formula, based on comprehensive consideration of the slipping and sticking friction on the contact arc between hot-rolled strip and work rolls. The Runge–Kutta method was applied to solve the improved differential equation, and then the distribution of rolling pressure on the contact arc was obtained. The roll force in the roll-bite can be calculated by integrating the vertical component of positive pressure and friction shear stress of every slice. This paper also analyzed the influence of friction condition, flow stress, roller distortion and other factors on the results of rolling pressure and force per unit width. Using 7 stands of hot strip mills as an example, this paper conducted actual industrial application verification. The computational results demonstrate that the proposed new model improves the setting precision of roll force and can be applied to online control of hot rolled strip.

In this paper, the steady-state response for a thermoelastic half-plane with voids is studied, in which, the surface of the half-plane is partly subjected to a surface harmonic force and a thermal source. The semi-analytical solutions and the numerical solutions of the half-plane problems with three different materials are obtained from a semi-analytical method and a developed differential quadrature element method in this paper, respectively. The corresponding numerical results are compared and the effects of parameters are considered as well. It can be seen that the semi-analytical solutions and corresponding numerical solutions coincide with each other. This means that the differential quadrature element method is a very efficient method for seeking the numerical solutions of the half-plane problems with discontinuity, and it has some advantageous properties, such as small computational amount, high accuracy, and better convergence.

Dynamics of axially moving material continuum is investigated for over 60 years. Research interest in this subject is still high due to more and more new fields of application. Alongside traditional applications, such as band saws’ blades and axially moving paper webs, appeared as flat objects moving at high speeds in space. Recently many studies on the dynamics of axially moving beam-like and plate-like systems were published. Inclusion in analysis viscoelastic properties of broad moving continua is also the subject of research. The paper presents a review of research in this field with particular emphasis on the axially moving plate-like elastic and viscoelastic systems. Unlike the string-like and beam-like systems, such review on the plate-like systems has not been published yet. A brief overview of the most important studies on the dynamics of moving string-like and beam-like systems is presented as well. This paper also includes a comparative analysis of some results of studies published by the authors in the field of dynamics of axially moving viscoelastic systems. Some suggestions on the directions of further research in this field end the paper.

In this paper, the classical and nonlocal elasticity are applied to investigate the deflection and instability of electrostatically actuated carbon nanotubes. The results are presented for different geometries and boundary conditions. They reveal that increasing radius and gap and decreasing length confine to increasing pull-in voltages of the carbon nanotubes. The results prove that application of the nonlocal elasticity theorem leads to stiffer structures with higher pull-in voltages. Thus, in order to obtain more accurate results about the mechanical and electromechanical behaviors of the carbon nanotubes, one should apply the nonclassical elasticity theories such as that applied in this paper.

In this paper, the stress relaxation tests were performed on Ti-6Al-4V alloy sheet with different initial total strains and different temperatures. Based on the experimental data, a novel empirical equation for predicting the stress versus time curves was proposed, and then an explicit constitutive model for describing the stress relaxation behaviours was further developed. The average prediction error was less than 8.27%, proving the proposed prediction curves were valid. In addition, two material constants which depend on initial stresses and total strains were introduced to accurately describe the stress relaxation curves. Then the methods for solving these two material constants under different initial conditions were presented. Finally, a simple method for dividing the stress relaxation stages was proposed. There are three stages in the stress relaxation curve studied in this paper, including the loading stage, stress relaxation-I and stress relaxation-II. A symbol called ηi which represents the normalized ratio of instantaneous stress to initial stress was used. When time τ is equal to α, the ratio of instantaneous normalized stress to normalized initial stress is reduced to η1 = 1/e, and the instantaneous plastic strain at this moment is (1 − η1) of the total plastic strain.

This paper discusses stochastic homogenization analysis of a periodic porous material fabricated using a rapid prototyping technique. A rapid prototyping system will be helpful to fabricate an order-made structure stably consisting of a porous material having a desired void distribution than a general porous material, but the influence of a geometrical random variation of pores should be still investigated, because some geometrical parameters are difficult to be perfectly controlled. In this paper, the stochastic homogenization analysis is performed for evaluation of the probabilistic characteristics of the homogenized elastic properties for a geometrical random variation in microstructure. The perturbation-based approach with the finite difference scheme is proposed for stochastic homogenization analysis of the porous material considering a parametric geometrical random variation. Influence of the random variations of microscopic geometry parameters on the homogenized elastic property is investigated, and accuracy of the finite difference based perturbation approach is discussed. In addition, a numerical result is compared to the experimental result, and applicability of the stochastic homogenization analysis to a practical problem is investigated.

In this paper, a nonlinear size-dependent finite element model incorporating surface energy effects is developed to study the mechanical behavior of tiny elastic functionally graded (FG) bodies. Here the classical elasticity theory is modified to incorporate the surface energy effects. Most of previous studies assumed that the surface energy depends only on the 2D symmetric infinitesimal surface strains and neglects the second-order products of surface strains/displacement gradients. These descriptions assume a small strain deformation of the surface and neglect the pre-strain that is, already, developed on the surface – before loading – due to the pre-tension stress σ 0 . Here in this paper, the pre-strain is considered which forces the surface to a state of large strain after loading instead of small strain. In this sense, in the presence of initial surface tension, the theory of surface elasticity is a hybrid formulation characterized by linearized bulk elastic material and second order finite deformation of the surface. In the proposed finite element model, the surface energy effect is taken into account in the derivation of the element stiffness matrix for the material elements located very close to the boundary surface. The proposed model is then used to study the effects of surface energy, including the 2nd order displacement gradient, on the mechanical behavior of plane-strain functionally graded elastic body.

One of the important failure criteria of press operations in industry for forming simulations is the Forming Limit Diagram (FLD). The complex loading effects on FLD, in particular the localized necking phenomenon under stretch-bending condition, have not been fully investigated and well understood. In practical sheet metal applications, the deformation is invariably three dimensional with a combination of stretching and bending. For most sheet materials under these complex loading processes used in industry, strong Bauschinger effect is observed, and the material hardening behavior tends to be anisotropic. This study aims to understand and evaluate such anisotropic hardening effect on the forming limit prediction under stretch-bending condition. The extended through-thickness Marciniak–Kuczynski (M–K) analysis is incorporated with Yoshida–Uemori (YU) two-surface kinematic hardening constitutive model, which has a more accurate description of the reverse loading behavior than that of the conventional isotropic hardening model. The material parameters used in this paper for YU model are calibrated with the experimental data from uniaxial large-strain tension-compression cyclic test. Both the isotropic hardening and YU kinematic hardening models with Hill'48 yield surface are employed in the analysis for the purpose of comparison. The Forming Limit Average Stress Diagram (FLASD) under stretch-bending condition is proposed to extend the understanding of Forming Limit Stress Diagram (FLSD) from in-plane to out-of-plane deformations. The “bending-ratio-dependent” phenomenon in forming limit diagram is predicted and observed in both stress/strain space with the proposed method. It suggests that the individual stress/strain state cannot represent system behavior. Forming limits under stretch-bending is suggested as an occurrence of system instability, not individual material instability. The system behavior of sheet metal deformation is reinforced as critical to the understanding of necking instability in stretch bending processes. The analysis shows that the Bauschinger effect provides positive effect in delaying the necking instability, predicting higher formability for sheet metals under stretch-bending. The insight obtained in this paper provides further understanding of the localized necking phenomenon under stretch-bending condition.

A central aspect of the linear vibration theory of rotating disks involves the concept of critical speeds. At such rotation speeds an axisymmetric disk can support a standing wave as recorded by a stationary observer. In such situations an applied space fixed constant force can give rise to a resonance in the disk. Such a response is of concern in industrial applications as diverse as circular saw blades and computer floppy disks. In such situations the magnitude of response may exceed the limits of linear theory. The present paper is concerned with the effects of large displacements upon the disk response in the neighborhood of such critical speeds. The effects of geometric nonlinearities and the influence of rigid body tilting and translation (caused by the boundary conditions) are considered. The equations of motion are based on Von Karman plate theory. The eigenfunctions of two self-adjoint eigenvalue problems, corresponding to the stress function and the transverse displacement, are determined and used as approximation functions in a numerically efficient Galerkin formulation. The coupled nonlinear ordinary differential equations of motion are solved using the Runge–Kutta method. Numerical results are presented for disks that are free to translate and rotate at their inner boundary and are constrained from lateral motion by space fixed linear springs. The effects of vibration magnitude on system response in the sub and super-critical speed regions are computed and the effects of large displacements on critical speed behavior and forced response are investigated. Experiments are conducted to verify the accuracy of the numerical results obtained in this paper.

This paper is concerned with the plastic energy absorption behavior of expansion tubes under axial compression by a conical–cylindrical die. The experiments and numerical simulation using FEM are presented in this paper. Experiments were conducted on circular 5A06 aluminum tubes with an internal radius fixed at 22.5 mm and different thicknesses between 1 and 5 mm; the tubes were pressed axially onto a series of conical–cylindrical dies each with a different semi-angle from 5° to 20°, where the radius of the cylindrical part was 24 mm. A numerical analysis was performed to investigate the tube deformation and the friction between the tube and die. A good fit of the experimental data was obtained by taking the value of the friction coefficient μ=0.05. Based on these experimental and numerical results, characteristics of driving force–stroke curves in different deformation modes are discussed in detail. Effects of tube dimensions and semi-angle of the die on steady-state force and energy absorption efficiency are also presented. Based on these experimental studies, a theoretical analysis to explain the deformation mechanisms of the tube expanded by a die is carried out and will be given in a subsequent paper [1] .

This paper analytically investigates plastic limit pressure of orthotropic strain-hardening cylindrical vessels under internal pressure. It is an interesting problem to illustrate the interaction between strengthening and weakening behavior during the deformation process. The Voce hardening law and Hill's yield criterion were adopted in the paper. A sequence of static and kinematic limit analysis problems were performed by updating the yield criterion and the deformed configuration. The equality relation between the greatest lower bound and the least upper bound was confirmed explicitly. Accordingly, exact solutions of plastic limit pressure were developed with an integral term. Particularly, exact closed-form solutions were obtained for certain values of the hardening exponent of the Voce hardening law. Finally, numerical efforts were also made for rigorous validations.

The dynamics of the workpiece of a lathe is simulated in the presented paper. A rotating Rayleigh beam is chosen as a simple model of the workpiece. The beam or the workpiece is subjected to forces from the cutting tool of the lathe. The external forces, in transverse and axial directions, are traveling in a repeating or periodic motion. The force in the axial direction is a large cutting force resulting in coupled bending deformation while forces in the transverse directions are the contacting forces. In this paper, the governing equations of the rotating Rayleigh beam are derived by Hamilton's principle. The external, periodic forces resulted from the tool are expressed in Fourier series. Galerkin's method is then chosen for disceretizing the partial differential equations. The instability regions of the responses are determined by using the method of multiple scales and the Floquet theory. Fast Fourier transform gives the frequency domain responses for examining the dynamic characteristics. The numerical results are discussed. Parametric studies are also performed.

Water drop erosion is regarded as one of the most serious reliability concerns in the wet steam stage of a steam turbine. The most challenging aspect of this problem involves the fundamental solution of the transient pressure field in the liquid drop and stress field in the metal substrate, which are coupled with each other. We solve the fundamental problem of high-speed liquid–solid impact both analytically and numerically. In Part I of this paper, the governing equations based on a nonlinear wave model for liquid are derived. Analytical and approximate solutions of one-dimensional liquid–solid impact are given for both linear and nonlinear models, which provide critical insights into the water drop erosion problem. Both continuous and pulsant impacts on rigid and elastic substrates are analyzed in detail. During continuous impact, the maximum impact pressure is always higher than the water hammer pressure. Upon pulsant impact and at a particular instant related with the impact duration, the maximum tensile stress appears at a certain depth below the solid surface, which can be readily related with the erosion rate. In Part II of this paper, two-dimensional (axisymmetric) liquid–solid impact is solved numerically, from which the most dangerous impact load/duration time and the most likely crack positions are deduced. Based on our recent solution of the water drop impact statistics (associated with the fluid flow in the blade channel), a comprehensive numerical study of the water drop erosion (fatigue) on a turbine blade is carried out.

This paper deals with the theoretical investigation of the double-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below saturating a porous medium subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis theory and normal mode analysis method have been carried out to study the onset of convection. The influence of various parameters like medium permeability, solute gradient, non-buoyancy magnetization and micropolar parameters (i.e. coupling parameter, spin diffusion parameter and micropolar heat conduction parameter) has been analyzed on the onset of stationary convection. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large values of buoyancy magnetization parameter M 1 and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and solute gradient. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and solute gradient, which were non-existent in their absence. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of overstability.

Sheet metal formability is conventionally assessed in a two-dimensional plot of principal strains or stresses in comparison to a forming limit curve. This method of assessment implicitly assumes that the forming limit is isotropic in the plane of the sheet. While the assumption of isotropy in the forming limit is perhaps a good engineering approximation, it is intrinsically inconsistent with the use of material models that are anisotropic. Since the trend today is to utilize models with full anisotropy in order to more accurately capture the physics of material behavior, the issue of anisotropy of forming limits must also be addressed. The challenge is that the forming limit is no longer defined by a curve but requires the definition of a surface in strain or stress space, and therefore it is no longer appropriate to view these limits with the convenience of two-dimensional diagrams. Furthermore, recent developments in the characterization of sheet forming limits under non-proportional loading suggest that is advantageous to view forming limit behavior in terms of stresses rather than strains, a view that is adopted in this paper. A solution to the challenge of assessing formability for an anisotropic material is proposed that rescales the stresses by a factor so that the scaled stresses have the same relationship to a single forming limit curve in a 2D plot in stress-space, as the actual stresses have to the true anisotropic forming limit in 3D space. The rescaling enables engineers to accurately view the formability of all the elements at the same time for a given finite element analysis of an application. This paper also discusses other challenges of using stresses in the assessment of formability, focusing on an analysis of the 2-Stage Forming Benchmark highlighted in the Numisheet ’99 Conference. Stresses are found in this application to unload to non-critical values after reaching critical levels earlier in a forming process, which suggests that a full integration of the stress-based forming limit criterion with FE simulation is required to detect critical states that may temporarily occur during the forming process.

This study focused on the variation of the friction coefficient during the upsetting process and the concept of treating the solution of unknown parameters as an inverse problem. Based on the experimental measurement data, the Levenberg–Marquardt method, a numerical optimization approach was used in conjunction with the constrained function, convergence criterion and the axial symmetry thermo-elastic-plastic finite element method to solve, inversely, the variation of friction coefficient during the upsetting process. The inverse calculation steps of the warm upsetting of the molybdenum proposed in this paper was based on the load values measured in the upsetting experiment. The inverse calculation procedures were taken to solve the variation of the friction coefficient during the warm upsetting process. The results related to stress distribution, strain distribution, temperature distribution and shape variation were then compared with those reported in other studies. The comparison further confirmed the justification of using the load to solve the friction coefficient inversely. By means of the inverse algorithm presented in this paper, physical phenomena that better approximated the reality could be obtained, and the entire upsetting forming simulation could be more complete.