Upper-bound Theory Research Articles

Rolling force model of V-shaped variable thickness rolling based on energy approach

V-shaped Variable Thickness Rolling (V-VTR) serves as a specialized cold rolling method for manufacturing rolled profiled strips, particularly useful for quickly preparing rolled profiled strips with large thickness ratios. In the process of variable thickness rolling, metal predominantly flows in the lateral direction. However, there is a scarcity of theoretical models for calculating the rolling force of variable thickness rolling, leading to a void in theoretical guidance for both device design and product thickness control. To fill this gap, we introduce a novel mathematical model specifically designed to calculate the rolling force, considering the lateral flow of the metal. Initially, the deformation pattern is investigated and subsequently modeled. Then, the models for kinematically permissible velocity and strain fields are developed, and their validity is confirmed through both experimental data and simulation analysis. Incorporating factors like strain energy and frictional work, a formula for rolling force of variable thickness rolling is derived based on the upper bound theory. The efficacy of this newly proposed rolling force model is confirmed through experimental testing, which indicates that the formula offers a high degree of accuracy.

Analysis of face stability for shallow shield tunnels in sand

The stability of the tunnel face is the key problem in shield tunnel construction. This paper focuses on the face stability of a shallow tunnel in sand. Numerical simulation and theoretical analysis are combined to study the limit support pressure and failure zone. Firstly, numerical simulation is employed to study the collapse of the tunnel face, obtaining the limit support pressure and collapse zone. A new failure model suitable for shallow tunnels is constructed based on these numerical simulations. Then, an analytic solution for the limit support pressure is derived using limit analysis upper bound theory. The accuracy and applicability of this proposed model are verified by comparing it with numerical results and classical analytical models. Through this research, it is found that the proposed model provides a more accurate description of situations where soil arches cannot be formed for shallow tunnels in sand, leading to higher accuracy in calculating the limit support pressure. The influence of various factors on stability of the tunnel face is analyzed, revealing mechanisms of tunnel face collapse.

Modeling and analysis of deformation characteristics for the two-layered metal clad plate produced by asymmetric rolling

Accurate prediction of deformation behavior of metal plate in deformation zone is the key point for rolling cladding process. At present, there is a lack of accurate analytical model about the deformation of metal plate in deformation zone. Therefore, an analytical model of the deformation characteristics in deformation zone for the two-layered metal clad plate produced by asymmetric rolling is established according to the practical requirements. Firstly, the stream function field model of the clad plate in the deformation zone is founded by using the stream function method. Based on the stream function field, the velocity field model of the clad plate in the deformation zone is established, and the precision of the velocity field model is demonstrated by the comparison between the finite element and the model analytical results. Based on upper bound theory, the layer thickness of the clad plate after rolling and the ratio of deformation power, friction power and shear power to the total power were calculated by minimizing the total rolling power. Finally, based on the thickness of clad plate after rolling and velocity field model, an analytical model is established to calculate the equivalent strain of the clad plate in the deformation zone. The accuracy of the analytical model is tested by the rolling cladding experiments and the finite element simulations. The results indicate that the analytical model has high precision and which makes it useful to predict the deformation behavior in the clad plate rolling process.

An Improved Approach of Classical Upper Bound Theory for Stability Analysis of Layered Slopes

The equations for the critical height of layered slope were derived by a related researcher, but they lack details and do not agree with the method of limit equilibrium. So this topic was reinvestigated. Considering the possible global or local failure modes of the layered slope, the equations for the critical height of the above modes are derived based on the upper limit theorem by using rotational failure mechanisms or rotational and translational failure mechanisms. Based on the above theory, the stability coefficients of layered slope are investigated. The results show that the presented upper bound approach has better performance than other methods mentioned in the literature and is consistent with the limit equilibrium method. In combination with the strength reduction method, the global and local stability of the layered slope are analyzed using the several examples. The high-calculation accuracy of presented method and the specific measures to improve the smoothness of the sliding surface are reviewed. From the perspective of the combination of sliding surfaces, the dimensionless parameter vector of the layered slope is introduced. The possible limitations of the upper bound limit approach with log spirals are pointed out and the conditions which must be fulfilled in order to achieve a higher computational accuracy are mentioned.

An Analytical Framework for Assessing the Unsaturated Bearing Capacity of Strip Footings under Transient Infiltration

The evaluation of the bearing capacity of strip footings generally assumes that the soil is either dry or fully saturated, which contradicts the actual condition in nature where the soil is often in a partially saturated state. Furthermore, infiltration has a significant impact on the shear strength of the soil. Following the upper bound theory of the limit analysis, this article provides a theoretical framework for assessing the bearing capacity under transient flow with linear variation in infiltration intensity for the first time. Firstly, the closed form of suction stress under linear transient infiltration is derived using Laplace transform and introduced into the Mohr–Coulomb criterion. A discrete failure mechanism with fewer variables and higher accuracy is provided to ensure kinematic admissibility. The upper bound solution for bearing capacity is obtained by solving the power balance equation. The present results are compared with results from the published literature and the finite element, confirming the validity and superiority of the theoretical framework provided. A parametric analysis is also conducted on three hypothetical soil types (fine sand, silt, and clay), and the results show that unsaturated transient infiltration has a positive influence on increasing the foundation bearing capacity. The magnitude of the influence is comprehensively controlled by factors such as soil type, saturated hydraulic conductivity, infiltration intensity, infiltration time, and water table depth. The increase in bearing capacity due to unsaturated transient infiltration can be incorporated into Terzaghi’s equation as a separate component presented in tabular form for engineering design purposes.

The development of a W-shaped channel extrusion for fabricating magnesium alloy shells by combining high amplitude shear strain with a shorter process

This paper introduced a novel single-pass severe plastic deformation method, W-shaped channel extrusion (WCE), for the direct manufacturing of magnesium alloy shells. Based on a strategically installed mandrel and die with a sloping asymmetric step structure, the WCE constructed a narrow severe plastic deformation gap specially reinforced by a backpressure-shear structure and an asymmetric shear-extrusion section. Therefore, slender bar with large height-to-diameter ratio (i.e. small diameter, large height) could easily be extruded into larger diameter shell in narrow gap through only one pass, and an impressive average plastic strain of over 4.1 was concurrently achieved, evidenced by upper bound theory calculation. The WCE process was further experimentally conducted on AZ80 alloy at 340 °C, demonstrating remarkable improvements in yield strength, ultimate tensile strength, and elongation of the extruded part by factors of 2.4, 2.2, and 4.3 over the as-cast condition. The consecutive multi-stage shear and high hydrostatic pressure catalyzed extreme grain refinement, favored high matrix dislocations and concurrently facilitated the early dynamic conception of granular β-Mg17Al12 particles in subgrains stage, all of which served the superior performance. Moreover, no prominent anisotropy detected in the extruded part due to the development of a special bimodal basal texture under shear stress, where its synergistic effect with grain refinement evidently promoted the balance of strength and plasticity. Finally, the mechanism of bimodal texture formation associated with specific slip activations during WCE and anisotropic weakening was revealed in detail.

Improve cardinality with two-dimensional unipolar (optical) orthogonal codes for multiple access interference

Abstract Optical code division multiple access (OCDMA) is one of the promising technologies to be implemented using all-optical networks. The OCDMA system performance is substantially determined by the particular encoding codes. Due to the different transmitter and receiver design considerations, selecting the right code family and code dimension is critical. In this research, 2D codes based on permutation vectors’ (PV) codes are presented to overcome the Multiple Access Interference (MAI) problem and system complexity. Consequently, the research work focused on two-dimensional unipolar codes to mitigate the probability error and cardinality of the optical CDMA system. A fast wavelength-hopping/time-spreading (FWHTS) is introduced as an encoding technique. The FWHTS is used for code construction design, and it achieves cardinality in upper bound theory without sacrificing good auto-and-cross correlation properties. Accordingly, hybrid codes with serial elimination interference receivers are presented to estimate the error probability of 2D optical codes. The proposed method is simulated using Python software. The performance metrics are flexibility, bit rate, minimum temporal length, computational complexity, cardinality and wavelength. The proposed method is compared with the existing 2D balanced incomplete block design (BIBD) technique and multidiagonal (MD) codes. The performance of the proposed method is approximately 4% higher than these existing methods. Subsequently, in the future work, a SAC-OCDMA code construction design shall be presented and the decoding techniques can be introduced, which can reduce the receiver noise and other phase intensity induced noise, respectively.

Closed-form solutions for collapse mechanisms of tunnel crown in saturated non-uniform rock surrounds

The crown stability of a deep-buried metro tunnel or station is evaluated by means of a kinematic analysis procedure, with considerations of non-uniform rock surrounds and pore water effects. Non-uniform rock strata are discretized to several layers with finite thickness, with varying rock parameters assigned to different layers and constant parameters within each layer. Pore water effect is considered as an external loading acting on rock mass and boundaries. Employing the nonlinear Hoek-Brown criterion, a preliminary study is presented to analyse tunnel crown stability in a uniform rock medium, in the realm of upper bound theory. After having obtained general solutions of 2D and 3D collapse mechanisms, non-uniformity of rock masses is considered and investigated in crown stability analyses. 2D and 3D closed-form solution are formulated based on variational principle and work rate balance equation. Apart from the collapse shape and dimension, the weight of collapsing block is calculated, which gives a quick estimate of the average overburden pressure and hence guides the design of tunnel lining. Some examples related to circular and rectangular (square) tunnels are provided to offer insights of the implication of influential factors on the crown stability of a tunnel deeply excavated in saturated non-uniform rock strata.

Estimation of the Critical Seismic Acceleration for Three-Dimensional Rock Slopes

An analytical approach for the estimating of critical seismic acceleration of rock slopes was proposed in this study. Based on the 3D horn failure model, the critical seismic acceleration coefficient of rock slopes was conducted with the modified Hoek–Brown (MHB) failure criterion in the framework of upper-bound theory for the first time. The nonlinear Hoek–Brown failure criterion is incorporated into the three-dimensional rotational failure mechanism, and a generalized tangent technique is introduced and employed to convert the nonlinear Hoek–Brown failure criterion into a linear criterion. The critical seismic acceleration coefficients obtained from this study were validated by the numerical simulation results based on finite element limit analysis. The agreement showed that the proposed method is effective. Finally, design charts were provided for exceptional cases for practical use in rock engineering.

Upper-Bound Analysis for Equal Channel Angular Pressing (ECAP) with an Intersecting Channel Angle of 120°

Abstract The exact calculations of the stress and strain distributions based on the controlling equations for a forming process with large deformation are often difficult. To circumvent such difficulties, some analytical methods such upper-bound analysis and slip-line field theory have been established by making a number of simplifying assumptions regarding the material properties and deformation modes. In this work an analytical model based on the upper-bound theory was successfully developed to predict material flow pattern and maximum process loads for an Equal Channel Angular Pressing (ECAP) die with circular cross-section and an intersecting channel angle of 120°. Based on the model, the power dissipated on all frictional and velocity discontinuity surfaces were determined and optimized in order predict the maximum process force as function of the channel geometry and the material plastic behavior. To validate the developed model, the ECAP die were produced and used to determine experimental load-displacement curves of AA6061-T6 specimens with different lengths. A good correlation between theoretical and experimental results was observed. In addition, the constant friction factor demonstrated to have a strong effect on the relative extrusion pressure.

Calculation Method of Underground Cavern Loosening Pressure Based on Limit Analysis Using Upper-Bound Theory and Its Engineering Application

From the point of view of the failure mechanism of the disturbed zone, this paper uses the limit analysis upper-bound theory to analyze the calculation formula of the loosening pressure, distinguish the difference between the vertical pressure and the horizontal pressure in the underground cavern, combine the loosening characteristics of the disturbed zone with the open-type disturbed zone and the annular disturbed zone, and construct the multirigidity slider translation and rotation failure mode to discuss the calculation method of surrounding rock loosening pressure of underground caverns in upper soft and hard rock stratum. The relevant calculation examples are given, and the application of the upper-bound theory of limit analysis is demonstrated in detail. Based on the actual engineering background, the calculation results of the calculation method of the loosening pressure of the cavity based on the upper-bound theory of the limit analysis are analyzed and compared for the different depths and different types of caverns. The difference, rationality, and applicability of the calculation results of this method are analyzed and discussed.

Limit Analysis of Basal Heave Stability in Unsaturated Soils Based on the Unified Strength Theory

An evaluation method was presented on the basal heave stability in unsaturated soils. A basal heave failure mechanism was established wherein the admissible velocity field was constructed in the light of the Prandtl failure mode. The shear strength theory based on two independent stress state variables was employed for unsaturated soils. The unified strength theory was introduced to account for the influence of intermediate principal stress. Analytical solution of safety factor (Fs) was then deduced based on the upper bound theory of limit analysis, and relevant numerical solutions were solved by means of the strength reduction method. It is found from parametric analysis that the intermediate principal stress and matric suction have a significant impact on Fs. Specifically, Fs is seriously underestimated if they are ignored. The theoretical framework in this work provides a new idea for analyzing the basal heave stability of unsaturated soils.

Collapse comparison of offshore platforms before and after fire using plastic analysis and genetic algorithm

PurposeOffshore structures are among the structures exposed to fire more often. Most of these structures are likely to be associated with flammable materials. In this research, some of the structures constructed on top of marine decks have been studied.Design/methodology/approachFor this purpose, the upper-bound theory of plastic analysis has been used to investigate its collapse behavior. In this way, genetic algorithm has been used for application of the combination of elementary mechanisms in the classic plastic analysis problem.FindingsThe studied structures are optimized by plastic analysis theory before and after the fire and their failure modes are compared with each other. The comparison of the results indicates significant changes in the load factor value, as well as the critical collapse mode of the structure before and after the fire.Originality/valueResults indicate that the combination of plastic analysis and a genetic algorithm can predict the collapse mode of the structure before and after the fire accurately.

Rayleigh quotient and orthogonality in the linear space of boundary functions, finding accurate upper bounds of natural frequencies of non-uniform beams

Instead of the traditional method of utilizing the extremal values of Rayleigh quotient, in this research we develop an upper bound theory to determine the natural frequencies in a linear space of boundary functions. The boundary function will have the following restrictions: boundary conditions are satisfied, fourth-order polynomial at least with unit leading coefficient. It proves that the maximum value of Rayleigh quotient in terms of kth-order boundary function builds a good upper bound when the $$(k - 3)$$th-order natural frequency is approximated. There are four boundary conditions of the beam, so the fourth-order boundary function is unique without having any parameter. On the contrary, the kth-order boundary function has $$k-4$$ free parameters, which leave us a chance to optimize them. We employ the orthogonality to derive the higher-order optimal boundary functions from the lower-order optimal boundary functions exactly, which are constructed sequentially by starting from the fourth-order boundary function. We examine the upper bound theory for uniform beams under three types of supports, and the single- and double- tapered beams under cantilevered support. Comparison is made between the first four natural frequencies with the exact/numerical ones, which proves the usefulness of the upper bound theory.

Convection driven by internal heat sources and sinks: Heat transport beyond the mixing-length or “ultimate” scaling regime

Thermal convection driven by internal heat sources and sinks was recently shown experimentally to exhibit the mixing-length, or "ultimate", scaling-regime: the Nusselt number $Nu$ (dimensionless heat flux) increases as the square-root of the Rayleigh-number $Ra$ (dimensionless internal temperature difference). While for standard Rayleigh-B\'enard convection this scaling regime was proven to be a rigorous upper bound on the Nusselt number, we show that this is not so for convection driven by internal sources and sinks. To wit, we introduce an asymptotic expansion to derive steady nonlinear solutions in the limit of large $Ra_Q$, the Rayleigh-number based on the strength of the heat source. We illustrate this procedure for a simple sinusoidal heat source and show that it achieves heat transport enhancement beyond the mixing-length scaling regime: $Nu$ increases linearly with $Ra$ over this branch of solutions. Using rigorous upper bound theory, we prove that the scaling regime $Nu \sim Ra$ of the asymptotic solution corresponds to a maximization of the heat flux subject to simple dynamical constraints, up to a dimensionless prefactor. Not only do 2D numerical simulations confirm the analytical solution for the sinusoidal source, but, more surprisingly, they indicate that it is stable and indeed achieved by the system up to the highest $Ra_Q$ investigated numerically, with a heat transport efficiency orders of magnitude higher than the standard mixing-length estimate.

The perfluoropolymer upper bound

AbstractPerfluoropolymers have fundamentally distinct thermodynamic partitioning properties compared to those of their hydrocarbon counterparts. However, current upper bound theory assumes hydrocarbon solubility behavior for all polymers. Herein, the fundamental presupposition of invariance in solubility behavior to upper bound performance is critically assessed for perfluoropolymers and hydrocarbon polymers. By modifying solubility relationships, theoretical perfluoropolymer upper bounds are established, showing a positive shift of the upper bound front as a result of beneficial solubility selectivities for certain gas pairs, including N2/CH4, He/H2, He/N2, He/CH4, and He/CO2. Within the framework of the solution–diffusion model, an analysis is presented to compare two independent approaches often pursued in efforts to surpass the polymer upper bound: (a) achieving solubility selectivity via perfluoropolymers and (b) improving diffusion selectivity via rigid hydrocarbon polymers. This analysis demonstrates the significant benefit that can be achieved by considering both the chemical composition and morphology of solid‐state macromolecules when designing membrane materials.

Coordinative optimization method of composite laminated structures based on system reliability

This paper develops the coordinative optimization method based on system reliability for laminated structures. The proposed method improves the rough RBO based on first layer failure (FLF) criterion for composite laminates, and the coupling optimization method of thickness and sequence in traditional RBO strategy based on last layer failure criterion (LLF) is improved. In this paper, the finite element analysis is used to obtain the response for the failure based on two-dimension Hashin failure criterion (the limit function). Obviously, the stiffness of composite materials will decline due to destruction of elements. Therefore, stiffness degradation is considered to describe the process of damage evolution. Subsequently, combining with the branch-bound method (B&B), we can complete the search of main failure sequences and calculate the system reliability with the help of the second-order upper bound theory. In order to guarantee the efficiency and accuracy of optimization, the adaptive GA algorithm is introduced in the whole optimization procedure. After the proposed optimization policy is given in detail, two laminated structures are presented and the results are compared with the traditional optimal method based on safety factor, which demonstrates the validity and reasonability of the developed methodology.

Stability Analysis for the Expansive Soil Slope Considering the Influence of Cracks Based on the Upper Bound Method

Crack is an important factor in slope stability analysis. The permeability of expansive soil is small, and its rainfall infiltration is limited during rainfall. Cracks provide a great channel for rainfall infiltration, which increases the infiltration depth and porewater pressure, further affecting the deep soil and ultimately leading to expansive soil landslides. A stability analysis method that considered the crack effect for the expansive soil slope is established, which based on the upper bound theory of limit analysis. And the effects of crack depth and crack angle on the stability of the expansive soil slope are explored and discussed. The results show that: The safety factor decreases with the increase of crack depth, and is the smallest under the consideration of the influence of earthquake action and deformation force. The effect of the crack angle on the slope safety coefficient are insignificant. When the crack angle greater than 70°, the safety factor becomes smaller. Compared with the crack angle, the crack depth exerts a greater influence on the slope safety factor.

Effects of helicity on dissipation in homogeneous box turbulence

The dimensionless dissipation coefficient $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D700}L/U^{3}$, where $\unicode[STIX]{x1D700}$ is the dissipation rate, $U$ the root-mean-square velocity and $L$ the integral length scale, is an important characteristic of statistically stationary homogeneous turbulence. In studies of $\unicode[STIX]{x1D6FD}$, the external force is typically isotropic and large scale, and its helicity $H_{f}$ either zero or not measured. Here, we study the dependence of $\unicode[STIX]{x1D6FD}$ on $H_{f}$ and find that it decreases $\unicode[STIX]{x1D6FD}$ by up to 10 % for both isotropic forces and shear flows. The numerical finding is supported by static and dynamical upper bound theory. Both show a relative reduction similar to the numerical results. That is, the qualitative and quantitative dependence of $\unicode[STIX]{x1D6FD}$ on the helicity of the force is well captured by upper bound theory. Consequences for the value of the Kolmogorov constant and theoretical aspects of turbulence control and modelling are discussed in connection with the properties of the external force. In particular, the eddy viscosity in large-eddy simulations of homogeneous turbulence should be decreased by at least 10 % in the case of strongly helical forcing.

Reduced modeling of porous media convection in a minimal flow unit at large Rayleigh number

Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number (Ra) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-Ra porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale cellular motions generally are localized within the thermal and vorticity boundary layers near the upper and lower walls, while in the interior, the flow exhibits persistent large-scale structures and only a few low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first strategy, the domain is decomposed into two near-wall regions and one interior region. Our results confirm that suppressing the interior high-wavenumber modes has negligible impact on the essential structural features and transport properties of the flow. In the second strategy, a hybrid reduced model is constructed by using Galerkin projection onto a fully a priori eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini et al. (2011) [45] to large Ra. The results indicate that the near-wall upper-bound eigenmodes can economically represent the small-scale rolls within the exquisitely-thin thermal boundary layers. Relative to DNS, the hybrid algorithm enables over an order-of-magnitude increase in computational efficiency with only a modest loss of accuracy.