Principle Of Minimum Energy Research Articles

Multi-scale pore model construction and damage behavior analysis of SiCf/SiC composite tubes

SiCf/SiC composite materials are essential for spacecraft thermal protection, determining the safety of spacecraft. However, their complex structures and intricate fabrication processes have led to an unclear understanding of their failure mechanisms, limiting their application. In this study, circumferential compression experiments were conducted, and X-ray computed tomography (X-CT) was used to analyze the distribution characteristics of pore spaces. The experimental results indicate that damage manifests as secondary fracture phenomena based on temporal evolution characteristics. Specifically, after experiencing performance degradation of 24.73–60.96%, the material still maintained basic stability, exhibiting a performance recovery of 31.55–36.1% under the applied load until failure. To predict the dynamic damage behavior of materials at multiple scales, a multi-scale method combining statistical and microscopic approaches was proposed. A micro representative volume element (RVE) random fiber pore distribution model was established based on random processes and random medium theory. Finally, considering the CT data and the principle of minimum potential energy, a macroscopic finite element model of the micro-pores' structural characteristics in a three-dimensional wound tube was reconstructed. The results indicate that cracks initiate around pore defects and gradually propagate to form crack bands. The model closely matches the macroscopic damage and microscopic characteristics of the material. This work provides a new approach for the numerical simulation of pore defects in such materials.

Material optimization of reinforced concrete structures in multiple limit states

AbstractA common limitation in current material optimization methods for reinforced concrete structures is that they often concentrate solely on either the serviceability limit state (SLS) or the ultimate limit state (ULS). In response, this paper introduces a novel approach to optimize the material layout in planar reinforced concrete structures in plane stress, accounting for both limit states simultaneously. The proposed method employs an iterative strategy, integrating criteria for serviceability and ultimate limit states while approximating the elastoplastic behavior of cracked reinforced concrete. Here, the criteria for the limit states are incorporated as limits on the strains. The methodology initiates with a finite element limit analysis for the preliminary layout. The minimum potential energy principle is utilized to determine the strains for the nonlinear material response, and subsequently, an iterative linearized method is applied for material optimization. The research showcases the successful application of this approach to simple examples and presents improved and optimized solutions that adhere to the limit criteria for more complex structural scenarios.

A method for predicting deformation field of deep foundation pit considering spatial effect

Urban underground spaces have experienced significant development and utilization due to the rapid progress of urban construction and continuous advancements in construction technology. The number of foundation-pit projects immediately adjacent to buildings, structures, and pipelines has been increasing. Ensuring the safety of construction and surrounding facilities, it is essential to predict deformation caused by foundation pit excavations. In this study, a simple calculation method was proposed for predicting deformation of an envelope structure and surface settlement caused by deep foundation excavation with spatial effects. Firstly, the principle of minimum potential energy was applied and the lateral displacement equation of the envelope structure was derived, considering spatial deformation effects. Secondly, based on linear elasticity theory, a surface settlement expression was formulated utilizing the plane strain boundary. Finally, the stratigraphic loss method was used to establish a connection between horizontal envelope deformation and surface settlement, allowing for calculations of surface settlement at any position behind the envelope structure. The effectiveness and practicability of the proposed method were demonstrated by comparison of two existing foundation-pit engineering examples.

Conglomerate rock interface debonding characteristics and damage evolution based on cohesive phase-field method

Rock failure often occurs due to uneven mechanical strength and discontinuous strain at the interface. However, due to the complex interaction mechanism between different phases of conglomerate, it is necessary to improve the crack growth model at the conglomerates' interface. In view of the physical structure and mechanical properties of conglomerate materials, this paper innovatively adopts the cohesive phase field model to study the material deformation and crack growth of conglomerate materials. Different from the traditional phase field model, this model characterises the real structure of the interface by adding an interface phase field, which is used to simulate the debonding of the interface during crack propagation and damage evolution. The regularization of the phase field is carried out to ensure the smoothness of the interface. The transition state of the interface region is determined by the element and the additional phase field. The energy equation is coupled with the crack propagation process using the crack surface density function. The evolution of the damaged phase field and displacement field is driven by the principle of energy minimization. The innovative assumption is that there is a virtual crack at the crack tip, and the interface related elements show a certain cohesive softening form during the damage evolution, which is the main way to realize the particularity of interface mechanics. By controlling the crack dispersion state and stiffness degradation of the material, several ideal softening models are innovatively assumed according to the cohesive force change law to satisfy the mechanical properties and stress function constraints. When cutting blocks n=2.5×104, the errors of the linear model, exponential model, and compound inverse model are 1.76%, 7.48%, and 5.19%, respectively. The results of the exponential model agree well with the stress strength (the error is 2%) and interface crack kinking angle (68.89°). The simulation results indicate that the deformation process of the conglomerate can be divided into three stages: elastic deformation stage, crack propagation stage, and stable load stage. The tensile strength of conglomerate is positively correlated with interface's fracture toughness and proportion of gravel area, respectively.

Three-dimensional calculation method on local slurry wall stability of diaphragm wall trench based on minimum potential energy principle

Because the soft weak soil layer rich in confined groundwater is prone to local instability and failure in the construction of diaphragm wall trenches, in order to check the local slurry wall stability in the construction of diaphragm wall trenches, a new three-dimensional (3D) calculation method on the local stability safety factor (LSF) of slurry walls is proposed based on the minimum potential energy principle. After the new method is applied to two engineering cases, the value of the LSF solved is 0.82 for engineering case 1, and the values of the LSF solved are 0.84 and 1.05 when the phreatic water levels are 2 and 4.5 m, respectively, for engineering case 2. These results are very close to the calculation results in the existing references, so the rationality of the new method is verified. Then the sensitivity of the LSFs obtained from the new method to various parameters has been analyzed and clarified on the basis of ten design examples. Finally, the new method is used to check the local slurry wall stability in the construction of diaphragm wall trenches for a practical project, and the obtained values of the LSF for three concerned weak soil layers are 1.37, 1.19, and 0.82, respectively. These results are in agreement with the actual situation in the construction site. This reveals that both the 3D model established and the 3D calculation method proposed in this paper are suitable for application in practical projects.

Formation Mechanism and Morphology Evolution of β-NaYF4 Single Microcrystals in Hydrothermal Process

Lanthanide-doped upconversion (Ln-doped UC) materials with predicable phase and geometry are attractive for their promising application in optics and biological. Herein, we report a facile strategy for deterministic synthesis of β-NaYF4 microcrystals by an ethylenediaminetetraacetic acid (EDTA) assisted hydrothermal process. The nucleation and growth process of the β-NaYF4 microcrystals have been experimentally revealed with different hydrothermal times. The phase and geometry of as-obtained crystals can be well manipulated by the experimental parameters such as pH values, precursors’ ratio, and reactant concentrations. The formation and morphology evolution mechanism of β-NaYF4 single microcrystals can be qualitatively analyzed using energy minimization principles under the thermodynamic and kinetic control. In addition, we presented the colour-changing up-conversion of Tm3+ and Yb3+ doped single β-NaYF4 microrod. Our work could help the understanding on crystal growth and design of rare earth fluoride mateirals at micro and nanoscale.

Static response of vertically loaded pile in inhomogeneous soil

In this study, an analytical model is developed to obtain the static response of the vertically loaded pile in the inhomogeneous soil with arbitrary properties varying along the depth. The soil displacement is expressed by the product of the axial displacement function and the shape function attenuating with the distance from the pile periphery. Through the minimum potential energy principle, the governing equations are established for the axial displacement function and the shape function. The governing equation for the shape function is independent of soil properties and it can be explicitly solved. The governing equation for the axial displacement function contains variable coefficients due to the inhomogeneity of soil and it is solved using the weighted residual method. An iterative procedure is introduced to implement the solution. The proposed model is validated by comparing the present solutions with those obtained by the available methods. The parametric study shows that the pile head stiffness is affected by the distribution of soil shear modulus heavily, in which the average shear moduli are the same for these distributions. Moreover, the relationship between the pile head stiffness and the pile slenderness ratio for a floating pile is different from that for an end-bearing pile.

Design Optimization of Flexural and Eigenvalue Characteristics of Perforated and Skewed Bio-Composite Functionally Graded Panels

The development of novel materials is of prime importance in the biomedical areas, especially for bone replacements and dental implants. This study uses finite element modeling of skewed functionally graded bio-composite structures with perforations to examine flexural and eigenfrequency responses. Here, the bio-composite structure comprises Titanium (Ti) and Hydroxyapatite (HAp). Voigt’s micromechanical material model via power-law distribution and Representative volume element (RVE) method is utilized to compute the in-homogenous material properties in the thickness direction. The strain field is based on the equivalent single-layer first-order shear deformation theory with five degrees of freedom, and the governing equations are obtained using principle of minimum potential energy. A computer algorithm is developed in a 2D finite element platform using eight-noded quadrilateral elements. The mesh stability of the skewed and perforated structure is confirmed through the smart mesh technique, and the present model’s accuracy is demonstrated by comparing it with the available reported results. Numerous examples reveal the significance of material gradation, skewness, aspect ratios, perforations, and support conditions on bio-composite functionally graded structures’ flexural and eigenfrequency responses. Furthermore, parametric optimization is conducted to minimize the flexural response and maximize the fundamental frequency by employing the response surface methodology (RSM), followed by determining optimal process parameters and valuable findings.

New insights into folding tape-spring

The tape-spring is widely used in aerospace, mechanical, and civil engineering due to its unique mechanical properties and compact size. This paper aims to establish a new analytical model to more accurately predict the folding behavior of the tape-spring, including the geometric configuration, folding moment versus longitudinal curvature curve, peak moment, and propagation moment. This model comprehensively considers the influence of stretching strain energy and bending strain energy on the total strain energy, and determines the folding behavior of the tape-spring through the minimum energy principle. The proposed analytical model was compared with Wuest’s model, Yee’s model, and Yao’s model. Experiments of tape-springs made from two different materials were utilized for validation. The study shows that the proposed model has the highest prediction accuracy compared with the classical models, with the error of approximately 10%. The reason for the significant difference in the prediction results between the proposed model and Yao’s model was given, and the relationship between stretching strain energy and bending strain energy during the folding process of the tape-spring was analyzed. The proposed model has broader applicability than classical models and provides new insights and solutions for accurate prediction of the folding behavior of the tape-spring.

Analyze the temperature-dependent elastic properties of single-walled boron nitride nanotubes by a modified energy method

The elastic properties of boron nitride nanotubes (BNNTs) were investigated utilizing an enhanced energy method. By considering small deformations and applying the principle of minimum potential energy, the variations in atomic bonds and bond angles within the nanotube structure were determined. The modified model incorporated the contribution of inversion energy to the overall potential energy of the system, leading to the derivation of analytical expressions for the Young's modulus, shear modulus, and strain energy of both armchair and zigzag BNNTs under varying temperatures. The results indicate that compared to zigzag BNNTs, the impact of inversion energy on the elastic constants of armchair BNNTs is more significant, especially at small diameters (<1 nm). In thermal environment, this study demonstrates that the change in Young's modulus of BNNTs is lower than that of carbon nanotubes (CNTs), confirming the superior thermal stability of BNNTs over CNTs. Furthermore, molecular structure mechanics (MSM) and continuum mechanics models were employed to analyze the strain energy of BNNTs. The effects of different bonds, bond angles, and inversion angles on strain energy were analyzed in a thermal environment, revealing distinct differences between the two types of BNNTs. These findings provide more accurate theoretical guidance for thermal applications based on the stretching of BNNTs.

Ispitivanje poprečne unutarnje sile sandučastog nosača primjenom principa varijacije energije

This study investigated the transverse internal force of a box girder with a trapezoidal cross-section under eccentric loading. The effects of the box girder distortion and frame shear difference on the transverse internal force were considered. An energy variation method based on the minimum potential energy principle was adopted for transverse internal force analysis of the frames. The variable shear difference of the top slab was considered an unknown quantity, and a fourth-order governing differential equation was established. The transverse bending moment of the frame was obtained by solving for the shear difference. Two examples were used to verify the proposed method and analyse the differences between the transverse internal force results calculated using different methods. The effect of the distribution of the box-girder distorted transverse bending moment on each slab was investigated for various stiffness ratios. The results showed that the transverse internal force of the box girder calculated using the proposed method agreed well with the finite element results, with a maximum error of less than 9.68 %. When the stiffness ratio of the box girder increased, the distribution of the distorted transverse bending moment on the top slab increased, whereas that on the bottom slab decreased.

Simplified nonlinear model and 3D printing model experiment verification of longitudinal joints of shield tunnel segments

The segment joints represent vulnerable and core-bearing areas within shield tunnel structures. Their flexural stiffness, starting from the contact surface of longitudinal joints to the bolt hole, exhibits continuous changes under bending loads, typically demonstrating nonlinear mechanical characteristics. This study investigates the influence of joint contact surfaces on flexural stiffness, extending from the joint contact surface to damaged concrete in the compressed zone. A new simplified model for the variable nonlinear stiffness of longitudinal segment joints in shield tunnels is proposed, considering the flexural stiffness of the joint contact surface and the width of the damaged concrete zone in the segment ring while simulating the flexural stiffness decay induced by the opening of segment joints. Based on the minimum potential energy principle, a design calculation method for joint flexural stiffness is derived, establishing the nonlinear bending stiffness coupling relationship between the segment and the joint. Through similarity theory, 3D printing technology, and indoor loading tests, the theoretical model's accuracy regarding tunnel segment joints is verified. This theoretical model provides a basis for designing and assessing the performance of tunnel-segment joints.

Modeling compliant bistable mechanisms: An energy method based on the high-order smooth curvature model

This paper applies an energy method based on the high-order smooth curvature model to address the challenges of kinetostatically modeling the complicated nonlinear post-buckling behavior of inclined compliant beams. The proposed energy method is grounded in the principle of minimum strain energy, implying that the total strain energy is minimized at the equilibrium configuration. In this work, the high-order smooth curvature model is adopted to accurately model the bending strain energy of large deformation beams. Subsequently, the Lagrange multiplier method is employed to ascertain the minimum of strain energy while concurrently determining the corresponding tip loads. Additionally, the deformation shape and maximum stress are determined via the smooth curvature model. The proposed method is introduced for the first time in modeling compliant bistable mechanisms, and it is proven that the method can be used for modeling compliant beams with inclined angles ranging from 0 to 90 degrees. Following the modeling, finite element analysis and experimental tests are conducted to verify the accuracy of the proposed energy method. A comprehensive comparative analysis between proposed method and existing methods is conducted. The comparison results prove that the model is more computationally efficient without compromising modeling accuracy The proposed modeling method can not only be used for modeling compliant bistable mechanisms but also has extendable applications in modeling initially-curved compliant beams, contact-aid design problems, and distributed load problems.The present work not only advances the practical application of the proposed energy method but also lays the foundation for future research in modeling compliant mechanisms.

Self-shaping of triglyceride and alkane drops: Similarities and differences

Small emulsion drops typically exhibit spherical shape at positive interfacial tension due to the energy minimization principle. However, in a series of studies (Denkov et al., Nature, 2015, 528, 392–395; Cholakova et al., Nature Phys., 2021, 17, 1050–1055) we showed that alkane droplets stabilized by appropriate saturated long-chain surfactants may spontaneously change their shape upon cooling, morphing into various polyhedra; hexagonal, tetragonal and triangular platelets; rod-like particles and even synthetic swimmers. These deformations are governed by the formation of thin plastic rotator phases adjacent to the drop surface. Although alkanes have numerous industrial applications, they cannot be used in food and pharma related products, in which most often triglyceride molecules are employed. The possibility for self-shaping of triglyceride drops has been demonstrated, but the detailed understanding of the process is currently missing. In the present study, we performed model experiments aimed to reveal the conditions under which the triglyceride emulsion drops may change their shape upon cooling. We show that most of the various non-spherical shapes known for alkanes can be reproduced with triglyceride droplets providing that the surfactant adsorption layer freezes before the nucleation of the oily molecules inside the drops. By comparing the behavior of triglyceride and alkane droplets, we draw unified picture and provide guiding principles which can be used for selection of appropriate surfactants enabling the spontaneous shape deformations upon cooling of oily drops of different chemical compositions.

Nonlinear bending characteristics of multidirectional functionally graded porous panels using higher-order finite element solution with parametric optimization

The current study aims to investigate the nonlinear bending behavior of unidirectional (1D) and multidirectional (2D/3D) functionally graded porous shell panels. Here, the material properties of the multidirectional functionally graded (FG)-panels are estimated by employing the representative volume element method and Voigt’s micromechanical homogenization scheme. The nonlinear mathematical model of the FG-panel is developed by incorporating Green-Lagrange type nonlinear strains in higher-order-midplane kinematics. The minimum potential energy principle is used to obtain the final form of the equilibrium equation for the multidirectional FG shell panel and further solved by using Picard’s iterative-based 2D-isoparametric finite element method via quadrilateral Lagrangian elements. The efficacy and robustness of the present model are checked by comparing its results with previously reported literature. Numerous examples are solved to analyze the impact of material parameters, such as volume fraction index and porosity index, and geometrical parameters, such as aspect ratio and support conditions, on the nonlinear deformation behavior of multidirectional FG shell panels. In addition, parametric optimization is carried out to minimize the flexural response by using the response surface methodology and, hence, determining the optimal values of multidirectional porous FG-panel. The results demonstrate that, within the defined ranges of FG-panel, lower values of porosity index and volume fraction index yield lesser deflection.

Analysis of multi-scale failure behavior of SiCf/SiC composite clad tube under flexible clamping damage

The intricate spatial fiber winding structure of SiCf/SiC ceramic matrix composite (CMC) cladding tubes makes it difficult to investigate in depth the inherent evolutionary mechanism of damage failure under specific operating conditions. Therefore, in this study, based on X-ray computed tomography (X-CT) reconstruction and cross scale numerical simulation methods, combined with clamping failure experiments under specific conditions for the cladding tubes, we conduct an in-depth analysis of the material's damage, failure, and mechanical behavior in different clamped regions. The research indicates that variations in the clamped regions have a minimal impact on the overall mechanical feedback of the material, and the maximum clamping load remains essentially around 500 N. The tube damage exhibits a unique " regional progressive damage evolution mechanism," where stress differentials of 108.49–115.33% appear between the inner and outer surfaces in different regions, leading to variations in the initiation and propagation of cracks within these areas. Furthermore, based on the minimum potential energy principle and semi-geodesic line theory, the intricate macroscopic tube structure formed by the spatial stacking and winding of SiCf/SiC cladding tube fibers is accurately reconstructed, which is combined with macroscale and microscale local feature models to capture the fibre/matrix morphology of the material with dynamic mechanical feedbacks, and experimentally validated in this work. The results indicate that the numerical model can effectively characterize the complex mechanical damage morphology and progressive failure lows of SiCf/SiC CMC at multiscale. Simultaneously, it is observed that stress concentrations in the fiber crossover region result in a load-bearing capacity 112–283% higher than in other areas. This phenomenon often leads to the first occurrence of failure in this region after clamping overload, exhibiting more pronounced damage characteristics.

Modelling beaches morphodynamic by Hadamard sensitivity analysis

The paper presents a morphodynamic model which can be coupled with any wave model capable of producing time/spectral averaged wave quantities. This model based on a wave energy minimization principle highlights the morphodynamic phenomenology, such as the sandbar creation. Such a model can be used in solving engineering optimization problems. It is also developed to illustrate the idea that beach sand transport can be thought as a non-local phenomenon. We used wave calculations from SWAN and XBeach in our model, and we compared the morphodynamic results to LIP and SANDS hydro-morphodynamic benchmark as well as open-sea simulations. Using supplementary mathematical development, we improved the minimization method using the Hadamard derivative.

Crack propagation in anisotropic brittle materials: From a phase-field model to a shape optimization approach

The phase-field method is based on the energy minimization principle which is a geometric method for modeling diffusive cracks that are popularly implemented with irreversibility based on Griffith’s criterion. This method requires a length-scale parameter that smooths the sharp discontinuity, which influences the diffuse band and results in mesh-dependent fracture propagation results. Recently, a novel approach based on the optimization on Riemannian shape spaces has been proposed, where the crack path is realized by techniques from shape optimization. This approach requires the shape derivative, which is derived in a continuous sense and used for a gradient-based algorithm to minimize the energy of the system. In this paper, the novel approach based on shape optimization is presented, followed by an assessment of the predicted crack path in both isotropic and anisotropic brittle material using numerical calculations from a phase-field model.

Multiscale mechanical analysis of 3D nanoparticle-reinforced metal-based composites considering interface effects

Nanoparticle-reinforced metal-based composites (NRMCs) have gained recognition for their remarkable interface effects and their demonstration of superior mechanical properties. To effectively account for interfacial effects and precisely assess the mechanical properties of NRMCs with arbitrarily complex internal structures, our study employs the principle of minimum energy and leverages the Gurtin-Murdoch interface theory to devise a finite element interface element that comprehensively considers the interface effect of NRMCs. Utilizing this novel interface element, we construct diverse NRMCs model and subject them to single-axis tension and compression simulations. Our findings reveal that the developed finite element model accurately captures the internal stress distribution of the NRMCs. Both the particle modulus and interfacial effects exert influence on the Young’s modulus of the model. When the particle modulus is small, interfacial effects have a large impact on Young’s modulus. Additionally, the effect of interface residual stress on Young’s modulus is strongly dependent on the direction of external loading. The developed finite element model offers a scientific approach for accurately predicting the mechanical performance of NRMCs.

Revisiting the Young’s model for ferrofluid droplets: Magnetowetting or magneto-dewetting?

Magnetic field-mediated wetting of ferrofluid droplets has emerged to be of fundamental importance because of its plethora of applications ranging from chemical and biomolecular analysis to soft robotics and medical imaging. However, while extensive studies continue to be reported on the underlying experimental facets, a fundamental theoretical depiction of the observed contact angle variation with the applied magnetic field remains grossly elusive, as attributable to a complex dependence of the induced magnetization on the droplet’s intrinsic magnetic properties. Circumventing these constraints, here we put forward a first-principle analytical model that delineates the variation in the contact angle as a function of an applied magnetic field on a sessile ferrofluid droplet, by appealing to energy minimization principles. One key feature of our analysis is the consideration of the local magnetization as a function of the magnetic field itself, resulting in a new contribution to the interfacial energy by virtue of normal traction at the droplet's contact with its non-magnetic surroundings. This theory is supported by experiments identified in prior research to elucidate the fundamental physics of relevance. Our results highlight the substantial influence of the saturation magnetization and the initial susceptibility of the ferrofluid on the resulting contact angle alterations, as well as possible non-trivial routes of highly selective and precise enhancement or attenuation in the extent of wetting. These findings are likely to be of fundamental importance towards establishing the design foundations of portable magnetic resonance imaging devices, micromixers, and microreactors, magnetically-manipulated automatic control of soft fluidic matters, among others.