Elastic Energy Density Research Articles

Modelling nematic liquid crystal in fractal dimensions

In this paper we present a fractal Berreman's model which account for azimuthal surface anchoring of nematic liquid crystals which play a leading role nowadays in biomedical field and pharmaceutical controlled release dosage forms. The model is based on the concept of "product-like fractal measure" in anisotropic media which permits to describe liquid crystals formation in terms of fractal dimensions. It was observed that fractal dimensions affect both the polar azimuthal angle and the Franck excess elastic energy density of the distortion. For fractal dimensions close to unity, our analysis reveals the emergence of fluctuations and large deformations in the dynamical variables of the theory which suggest the presence of non-linear elastic instabilities or emergence of defect structures in liquid crystals. These fluctuations fade away for fractal dimensions much less than unity. We have evaluated the elastic energy per unit of area which is found to depend on the square of the fractal dimension. This leads to a decrease of the saturation voltage which is necessary to reduce the elastic energy of liquid crystals films in order to minimize the elastic energy. This reduction may describe nematic liquid crystals free from deformations, singularities or weak anchoring surfaces.

Ex-situ observation of ferrite grain growth behavior in a welded 9Cr-1Mo-V-Nb steel during aging at 740 °C

It has recently been reported that ferrite grains coarsened to several hundred micrometers were occasionally observed in long-term serviced 9Cr-1Mo-V-Nb steel welds. To clarify the factors that cause ferrite grain growth in martensite during high-temperature exposure, alternating aging heat treatment and observation of the same field of view was performed using a welded 9Cr-1Mo-V-Nb steel. Such ex-situ observations revealed that the rapid grain growth of ferrite by consuming martensite occurred in the weld metal during aging at 740 °C. In the test material used in this study, some δ-ferrite grains were observed in the weld metal near the fusion line. In the region where δ-ferrite grains were observed, the concentrations of austenite-forming elements such as Mn and Ni were locally decreased in the matrix due to dilution by the base metal, promoting δ-ferrite retention after welding. Ex-situ observation indicated that no significant grain growth of δ-ferrite occurred during aging. Therefore, it was suggested that a new ferrite grain formation followed by rapid grain growth consuming martensite occurred during aging. The elastic strain energy density of the dislocations PMdis and the interfacial energy density PMsurf in martensite can affect the driving force for ferrite grain growth by consuming martensite. Based on the evaluation results for PMdis and PMsurf after 500 h of aging, PMsurf was considered to be the main driving force for ferrite grain growth. Although the ferrite-formation process could not be directly observed, it is possible that ferrite was formed by the recrystallization of martensite through the bulging mechanism.

A method for assessing the risk of rockburst based on coal-rock mechanical properties and In-Situ ground stress

With the increase in mining depth and intensity, dynamic disasters such as rockburst in mines are becoming more severe. Deep resource extraction is characterized by a high in-situ stress geological environment, closely associated with geological dynamic disasters. However, there is currently no quantitative analysis method for the correlation between the two. In this study, an elastic energy density calculation method is employed, considering the dissipative effect of the self-weight stress field on the tectonic stress field. The remaining energy, referred to as impact energy, is used to classify the risk of coal seam impact, providing a computational method for rapid assessment of impact risk before mining production. The proposed calculation method is compared with 22 mine impact engineering practices in the literature, showing accurate predictions for 21 mines. Since measuring in-situ stress and coal seam physical and mechanical properties is a preliminary work in coal seam extraction, the comprehensive analysis of these data holds significant research and practical value.

Ultrahigh Elastic Energy Storage in Nanocrystalline Alloys with Martensite Nanodomains.

Elastic materials that store and release elastic energy play pivotal roles in both macro and micro mechanical systems. Uniting high elastic energy density and efficiency is crucial for emerging technologies such as artificial muscles, hopping robots, and unmanned aerial vehicle catapults, yet it remains a significant challenge. Here, a nanocrystalline structure embedded with elliptical martensite nanodomains in ferroelastic alloys wasutilized to enable high yield strength, large recoverable strain, and low energy dissipation simultaneously. As a result, the designed Ti-Ni-V alloys demonstrate ultrahigh energy density (>40 MJ m-3) with ultrahigh efficiency (>93%) and exceptional durability. This concept, which combines nano-sized embryos to minimize energy dissipation of psuedo-elasticity and employs a fine-grained structure to enhance yield strength, can be applied to other ferroelastic materials. Furthermore, it holds promise for the development of phase transformation-involved functionalities such as high-performance dielectric energy storage, ultralow-hysteresis magnetostrain, and high-efficiency solid-state caloriccooling.

Experimental study on physico-mechanical responses and energy characteristics of granite under high temperature and hydro-mechanical coupling

Thermal and hydro-mechanical coupling effects on granite's physico-mechanical responses and energy characteristics can influence its carrying capacity, hydraulic and heat transfer performance. The evolution of these properties is crucial for geothermal reservoir stability assessment and the optimization of deep geothermal energy development techniques. Hence, a variety of physical tests and rock mechanics experiments were conducted. Key findings include: (1) As temperature rises, granite undergoes thermal expansion and structural integrity degradation. The peak strength σp, elastic modulus E, total energy density U, and elastic energy density Ue initially rise at 25 °C–150 °C and subsequently decrease above 150 °C, while the proportion of dissipated energy ηd is opposite. Granite also initially experiences hardening, then turns to brittle-ductile transition. (2) The crystallinities of quartz, albite, and orthoclase demonstrate substantial deterioration beyond 450 °C. (3) 150 °C and 450 °C are regarded as the temperature thresholds for mechanical properties. (4) As confining pressure rises, granite experiences hardening, with σp, E, U, and Ue increasing, and ηd decreasing. (5) Pore water pressure increases ηd and decreases σp, E, U, and Ue, and its effect on the mechanical responses is pronounced when it reaches 80 % of confining pressure or exceeds 450 °C.

Performance evaluation of a novel adaptive temperature-controlled lining technology based on thermo-hydro-mechanical coupling analysis

To mitigate the hazards of high geotemperatures in hydraulic tunnels, the performance of a novel adaptive temperature-controlled lining was investigated across various working phases of high geothermal tunnels using the thermo-hydro-mechanical coupling analysis with the finite element software COMSOL. The results indicated an error margin of only 2% to 6% compared to the field measurements, proving the feasibility of the method. Compared with the conventional concrete lining, the adaptive temperature-controlled lining reduced the average inner wall temperature by approximately 55% during construction, decreased the maximum thermal deformation by approximately 37%, transferred the localized energy elevation from the inner wall to the outer lining, and lowered the maximum elastic strain energy density by approximately 62%. During the water-filled operational phase, the radius of temperature influence in the low-temperature zone of the adaptive temperature-controlled lining expanded by approximately 20%, whereas the maximum thermal stress decreased by approximately 76%, the thermal deformation by 50%, and the elastic strain energy density by 88%. Throughout both the construction and operational phases, the adaptive temperature-controlled lining significantly enhanced the temperature control, improved the stress distribution, and reduced the risk of concrete cracking. These findings could offer the innovative engineering insights into the design, construction, and operation of high-geothermal tunnels.

A phase-field gradient-based energy split for the modeling of brittle fracture under load reversal

In the phase-field modeling of fracture, the search for a physically reasonable and computationally feasible criterion to split the elastic energy density into fractions that may or may not contribute to crack propagation has been the subject of many recent studies. Within this context, we propose an energy split – or energy decomposition – aimed at accurately representing the evolution of a crack under load reversal. To this purpose, two key assumptions are made. First, the damage gradient direction is interpreted as being representative of the normal-to-crack direction, as already assumed in previous works in the literature. The second assumption consists of considering the sign of the projection of the stress tensor onto the damage gradient direction at a point as an indicator of whether this point should behave as an opening or as a closing crack. We associate the latter case (crack closing) to both (a) a complete recovery of elastic energy density of the intact material (i.e., perfectly rough crack surfaces) and (b) a zero crack driving force at that point. The first case (crack opening) is treated classically as a damageable material point at which damage can increase. The implementation of the proposed approach turns out to be remarkably simple and computationally robust. For the evaluation of the displacements and damage gradients at nodes, the classical Z2 technique is used, and a new effective and computationally convenient iterative strategy is implemented to guarantee convergence of the staggered scheme. Four examples are presented in order to assess the suitability of the present model by using both AT1 and AT2 regularization models. Results show the desired effect of limiting crack propagation to prevailing tensile states, as well as of recovering the initial intact stiffness upon load reversal, even when two of the most common energy splits fail.

Oxygen-induced decomposition of the body-centered cubic HfNbTaTiZr high-entropy alloy

The factors affecting the stability of an initially single-phase body-centered cubic (BCC) HfNbTaTiZr solid solution are investigated by heat treatments at 900 °C up to 1000 h in different atmospheres. The BCC solid solution is stable but oxygen (O) contamination originating from the aging atmosphere leads to the co-precipitation of an O-enriched hexagonal close-packed (HCP) Hf-Zr-rich phase and a second Nb-Ta-rich BCC phase. As O from the atmosphere diffuses from the surface to the core of the sample at a much faster rate along grain boundaries than through the grain interior, this results in an inhomogeneous distribution of precipitates, which formed at grain boundaries by a monotectoid reaction. In contrast, coincidence-site-lattice boundaries are not affected by this transformation owing to their lower diffusivities and lower capacity for heterogeneous nucleation. Similar phases were observed in a HfNbTaTiZr enriched with 3 at.% O, thus confirming the role of oxygen in phase stability.In both alloys, the orientation relationship (OR) between the HCP and BCC phases is predominantly of Burgers type but our analyses further revealed the presence of two minor ORs. More interestingly, both alloys exhibit rod-shaped HCP precipitates in contrast to the plate-shaped precipitates observed in conventional Ti and Zr-based alloys. This result can be rationalized by calculations of elastic strain energy density, which suggest that the rods should grow along 〈100〉BCC to minimize the elastic strain energy. Deviations of up to 20° from this theoretical direction were observed and are likely related to relaxation by misfit dislocations or by plastic deformation.

A multiscale modeling to determine in vitro mechanical responses of different cells at the cell-substrate interface under fluid perfusion.

Investigating the influence of different cellular mechanical and physical properties on cells in vitro is important for assessing cellular activities like differentiation, proliferation, and migration. Evaluating the mechanical response of the cells lodged on a scaffold due to variations in substrate roughness, substrate elasticity, fluid flow, and the shapes of the cells is the main goal of the study. In this comprehensive analysis, a combination of the fluid structure interaction method and the submodeled finite element technique was employed to anticipate the mechanical responses across various cells at the interface between cells and the substrate. Fluid inlet velocity, substrate roughness, and substrate material were varied in this analysis. Different cell shapes were considered along with various components such as cell membrane, cytoplasm, nucleus, and cytoskeletons. This analysis shows the effect of these individual parameters on the elastic strain and strain energy density of cells at the cell-substrate interface. The results highlight that substrate roughness has a more significant impact on the mechanical response of cells at the interface than substrate elasticity. However, effect of the substrate elasticity becomes crucial for extremely soft substrate materials. The results of this research can be applied to identify the optimal parameters for fluid flow and create a suitable condition for cell culture.

Mechanical behaviors and progressive fracture processes of dry basalt after Martian cryogenic freeze-thaw cycles

The steep Martian terrain features numerous notches and alcoves that could potentially be adapted as temporary shelters. By sealing their entrances and modifying the interior space, these alcoves can be transformed into habitable structures. Given their prolonged exposure to Martian temperature fluctuations, it is crucial to understand the mechanical properties of alcove rocks to design appropriate shelter parameters. This study investigates the mechanical behavior of basalt specimens subjected to Martian cryogenic freeze-thaw (F-T) cycles (−143∼35 °C) and subsequent uniaxial cyclic loading tests. The test results indicate that the longitudinal wave velocity, peak stress, and elastic modulus of basalt specimens degrade to varying degrees with an increasing number of F-T cycles. At higher stress levels, the ratio of dissipated energy density to elastic energy density in basalt specimens subjected to different F-T treatments converges to approximately 0.25. During the uniaxial cyclic loading tests, both cumulative acoustic emission (AE) counts and average maximum principal strains of the specimens increase stepwise. For specimens after F-T treatment, the spectral-domain characteristics can be classified into three frequency bands, with the middle and high frequencies predominating. The application of the Gaussian Mixture Model (GMM) clustering algorithm enhances the analysis of AE results, facilitating the identification of tensile and shear cracks. As the number of F-T cycles increases, the percentage of shear cracks in basalt specimens decreases from 76.5% (0 cycle) to 53.5% (7 cycles).

Energy and deformation field evolution characteristics of layered cemented paste backfill under cyclic loading and unloading

Due to the mining environment, layered cemented paste backfill (LCPB) is usually subjected to the cyclic pressure. Therefore, it is of great significance to investigate the mechanical behavior, failure mode and energy evolution characteristics. LCPB samples were prepared and subjected to constant lower limit cyclic loading and unloading test. The digital image correlation (DIC) technology was applied to monitor deformation field of LCPB. The obtained results indicate that: (1) The area of the hysteresis loops increases sequentially with increasing unloading stress level, while the spacing between the hysteresis loops gradually increases, showing the typical “dense-sparse” two-stage variation law. (2) Energy density and the unloading stress level is in agreement with the power function. The limiting elastic energy density (uemax) shows a linear decrease with the increase of h. The exponential function can well characterize the quantitative relationship between uemax and c/t ratio. (3) The damage mode of LCPB gradually tends to x-shape shear-dominated damage with the increase of h. The damage mode of LCPB is gradually dominated by shear-type damage with the decrease of c/t ratio. Ultimately, the findings of this work can provide a reference for the design of LCPB.

Experimental study on failure mechanism of soft rock-coal bodies in abandoned mines under cyclic dynamic loading

During the construction and operation of a pumped storage power station in an abandoned mine, the soft rock-coal body structure, comprising the roof and the residual coal pillars, encounters a complex stress environment characterized by cyclic loads. The study of its failure mechanism under cyclic dynamic loading holds significant theoretical and practical importance to stay the safety and stability of the abandoned mine pumped storage power station. In this paper, we take “roof-residual coal pillar” soft rock-coal combinations with different percentages of rock as the research object, and study their mechanical properties, failure mechanism, energy evolution characteristics and acoustic emission distribution characteristics through cyclic dynamic loading experiments. The results of the experiment indicate that: (1) Both weak cyclic dynamic loading and high rock percentage enhance the deformation resistance of soft rock-coal combinations. Under low-disturbance horizontal cyclic loading, its peak strength and modulus of elasticity increase with increasing rock percentage. (2) Under low-disturbance horizontal cyclic loading, an increasing trend is observed in the average total strain energy density, dissipation energy density, and elastic energy density of the combinations as the percentage of rock increases. (3) Under low-disturbance horizontal cyclic loading, as the percentage of rock increases in the soft rock-coal combinations, the degree of failure in the rock body part progressively intensifies, while the destruction of the coal portion progressively decreases. (4) The large number of acoustic emission signals are generated at the instant of destabilization and destruction of the coal-rock combinations, mainly dominated by the signals generated by the destruction of the coal body. Acoustic emission counts and absolute energy at key point N2 decrease as the percentage of rock increases. The b value is primarily distributed in the cyclic dynamic loading stage and the failure stage, both displaying zones of sudden increase and sudden decrease in b value.

Singularity distance computations for 3-RPR manipulators using intrinsic metrics

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization.

Long-term elasto-static compressive loading drives rejuvenation of a metallic glass

The current work focuses on the impact of mechanical protocol which can lead to either rejuvenation or relaxation of metallic glasses (MGs) through elasto-static compressive loading (ECL) treatment. Here we demonstrate that, even after imposing a long-term ECL process, the Zr52.5Cu17.9Ni14.6Al10Ti5 MG subjected to an elasto-static stress of 85% of the yield stress at room temperature shows varying degrees of rejuvenation without any relaxation. The MG undergoes an increasing trend in structural rejuvenation with an increased stored energy and a more disordered structure. Nanoindentation results reveal that the variation of mechanical responses emerges from the complementary effects of the thermal activation process and the structural heterogeneity of glassy phase. The spital heterogeneities can effectively reduce the activation barriers of the shear transformation zones (STZs) and widen their distributions, which will perturb the shear-banding processes from single motion to collective movements, leading to shear band multiplication/branching and an increase in the elastic energy density cut-off. As a result, the plasticity is significantly enhanced and the yield strength is decreased. These findings shed light on that ECL process is an effective way to enhance the structural heterogeneity and the level of rejuvenation of a monolithic MG, which may allow the design of MGs with desirable mechanical properties.

Exploring the structural mechanics of titanium nickel solid alloy using COMSOL Multiphysics: A Poisson equation and continuity equation perspective

This study investigates the structural mechanics of titanium-nickel (Ti-Ni) alloy thin film using computational modelling through COMSOL Multiphysics based on Poisson’s equation and continuity equation for stress check by considering its linear elastic, conservation of charge and providing insight into nanomaterial deformation. In the COMSOL environment the parameters for titanium nickel (Ti-Ni) are embedded in the COMSOL Simulink interface. The Thin film layer was designed by defining the layer geometry of the size and shape of the layer with a width of 500 μm, depth of 200 μm and height of 3 μm subjected to boundary conditions such as von – mises stress, surface temperature, iso-surface temperature, multi-slice electric potential, displacement component, surface elastic strain energy density and total enthalpy. The results displayed a trend that is, as the surface temperature increases there will be an increase in the current densities associated with high electrical conduction. On the same note, the designed thin film layer will pass the percolation threshold. The results of surface elastic strain energy density and total enthalpy imply that the designed thin film layer is effective and efficient as a structural pseudopotential device (photodiode).

Study on rock energy evolution and constitutive model under water–rock interaction

To explore the degradation in rock mechanical properties due to water–rock interactions and formulate a nonlinear damage constitutive model for the entire stress–strain curve of rocks, experiments involving saturated, dry-wet cycling, and uniaxial compression tests were conducted on granite and siltstone. Based on energy theory, the energy evolution of the rock throughout the testing process was scrutinized. The range of the compaction stage was identified through the analysis of the dissipation energy change curve. A nonlinear damage constitutive model for rocks subjected to water-rock interactions was then devised, drawing on concepts from statistical damage mechanics. The findings indicated a progressive reduction in uniaxial compressive strength and elastic modulus due to water–rock interactions, while the Poisson’s ratio was observed to increase. The energy density curve under these conditions delineated four distinct phases: compaction energy dissipation, linear energy accumulation, pre-peak gradient increase, and post-peak sudden change. The introduction of the unit elastic strain energy density metric underscored the deteriorating effects from an energy standpoint. A nonlinear damage constitutive model, incorporating the compaction stage based on a coupled damage variable, was formulated. The predictions of this model closely matched the empirical data, thereby affirming its validity. This model provides an enhanced depiction of rock deformation and failure mechanisms under the influence of water–rock interactions.

Creep Energy Evolution of Red-Bed Soft Rocks in South China under Chemical-Stress-Seepage Coupling

The red-bed soft rocks in South China have obvious creep characteristics and are prone to engineering geological disasters such as landslide and foundation settlement under the action of rainfall, groundwater, and load. In order to reveal its creep characteristics and mechanism under complex conditions, a step-loading creep test was carried out under chemical-stress-seepage coupling, and the energy evolution law of the whole creep process was analyzed based on linear energy storage and energy dissipation theory. The results also show that the acid chemical solution has the greatest influence on the triaxial strength and creep strength, and the creep damage and energy evolution of red-bed soft rock are universal. The creep damage and total strain increase with the increase of acidity, the decrease of confining pressure, and the increase of seepage pressure. The evolution law of creep damage shows the characteristics of slow-acceleration-rapid growth, and with the increase of load level, it has obvious transfer and accumulation. After entering the constant velocity creep stage, the damage rate begins to accelerate. The proportion of instantaneous strain and creep strain in the total strain increment is about 50%, and confining pressure has little influence on their respective proportions. The instantaneous strain is more sensitive to the acidity of the chemical solution, and the proportion of creep strain increases gradually with the increase of seepage pressure. The relationship between elastic energy density and total energy density is linear. The elastic energy density and dissipated energy density in the loading stage and creep stage all increase nonlinearly with loading time. The density of dissipated energy in the creep phase is lower than that in the loading phase, but the opposite is true in the higher stress phase, and the law of energy dissipation can explain the hardening damage effect in the creep process of soft rock samples. The research results provide a new perspective for us to reveal the mechanical properties and failure mechanism of red-bed soft rocks and provide an important theoretical basis for predicting and evaluating the creep instability and long-term stability of such rocks.

On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discussing the evolution of the elastic domains in the strain and stress spaces as damage evolves, we highlight the links between the nucleation and propagation conditions and the modulation of the elastic energy with the phase-field variable. In light of the identified requirements, we review some of the existing energy decompositions, showcasing their merits and limitations, and conclude that none of them is able to fulfil all requirements. As a partial remedy to this outcome, we propose a new energy decomposition, denoted as star-convex model, which involves a minimal modification of the volumetric-deviatoric decomposition. Predictions of the star-convex model are compared with those of the existing models with different numerical tests encompassing both nucleation and propagation.

Fatigue characteristics and energy evolution analysis of red sandstone under the coupling of freeze–thaw and cyclic loading

In order to improve the safety of rock mass engineering in cold regions, this paper discusses the fatigue characteristics and energy evolution of red sandstone under the coupling of freeze–thaw and cyclic loading. Firstly, the correlation between the fatigue life, failure stress, and the number of freeze–thaw cycles of the red sandstone samples was calculated using the Pearson correlation coefficient method. Secondly, the energy evolution of the samples was analyzed using the image integration method and the least squares method. Finally, the damage mechanism of the samples under the coupled effect of freeze–thaw and cyclic loading was discussed using a conceptualized model. The results show that freeze–thaw cycles are negatively correlated with fatigue life and failure stress. The increase in freeze–thaw and fatigue damage leads to an increase in ductility and a decrease in the stiffness of the samples. The total energy density, elastic energy density, and dissipated energy density of the samples all show a quadratic polynomial relationship with the stress level. The energy storage and dissipation coefficients are negatively and positively correlated with the number of freeze–thaw cycles, respectively.

Effect of thermal mismatch strain on the self-assembly of nanorods in YBCO/NRs nanocomposite films

This study presents a thermoelastic model to investigate the impact of thermal mismatch strain, resulting from distinct thermal expansion coefficients between the film matrix and nanorods, on the self-assembly of nanorods in YBCO/NRs nanocomposite films. An accurate understanding of thermal mismatch strain generated during nanocomposite film preparation is vital for controlling the self-assembly of artificial nanorods in such films. In this model, we calculated the total elastic energy density of the film matrix and nanorods, considering both lattice and thermal mismatch strains, to analyze the effect of thermal strain on the diameter of c-oriented nanorods in YBCO nanocomposite films. The calculated nanorod diameter values match experimentally measured values for BaZrO3 and BaSnO3 nanorods. Furthermore, compared to results that only consider lattice mismatch strain, the proposed model yields closer calculation values to experimental measurements by incorporating the impact of thermal mismatch strain. The thermal mismatch strain also affects the spacing between two adjacent nanorods, having the most profound effect on BSO nanorods compared to BZO and BHO nanorods. These findings contribute to understanding the mechanisms involved in nanorod self-assembly and provide insights to optimize the design of YBCO/NRs nanocomposite films.