Strain Energy Research Articles

Non-linear analysis of anisotropic coated fabric utilizing Variational Asymptotic Method

The dimensional reduction of a hyperelastic coated fabric from 3D to 2D is accomplished asymptotically using the Variational Asymptotic Method (VAM). This method integrates constrained calculus of variations and asymptotics. The VAM bifurcates the analysis into two: a 1D through-the-thickness analysis and a 2D reference surface analysis. The 1D analysis leads to the derivation of an asymptotically correct 3D warping functions and a 2D non-linear constitutive law. The 2D non-linear reference surface analysis utilizes the derived 2D non-linear constitutive law to obtain 2D displacements and strains through the 2D non-linear FEA. The classification of 3D strain energy density into distinct orders is enabled by introducing two intrinsic small parameters: 1) a geometric small parameter denoted by the ratio of thickness to characteristic length (h/l≪1), and 2) a physical small parameter that ensures the largest component of 3D strain is restricted to 20 percent, which is less than 1. The model takes into account both geometric and material nonlinearities. The strain energy function, which describes the anisotropic characteristics of the coated fabric has contributions from the strain energies of fiber, matrix, and fiber-fiber interaction. The findings of the study include analytically derived 3D warping functions, a 2D nonlinear constitutive law, and the prediction of warp and weft stresses and strain for a biaxial loaded tensile specimen. These findings align with the experimental outcomes.

Effect of extrusion conditions on microstructural characteristics and mechanical properties of SEN6 magnesium alloy

This study investigates the influence of extrusion conditions on the microstructure and tensile properties of Mg–6Al–0.3Mn–0.3Ca–0.2Y (SEN6) alloy, which is processed at various extrusion temperatures and ratios. An increase in both the extrusion temperature and ratio facilitates dynamic recrystallization during hot extrusion by enhancing thermal and strain energies, increasing the area fraction of recrystallized grains within the extruded material. As either the extrusion temperature or ratio increases, lattice rotation within the recrystallized grains intensifies, resulting in a stronger basal texture in the extruded material. During extrusion, the undissolved Al2Y, Al8Mn4Y, and Al2Ca particles within the billets are partially fragmented and realigned along the extrusion direction. However, the quantity, size, and dispersion of these particles are largely unaffected by variations in extrusion temperature and ratio. The tensile yield strength of the extruded material shows a positive correlation with increasing extrusion temperature and ratio, predominantly due to enhanced texture hardening. Specifically, the tensile yield strength of the material extruded at 400 °C with an extrusion ratio of 25 is 23.2 MPa higher than that of the material extruded at 300 °C with an extrusion ratio of 10. Despite the presence of non-recrystallized grains in materials extruded at the lower extrusion ratio of 10, compression and double twins form uniformly throughout the material during tension, as the widths of non-recrystallized grains are similar to those of recrystallized grains. Consequently, all extruded materials exhibit high tensile elongations, exceeding 20%.

Bending and twisting rigidities of 2D materials

A novel torus-based surface parametrization is proposed and used to evaluate the bending and twisting rigidities of 2D materials of arbitrary symmetries. A generalized constitutive model is used to capture the effects of simultaneous bending and twisting curvatures on the strain energy density of a 2D material. The flexural rigidities are evaluated by subjecting the 2D material specimens to different curvature states, including simultaneous bending and twisting, evaluating the corresponding strain energy densities and curve fitting the constitutive model to the data. The proposed methodology is independent of the method used to evaluate the strain energy of the 2D materials, e.g., density functional theory or classical interatomic potentials. The flexural rigidities of graphene, molybdenum disulfide, and black phosphorus are evaluated and compared to those published in the literature. The directional dependence of the flexural rigidities is investigated. Dimensionless coefficients are presented for quantifying the bidirectional bending and bending-twisting coupling effects in 2D materials. Lastly, this study includes an analysis of the flexural anisotropy and flexure-thickness distension coefficients of black phosphorus.

Visco-Hyperelastic Constitutive Modeling for High-Damping Rubber Materials During Combined Quasi-Static Compression–Cyclic Shear Deformation Process

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Tensile fracture of blunt V-notches in rectangular plates additively manufactured from Acrylonitrile Butadiene Styrene with different raster angles

In this research, the tensile fracture behavior of rectangular specimens made of Acrylonitrile Butadiene Styrene (ABS) containing double-edge blunt V-notches is evaluated. The specimens are printed using the fused deposition modeling (FDM) technique with five different raster angles and three different notch tip radii. To simplify the complex calculations related to the anisotropic behavior of these specimens, the Virtual Isotropic Material Concept (VIMC) is used in conjunction with three well-known linear elastic notch fracture mechanics (LENFM) criteria, i.e., the mean stress (MS), point stress (PS), and averaged strain energy density (ASED) criteria as well as the extended finite element method (XFEM). It is shown that all these fracture models can predict the load-bearing capacity of the 3D-printed notched specimens by acceptable accuracies, where the best predictions belong to the MS criterion with the critical distance obtained directly from the finite element method, and the least accurate predictions belong to the XFEM method.

A Constitutive Framework for Modeling of the Visco-Hyperelastic Materials: Application to the Mechanical Behavior of Cylinders Made of the Rate-Dependent Elastomers

In this paper, the formulation of the mechanical behavior of visco-hyperelastic materials is developed in a thermo-dynamically compatible framework. In this viewpoint, the stress power is assumed to be equal to the sum of the reversible (elastic) energy rate stored and the irreversible energy rate dissipated. Based on this assumption and using the second law of thermodynamics, the constitutive modeling framework is obtained for the rate-dependent materials. Inspired by the constitutive law of a Newtonian fluid and also, models of density strain energy of an elastic material, a framework for the dissipative energy rate related to the irreversible part is proposed for viscoelastic materials. In this framework, the reversible energy part is considered a function of the right Cauchy–Green deformation tensor, and the irreversible energy part is considered a function of the same tensor’s rate. To evaluate the proposed model for nonlinear viscoelastic behavior modeling, the results of tests performed on non-compressible materials at different rates are used, and the proposed model provided acceptable results. Finally, the proposed model is implemented to find a closed-form analytical solution for mechanical behavior modeling of the thick-walled cylindrical shells made of visco-hyperelastic materials, also, stress and stability analyses are assessed for these types of cylinders. To achieve this goal, the effect of pre-stretch on the stability of cylinders has been substantially investigated. Furthermore, based on the proposed model, the effect of the parameters such as the wall thickness and behavior of the material used in the cylinder are examined on the stability of open-ended and closed-ended cylinders.

Energy characteristics and damage constitutive model of pre‐cracked granite with or without filling materials

AbstractIn order to study the effects of pre‐crack inclination angle and filling material on granite, uniaxial compression tests were performed on pre‐cracked granite specimens without filling material and specimens filled with epoxy resin, gypsum, and organosilicon. Results indicate that the peak strength and apparent stiffness of pre‐cracked granite without filling materials increase with increasing pre‐crack inclination angle. The strain energy absorbed by the filling material improves the energy storage limit, inhibits the energy release effect when microcracks are connected to each other, and ultimately improves the compressive resistance and energy storage capacity. The strengthening effect of gypsum and epoxy resin is relatively better, which is more significant when the pre‐crack inclination angle equals 0°–60°. Finally, a damage constitutive model is established, which is in good agreement with the test and numerical simulation data. The results can provide reference for the research of grouting reinforcement of rock masses.

Correlation analysis of evaluation methods and indicators for low-temperature performance of epoxy asphalt

Selecting an epoxy asphalt (EA) material with excellent low-temperature performance is a challenging task due to that the traditional test methods and indicators for thermoplastic asphalt may not be appropriate for thermosetting EA, and the binders’ low-temperature performance does not always match that of mixtures. The primary purpose of this study is to determine the appropriate low-temperature evaluation methods and indexes of EA materials and the correlation between the low-temperature performance of EA binder and EA mixture (EAM). First, the low-temperature performance of four EA binders was analyzed using tensile, bending beam rheometer (BBR), and dynamic mechanical analysis (DMA) tests. Then, the low-temperature cracking resistance of the EAMs was evaluated through beam bending and semi-circular bend (SCB) tests. Finally, Pearson correlation analysis was performed separately on the low-temperature indicators of EA binders and EAMs. Moreover, the Grey correlation analysis was employed to determine the correlation between the low-temperature properties of the two. The results showed that the correlation between fracture elongation (ε) and other test indicators was not significant. The multiple indicators of the BBR, such as viscoelastic parameters, viscous deformation ratio (JR), along with the glass transition temperature (Tg) of the DMA, showed consistent evaluation results for the low-temperature performance of the four EA binders. However, the correlation of these indicators with the cracking resistance of the EAM was notably different. The flexural strain energy density of the beam bending test and SCB fracture energies demonstrated excellent correlation. The grey correlation analysis results of these two indicators as reference series were highly similar. Specifically, the novel JR in this study and the Tg exhibited a better correlation with the low-temperature cracking resistance of EAM. These two indicators were preferentially recommended for the evaluation of the low-temperature performance of EA and the prediction of the low-temperature cracking resistance of EAM.

Quantitative evaluation of PI-RDL interfacial delamination in fan-out wafer-level packaging during unbiased highly accelerated stress test

Fan-Out Wafer-Level Packaging (FOWLP) is increasingly utilized for its superior performance, facilitated by flexible heterogeneous integration. Despite its advantages, the interface between Polyimide (PI) and Redistribution Layers (RDL) is susceptible to significant vulnerabilities, notably delamination. This study introduces a quantitative method to assess PI-RDL interface delamination in FOWLP under unbiased Highly Accelerated Stress Tests (uHAST), a standard approach in accelerated reliability testing characterized by high humidity and temperature. Central to our methodology is the development of a time–temperature-moisture equivalence-based creep constitutive model, which significantly enhances our understanding of the impact of temperature and moisture on creep behavior. Furthermore, we have developed an innovative “open moisture absorption” method for preparing samples for four-point bending tests, enabling precise quantification of the time-induced degradation of PI-RDL interfacial fracture toughness under saturated conditions. Our comprehensive finite element model integrates moisture diffusion, the newly developed creep constitutive model, and solid mechanics to accurately simulate the accumulation of strain energy release rate during uHAST, facilitating precise predictions of delamination initiation times in FOWLP. Experimental validation under two uHAST conditions—130 °C/85 %RH for 96 h and 110 °C/85 %RH for 264 h—demonstrates the effectiveness of our approach, predicting the delamination initiation time at the PI-RDL interface with over 80 % accuracy. This predictive methodology is pivotal in enhancing the structural optimization and extending the service life of FOWLP in varied environmental conditions.

Preparation and thermal performance testing of diatomite-modified asphalt pavements

In this study, the efficacy of diatomite as a modifier in asphalt pavements is thoroughly examined. The research primarily focuses on evaluating the thermal performance of asphalt mixtures modified with varying diatomite contents. Key findings indicate significant improvements in the rutting resistance, with the dynamic stability of the 15% diatomite mixture being 3.4 times higher than the control mixture. Additionally, the 10% diatomite mixture shows a 20% increase in bending strain energy density, enhancing low-temperature performance. However, a diatomite content beyond 10% tends to diminish these benefits. Fatigue life at 10% diatomite content is improved by 18%–24%, demonstrating the material’s potential in extending pavement longevity. Water stability tests also reveal notable improvements, with the 15% diatomite mixture exhibiting a 15% increase in tensile strength ratio post-water conditioning. These results collectively establish diatomite as a cost-effective and efficient modifier for asphalt pavements, promoting enhanced durability and performance.

A discrete-to-continuum model for the human cornea with application to keratoconus.

We introduce a discrete mathematical model for the mechanical behaviour of a planar slice of human corneal tissue, in equilibrium under the action of physiological intraocular pressure (IOP). The model considers a regular (two-dimensional) network of structural elements mimicking a discrete number of parallel collagen lamellae connected by proteoglycan-based chemical bonds (crosslinks). Since the thickness of each collagen lamella is small compared to the overall corneal thickness, we upscale the discrete force balance into a continuum system of partial differential equations and deduce the corresponding macroscopic stress tensor and strain energy function for the micro-structured corneal tissue. We demonstrate that, for physiological values of the IOP, the predictions of the discrete model converge to those of the continuum model. We use the continuum model to simulate the progression of the degenerative disease known as keratoconus, characterized by a localized bulging of the corneal shell. We assign a spatial distribution of damage (i.e. reduction of the stiffness) to the mechanical properties of the structural elements and predict the resulting macroscopic shape of the cornea, showing that a large reduction in the element stiffness results in substantial corneal thinning and a significant increase in the curvature of both the anterior and posterior surfaces.

Structural damage identification based on dual sensitivity analysis from optimal sensor placement

The structural damage identification (SDI) methods based on incomplete modal information can avoid the extension for unmeasured degrees of freedom, but the absence of essential damage information often leads to the failure of SDI. To address this problem, a novel SDI method based on dual sensitivity analysis and optimal sensors placement technique is proposed in this study. Firstly, in the optimal sensor placement technique, an improved eigenvector sensitivity method combined with weighted modal kinetic energy is proposed, which enables the acquisition of eigenvector information related to damage sensitivity, and incorporates it into the modal strain energy sensitivity matrix to obtain the dual sensitivity analysis matrix. Then, the sparsity of structural damage is considered, and the L1 sparse regularization is selected and introduced into the dual sensitivity analysis damage equation for better SDI results. Finally, to assess the effectiveness of the proposed method, a series of numerical simulations and experimental verifications were carried out under different structural damage scenarios. The results indicate that the proposed method can efficiently localize and quantify the structural damage with minimal modal information in one single step.

The role of microfaceting in heteroepitaxial interfaces

We investigate the effects of microfaceting on the orientation and interface structure of fcc films on fcc substrates by molecular dynamics simulations and atomic resolution electron microscopy. For (110) substrates, the simulations reveal a misorientation between film and substrate lattices that undergoes a sudden change when interfacial facets approach the “magic size”, where a lattice dislocation minimizes the strain energy by compensating the misfit between film and substrate. For a large misfit, the angle of misorientation varies between zero and several degrees, depending on the size and sequence of microfacets. Experimental observations of interfaces in Ag/Ni near the Ni(110) orientation uncover the presence of magic-size {111} facets and show how microfaceting controls the partitioning of misfit dislocations. For facets smaller than the magic size, misfit may be compensated by partial rather than perfect dislocations. In vicinal {hhl} interfaces, made of {111} terraces separated by single-layer steps, a partial dislocation per terrace leads to film growth in the heterotwin orientation. This concept explains a range of previous results on interface structures in a variety of heteroepitaxial systems.

Equivalent stress concept to account for the effect of local cyclic stress ratio on transverse cracking in tension–tension fatigue

Presented test results on transverse cracking in cross-ply laminates upon tension–tension cyclic loading show that the increase of crack density depends not only on the maximum transverse stress in the cycle but also on the local cyclic stress ratio RTloc in the analyzed layer. To include the effect of the RTloc in the model with statistical failure stress distribution for crack initiation (based on Weibull distribution) adapted for fatigue, an equivalent stress is introduced in a similar manner as the equivalent strain energy release rate has been used for delamination crack propagation. The equivalent stress in the layer is defined as a power function of the maximum stress and the stress ratio in the layer. It was found, testing laminates with two different fiber contents that higher the local stress ratio in 90-layer, higher the transverse cracking resistance. Transverse crack density simulation using the developed equivalent stress model has been validated against test results.

The Integration of Mechanical Energy Absorbers into Rollover Protective Structures to Improve the Safety of Agricultural Tractors in the Event of Rollover

The combination of safety belts and rollover protective structures (ROPSs) is key in improving the safety of agricultural tractors in the event of rollover. However, we also have the opportunity to enhance the security provided by each ROPS; one such example is the combination of this safety device with adequate mechanical energy absorbers (MEAs). Inexpensive disc-shaped MEAs can be included in the anchoring points of a ROPS onto the chassis of a tractor. Three configurations of ROPS combined with MEAs were tested during the application of loads that simulated the effects of side rollover in the vehicle. The tested configurations included a blank MEA as a reference case alongside a single MEA and a stack assembly containing both elements. The results of the tests show that both the deformation of the ROPS itself and the strain energy are larger in the case of blank MEAs; thus, there is also a risk that the clearance zone will be infringed upon and that the protective structure will collapse. We can conclude that the implementation of an appropriate MEA in ROPS reduces the deformation of the ROPS itself and its strain energy in cases of vehicle rollover; hence, the safety provided by such protection systems may be improved at a low cost.

Mechanical responses of chemically corroded sandstone under cyclic disturbance: Insights from fatigue properties and macro-micro fracturing mechanism

Engineering rock masses are likely to be subjected to coupled chemical corrosion and cyclic load disturbance, deteriorating the internal structure of rocks and further threatening the long-term stability of engineering structures. In this study, to investigate the combined effect of cyclic loading and chemical corrosion, the cyclic uniaxial compression tests are conducted on sandstone specimens chemically corroded by different corrosion times (0, 20, 40, 60 days) and pH values (1, 3, 7, 11, 13). Our results indicate that chemically corroded sandstone specimens featured by an increase in irreversible strain and hysteresis energy density compared with natural rocks. A coupling damage variable is proposed to reflect the initial damage induced by chemical corrosion and the mechanical damage induced by cyclic disturbance, and the accumulation rate of coupled chemical-fatigue damage variables significantly increases with increasing corrosion time or enhancing acidity and alkalinity. Fatigue failure modes of tested specimens transform from splitting failure to unidirectional shear failure as corrosion time increases, and chemically corroded specimens are featured by ductile failure characteristics compared to natural state. Moreover, X-ray diffraction (XRD) and Scanning Electron Microscope (SEM) are employed to analyze the micro-fracturing mechanism of sandstone. Cement minerals dominate the reactions in acidic solutions, whereas quartz takes the lead in alkaline conditions. Due to pronounced influence of cement minerals on the macroscopic mechanical properties of sandstone, acid corrosion specimens tend to experience more severe damage. Micro-cracks inside rocks under cyclic loading have sufficient time to pass through weakly bonded particles, and chemical reactions markedly weaken inter-particle bonding degree, and thus the fatigue fracture of chemically corroded specimens predominantly exhibits intergranular (IG) fracture.

Deep Learning in Rockburst Intensity Level Prediction: Performance Evaluation and Comparison of the NGO-CNN-BiGRU-Attention Model

Rockburst is an extremely hazardous geological disaster. In order to accurately predict the hazardous degree of rockbursts, this paper proposes eight new classification models for predicting the intensity level of rockbursts based on intelligent optimisation algorithms and deep learning techniques and collects 287 sets of real rockburst data to form a sample database, in which six quantitative indicators are selected as feature parameters. In order to validate the effectiveness of the constructed eight machine learning prediction models, the study selected Accuracy, Precision, Recall and F1 Score to evaluate the prediction performance of each model. The results show that the NGO-CNN-BiGRU-Attention model has the best prediction performance, with an accuracy of 0.98. Subsequently, engineering validation of the model is carried out using eight sets of real rockburst data from Daxiangling Tunnel, and the results show that the model has a strong generalisation ability and can satisfy the relevant engineering applications. In addition, this paper also uses SHAP technology to quantify the impact of different factors on the rockburst intensity level and found that the elastic strain energy index and stress ratio have the greatest impact on the rockburst intensity level.

A numerical study to discriminate between the hydrostatic and deviatoric strain energies impact on cells activity during bone remodelling

Bone remodelling is regulated by specific cellular activities. In particular, the osteocytes are able to transform a mechanical signal into a chemical one and transmit it to the osteoclasts and osteoblasts cells. Additionally, osteocytes are more sensitive to fluid shear than pressure stress. Thus, the mechanical environment, especially the type of external loads applied to the bone structure, may highly influence its evolution over time. In this paper we propose a strain energy based model in which we distinguish between the deviatoric and hydrostatic energy contributions. We study here simple loads and geometries for which the strain energy distribution is homogeneous. From these two cases, we are able to show that when considering more complex structures such as the bone microstructure where the energy distribution is heterogeneous, it is necessary to discriminate between the deviatoric and hydrostatic energy parts.

Fractal Characteristics and Energy Evolution Analysis of Rocks under True Triaxial Unloading Conditions

To investigate the mechanical properties and energy evolution laws of rocks under true triaxial unloading conditions, a study was conducted using a true triaxial rock testing system on three different types of rocks: coal, sandy mudstone, and siltstone. The study examined the mechanical behavior, failure patterns, and fractal dimensions of these rocks under true triaxial unloading conditions. The tests revealed significant variations in stress–strain curves and failure patterns among the different rock types. Observation indicated that rocks with lower peak strength exhibited higher fractal dimensions and increased fragmentation upon failure. Subsequently, based on the experimental data of siltstone, the impact of the unloading rate and particle size distribution on the energy evolution under true triaxial single-sided unloading paths was further investigated using the three-dimensional particle flow software PFC3D 6.0, revealing the micro-mechanisms of rock energy evolution. The study revealed that when the initial stress unloading level was low, the total energy and strain energy at the peak strength exhibited a strong linear relationship with the unloading rate. Before the stress peak, the dissipative energy was mainly composed of frictional energy. After the stress peak, the dissipative energy consisted of frictional energy, damping energy, and kinetic energy. The heterogeneity of rock significantly affected the distribution of dissipative energy, with an increase in rock heterogeneity leading to a decrease in frictional energy and an increase in kinetic energy.

The influence of invariant horizontal projection area constraint on shape optimization of free-form surface structure with openings

Free-form curved surface shell roof is favored by architects, but the openings for daylighting will decline its mechanical properties. In the shell shape optimization under the vertical load, when the outer contour of the openings or the shell changes, the total load of the structure also changes with the changes in its horizontal projection area. This contradicts the premise of structural optimization, that is, to maintain the principle of unchanging total loads. To solve this problem, the invariant horizontal projection area of the shell was taken as constraints in the shell shape optimization in which the objective function was structural strain energy and optimization design variables were internal and external boundary points of the curved surfaces with openings, and the key points on the curved surface which were design variables of the overall shape of the curved surface. The numerical examples verified the validity of the method, and the influence of the invariant horizontal projection area constraint of the curved surface with openings on the optimization results of the minimum strain energy, optimization with the frequency constraint, and the initial surface with the different support distribution were discussed. The results of the numerical examples showed that the invariant horizontal projection area constraint has a greater impact on the results of the curved surface with opening optimization, and its influence should be considered in the engineering design.