Equivalent Strain Energy Research Articles

A non-classical computational method for modelling functionally graded porous planar media using micropolar theory

The current study proposes a computational-based method to employ the non-classical micropolar continuum for modelling plates with in-plane functionally graded porosities. Initially, a homogenisation method is developed to derive the micropolar parameters of porous heterogenous plates based on strain energy equivalence in various designed deformations simulated via finite element analysis. The modelling procedure is further augmented to accommodate structures with functionally graded porosities. The established method offers an effective framework for studying the mechanical behaviour of porous plates with various porosity distributions and a wide range of aspect ratios. Results indicate that the micropolar theory-based modelling surpasses traditional Cauchy theory in accurately predicting the stiffness and displacement distribution of the FG porous structures. The novelty of this study lies in the integration of micropolar theory with the homogenisation of graded porosity patterns, offering enhanced predictions for materials with microstructural features. Additionally, a custom finite element formulation is developed in COMSOL to implement micropolar elasticity, significantly improving the computational efficiency to account for complex geometry, loading, and boundary conditions. To show the applicability of the method, the modelling is used to design a dental implant with its functional property mimicking that of the natural bone to avoid stress-shielding while ensuring proper occlusivity.

Study on the equivalent model considering in-plane and out-of-plane mechanical properties for a curved sandwich structure with square honeycomb core

The curved honeycomb sandwich structure has larger surface area, effectively reducing the number of fasteners and parts of civil airplane fuselages, engine cowls, and ship’s hulls. Its equivalent model building can significantly shorten the simulation analysis time and improve the efficiency of optimal design and performance prediction, but the reliability and precision of the model are closely related to the calculation accuracy of equivalent elastic parameters. In the present study, an equivalent model considering the in-plane and out-of-plane mechanical properties of the core is developed for a curved square honeycomb sandwich structure. The in-plane and out-of-plane equivalent elastic parameters of the honeycomb core composed of hollow quadrangular frustum cells with symmetrical rectangular sides (i.e. square cell) are derived by analytical method, equivalent strain energy principle, and equivalent stiffness principle. And then combined with the sandwich panel theory, an equivalent model of the curved sandwich structure is established. The three-dimensional (3D) geometric models of planar honeycomb cores with hollow quadriprism cells are built by SolidWorks software, and the 3D printing is performed using Polylactic Acid (PLA) material. The three-point bending tests are conducted on the planar honeycomb sandwich structures with carbon fiber composite face sheets, and the test data are compared with the numerical results of this structure and its equivalent model. Based on ANSYS Workbench, the static analysis under bending and compression conditions, modal analysis, and harmonic response analysis are implemented on honeycomb sandwich structures with different bending degrees and their equivalent models. The results show that the load-displacement test curves of planar sandwich structures agree well with the finite element results of these structures and their equivalent models, and the equivalent models of curved sandwich structures accurately reflect the bending and vibration characteristics, fully verifying the feasibility of the presented equivalent model. This study provides important references and guidance for establishing the equivalent model of curved honeycomb sandwich structures.

Anand Model Constants of Sn-Ag-Cu Solders: What Do They Actually Mean?

Abstract Finite element simulations are broadly utilized to calculate the thermally induced mechanical deformations of interconnecting solder materials in electronics. The Anand unified viscoplasticity model is a prevalent choice for simulating the mechanical responses of solder joints, comprising nine parameters whose individual effects are not fully addressed in the literature. For this reason, this study examines the influence of each Anand parameter on the mechanical response of leadless Tin-Silver-Copper solder alloys with varying silver content through comprehensive nonlinear finite element simulations. Anand model coefficients for different SAC alloys, including SAC105, SAC205, SAC305, and SAC405, were sourced from existing literature and used to establish a systematic study matrix for each parameter. These coefficients were then integrated into thermomechanical simulations to induce inelastic deformations in the solder interconnections. The response of the solder interconnects is analyzed using equivalent inelastic strains and inelastic strain energy density. Results indicated that specific Anand parameters could skew the solder joint stress-strain response towards brittle behavior, while other parameters could lead to a more ductile response. Through statistical factorial analysis, the significance of each parameter is found to vary considerably, ranging from negligible to highly significant. The findings of this study are crucial for understanding the behavior of different SAC solder configurations and their expected thermal fatigue performance based on Anand creep constants. Furthermore, this paper provides foundational insights into the interpretation of Anand coefficients and their influence on the mechanical response of solder materials, a topic not yet explored in the existing literature.

Research on compression failure criteria and characteristics of rock-concrete assemblies with rough interfaces

The combination of rock and concrete lining structures is a typical composite structure in the field of engineering. This study is based on the concept of equivalent strain energy and establishes a mechanical equivalent model for rock-concrete assemblies (RCA). Assuming that both rock and concrete satisfy the Mohr-Coulomb criterion, we derive the shear failure criterion of the equivalent model considering the roughness of the rock-concrete interface. The applicability of the model was verified through uniaxial and triaxial tests on eight different types of RCA structures. The research results indicate that an increase in confining pressure enhances the strength of the RCA. When the confining pressure reaches a certain value, concrete only experiences shear failure, and no macroscopic cracks appear in the rock. The structure of the RCA tends towards isotropy. As the height ratio of the RCA increases, its strength decreases. At minimal concrete height ratios, the strength of the RCA gradually approaches that of concrete. This study can provide valuable insights for designing and evaluating stability in engineering rock bodies within diverse geological environments.

Corrected Method for Scaling the Structural Response Subjected to Blast Load

In scale-down tests of ship structures subjected to a blast load, the accuracy of the predicted response of a prototype is affected by the material substitution and geometric distortion between a scaled model and a full-size structure; this is known as incomplete similarity. To obtain a more accurate response from a prototype during small-size tests, a corrected method for scaling the response of thin plates and stiffened plates under a blast load was derived. In addition, based on numerical simulations of explosion responses by employing the elastic–plastic model and the Johnson–Cook constitutive model, it was found that using the average yield stress derived from the equivalent plastic strain energy in the ideal elastic–plastic model can obtain consistent structural responses. Moreover, a method for calculating the distortion factor caused by the yield stress of different materials was proposed. Furthermore, it was demonstrated that the average effective plastic strain between the prototype and the corrected model is equal, and based on this, a similarity prediction method was established to correct the distortions caused by yield stress and the thickness of blast loaded plates. The results indicate that the proposed correction method can compensate for the differences caused by distorted factors of yield stress and thickness, with the maximum error in the structure’s peak displacement being less than 3%.

Re‐derivation and mathematical analysis for linear peridynamics model for arbitrary Poisson ratio's material

AbstractThis paper is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson ratio's material. Based on the fundamental laws of dynamics, we re‐derive the bond‐based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well‐posedness of time‐dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.

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.

Comparison of analytical and numerical methods for estimating notch strains**

AbstractIn fatigue assessments, local strains and stresses can be easily evaluated using accurate finite element analysis in a case of an individual fatigue‐critical notch or structural detail. In practical design and analysis of global models, the consideration of local notch effects using suitable estimation rules in the combination with linear elastic analysis is reasonable. Various notch rules exist but the current knowledge lacks understanding how applicable different notch rules are for estimating notch strains from elastic stresses. This work studies the Neuber′s rules for sharp and mild notches, Glinka′s equivalent strain energy density method, the correction of Neuber′s rule proposed by Sonsino for mild notches, as well as the Seeger‐Beste method. The application of different estimation rules for notch strains, applied to the elastic‐perfectly plastic steel material and to Ramberg‐Osgood material model for the aluminum material, including work hardening effect, was studied in this work by comparing the estimations with the corresponding experimental data and finite element analysis results. The Neuber′s rule for mild notches and Sonsino′s averaging method gave best correlations with the experimental data and numerical (finite element analysis) results and can be thus recommended for such analyses in compatible cases studied in this work.

Closed-form effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates

This study derives closed-form expressions for the effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates. The derivation utilizes the homogenization method based on equivalent strain energy and the Kirchhoff plate theory. Rigorous accuracy assessments of the closed forms are conducted through comparisons with the existing literature and direct finite element analysis. A detailed parametric study is also conducted using the obtained closed forms to provide insights into the relationships between the effective elastic properties and the unit-cell shape. Lastly, general forms for some effective properties applicable to out-of-plane and in-plane situations are presented for hexagonal lattices.

A tensorial energy-release-rate based anisotropic damage-plasticity model for concrete

This paper proposes an anisotropic damage model for concrete to capture its plasticity and damage behavior effectively. Utilizing the framework of continuum damage mechanics, the model considers the influence of plastic Helmholtz free energy on damage evolution. This approach explicitly deduces damage-related thermodynamic forces without resorting to assumptions such as strain equivalence or energy equivalence. The proposed model employs tensile and compressive damage tensors to describe the development of anisotropic damage in concrete. The solution for plastic strain utilizes the empirical plasticity model. Furthermore, a unified evolutionary criterion for isotropic and anisotropic damage is also established, incorporating damage yield functions and orthogonal flow laws. Numerical simulations are conducted to validate the model’s ability to reproduce typical nonlinear behaviors of concrete structures under various load conditions.

Equivalent micropolar model for porous guided bone regeneration mesh: Optimum design for desired mechanical properties

In the present work, a micropolar continuum model is adopted to homogenise a heterogeneous porous model of Guided Bone Regeneration (GBR) meshes. GBR meshes are used in dentistry as mechanical barriers to isolate and protect the area of bone loss from the surrounding tissue while allowing new bone growth. The mechanical constants of the continuum are derived based on the strain energy equivalence of a periodic porous plate with its equivalent ortho-tetragonal micropolar model under prescribed boundary conditions. The effects of various architectural features such as pore shapes, patterns and sizes on the material parameters are investigated. The results show that the micropolar theory provides a better prediction of the response of the 2D porous geometries considered for the GBR mesh, compared to the classical Cauchy theory. The collected equivalent material parameters are further used for GBR mesh design, considering both mechanical and biomedical requirements. As an example, different materials and arrangements are analysed to find micropolar constitutive parameters that are comparable to bone parameters reported in the literature. This allows the GBR mesh to possess the mechanical performance that matches the adjacent bones.

Equivalent plate model of shape memory polymer composite based variable height corrugated structure

In this work, an equivalent plate model for a shape memory polymer composite based corrugation structure with varying height, for morphing application, is developed. A thermodynamically consistent constitutive relation based on temperature dependent phase transformation concept for shape memory polymer matrix is used in the composite model. The homogenization model is based on the equivalence of strain energy of a representative unit volume. Following validation, as an application, commonly used airfoil profile NACA 63210 was considered to demonstrate the model utility. Parametric studies have been undertaken to investigate the influence of corrugation parameters like thickness and number of corrugations.

Design and Manufacturing of a Low-Cost Prosthetic Foot

Below-knee prosthetics are used to restore the functional activity and appearance of persons with lower limb amputation. This work attempted to design and manufacture a low-cost, novel, comfortable, lightweight, durable, and flexible smart below-knee foot prosthesis prototype. This prosthesis foot was designed according to the natural leg measurement of an adult male patient. The foot is composed of rigid PVC layers interspersed with elastic strips of PTFE, and the axis of the ankle joint is flexible and consists of metal layers and a composite of polymeric damping strips with different mechanical properties, making it flexible and allowing it to absorb shocks and store and release energy. The design, modeling, and simulation of the manufactured prosthetic foot were performed via the ANSYS 18.0 software and the finite element method (FEM), where a large number of parallel and oblique planes and sketches were created. This work included four adult patients weighing 50, 75, 90, and 120 kg with different walking cycles. The results show that the highest equivalent von Mises stress and total deformations for the prosthetic limb occur at the beginning of the walking step, while the highest equivalent elastic strains and strain energy release rates are observed at the end of the walking step, regardless of the weight. This prototype can satisfactorily perform the biomechanical functions of a natural human foot, and it can be produced in attractive sizes, models, and shapes to suit different levels of below-knee amputations for different ages and weights, especially for patients with limited income.

Study of the effect of crystal orientation on the mechanical response and fatigue life of solder joints under thermal cycling by crystal plasticity

Electronic products are subjected to thermal cycling caused by power cycling in service processes. Owing to the difference in the coefficients of thermal expansion (CTEs) of solder joints and other packaging materials, solder joints suffer from thermomechanical fatigue, and cracks appear at their bonding interfaces. Solder joints are usually a single grain or three grains with Beachball morphology, and their composition is Sn-based Pb-free SAC305 (96.5Sn–3.0Ag–0.5Cu wt.%) solder, which exhibits obvious anisotropic mechanical behavior. This study investigated a thermal cycling simulation of Plastic Ball Grid Array (PBGA) packaging, in which the Anand model was used for solder joints, and the outermost solder joint was found to be a dangerous point. Sub-models were then established for this solder joint with several typical crystal orientations. The effect of crystal orientation under thermal cycling was simulated using thermomechanical crystal plasticity. When the c-axis of Sn crystal was perpendicular to the substrate, the deformation resistance was the largest, and the accumulation of equivalent plastic strain and strain energy dissipation was the slowest; hence, the fatigue life was the longest. As the angle between the c-axis and the substrate decreased, the deformation accumulated faster. When the c-axis was parallel to the substrate, the deformation of the solder joint was the fastest, and thus this joint was the first to fail and had the shortest fatigue life. In addition, the simulation of three grains indicated that orientations and their distributions had an influence on the shear strain accumulation of slip systems and fatigue life.

Application of modified first order shear deformation theory to vibration analysis of SSSS and CCSS thick anisotropic rectangular plates

This research presents a vibration analysis of a thick anisotropic rectangular plate using modified first shear deformation theory. Modified first shear deformation theory, which is not built upon the classical plate theory, was used to develop the kinematic equations and constitutive relations of a deformed section of thick anisotropic rectangular plate from which the generalized stress equations were determined. Using the assumptions of the theory, the strain energy and external work equations were formulated, and by employing the principles of total minimum potential energy, the total potential energy functional of a thick anisotropic plate was developed. Minimization of the total potential energy functional with respect to the deflection function (w) and with respect to the shear rotations (Φx) and (Φy) respectively, resulted in the governing differential equation and the two compatibility equations of the plate. The displacement functions that satisfy the governing and compatibility equations were obtained by solving the governing and compatibility equations. From the general displacement function, the peculiar deflection equations (shape functions) were obtained for the boundary conditions considered, which are simply supported on its four edges (SSSS), clamped on two adjacent edges, and simply supported on the other two (CCSS). Using the displacement equation and the equation for rotation in the x-direction (Φx) and equation for rotation in y-direction (Φy) the direct governing and two direct compatibility equations were obtained, from which coefficients that enable the formula for calculating fundamental natural frequencies to be obtained. For the boundary condition analyzed in this work, the stiffness coefficients (kR, kQ, kRQ, kq, kRRQ , kRQQ , kNR, kNQ, kNRQ and kλ ) were computed and used in determining the fundamental natural frequency parameter values at various span to depth ratios (5,10,20, 25 and 100), aspect ratios (1 to 2 at the increment of 0.1) and angle of fibre orientation (0o, 15o, 45o). The solutions of this study were compared with those from previous researchers. The fundamental natural frequency parameter values obtained in this study were compared with the work of Reddy (1984) for 0o angle of fibre orientation at span to depth ratios of 5,10, 20, 25, 50 and 100 at an aspect ratio of 1. The percentage difference values were 6.073%, 3.197%, 1.132%, 0.788%, 0.255%, and 0.112%, respectively. These differences revealed the closeness of the results of this present study to the results of Reddy (1984). This shows that the present theory provides good and acceptable solutions to the vibration problems of thick anisotropic rectangular plates.

Analyzing the Stability of Rock Surrounding Deep Cross-Tunnels Using a Dynamic Velocity Field

With the increasing number of deep rock engineering projects, many different types of tunnels have emerged, such as cross-tunnels. These tunnels intersect with each other in rock, which causes potential safety hazards. We must analyze the stability of the surrounding rock, to ensure worker safety. This article presents a method for dynamically assessing the stability of the surrounding rock in deep-buried cross-tunnels. The method consists of two main analysis steps: (1) P-wave velocity field inversion; and (2) Stability analysis of the surrounding rock. The P-wave velocity field inversion involves inverting the S-wave velocity field by Rayleigh wave and inverting the P-wave velocity field by adjoint state traveltime tomography. Then, a method of stability analysis is proposed which is used to update the mechanical properties of the rock (based on the continuously updated wave velocity field). The elastic modulus of the surrounding rock is approximated throughout the excavation process. CASRock V1.0 (Cellular Automation Software for engineering Rockmass fracturing processes) is used to assess rock damage via the equivalent plastic shear strain and local energy release rate. The new method is used to analyze the stability of a new tunnel excavated in Jinping (in China). The results reveal the severity and spatial distribution of the damage caused. The yield depth is concentrated near the sidewalls, while the top and bottom of the tunnel exhibit a smaller depth. The yield depths present a particular pattern of change (high–low–high–low) with increasing distance from tunnel #2. Finally, this research enriches our understanding of excavating deep cross-tunnels and makes an important contribution to improving worker safety in deep cross-tunnels.

Mechanistic modeling of fatigue crack growth in asphalt fine aggregate matrix under torsional shear cyclic load

Fatigue crack growth in asphalt pavements under vehicle load may primarily occur within the asphalt fine aggregate matrix (FAM) phase of the mixture. This study aims to predict fatigue crack growth in FAM by developing a mechanistic-based fatigue crack characterization model and fatigue crack growth model under torsional shear cyclic load. Initially, a fatigue crack characterization indicator, fatigue damage density, was derived using principles of torque and dissipated strain energy equivalence. Afterwards, a fatigue crack growth model was established based on the pseudo J-integral Paris’ law. Finally, the viscoelastic damage constitutive model was further constructed by coupling the fatigue crack growth model with a viscoelastic model, which was subsequently implemented in COMSOL Multiphysics. The results show that the fatigue damage density can be determined by the initial shear modulus and phase angle, as well as shear modulus and phase angle under fatigue conditions. Additionally, a logarithmic-linear correlation exists between the fatigue crack growth rate and the dissipated pseudo strain energy rate. The parameters of the fatigue crack growth model exhibit minimal variation across different shear strain levels and temperatures. Overall, the proposed numerical model can effectively simulate damaged torsional shear cyclic tests of FAM.

Design and analysis of ultra-low moment noise flexure pivot in space gravitational wave detectors

To control the pointing accuracy of the telescope in the space gravitational wave detector at the nano-radian level, it is essential to design a flexure pivot with extremely low rotational moment nonlinearity during rotation. This study derives the theoretical equation of the nonlinear rotational moment of flexure pivot based on the beam bending strain energy equation and completes the design of ultra-low moment noise flexure pivot by integrating the structural form of conventional cross-spring flexure pivot. The nonlinear moment noise distribution in the frequency domain of three kinds of flexure pivots is then compared regarding TianQin’s circular high orbits. Then the relevant finite element simulation analysis will be performed for the flexure pivot structures, respectively, to ensure the accuracy of its rotational stiffness model and verify the correctness of nonlinear moment theory optimization through physical experiments. The results show that the optimized flexure pivot can effectively reduce the nonlinearity, and basically meet the nonlinear moment requirements in space gravitational wave detection.

An equivalent strain energy density model for fatigue life prediction under large compressive mean stress

In this work, a modified Smith-Watson-Topper (SWT) model is proposed for the consideration of large compressive and tensile mean stresses. Firstly, the maximum normal stress in the original SWT model is replaced with equivalent stress amplitude, and the SWT parameter is thereafter interpreted as equivalent strain energy density (ESED). The newly proposed ESED model gives improved prediction results in fatigue life, especially under conditions with large compressive mean stress. Secondly, the applications of different mean stress correction (MSC) models are investigated with their limitations carefully examined. In order to improve the ESED model with higher accuracy, a new exponential MSC model is proposed. Moreover, an additional parameter concerning the dependency of material on mean stress is proposed. With the exponential MSC model incorporated, the ESED model is validated with experimental data from several different materials, and shows obviously improved prediction results compared with SWT model and some recent life prediction models.

Enhanced microstructure and mechanical properties of cementless ultra-high-performance fiber-reinforced alkali-activated concrete with silicon dioxide nanoparticles

This study investigates the effect of silicon dioxide nanoparticles (nano-SiO2) on the microstructure and mechanical properties of eco-friendly, cementless ultra-high-performance fiber-reinforced alkali-activated concrete (UHP-FRAAC). Various amounts of nano-SiO2 were adopted in a range of 0%-5% of the mass of silica fume (SF). The experimental results showed that the addition of nano-SiO2 improves the packing density of ultra-high-performance alkali-activated concrete (UHP-AAC) and generates abundant calcium (alumino)silicate hydrate (C-(A-)S-H) gels to increase its density. Therefore, the interfacial bond strength and pull-out energy of steel fibers from UHP-AAC could be enhanced. At 2% nano-SiO2 replacement, the compressive strength of UHP-FRAAC was the highest (184.2 MPa), and its total porosity was the lowest, decreasing from 9.72% to 7.53%. The highest equivalent bond strength and strain energy density of UHP-FRAAC occurred at 2% nano-SiO2 replacement, i.e., 10.9 MPa and 72.5 kJ/m3, respectively. The maximum tensile strength of 14.5 MPa was observed at a nano-SiO2 content of 3%. Higher dosages of nano-SiO2, 3% or more, negatively affected the interfacial bond and tensile properties of UHP-FRAAC due to its agglomeration, but still exhibited higher performance than that of the plain UHP-FRAAC up to the dosage of 5%. Therefore, the optimal dosage of nano-SiO2 in UHP-FRAAC was suggested to be 2% by mass of silica fume.