Equivalent Elastic Modulus Research Articles

Tensile behavior and durability prediction of GFRP-steel composite bars under chloride environments

A fiber-reinforced polymer (FRP)-steel composite bars (FSCBs), with an inner steel core and outer FRP jacket, is a novel type of reinforcement designed to inherit good durability and ductility suitable for seawater sea sand concrete structures. The long-term tensile behavior of FSCBs in chloride environments is important. The long-term tensile behavior of glass-type FSCBs with different fiber layer thicknesses under chloride environment attack was investigated in this study. FSCBs with an inner steel bar of fixed diameter (8 mm) and an outer FRP jacket of three thicknesses (2, 4, and 6 mm) were used for research. Seawater-immersion accelerated-aging experiments at three different aging temperatures (23, 40, and 60 °C) and aging times (1, 6 and 12 months) were conducted. The failure modes, equivalent elastic modulus, pseudo strain hardening moduSTlus, and tensile strength retention of the FSCBs under the above mentioned aging temperatures and times were analyzed. The experimental results determined that the chloride environment had no significant effect on the equivalent elastic modulus of the FSCBs. However, the pseudo-strain hardening modulus of the FSCBs decreased with increasing aging time. Moreover, the tensile strength retention of the FSCBs decreased with increasing temperature under the same aging time and FSCBs diameter. A prediction model for the tensile strength retention of FSCBs with different FRP layer thicknesses at different aging temperatures and times was proposed and validated. The results provided a better understanding of the durability of FSCBs in marine hydrothermal environments to establish the criteria and standards of design.

Calculating the equivalent elastic moduli and their influence on modelling the sound insulation of softwood cross laminated timber (CLT)

There are characteristics unique to natural materials such as timber that must be dealt with carefully to develop successful sound insulation prediction models. Timber has low internal losses typical of low-density building materials but with material properties that vary according to growth conditions. To study these factors in cross laminated timber (CLT) products, orthotropic thin plate and isotropic thick plate models are investigated. Equivalent elastic moduli are obtained using two methods from values and limits provided in the timber standards and literature, or by the manufacturer. The equivalent moduli are used to calculate the sound insulation of three- and five-layer CLT plates. Appropriate upper and lower limits are calculated from the combined material parameters and compared with sound insulation results measured in accredited laboratories for the first time. EN ISO 10140 test standards require that surface density of the sample be documented in the laboratory; additional precision information such as the fifth percentile of the material density is not required. However, considering the limits on the internal loss factors and elastic moduli only in the thin plate sound insulation model, gives a poor overall prediction performance (23.6% within the upper and lower limits). In this paper we combine the limits on elastic moduli, internal loss factors, and density. With these combined parameters the prediction is improved; less than half of the measured sound insulation values fall within the upper and lower limits of a typical thin plate model (44.0%). The single figure sound reduction index values are underestimated in seven out of nine cases. The prediction is further improved when a thick plate model is considered. There are several layers of complexity in the measurement and modelling conditions that must be carefully considered. The contributions from these compounding factors are discussed, and additional moduli and density parameters are proposed for inclusion in the literature and standards.

A characterization method for equivalent elastic modulus of rock based on elastic strain energy

Energy is an internal variable during rock deformation and failure, and its dissipation and conversion law can reflect the rock’s internal damage and deterioration state. Analysis of rock deformation and failure process from the perspective of energy is helpful to deeply understand the mechanism of rock damage, fracture and instability failure, and has important theoretical and practical significance for the stability evaluation and support control of surrounding rock. In this study, through single cyclic loading and unloading (SCLU) experiments, cyclic triaxial loading and unloading (CTLU) experiments and conventional triaxial compression (CTC) experiments, the equivalent elastic modulus method based on elastic strain energy is proposed to analyze the energy conversion of rock. The results show that the error of the elastic strain energy calculated by the strain energy formula method is generally higher than 10% with the secant and tangent modulus of the loading and unloading curve as input parameters. Taking the equivalent elastic modulus proposed in this study as an input parameter, more accurate elastic strain energy can be obtained by the strain energy formula. During the rock failure process, the equivalent elastic modulus shows a three-stage characteristic of increase, steady and decrease. The equivalent elastic modulus can be estimated by the quadratic function between the equivalent elastic modulus and confining pressure and axial strain. Under the same deformation and deviatoric stress, the elastic strain energy stored in rock increases with increasing confining pressure. The local maximum energy dissipation rate corresponds to stress drop, and the peak energy dissipation rate appears near the peak strength. High energy dissipation mainly occurs in a short time after peak strength, and energy release and dissipation are more sudden and severe under high confining pressure.

Rock physics and seismic reflectivity parameterization and amplitude variation with offsets inversion in terms of total organic carbon indicator

Total organic carbon (TOC) prediction with elastic parameter inversions has been widely used in the identification and evaluation of source rocks. However, the elastic parameters used to predict TOC are not only determined by TOC but also depend on the other physical properties of source rocks. Besides, the TOC prediction with the elastic parameters inversion is an indirect method based on the statistical relationship obtained from well logs and experiment data. Therefore, we propose a rock physics model and define a TOC indicator mainly affected by TOC to predict TOC directly. The proposed rock physics model makes the equivalent elastic moduli of source rocks parameterized by the TOC indicator. Combining the equivalent elastic moduli of source rocks and Gray’s approximation leads to a novel linearized approximation of the P-wave reflection coefficient incorporating the TOC indicator. Model examples illustrate that the novel reflectivity approximation well agrees with the exact Zoeppritz equation until incident angles reach 40°. Convoluting the novel P-wave reflection approximation with seismic wavelets as the forward solver, an AVO inversion method based on the Bayesian theory is proposed to invert the TOC indicator with seismic data. The synthetic examples and field tests validate the feasibility and stability of the proposed AVO inversion approach. Using the inversion results of the TOC indicator, TOC is directly and accurately estimated in the target area.

Finite Element Analysis of Residual Stress Distribution Patterns of Prestressed Composites Considering Interphases

New finite element analysis procedures are developed in this study to obtain the precise stress distribution patterns of prestressed composites. Within the FEM procedures, an equivalent thermal method is modified to realize the prestress application, and a multi-step methodology is developed to consider coupling effects of polymer curing and prestress application. Thereafter, the effects of interphases' properties, including the elastic modulus and coefficient of thermal expansion (CTE), on the stress distribution patterns are revealed. Analytical methods for residual stress prediction are modified in this study to demonstrate the finite element analysis procedures. From the residual stress results, it is found that the increase in the prestress level tends to contribute to the initiation of interphase debonding. The increase in the elastic modulus or CTE of the interphase results in very large circumferential and axial stress values appearing in the interphase. When the elastic modulus in the interphase is heterogeneous, the predicted stress values in the fiber and matrix are similar to the results predicted with the equivalent elastic modulus of the interphases. However, the heterogeneous elastic modulus results in serious circumferential and axial stress gradients in the interphase.

A 3D re-entrant structural metamaterial with negative Poisson’s ratio reinforced by adding arrow structures

Structural metamaterial with negative Poisson’s ratio often has low stiffness due to its porous structure. For engineering applications, the various strategies have been developed to enhance its stiffness but also weaken the negative Poisson effect. In this work, we propose a 3D reinforced re-entrant structure (RRS) by adding arrow structures to the classical re-entrant structure (RS), which is fabricated by photocuring 3D printing of photosensitive resin. The structure-properties correlative mechanism of RRS is systematically analyzed by combining experiment (quasi-static compression experiment) and finite element simulation (ABAQUS). It is found that the equivalent elastic modulus and negative Poisson’s ratio of RRS can be tuned by various structural parameters (i.e. the thickness ratio η, the length ratio of the slant rod α, the length ratio of the vertical rod β, and the angle θ of the re-entrant cells). By optimizing the structure parameters, the equivalent elastic modulus of RRS with the negative Poisson’s ratio of −0.28 can be significantly increase to 13.67 MPa, which is 12.32 times larger than that of RS with the same negative Poisson’s ratio. Due to the flexible design of stiffness and negative Poisson ratio, our proposed 3D reinforced re-entrant structural metamaterial provides a potential way for designing engineering materials with desirable performance.

The optimal design of 3D-printed lattice bone plate by considering fracture healing mechanism.

The biomechanical stimulus is the most important factor for fracture healing and mainly determined by the structural stiffness of bone plate. Currently, the materials commonly used in bone plates are stainless steel and titanium, which often lead to stress shielding effects because of their higher elastic modulus compared with the bone. This article suggests an optimal design method of lattice bone plate based on fracture healing theory. First, the mechanical regulation model with deviatoric strain is established to simulate the tissue differentiation process during fracture healing process. The ratio of the average elastic modulus of callus at the 120th day to the elastic modulus of mature bone is used to characterize the fracture healing rate. Second, the optimal elastic modulus of the design domain is obtained by the optimization mathematical model with the maximum fracture healing rate. Then, the design domain is filled with microstructures, the porosity of which is adjusted to make it possible that the equivalent elastic modulus is equal to the optimized value. And the finite element analysis of the bone plate with microstructure is executed. Finally, the designed lattice bone plates are manufactured through 3D printing, and the mechanical test is carried out. The simulation results indicate that the fracture healing rate is maximum when the elastic modulus of material in design domain is 38 GPa under the constraints of fixation stability. And both the finite element analysis and experiment results show that the designed lattice bone plate meet the strength requirements of fracture internal fixation implants.

PECULIARITIES OF TRANSPORTATION OF GRAIN PRODUCTS BY CONVEYOR SPIRALS

Under the current conditions of price growth, the expansion of agricultural enterprises and the creation of new ones based on them, the economic feasibility and practical possibility of purchasing the necessary equipment has significantly decreased. In connection with this and with the aim of expanding the functional capabilities of the technical means available in farms, the application of devices to serial machines is becoming more and more effective. The article presents an overview of modern attached screw equipment for quick unloading of motor vehicles. When designing screw equipment, it is necessary to take into account the peculiarities of their transport properties. The transportation of bulk goods by angle screw equipment depends on the frequency of rotation and the level of filling of the conveyor with the product. Structural and mechanical properties primarily depend on the density of packing of particles of loose material, their relative mobility. The mechanical properties of such a structure can be described by the equivalent modulus of elasticity and the equivalent Poisson's ratio. The nature of energy transfer in a loose material is largely determined by the forces of friction and adhesion at the points of contact of its particles. A characteristic feature of screw conveyors is that the screw working body is freely (without supports) located in an elastic or rigid casing and along its inner surface transports the material to the unloading area. The article presents the main theoretical ratios for determining the effectiveness of hinged screw products. A study was conducted to determine the influence of the main technical parameters on the performance of high-speed conveyors at a large angle.

Experimental study on tensile mechanism of UHPC grouted sleeve splice

The grouted sleeve splice is a competitive splice in the prefabricated structure that plays a key role in ensuring the stability of the whole structure. Despite the large amounts of experiments performed, its force mechanism has rarely been explored. In this study, a novel method was developed to calculate the constitutive model of grouted sleeve splice, and a comprehensive understanding of the force mechanism was analyzed. Tensile tests of 24 specimens were conducted to explore the mechanical properties of the grouted sleeve splice. The results showed that the anchorage length of rebar with ultra-high-performance grout material can be shorter than that recommended in design standards. The anchorage length and eccentricity of rebars have a significant impact on the mechanical performance of the grouted sleeve splice, which deserves special attention in the structure design and construction. The non-anchorage section of the grouted sleeve splice offers little contribution to the bearing capacity. Based on the concept of equivalent section elastic moduli, a theoretical calculation method was established to reveal the stress–strain relationship of the grouted sleeve splice. The errors between the theoretical model and test results vary from 0.20% to 18.90%. This study provides new perspectives for theoretical research on the mechanism of grouted sleeve splice.

A concrete sandwich wallboard damage identification method based on strain energy density increment

During the loading process of concrete members, the identification of characteristic damage states is not sensitive enough. This paper investigates the damage evolution of four concrete sandwich wallboards subjected to vertical compression by employing the strain energy density increment to enhance identification sensitivity. First, the equivalent elastic modulus model for a wallboard subjected to vertical compression is established based on the material stress–strain relationship. Then, the strain data of concrete, obtained from experiments and numerical simulations, are modeled as the normalized unit approximate strain energy density increments (NUASEDIs), which follow the damage evolution of the wallboard as a more sensitive index. The first cracking formation of wallboard corresponds to the first high NUASEDI. The failure damage state corresponds to the final high NUASEDI, which reflects the process changing from the local to the global failure. Finally, the strain energy density increment theory improves averagely the cracking identification sensitivity of concrete members by 13.32% and the failure identification sensitivity by 3.01%.

Research on Inherent Frequency and Vibration Characteristics of Sandwich Piezoelectric Ceramic Transducer

Great progress has been made in the field of ultrasonic processing in recent years, and piezoelectric ceramic transducers have been widely used as drive sources. In this paper, a sandwich piezoelectric ceramic transducer is designed, and the vibration of each part of the transducer is analyzed by elastic mechanics and piezoelectric theory. According to its mechanical and electrical boundary conditions, the vibration model of the piezoelectric transducer was established. Based on the equivalent elastic modulus method for simplifying the pre-stressed bolts into a one-dimensional transducer vibration model, the relationship between the one-dimensional axial response frequency of the transducer and the length of each component was obtained. Based on the half wavelength theory, a transducer with the vibration node in the crystal stack and an inherent frequency of 15 kHz was designed and fabricated. In order to verify the natural frequency and vibration characteristics of the piezoelectric transducer, a laser vibration measurement system was built in this study. The vibration characteristics of the transducer under different parameters such as voltage and frequency were analyzed, and the accuracy of the vibration model was verified. The vibration states of the end surface of the transducer and the radial surface were evaluated at the first-order inherent frequency and second-order inherent frequency. The results show that the equivalent simplified model established in this study can effectively design the inherent frequency of the transducer, and the operation at the first-order inherent frequency meets the one-dimensional assumptions of this study. The transducer operating conditions measured in this study also provide a more detailed reference for ultrasonic processing applications.

Discussion of “Contact Unloading Behaviors of Elastic-Power-Law Strain Hardening Material Considering Indenter Elasticity Effect” (Chen, C., Wang, Q., Wang, H., Ding, H., Hu, W., Xie, W., Weng, P., Jiang, L., and Yin, X., 2022, ASME J. Tribol., 144(12), p. 121501)

Discussion of “Contact Unloading Behaviors of Elastic-Power-Law Strain Hardening Material Considering Indenter Elasticity Effect” (Chen, C., Wang, Q., Wang, H., Ding, H., Hu, W., Xie, W., Weng, P., Jiang, L., and Yin, X., 2022, ASME J. Tribol., 144(12), p. 121501)

Parametric analysis of the stress-strain state for the unsupported and supported horizontal underground workings

An analysis of analytical and numerical approaches to the problem of determining the stress-strain state of underground workings has been carried out. A system of parametric analysis of the stress-strain state of unsupported and supported horizontal underground workings has been developed. This system proved that to determine the stress-strain state of horizontal unsupported workings of a certain diameter, it is enough to perform one numerical calculation of a finite-element model with unit parameters, and using simple analytical formulas, extrapolate the stress-strain state of this model to all possible cases of real workings. Finite element models of underground workings were developed to justify the author’s system. A method of parametric analysis of the strained state of a supported working by using models with equivalent bending stiffness has been developed and substantiated. On its basis, it is possible to perform a numerical analysis of a finite-element model with unit parameters of the frame, selecting the equivalent modulus of elasticity.

A Revised Continuous Observation Length Model of Rough Contact without Adhesion

The real contact area of rough surfaces has significant importance in many engineering applications, such as tribology, wear, lubrication and seals. A continuous observation length-dependent mechanic model of rough contact without adhesion is proposed, which assumes that the rough surface is divided into ideal subplanes. However, the model ignores the elastoplastic deformation of asperities, and the standard deviation of ideal subplanes’ heights is assumed to vary linearly with continuous observation length, which is not precise for all the surface fractal dimensions. In this work, a revised continuous observation length model is proposed with elastic, elastoplastic and fully plastic stages. The expressions of force and real contact areas are derived. For surfaces with different fractal dimensions, the quadratic polynomial, cubic polynomial and power relationships between standard deviation and observation length are proposed, respectively. In addition, the influences of the dimensionless observation length, fractal dimension and equivalent elastic modulus on the real contact areas in different contact stages are also analyzed. It can be concluded that the quadrate real contact area decreases as the dimensionless observation length decreases, which can be applied to the percolation theory for leak seal problems.

Time-Dependent Sensitivity of Structural Reliability Assessment to Multi-Source Uncertainties Using Lamb Wave

Abstract This article investigates the time-dependent sensitivity of structural reliability assessment to multisource uncertainties using Lamb wave. To precisely model the influence of local damage on the structure in the course of damage growth, a surface damage effect model is proposed to obtain the equivalent elasticity modulus, which can be coupled with the structure model. The evolution of the surface damage is modeled using the fatigue crack propagation model. Furthermore, by setting up the component and structure failure criteria, the time-dependent reliability model of the structure under multisource uncertainties from Lamb wave detection and material properties is established. The method of score function is employed to evaluate the sensitivity index, which is defined as the derivative of the reliability with respect to the distribution parameters of uncertain variables. A spatial truss structure is used to demonstrate the overall procedure. Numerical results show that the sensitivities indices are time and damage size dependent. The sensitivity contributions from Lamb wave quantification model and the material properties are comparable when the crack size is relatively small. When the crack grows to a larger size, the reliability assessment result is much more sensitive to uncertainties associated with material properties.

Study of the Vibrations of Some Composite Bars with Polypropylene Honeycomb Core and Carbon Fiber and Fiberglass Fabric Faces

The paper generalizes the Timoshenko model for thick bars, using a new model that is applied to the vibration study of multilayer composite bars. In the proposed mathematical model, three coefficients are introduced that take into account the non-uniformities of the tangential and normal stresses in the bar section. The vibrations of some composite bars with a polypropylene honeycomb core with a thickness of 10 mm, 15 mm and 20 mm are experimentally studied, on the faces of which one or two layers of carbon fiber, respectively glass fiber was poured. For each analysed bar, the stiffness and the equivalent modulus of elasticity are determined and the variation of the damping coefficient according to the length of the bar is studied.

Equivalent characteristic model of a rough contact interface based on virtual material method

The characteristic of rough contact interface has a major influence on overall performance of machines. To facilitate the setting of contact interface characteristics during the performance analysis of the whole machine, an equivalent characteristic model of a rough contact interface based on virtual material method is established. The normal and tangential properties of the rough contact interface per unit area are first experimentally conducted. Furthermore, according to the Greenwood-Williamson (GW) model and stress wave propagation conditions of discrete elements, the relationship between normal and tangential contact characteristics is explored by assuming the rough contact interface per unit area as a thin virtual layer. Based on virtual material method, the equivalent characteristic model of a rough contact interface is established, including equivalent virtual elastic modulus, Poisson’s ratio, thickness, and density. Finally, taking two step-shaped steel plates connected by four bolts for example, a numerical simulation of structural characteristics under different preloads is carried out. Numerical analysis results are then compared with tested results to verify the validity of the equivalent virtual material model of rough contact interface, and a good agreement between them is obtained.

Prediction of Equivalent Elastic Modulus for Metal-Coated Lattice Based on Machine Learning

Prediction of Equivalent Elastic Modulus for Metal-Coated Lattice Based on Machine Learning

Equivalent in-plane dynamic elastic moduli of lattice structures with Plateau borders

Lattice structures with Plateau borders (LSPB) have attracted increasing interests recently due to the improved stiffness, strength, energy absorption properties. Undoubtedly, vertexes with Plateau borders (VPB) play a significant role on the dynamic elastic moduli. This paper proposes an analytical framework of the frequency-dependent equivalent in-plane dynamic elastic moduli (Young’s moduli E1(ω), E2(ω), Poisson’s ratio ν12(ω), ν21(ω) and shear modulus G12(ω)) of LSPB. First, dynamic stiffness (DS) matrix of a lattice cell edge (based on rod and Timoshenko theories) connected to VPBs (modelled as rigid bodies) at both ends is formulated. Then, based on the above DS matrix and the unit cell method, closed-form expressions of equivalent in-plane dynamic elastic moduli are proposed, which are sufficient general to be applied to four types of lattices. The effects of mass, inertia moment and size of VPB on the equivalent dynamic elastic moduli are studied, with both physical and mathematical interpretations. Furthermore, the proposed expressions are applied to honeycomb, rectangular, auxetic and rhombus LSPB and some interesting and important observations are made. This research provides analytical expressions for broadband dynamic elastic moduli of LSPB, which can be directly used in the design and optimization of composite structures with lattice cores.

Research on Mechanical Properties of V-Type Folded Core Sandwich Structures

A folded core is a three-dimensional configuration formed by folding a flat sheet of paper. Similar to foam and honeycomb, folded cores are widely used in aerospace as the cores for sandwich structures due to their excellent mechanical properties and light structural weight. In general, the core’s configuration is heterogeneous and anisotropic, while from a macroscopic point of view it is considered homogeneous. Analytical solutions of equivalent mechanical properties are generally obtained by equivalent methods and modified by numerical simulations due to the complexity of the configuration. We can find the best combination of a core’s parameters by using analytical solutions. The faceted regular folded configuration (RFC), consisting of repetitive identical cells, can be expanded to a plane. According to the principle of equivalent load-deformation, this article investigates the equivalent mechanical properties of the V-type folded core (VFC). The analytical solutions of Ez—equivalent elastic modulus in Direction z, equivalent shear modulus in Plane xz—Gxz, and equivalent shear modulus in Plane yz—Gyz have been obtained and verified by numerical simulations. In addition, the reliability of the equivalent mechanical properties is further demonstrated by the case study. By applying the analytical solutions, the optimization process of VFC can be simplified and sped up, which is of great engineering significance to the aircraft design process.