Stress Equilibrium Equations Research Articles

Deep neural network homogenization of multiphysics behavior for periodic piezoelectric composites

We present a physically informed deep neural network homogenization theory for identifying homogenized moduli and local electromechanical fields of periodic piezoelectric composites. The theory employs a multi-network model to represent solutions to stress equilibrium and electrostatic partial differential equations in the inclusion and the matrix phases, leading to a more accurate depiction of field variables. Satisfaction of periodicity conditions for hexagonal and cubic symmetries are efficiently tackled using a series of sinusoidal functions with known periods and adjustable parameters. Additionally, the theory applies fully trainable weights to the collocation points. These trainable weights, trained concurrently with the neural network weights, compel the neural networks to enhance their performance when faced with large local stress/deformation gradients. The predictive capability of the proposed theory is illustrated by comparison with finite-element solutions for composites reinforced with unidirectional fiber, spherical particle, or weakened by an ellipsoidal cavity over a wide range of volume fractions.

Free vibration analysis of fiber-reinforced composite multilayer cylindrical shells under hydrostatic pressure

With the development and application of composite materials in deepwater submersibles, the influence of high external pressure from deepwater environments on the vibration characteristics of composite pressure-resistant cylindrical shells cannot be ignored. In this paper, the free vibration characteristics of prestressed composite multilayer shells are methodically examined by establishing partial differential equations based on the three-dimensional elasticity theory. To solve these equations, the state-space technique and double Fourier series expansion are employed to transform them into state-space ordinary differential equations. These equations are utilized to analyze vibration problems of simply supported composite multilayer cylindrical shells with arbitrary layered angles. The initial stress equilibrium equation is solved to determine the prestress induced by the hydrostatic pressure. This overcomes the limitation of membrane stress and considers non-uniform distributions of radial and interlaminar prestresses. The proposed theoretical model and solution strategies are validated by comparing with theoretical and experimental results from the existing literature and the finite element method. Additionally, the present investigation reveals that the natural frequencies of thick shells can be calculated more accurately using the prestress derived by the proposed approach, especially subjected to noticeable hydrostatic pressure. Finally, the performed investigations indicate that the circumferential prestressing has a more pronounced effect on the shell's natural frequencies than axial prestressing.

Torsional vibration of tapered pile including circumferential support of surrounding soil

The tapered pile is usually divided into finite segments along the vertical direction due to its variable cross-section when investigating its torsional dynamic characteristics. However, an annular projection would be formed at the contact surface of the adjacent pile segments due to the difference in radius, through which the surrounding soil will apply the circumferential support on the pile segment. In this study, the Voigt model is adopted to model the circumferential support of the surrounding soil acting on the annular projection. On this basis, an amended impedance function transfer method is established combined with the circumferential stress equilibrium equations and circumferential displacement continuity conditions at the contact surface of the adjacent pile segments. The shear complex stiffness transfer method is employed to consider the construction disturbance during pile installation. Then, an improved analytical solution for the torsional vibration of the tapered pile considering the circumferential support of the surrounding soil is established by solving the equations for the soil and pile. Based on the solution, the torsional vibration of the tapered pile is investigated and the influence of the circumferential support of the surrounding soil is revealed under different pile parameters and construction disturbance conditions.

Adaptive deep homogenization theory for periodic heterogeneous materials

We present an adaptive physics-informed deep homogenization neural network (DHN) approach to formulate a full-field micromechanics model for elastic and thermoelastic periodic arrays with different microstructures. The unit cell solution is approximated by fully connected multilayers via minimizing a loss function formulated in terms of the sum of residuals from the stress equilibrium and heat conduction partial differential equations (PDEs), together with interfacial traction-free or adiabatic boundary conditions. In comparison, periodicity boundary conditions are directly satisfied by introducing a network layer with sinusoidal functions. Fully trainable weights are applied on all collocation points, which are simultaneously trained alongside the network weights. Hence, the network automatically assigns higher weights to the collocation points in the vicinity of the interface (particularly challenging regions of the unit cell solution) in the loss function. This compels the neural networks to enhance their performance at these specific points. The accuracy of adaptive DHN is verified against the finite element and the elasticity solution respectively for elliptical and circular cylindrical pores/fibers. The advantage of the adaptive DHN over the original DHN technique is justified by considering locally irregular porous architecture where pore–pore interaction makes training the network particularly slow and hard to optimize.

The effect of the stress equilibrium factor on the composite case of the solid rocket motor

In this paper, based on the composite structure design and analysis software ANSYS ACP, the layup design for the composite cases of the equal pole hole solid rocket motor with different stress equilibrium factors is carried out. The finite element software ANSYS Workbench is used to study in detail the effects of stress equilibrium factor on the stress level, failure degree, and burst pressure prediction of the case under a certain internal pressure. The results show that reducing the value of the stress equilibrium factor can reduce the stress level of each winding layer along the main direction of the material in the motor case and gradually transfer the maximum stress of the case along the fiber direction from the weak stress zone to the cylinder section. This is beneficial for improving the burst pressure of the case and making the burst point located in the cylinder section. However, whether the case failure is related to the selected failure theory and strength criterion, the failure results and burst pressure obtained using different failure theories and strength criteria may be different, which indicates that each failure criterion has its scope of application and limitations. Under the Tsai-Wu criterion, using the stress equilibrium equation to calculate the stress equilibrium factor can just ensure that each winding layer of the motor case does not fail under the working pressure. In summary, when designing a composite case and predicting its failure, in addition to considering whether the stress equilibrium factor taken is conducive to meeting the design requirements for the strength of the case, it is also necessary to consider the use of multiple failure criteria to study the failure of the case. Obviously, arbitrarily selecting the stress equilibrium factor is not appropriate in the above process. By analyzing the influence of the stress equilibrium factor on the strength and failure condition of the motor case, it can provide an effective reference for the design of a composite case for SRM.

A parallel and performance portable implementation of a full-field crystal plasticity model

We have developed a parallel implementation of an Elasto-Viscoplastic Fast Fourier Transform-based (EVPFFT) micromechanical solver to enable computationally efficient crystal plasticity modeling for polycrystalline materials. Our primary focus lies in achieving performance portability, allowing a single EVPFFT implementation to run optimally on various homogeneous architectures, including multi-core Central Processing Units (CPUs), as well as on heterogeneous computer architectures comprising multi-core CPUs and Graphics Processing Units (GPUs) from different vendors. To accomplish this goal, we have leveraged MATAR, a C++ software library that simplifies the creation and utilization of multidimensional dense or sparse matrix and array data structures. These data structures are designed to be portable across diverse architectures through the use of Kokkos, a performance-portable library. Additionally, we have employed the Message Passing Interface (MPI) to efficiently distribute the computational workload among processors. The heFFTe (Highly Efficient FFT for Exascale) library is used to facilitate the performance portability of the fast Fourier transforms (FFTs) computation. The computational performance of EVPFFT is evaluated and presented in terms of parallel scalability and simulation runtime on different high-performance computing (HPC) architectures. The utility of the developed framework to efficiently simulate the micro-mechanical fields in polycrystalline microstructures in engineering applications is discussed. Program summaryProgram Title: EVPFFTCPC Library link to program files:https://doi.org/10.17632/2k8579fyyv.1Developer's repository link:https://github.com/lanl/FierroLicensing provisions: BSD 3-Clause LicenseProgramming language: C++External routines/libraries: MPI, Kokkos, MATAR, HeFFTe, HDF5Nature of problem: EVPFFT is a crystal plasticity code designed to compute micro-mechanical fields within a polycrystalline representative volume element (RVE) and predict the macroscale response of the RVE.Solution method: EVPFFT uses the periodic Green's function method in Fourier space to solve the field equations of static stress equilibrium in a periodic spatial domain.

A new method of failure analysis

The present paper develops a new failure analysis method under plane strain conditions considering a generalized linear yield criterion. The yield criterion and the stress equilibrium equations constitute a hyperbolic system of equations. It is shown that two auxiliary variables satisfy the equation of telegraphy. Simple analytical relationships connect these variables and the radii of curvature of the characteristic curves. The calculated radii of curvature allow for the corresponding characteristic net to be constructed. Then, the stress field is determined using another set of analytical relationships. Thus, a numerical procedure is only necessary for solving the equation of telegraphy. This equation can be integrated by the method of Riemann. In particular, Green’s function is the Bessel function of zero order. A simple example illustrates the general method.

Shear Bearing Capacity Prediction of Steel-Fiber-Reinforced High-Strength Concrete Corbels on Modified Compression Field Theory

In order to analyze the shear mechanism of the steel-fiber high-strength concrete corbels, a calculation model for the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels was proposed based on the modified compression field theory. Considering the existence of residual tensile stress in steel-fiber-reinforced concrete at crack locations, the cracked steel-fiber-reinforced concrete was treated as a continuous material. The constitutive relation of cracked steel-fiber-reinforced concrete and the local stress equilibrium equation were modified. It was compared with the results of 34 steel-fiber high-strength concrete corbels, including those in this paper. The predicted results were compared with the experimental values and the predictions of the Fattuhi model, Campione model, and Russo model to validate the rationality of the proposed model. The results revealed that the mean value between the experimental values and the predicted results of the proposed model is 1.104, with a variance of 0.003, showing good agreement. The proposed model can accurately predict the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels.

A new C0 continuous refined zigzag {1,2} finite element formulation for flexural and free vibration analyses of laminated composite beams

This study presents a novel C0 continuous refined zigzag finite element formulation {1,2}, namely RZE{1,2}, for the bending and free vibration analyses of laminated composite beams. The Refined Zigzag Theory (RZT) effectively combines accuracy and computational efficiency, making it a robust approach for thin and thick laminated composite structures. The RZT eliminates the need for shear correction factors, thereby enhancing the overall streamline of the analysis process. The present RZE{1,2} formulation takes into account the transverse stretching by introducing quadratic through-thickness variations of deflection components. The governing equations of the RZT are derived by means of Hamilton’s principle. The through-the-thickness variations of the transverse shear and normal stresses are calculated by integrating the stress equilibrium equations in a post-processing step. Therefore, the Peridynamic Least Squares Minimization (PDLSM) approach is utilized to obtain precise derivatives of axial stresses in the stress equilibrium equations. A new finite element formulation is built up with 3-nodes with a total of 15 DOF. In order to study the influence of transverse stretching, different types of boundary conditions and material variations are applied for laminated composite beams for the bending and free vibration analyses. The distribution of displacements as well as the axial and transverse stresses of the thick beams are extensively examined. The outcomes of RZT are in good agreement with the reference solutions in the literature. The findings of the present study reveal that the presence of the transverse stretching produces more realistic predictions for thick beams where the shear deformations are influential.

Stress analysis of laminated and sandwich beams subjected to concentrated load by using quasi-two-dimensional theory

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

Numerical solution for nonlimit-state earth pressure considering interlayer shear stress and the soil arching effect

The earth pressure of multilayer cohesive soils is affected by interlayer interaction and the soil arching effect. However, only the soil arching effect is considered in the existing methods for calculating earth pressure, and the interlayer shear stress caused by principal stress deflection is ignored. For a vertical rigid retaining wall with a horizontal backfill surface, the trajectory of the minor principal stresses in the soil behind the wall takes the form of an arc of a circle. On this basis, the horizontal and vertical stress equilibrium equations of the differential soil elements behind the wall are derived. The interlayer shear stress, wall displacement and soil arching effect are incorporated into the horizontal slice model, and the numerical scheme of the nonlimit-state earth pressure, the resultant force and the location of the action point are determined with the numerical solution method. Furthermore, the rationality of the proposed solution is demonstrated through comparison with experimental data and other solutions.

Mechanics and Energetics of Kink Deformation Studied by Nonlinear Continuum Mechanics Based on Differential Geometry

This study conducts dislocation-based modeling and numerical analysis for kink deformation using the theory of nonlinear continuum mechanics based on differential geometry. The kinematics of the continuum is represented by the reference, intermediate, and current states, which are related to the multiplicative decomposition of the deformation gradient into plasticity and elasticity. The intermediate state is determined by the Cartan first structure equation, whereas the current state is obtained from the stress equilibrium equation. Kink deformation is modeled by a planar array of edge dislocations, and the governing equations are solved numerically using the finite element method. Quantitative verification of the model is conducted by comparing the bending angle at a kink interface and the theoretical prediction given for the tilt grain boundary. We construct two types of ortho-type kink models for several interface lengths to understand the energetics and mechanics of the kink growth process. The present numerical analysis demonstrates the significant stress concentration at the growth front due to the formation of wedge disclination. We also discuss the material strengthening mechanism from elastic strain energy, nonsingular stress fields, and nonlinear elastic interaction between the disclinations.

Технологические режимы раздачи и обжима при локальном нагреве

The processes of spread and reduction of a heated rough piece under visco-plasticity conditions are viewed. The ratio for force stress calculating in operations, damage to the material of the rough pieces is obtained. In the branches of special en-gineering, high-strength alloys based on titanium and aluminum are used. In the branches of special engineering, high-strength alloys based on titanium and aluminum are used. These alloys have mechanical properties anisotropy. Processing of these alloys is difficult. For this reason, the pressure treatment operation is performed with heating of the deformation zone. The material in the deformation zone exhibits viscous properties. Deformation hardening and softening (stress relaxa-tion) of the material take place simultaneously. Besides, the lower the deformation rate, the greater the softening. In this regard, a constitutive equation representing these processes is found. The factor of hardening and softening creates condi-tions for reducing the power mode of pressure treatment operations and increasing the degree of primary part forming. Stress relaxation calculation with the help of analytical dependencies is necessary at the stage of expansion and pressing development. The calculated ratios are recorded as a function of the speed of these operations. In this case, the specified deformation (the change in the degree of forming) is taken into account, adjusted depending on the speed and mechanical characteristics of the bearing alloy anisotropy. The calculated ratios are obtained under conditions of a flat voltage scheme, which corresponds to expansion and pressing. Stress equilibrium equation and yield condition of anisotropic material are used. The joint solutions of this equation and yield conditions determine values of the meridional and circumferential stress-es arising in the piece part material. The values of the stresses allow calculating the forces of operations. It is shown that the speed of expansion and pressing and mechanical properties anisotropy affect the damage to the material of the «green body». Dependences for the calculation of damage are obtained on the basis of energy and deformation strength criteria. These dependencies allow predicting the quality of products. It is also shown that anisotropy affects the technological modes of expansion and pressing. As the anisotropy coefficient increases, the stresses and forces of operations decrease. Calculations of stresses, forces and material damage in the process of expansion of anisotropic titanium alloy VT14 at 875 ℃ are made.

Study on Stress and Displacement of Axisymmetric Circular Loess Tunnel Surrounding Rock Based on Joint Strength

The development of an effective evaluation method suitable for loess-tunnel excavation is necessary to avoid the collapse accidents caused by tunnel excavation and any secondary disasters. Although the Fenner formulas and the modified Fenner formulas are widely used in tunnel engineering, a defect still exists in these formulas because the Mohr–Coulomb (M–C) criterion exaggerates the tensile strength of the surrounding rock of the loess tunnel. A newly modified Fenner formula was derived based on joint strength to overcome this deficiency. First, the expressions of stress and the radius of the plastic zone of the surrounding rock of the loess tunnel and the expressions of radial displacement were derived based on the stress-equilibrium equation of the axisymmetric plane and the joint strength. Then, the difference in the modified Fenner formulas based on the two kinds of strength criteria for the loess tunnel were compared. The results showed that the radius of the plastic zone and the radial displacement of the loess tunnel determined by the modified Fenner formula based on joint strength were larger than those determined by the modified Fenner formula based on M–C strength. However, the plastic stress of the plastic zone determined by the modified Fenner formula based on joint strength was smaller. The comparative analysis reveals that the modified Fenner formula based on joint strength can evaluate the stress and plastic-displacement field of the surrounding rock of a loess tunnel more reasonably.

Stress Distribution in Direct Shear Loading and its Implication for Engineering Failure Analysis

Shear stress concentrations may promote damage and failure processes. Quantities of studies have focused on the direct shear loading test, while the analytical model has not yet been studied in depth. Aiming to fill the knowledge gap, the theoretical and numerical analyses of the shear stress distribution in the shear band were investigated. In order to reflect the variation in the stress state, the differential element method was first used. The shear stress distribution equation was derived from the stress equilibrium, geometric and physical equations. The shear stress distribution was plotted, using the proposed equation. After that, the ratio of yield strength to crack initiation strength was calculated. The analytical model was analyzed with FDEM simulation, and the results were compared with those obtained from the laboratory tests. Using the elastoplastic theory, the damage evolution and process in rock were characterized from laboratory scale. The implication for underground engineering analysis was finally discussed with a case study of strain rockburst in hard rock. The analytical model and results could provide a fundamental basis for stability analysis in geotechnical engineering.

Stability Analysis of Landfills Contained by Retaining Walls Using Continuous Stress Method

An analytical method for determining the stresses and deformations of landfills contained by retaining walls is proposed in this paper. In the proposed method, the sliding resisting normal and tangential stresses of the retaining wall and the stress field of the sliding body are obtained considering the differential stress equilibrium equations, boundary conditions, and macroscopic forces and moments applied to the system, assuming continuous stresses at the interface between the sliding body and the retaining wall. The solutions to determine stresses and deformations of landfills contained by retaining walls are obtained using the Duncan-Chang and Hooke constitutive models. A case study of a landfill in the Hubei Province in China is used to validate the proposed method. The theoretical stress results for a slope with a retaining wall are compared with FEM results, and the proposed theoretical method is found appropriate for calculating the stress field of a slope with a retaining wall.

New unified family of GBT deformation modes for the analysis of thin-walled cylinders

Generalised Beam Theory assumes the displacement field of a thin-walled member as a linear combination of deformation modes. The main family of deformation modes is constructed based on two simplifying assumptions: null membrane shear strain and null membrane transverse extension. If these modes are not sufficient for an accurate representation of the structural behaviour, two other families of modes are derived (from the main family) for which the two simplifying assumptions are eliminated. In this paper, for the special case of thin-walled cylinders, all three families of deformation modes are replaced by a new unified family, for which the membrane shear strain and transverse extension are not neglected. The derivation of the proposed modes is based on the stress equilibrium equations, and it involves the trigonometric series representation of the longitudinal modal amplitudes. The advantages of the proposed modes are highlighted throughout several linear buckling analyses with various loading and boundary conditions.

Analytical analysis for non-homogeneous two-layer functionally graded material

Abstract In this study, the nonlinear analytical analysis of a two-layer geometry made of functionally graded materials (FGMs) is examined. FGMs can be used in various engineering applications, such as building materials in civil engineering, due to the advantages of smoothly varying properties. The equations of stresses and displacements in the radial and circumferential directions (r, θ ) have been found by extracting the governing equations and defining them in the form of power-exponential functions. In the present paper, modulus of elasticity and heat conductivity coefficient (except for Poisson’s coefficient) are assumed to be expressed by power-exponential functions in radial and circumferential coordinates. The temperature distribution is also considered as a function of radius (r) and angle (θ). The analysis is implemented based on the theory of small elastic deformations and with the assumption of a very large length in plane strain mode. To analyze the governing equations, first, the heat transfer equations are obtained, and then the Navier’s equations are derived by combining the stress–strain, strain–displacement, and stress equilibrium equations. Then, the displacement equations and stress equations are obtained by solving the Navier’s equations. A direct method is presented to solve these equations analytically.

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds.Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

Effect of body weight on the L3 lumbar vertebra with lumbar spinal stenosis in different body-bending degrees: A finite element simulation

Lumbar spinal stenosis (LSS) is a common disorder of the human lumbar spine found in elderly people. The disease can cause severe pain in patients, especially when people bend down. The main purpose of this paper is to develop the finite element simulation to study the effect of the weight acting on the lumbar spine when people bend down with various degrees. The computational domains of the three-dimensional L3 lumbar spine were constructed from CT scan images from the intact spine and the spine with LSS. The mathematical model was developed in order to compare the displacement and the von Mises stress between both lumbar spines. The simulations based on the stress equilibrium equation, the strain–displacement relation, and the generalized Hooke’s law were achieved using the finite element method. The result reveals that a high level of displacement occurs around the transverse process and the spinous process of both lumbar vertebrae when a person bends down. Moreover, the investigation shows the von Mises stress that occurs in a patient with LSS is higher than in a normal person. In addition, the von Mises stress increased with an increase in the bending degree.