Analytical Solutions For Stresses Research Articles

Analytical solutions for film stress and bending deformation of coated optical lenses

Axisymmetric film/substrate systems are irreplaceable for many applications, owing to special structural requirements. Stoney's formula is widely used to analyze the thin plate film/substrate system. Many researches have been presented to relax limitations of Stoney's formula, which considered the substrate as a thin plate without any curvature. For the optical lens, the axisymmetric thin shell substrate with different curvatures on its upper and lower surfaces is common due to the need of optical focusing. The film stress of the optical lens needs to be optimized and measured due to its effects on system deformations and optical performances. However, the optical lens cannot be solved by Stoney's formula and its extensions due to the curvatures. In this paper, analytical solutions for the thin shell optical lens with different curvatures on upper and lower substrate surfaces are developed by the thin shell theory with the axisymmetric distribution film stress resultant (FSR) in consideration. The FSR is modeled by thermal stress equivalent method, which allows the calculation for the axisymmetric distribution FSR induced by many factors. The forward and inverse solutions between the FSR and the bending deformation are provided in the form of simple formulas with correction coefficient vectors depending on the global substrate thickness, which enables its feasibility in practical applications. Moreover, the two solutions are verified by the finite element method considering the deformation normal to the substrate surface measured by the interferometer in inverse solution, and small errors are found when the optical lens conforms to the assumptions of the method in this paper. The proposed method is efficient to predict the optical lens’ bending deformations, and extends the usage of curvature-based techniques from plates to shells with different curvatures on upper and lower surfaces in the film stress measurement, which makes great significance to the thin film/substrate application.

Stress and Displacement of Deep‐Buried Composite Lining Tunnel under Different Contact Conditions

Stress and displacement of the composite lining are important factors to be considered during tunnel design. By the complex variable method, analytical solutions for stress and displacement of surrounding rock, primary support and secondary lining satisfying the interface continuity and boundary conditions under far‐field stresses are derived. Taking the railway composite lining tunnel as an example, the analytical distributions of stress and displacement along boundaries are given, which was in good agreement with the numerical solution calculated by finite element software. The results show that the maximum normal stress ratio (load sharing ratio) of the outer boundary between the secondary lining and the primary support is 0.74. The radial displacement of the inner boundary of surrounding rock, primary support, and secondary lining change consistently. The maximum settlement and uplift occur at the vault and bottom, respectively. The tangential stress of secondary lining is compressive stress, while the tangential stress of primary support is tensile stress and compressive stress. The maximum tangential stress of primary support and secondary lining is smaller than the allowable stress of concrete.

Theoretical study about influence of geometry and mechanical load on the delamination in tungsten disulfide/poly(methyl methacrylate) nanocomposite structure under axial load

The influence of the geometry and the magnitude of axially applied mechanical load on the delamination in three-layer tungsten disulphide (WS2)/SU-8/poly(methyl methacrylate) (PMMA) nanocomposite, is investigated theoretically. First, for considered nanostructures with thinner and thicker PMMA layer two different analytical solutions (Case 1 and Case 2) for the interface shear stress (ISS) in the middle layer of the structure are obtained, based on the application of two-dimensional stress-function method and minimization of the strain energy. Second, the theoretical criterion for delamination in the interface layer, based on the model ISS, is formulated and the obtained non-linear equation in respect to debond length is solved numerically, for both solutions, at different values of mechanical load and geometry of the structure layers. It was found that delamination doesn't appear at fixed WS2 length of 10 μm, if the loading is up to 5 GPa for Case 1 and up to 1.175 GPa for Case 2, respectively. With increasing WS2 length, the delamination occurs at increasingly higher values of the applied external load, for Case 2. For Case 1 a delamination is not occurs. At fixed applied load, it was found for Case 2, that as the length of WS2 increases, the debonding length increases, too. The obtained results could be used for fast prediction of delamination in similar nanostructured devices.

Equivalent stiffness calculation of composite hat stiffened laminate

The stiffness is the main reason that restricts the large-scale of composite ships, the basis to solve this problem is to solve the stiffness of composite ships’ skeleton structure. Based on classical laminates theory, this paper calculated the equivalent in-plane stiffness and bending stiffness of composite hat stiffened laminate using equivalent stiffness principle. The hat stiffened laminate can be simplified and equivalent to orthotropic laminates using this solution idea, which solves the problem that it is difficult to obtain the analytical solutions of deformation, stress and strain of stiffened laminate. The research results can be used to calculate the stiffness of composite stiffened laminate and the local or global stiffness of the hull girder with such a skeleton structure.

Stability Analysis of the Surrounding Rock-Lining Structure in Deep-Buried Hydraulic Tunnels Having Seepage Effect

To clarify the factors affecting the stability of deep-buried hydraulic tunnels containing pore water, the elastoplastic theory and the Mogi-Coulomb strength criterion were used to derive the analytical solutions of stress on the surrounding rock-lining structure, tunnel wall displacement, and plastic zone radius in surrounding rock under different operating conditions. During this process, the seepage effect and surrounding rock-lining interaction were considered. The influencing rules of seepage action, intermediate principal stress coefficient, lining permeability coefficient, and lining thickness on the stability of the surrounding rock-lining structure were investigated in depth. The results show that the seepage effect significantly changed the stress distributions in the lining structure and surrounding rock, reduced the bearing reaction force, and lowered the tunnel stability. The bearing reaction force was decreased considerably from the intermediate principal stress, and the plastic zone radius in the surrounding rock and the tunnel wall displacement was obviously reduced. Moreover, the bearing reaction force was reduced, and the plastic zone radius in the surrounding rock and the tunnel wall displacement was increased with the decrease of the lining permeability coefficient. With increasing the lining thickness, the bearing reaction force was enhanced, and an apparent restriction on the development of plastic zone in the surrounding rock appeared at the beginning, but the restriction effect weakened subsequently. This research can theoretically provide references for analyzing the stability of hydraulic tunnels containing pore water.

Analytical stress solution for cold region tunnels with unequal ground stress and support delay under different frost heave conditions of surrounding rock

Stress response analysis of tunnels under the frost heave state is a crucial problem in cold region tunnel construction. Hence, in this study, an analytical stress solution for cold region tunnels with unequal ground stress and support delay under different frost heave conditions is presented. The solution simultaneously considers the unequal ground stress along the horizontal and vertical directions, the rock displacement occurring before support installation, and the different frost heave modes of surrounding rock. To resolve the unequal ground stress boundary condition, the complex function method is utilized, and the stress solution is directly acquired by the method under isotropic frost heave (IFH) condition, whereas the solution under transversely isotropic frost heave (TIFH) condition is acquired through combination with the superposition method. In addition, a displacement release coefficient is adopted to depict the support delay. The solution is validated by existing solutions in degenerate cases. Finally, the stress distributions and the influence of key factors for cold region tunnels are discussed with a case study. In the lining, the radial stress increases with the radius and decreases from 0° to 90°, and the maximum of tangential stress occurs at the inner edge. Moreover, the normal contact stress between the lining and surrounding rock is much greater under TIFH condition; however, the tangential contact stress is identical under the two different frost heave modes. The lateral pressure coefficient shows greater impact on the stress in the lining along the vertical direction, whereas the displacement release coefficient exhibits greater effect on the stress along the horizontal direction. The stress in the lining under TIFH condition is much greater than that under IFH condition, and increases significantly as the degree of anisotropy increases, which implies that the TIFH condition is a much severer state.

A general analytical solution of stresses around circular holes in functionally graded plates under various in-plane loading conditions

A general analytical solution of stresses around circular holes in functionally graded plates under various in-plane loading conditions

Stability Analysis of Shield Excavation Surface with Curved Body Model

Currently, the limit equilibrium method of wedge shape is generally adopted for the stability of the excavation surface in shield tunnel construction. In this method, the sliding surface is approximated by a plane, and the lateral friction force of the sliding surface is estimated and assumed to be a known quantity to derive the equilibrium equation and calculate the support force of the excavation surface. However, it is difficult to accurately estimate the lateral friction force of the sliding surface because of many factors that affect such friction, for example, soil arching. In this paper, a sliding block model of curved surface body composed of a semicircular platform and a partial sphere is proposed for the stability of the excavation surface in shield tunnel construction. By using the symmetry of the surface body model, there is no need for estimating the friction force of the sliding surface during the stability analysis of the excavation surface, and the calculation accuracy of the limit support pressure is improved. Furthermore, the force analysis of the semi-circular platform surface model above the tunnel vault is carried out. Considering the variation of the lateral pressure coefficient caused by the deflection of the principal stress under the soil arch effect, the analytical solution of the vertical stress in the soil is derived when the sliding crack surface is curved. Finally, we present a sensitivity analysis of various parameters affecting the support force of the excavation face through the study of examples. The influencing factors and changing rules related to the support force of the excavation face are also discussed.

Analytical Solutions for Thermomechanical Soil–Structure Interaction in Single Semifloating Energy Piles Embedded in a Layered Soil Profile

Analytical solutions for axial displacement, strain, and stress in a single semifloating energy pile embedded in a layered soil profile subjected to thermal and mechanical loads were derived. An analytical solution for the location of zero displacement, also known as a null point, was derived for a pile subjected to thermal load. It was shown that the location of the thermal null point corresponds to the location of the maximum magnitude of thermal axial stress. It also was shown that the location of the null point moves toward the pile tip as the stiffness of the bedrock below the pile tip increases. The analytical solutions were validated against in situ full-scale energy pile tests. The solutions along with the validation process delineated the load transfer mechanism in energy piles subjected to thermal, mechanical and combined thermomechanical loads embedded in a four-layer soil profile. Flowcharts delineating the procedures for obtaining the analytical solutions for a single energy pile embedded in an arbitrary number of layers, and subjected to thermal and mechanical loads are provided. Although the continuity of stresses and displacements at the interface of different soil layers is maintained, displacement, strain, and stress diagrams exhibit a lack of smoothness, the amount of which depends on the difference in the stiffness of these layers. In summary, the presented solutions provide a rational, mechanics-based framework for advancing the understanding of thermomechanical response of energy piles that not only is essential for analysis and design, but ultimately contributes to a wider use of energy piles and increased sustainability of civil engineering infrastructure.

Investigation of One-dimensional Consolidation of Fractional Derivative Model for Viscoelastic Saturated Soils Caused by the Groundwater Level Change

The Fractional calculus theory is introduced into the Merchant model to study the mechanical characteristics of saturated soils. Laplace transform is applied to the 1D consolidation and the fractional derivative Merchant model constitutive equations of saturated soils. In the transform domain, the analytical solutions of effective stress and settlement are obtained by solving the simultaneous equation. The Crump's method is used to the numerical inversion of Laplace transform, and the semi-analytical solution of the one-dimensional consolidation is obtained. A series of parameters, such as the fractional order, compression modulus, coefficient of viscosity, permeability coefficient and the variation models of groundwater level, are studied. Considering the continuity conditions, the 1D consolidation equation of single-layer saturated soil is generalized to multi-layer soils, and then the influence of different boundaries on several kinds of consolidation behavior was analyzed.

Strength criterion of graphene GBs combining discrete bond strength and varied bond stretch

The strength criterion of graphene grain boundaries (GBs) under complex stress states plays an important role in guiding the design of high-performance graphene-related materials and devices under severe mechanical environments like ultrastrong structural materials or flexible electronics. However, finding a unified strength criterion that can adapt to different GB angles and loading states is still a challenge. In this work, we proposed a general method to construct the strength criterion of graphene GBs that includes the atomic details of defect. Firstly, systematic molecular dynamics (MD) simulations were carried out to explore the fracture behaviors of graphene GBs with different GB angles and loading states. Then, bond strength of heptagon ring σ t h R was obtained by projecting the applied stress field to the fractured bond direction, which gives the same value for the same type of C-C bond of same GB angle under different loading states. Besides, to further unify the bond strength of different GB angles, the residual stress caused by other pentagon–heptagon defects was considered by using the disclination dipole theory, in which the intrinsic bond strength σ t h 0 for different GB angles collapses to the same value. With this scenario, the strength criterion of graphene GBs is presented as the bond stretch stress generated by external load exceeds any bond strength of the hexagon–heptagon defect. Furthermore, a unified bond strength is found, linearly decreasing with the equilibrium bond length, which can further be applied to the strength criterion of other defective structures like two types of SW defects (without analytical solution of residual stress) and even pristine graphene. The present results provide a comprehensive understanding of the failure of graphene by considering the balance of bond strength of defective structure and bond stretch stress generated by external load, which may pave the way to construct the atomic structure based strength criterion of other two-dimensional materials.

Analytical Solutions for a Deep-Buried Tunnel with Noncircular Cross Sections in Orthotropic Rock Mass

For the generalized plane strain problem of a deep-buried tunnel excavated in orthotropic rock mass under far-field biaxial compression, the stress and displacement are the most important basis for determining tunnel support design. First, by using the complex variable method, the conformal mapping and affine transformation functions are represented by the same variable. Two stress functions are obtained by the integral method, and thus, the analytical solutions for stress and displacement in orthotropic rock mass are obtained. Second, taking a horseshoe-shaped tunnel as an example, the stress and displacement distributions around the tunnel are given by using the analytical and the finite element method software ANSYS, respectively. The analytical and numerical results are in good agreement, especially at the tunnel edge. Finally, compared with isotropic rock mass, the results show that material constants of the rock mass have little effect on the stress and significant effect on the displacement.

The Analytical Solution of Stress and Displacement for Multi-layer Cylinder Under Radial Seepage

The Analytical Solution of Stress and Displacement for Multi-layer Cylinder Under Radial Seepage

An analytical analysis for the mechanical performance of fully-grouted rockbolts based on the exponential softening model

This paper presented a novel bond-slip model to better reveal the mechanical behavior of the bolt-grout interface for fully-grouted rockbolts under tensile loads by considering the non-linear response in the softening stage. The exponential decay function is adopted for describing the non-linear response in the softening stage. Based on the improved bond-slip model, the corresponding analytical solutions for the interfacial shear stress and the axial force of the bolt under different loading stages are solved. Then, the validity of this proposed model was verified by comparing with the experimental results. The results show that compared with the linear softening model, the proposed model is more suitable for predicting the mechanical performance of fully-grouted rockbolts. Finally, a series of parametric studies are conducted to explore the effect of model parameters on the mechanical properties of fully-grouted rockbolts. The results indicate that compared with the anchor length, the bolt diameter and the bond strength of the bolt-grout interface have a significance influence on the ultimate load of bolt, especially for the elastic and softening stage. Moreover, it can be found that using the linear softening model maybe overestimates the supporting performance of grouted bolt, resulting in an unsafe design for bolt.

Thermo-mechanical analysis of laminated cylindrical shell with viscoelastic bonding interlayers

This work proposes analytical solutions to present coupling thermo-mechanical analysis of laminated cylindrical shells bonded by viscoelastic interlayers, considering the temperature effects on both thermal expansion difference of adjacent layers and temperature-dependent viscoelastic property of bonding interlayer. In the analytical model, according to the heat transport theory, the solutions of non-uniform temperature fields in the laminated cylindrical shells are solved by means of the transfer matrix method. Then, based on the thermo-elasticity theory as well as the viscoelasticity theory, the governing equations for the laminated cylindrical shell are established. The standard linear solid model is employed to simulate the temperature-dependent viscoelastic property of the interlayer, in which the viscoelastic constants are determined according to the time–temperature equivalence principle. At last, the analytical solutions of time-dependent stresses and displacements are determined by the Laplace transform and the Fourier series expansion. The results indicate the present solutions are consistent with the finite element solutions. The influences of temperature, material and geometric parameters on the long-term bending behavior of the laminated cylindrical shell are studied.

Stress intensity factor for a circumferential crack in a coke drum under two-dimensional transient temperature distribution

Coke drums are subjected to extreme temperature cycling in service, leading to circumferential cracking in the drum wall. For a deeper understanding in the cracking mechanism, stress intensity factor (SIF) for a circumferential crack in a coke drum was investigated in this paper. Based on two-dimensional heat conduction theory and classical thin shell theory, analytical solutions for the transient temperature distribution, displacement, and thermal stress in the drum wall during hot feed filling and water quench steps were determined. The weight function method (WFM) was applied to investigate the mode I SIFs based on the analytical thermal stress solutions. Numerical simulations using finite element method (FEM) were also performed to validate the analytical solutions for both the thermos-elastic field and the SIFs. The effects of shell geometry, operating parameters and material properties on the SIF during the water quench step were discussed. It is shown that the SIF increases significantly as crack depth increases. A greater shell thickness or lower diameter to thickness ratio results in lower SIFs. Lower rising rate and higher temperature of quench water can reduce the SIF. The SIF increases as the thermal conductivity and thermal expansion coefficient of the drum wall materials increase. Finally, the SIFs for Cr–Mo steel and carbon steel coke drums are also discussed.

The effect of the ratio of the radius of the hole and crack length on the multi‐field behaviors of thermoelectric materials under oblique loads

AbstractThis paper establishes a defect model of thermoelectric materials that the oblique multi‐field loads are applied along an arbitrary direction with the crack initiated from a circular hole, and gives both analytical solutions and finite element method (FEM) simulation. On the one hand, in virtue of the conformal mapping function and the Cauchy integral formula, the analytical solutions of electric current density, energy flux, and stress are obtained. Furthermore, the thermoelectric field intensity factors (TEIFs) and the stress intensity factors are also attained. Most notably, we apply the extreme value theorem of multivariate function to explore the effect of the ratio of the radius of the hole, R, and crack length, L, on the fields’ intensity factors of thermoelectric materials under oblique multi‐field loads. Importantly, the maximum value of the mode‐II stress intensity factor (SIF) is obtained when . On the other hand, the finite element simulation is used to explore the fracture behaviors of thermoelectric materials. The analytical solutions and the finite element simulation results are consistent. The effects of the oblique multi‐field loads on the thermoelectric field and the stress field at the crack tip are detailed. The concentration phenomenon of the thermoelectric field and the stress field at the crack tip are influenced by the radius of the circular hole, the crack length, and the directions of the loads.

Note on the Early Thermoelastic Stage Preceding Rayleigh–Bénard Convection in Soft Materials

In this paper, we focus on the first stage of transition to Rayleigh–Bénard convection in soft-jammed systems (yield stress fluids) confined in a parallelepiped box heated from the bottom. Up to yielding, the material is in a solid-state with a constant elastic modulus. By means of a linear thermoelastic model, an analytical solution for stresses and strains induced by the gravity and the temperature gradient is derived. The analytical solution allows us to emphasize the appropriate dimensionless parameters. The onset of plastic deformation is then investigated using the classical yield criteria (Tresca, von Mises and Drucker–Prager). This analysis is subsequently applied to experimental data of the literature dealing with Rayleigh–Bénard convection in Carbopol micro gels.

Thermal analysis for clamped laminated beams with non-uniform temperature boundary conditions

The focus of this study is thermal responses of a clamped composite laminated beam with arbitrary layer numbers under non-uniform temperature boundary conditions. Analytical solutions of temperature, stresses, and displacements are derived based on the theory of thermoelasticity. The temperature distribution in the laminated beam is divided into two parts. The first part is constructed to satisfy the inhomogeneous temperature boundary conditions, while the second part is obtained on the basis of the Fourier law of heat conduction and the temperature environment. On the other hand, the unit pulse function and the Dirac delta function are introduced to translate the clamped support into the simply support with an unknown horizontal stress. According to the continuities at the interface and the state space method, the relationships of displacements and stresses between the top and bottom surfaces of the laminated beam are derived. Finally, the unknown coefficients of displacements and stresses are determined by the mechanical boundary conditions. It can be observed from the numerical results that this method has excellent convergence performance. The accuracy of this method is verified by comparing with the results of the finite element method. Furthermore, the effects of surface temperature, material properties, length-to-thickness ratio and layer numbers on the distributions of temperature, displacements and stresses in the laminated beam are in depth investigated.

Elastic Stress and Plastic Zone Distributions around a Deeply Buried Tunnel under the Nonhydrostatic Pressure

The elastic stress and the plastic zone are the important mechanical parameters to determine the tunnel support design. Based on Muskhelishvili’s complex variable function, the analytical solution for the elastic stress around a deeply buried noncircular tunnel under the nonhydrostatic pressure is firstly derived. The shape and size of the plastic zone of the surrounding rock mass are then determined by substituting the elastic stresses into the Drucker-Prager yield criterion. Finally, taking a horseshoe-shaped tunnel as an example, the analytical solutions of elastic stress distribution and plastic zone shape around the tunnel under different lateral pressure coefficients are in good agreement with ANSYS numerical solutions. The calculation results show that if the vertical in situ stress exceeds the critical value, with increasing lateral pressure coefficient, the hoop stress at the roof and floor of the tunnel increases significantly and the shape and size of the plastic zone change obviously.