Shear Stress Components Research Articles

Permanent deformation and particle crushability of calcareous sands under cyclic traffic-induced loadings through simple shear apparatus

This study focuses on investigating permanent deformations, encompassing normal and shear strains, of calcareous sandy soils subjected to drained cyclic traffic-induced loadings. The investigation utilizes a Simple Shear (SS) apparatus allowing for variations in both normal and shear stress components over each cycle and incorporates Principal Stress Rotation (PSR), a feature not replicable in conventional cyclic uniaxial triaxial tests which is the key aspect of this research. The study also accounts for the harmonically changing the effective horizontal stress resulting from cyclic variations of effective vertical stress, conducted under a horizontally constrained boundary condition with zero lateral strain. A series of drained cyclic simple shear experiments is carried out, implementing a heart-shaped stress path, encompassing up to 1000 cycles. The objective of the study is to analyze permanent normal and shear deformations, along with associated total particle crushing, using both sieving analyses and 2D image processing of particles. The study also evaluates the impact of an initial static shear stress originating from the longitudinal slope of roads. The findings highlight the influences of induced cyclic amplitudes of stress components, principal stress and strain rate rotation, and initial static shear stress on the development of permanent strains. Furthermore, the research characterizes particle crushability in terms of total crushability over such loading, examining its dependency on relative density and variations in both shear and normal stress components.

Comparative study of a femoral amputated limb with and without implant

The contact pressure at the socket–residual limb interface is the most critical parameter for evaluating the comfort of a leg prosthesis. Experimental studies have analyzed this parameter for typical postures and during walking on a flat surface of an amputee; however, experimental tests require a real socket prototype equipped with transducers. To optimize socket design, this work presents a virtual approach based on a digital avatar of a patient wearing a lower limb prosthesis. Our study integrates two different types of simulations: the first concerns the femur with an implant, the second without an implant, and includes the evaluation of pressure at the socket–residual limb interface using finite element analyses (FEA). The objective of this study is to understand the distribution of loads and stresses at the interface between the residual limb and the prosthesis. The lower contact pressure is represented by CPRESS, while CSHEAR1 is the frictional shear stress component in the first local tangent direction, and CSHEAR2 is the frictional shear stress component in the second local tangent direction.

Direct numerical simulation of turbulent, generalized Couette–Poiseuille flow

We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette–Poiseuille flow. A particular example is specified by spherical coordinates $(Re,\theta,\phi )$ , where $Re = 6000$ is a global Reynolds number, $\phi$ denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and $\tan \theta$ specifies the ratio of the mean volume-flow speed to the plate speed. The limits $\phi \to 0^\circ$ and $\phi \to 90^\circ$ give alignment and orthogonality, respectively, while $\theta \to 0^\circ,\ \theta \to 90^\circ$ correspond respectively to pure Couette flow in the $x$ direction and pure Poiseuille flow at angle $\phi$ to the $x$ axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of $(Re,\theta,\phi )$ . The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with $\phi =90^\circ$ . With increasing $\theta$ , the energy peak gradually moves in the direction of increasing $k_x$ and decreasing $k_z$ . Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the $Q$ -criterion visualization of a three-dimensional time-averaged flow.

New analytical model for multi-layered composite plates with imperfect interfaces under thermomechanical loading

AbstractA new analytical model for thermoelastic responses of a multi-layered composite plate with imperfect interfaces is developed. The composite plate contains an arbitrary number of layers of dissimilar materials and is subjected to general mechanical loads (both distributed internally and applied on edges for each layer) and temperature changes, which can vary from layer to layer and along two in-plane directions. Each layer is regarded as a Kirchhoff plate, and each imperfect interface is described using a spring-layer interface model, which can capture discontinuities in the displacement and stress fields across the interface. Unlike existing models, the governing equations and boundary conditions are simultaneously derived for each layer by using a variational procedure based on the first and second laws of thermodynamics, which are then combined to obtain the global equilibrium equations and boundary conditions for the multi-layered composite plate. A general analytical solution is developed for a symmetrically loaded composite square plate with an arbitrary number of layers and imperfect interfaces by using a new approach that first determines the interfacial normal and shear stress components on one interface. Closed-form solutions for two- and three-layer composite square plates are obtained as examples by directly applying the general analytical solution. Numerical results for two-, three- and five-layer composite plates under different loading and boundary conditions predicted by the current model are provided, which compare well with those obtained from finite element simulations using COMSOL, thereby validating the newly developed analytical model.

Physical aspects of epithelial cell-cell interactions: hidden system complexities.

The maintenance of homeostasis and the retention of ordered epithelial cell self-organization are essential for morphogenesis, wound healing, and the spread of cancer across the epithelium. However, cell-cell interactions in an overcrowded environment introduce a diversity of complications. Such interactions arise from an interplay between the cell compressive and shear stress components that accompany increased cell packing density. They can lead to various kinds of cell rearrangement such as: the epithelial-to-mesenchymal cell state transition; live cell extrusion; and cell jamming. All of these scenarios of cell rearrangement under mechanical stress relate to changes in the strengths of the cell-cell and cell-matrix adhesion contacts. The objective of this review study is twofold: first, to provide a comprehensive summary of the biological and physical factors influencing the effects of cell mechanical stress on cell-cell interactions, and the consequences of these interactions for the status of cell-cell and cell-matrix adhesion contacts; and secondly, to offer a bio-physical/mathematical analysis of the aforementioned biological aspects. By presenting these two approaches in conjunction, we seek to highlight the intricate nature of biological systems, which manifests in the form of complex bio-physical/mathematical equations. Furthermore, the juxtaposition of these apparently disparate approaches underscores the importance of conducting experiments to determine the multitude of parameters that contribute to the development of these intricate bio-physical/mathematical models.

Prediction of fatigue life of a bolted joint in railway steel arch bridge using multiaxial fatigue criteria

Fatigue represents a critical condition for infrastructure subjected to repeated cyclic loads, such as steel railway bridges. The literature offers several methods to estimate fatigue life, consisting of three main phases: cycle counting, fatigue damage criterion, and damage accumulation criterion. Applying S-N curves might not be conservative in the case of railway bridges subjected to traffic because the stress-time history is complex and cannot be reduced to a sinusoidal history. Additionally, the presence of a multiaxial stress state must be considered. Therefore, this work compares eight multiaxial fatigue damage criteria for evaluating fatigue life in a scenario between high and low-cycle fatigue. Specifically, the authors considered four low-cycle fatigue criteria, namely Smith–Watson–Topper (SWT), Kandil, Brown and Miller (KBM), Glinka, Fatemi and Socie (FS), and four high-cycle fatigue criteria, Crossland, Basquin and the methods recommended by Eurocode 3 and British Standard. The rainflow counting method in ASTM E1049-85 (2011) and Miner’s rule for damage accumulation were used. The Polcevera railway steel bridge was selected as a case study. A 3D numerical model was developed for this purpose using Midas Civil software, taking into account the material non-linearity of the bridge’s elements. Once the area with the highest stress concentration was identified, a detailed analysis was conducted to estimate the stress and strain time histories induced by train traffic. A sensitivity analysis was conducted after critically comparing the eight methods for predicting fatigue life to assess the impact of traffic parameters, train velocity, axle load, and convoy length on fatigue life. It has been found that the criteria considering axial stress tend to overestimate the number of fatigue cycles to failure compared to the criteria, including the effect of shear stress components.

Experimental investigation of flow and sediment transport on an equilibrium beach formed by surging breaker

Flow patterns and sediment transport induced by breaking waves on a beach profile have been studied in a laboratory wave flume. Two parallel experiments were conducted on the same profiled rigid bed and sediment (mobile) bed, each subjected to identical wave forcing. The rigid bed experiments include measurements of bed shear stress and water-surface elevation, while the sediment bed experiments involve the measurement of pore-water pressures. The beach profile features a beach step leading to the generation of a hydraulic jump at the end of the rundown phase and accordingly forms a backwash vortex. The bed shear stress measurements revealed that the maximum onshore directed mean bed shear stress occurs after the wave breaking whereas the offshore directed one occurs during rundown at the same section. The increase of the bed shear stress with respect to that in the approaching wave boundary layer during the wave breaking and hydraulic jump processes can be by as much as a factor of 13 and 4, respectively. The effect of the wave breaking on turbulence is also more pronounced by a factor of 2.2 than the hydraulic jump in terms of r.m.s values of the fluctuating component of the bed shear stress. The pore pressure measurements showed that the vortex generated during the wave breaking creates an impulsive, upward-directed pressure gradient force that can be up to 1.2 times the submerged weight of the sediment. The sediment transport occurs in the sheet flow regime in a large part of the beach profile. The turbulence generated by the backwash vortex and the hydraulic jump at the beach step directly impacts the local sediment suspension which serves as the source for the advected sediment in the swash zone. The results are compared with a previous study that used the same methodology but involved plunging waves and the initial bed profile.

A refined model for functionally graded graphene nanoplatelets reinforced composite beams under thermal loads

Graphene is highly regarded as a promising material for various engineering purposes due to its remarkable physical characteristics. However, existing higher-order shear deformation theories may struggle to accurately capture the interlaminar mechanical response of functionally graded graphene nanoplatelets (FG-GNP) reinforced composite beams subjected to thermal loads. This arises from the pronounced discrepancies in material properties and thermal expansion coefficients among the various layers, making the interlaminar mechanical behavior of composite structures exceedingly complex under thermal loading. As a result, a meticulous analysis of interlaminar stresses becomes necessary for the structural assessment of GNP-reinforced composite beams under temperature variations. To overcome this limitation, we propose an advanced model incorporating transverse normal thermal deformation, tailored specifically for FG-GNP reinforced composite beams subjected to thermal loads. The developed model primarily boasts two advantages. Firstly, although transverse normal deformation is introduced into the displacement field, no additional displacement variables are increased, thereby maintaining model simplicity and efficiency. Secondly, the model achieves a high-precision interlaminar shear stress field taking into account the interlaminar continuity conditions and temperature effects. Furthermore, the elimination of second-order derivatives of in-plane displacement parameters from the transverse shear stress components leads to a simplification in the finite element implementation. Based on the proposed model, a simple three-node beam element is constructed. The 3D elasticity solutions and the results from alternative modeling frameworks are used to assess the performance of the developed model. Additionally, a comprehensive investigation is conducted to explore the effects of various parameters on the deformations and stresses of GNP-reinforced composite beams.

Stagnant rings and uniform film analysis of Phan-Thien Tanner fluid film flow on a vertically upward moving tube

Analyzing Phan-Thien Tanner fluid film behavior on a vertically upward moving tube can help predictive models in engineering, notably in coating and lubrication operations. This paper provides a theoretical analysis of the dynamics of stagnant rings and uniform Phan-Thien Tanner fluid film adhered to a vertically upward moving tube. Formulated ordinary differential equations are solved to get exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant rings. Highly nonlinear algebraic equations are solved using Newton's method in MAPLE to find the linear and exponential Phan-Thien Tanner film thicknesses. The uniform film thickness widens with increasing constant tube velocity, while it decreases with increasing Deborah number, Stokes number, and elongation parameter. The analysis delineates that stagnant rings shrink around the tube as the Stokes number, Deborah number, and elongation parameter increase. The minimal Stokes condition for the existence of a realistic stagnant ring is also determined. It has been established that whenever the Stokes number is less than the minimal Stokes condition, stagnant rings do not form. A comparison between the exponential Phan-Thien Tanner, linear Phan-Thien Tanner, upper convected Maxwell, and Newtonian fluids is also provided for stagnant rings and fluid film thickness. Following the application of an efficient approximation on tube geometry, plate geometry is approximated, and the outcomes are consistent with existing literature. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

Wall Shear Stress Characteristics Of A Turbulent Boundary Layer Obtained With Multi-Aperture Defocusing Micro-PTV And High-Speed Stereo Profile-PIV

A multi-aperture 3d 3c micro PTV system is presented which relies on single window optical access for both high-speed tracer illumination and image recording. Similar to the “defocusing” concept described by C. Willert & Gharib (1992), the wall distance of individual particles is obtained from the size of projected particle image triplets formed by a triplet of apertures on the entrance pupil of the microscope lens. The present article extends upon previously published material Klinner & Willert (2022) by applying multi-aperture micro particle tracking velocimetry (MA-micro PTV) on a canonical turbulent boundary layer (TBL) to track the near-wall motion of tracer particles with the aim of estimating the unsteady wall-shear stress from particle tracks. The technique is validated with measurements of a TBL inside the closed test section of the 1 m wind tunnel of DLR in Göttingen (1mWK) at free- stream velocities of 5.2 to 20 m/s with corresponding shear Reynolds numbers between 560 and 1630. In the viscous sublayer, the probability density distributions of stream- and span-wise wall shear stress (WSS) components could be reliably captured down to probability densities of 10E−3. As far as we know, this has not been achieved in the past, particularly not for the span-wise component. For the stream-wise component the measured Reynolds-number dependency of the root mean squared fluctuations agrees to correlations by Örlü & Schlatter (2011). Furthermore, the skewness of the stream-wise WSS agrees well to values from DNS. On the other hand, the span-wise WSS fluctuations consistently underestimate the predicted DNS values, for which it is assumed that the deviation can be explained by the rapid decrease of the span-wise velocity fluctuations with increasing wall distance. Wall-normal profiles of 3c velocity statistics were obtained by bin averaging of particle velocities. Up into the buffer layer, these profiles agree well with profiles from stereo PIV as well as from DNS and LES, indicating an accurate determination of both near-wall velocities and inner scaling.

Analysis of unsteady internal flow characteristics in axial pump with varying number of blades using computational modelling and vibration techniques

Axial pumps were extensively applied in many varying applications because of their large flow and low head. In this research investigation, the comparative analysis of the findings unveiled that the predominant source of hydraulic-induced vibration in the pump stemmed from pressure pulsations at the impeller inlet. Notably, similar patterns in amplitude were observed as flow rates increased. Furthermore, time-domain analysis confirmed that pressure pulsations and vibrations are highly correlated under high flow rates. Pressure pulsation's Frequency-domain analysis also revealed that it was a multiple of shaft frequency and changed from one multiple to three multiples of vibration. While analyzing the flow rate characteristics pertaining to pressure pulsation and vibration, it was determined that pressure pulsations at the impeller inlet had the potential to generate frequency components across a broad spectrum, encompassing both low and high flow rates, as well as their respective multiples. This phenomenon was particularly pronounced in regions characterized by unstable and high flow rates. Vibration ingredients likely influenced by pressure pulsation at the impeller inlet could be as low as design flow rates and as high as high flow rates. A vibration frequency with a multiple did not seem to be influenced by pulsating pressure at the impeller entrance. The present study focuses on investigation of flow behaviors in an axial pump with varying numbers of blades. It is an important geometric parameter that significantly affects the pump's performance. Therefore, the dynamic flow patterns in a pump, considering changed flow and impeller blade configurations, are investigated by employing the sliding mesh technique in combination with SST (k-ω) turbulence model. The numerical results exhibit a commendable alignment with the existing experimental data, enhancing the predictive accuracy of pump performance. Qualitative analyses encompass static pressure, shear stress, and various velocity components. Concurrently, quantitative investigations delve into pressure fluctuations and average pressure across a spectrum of operating conditions and impeller blade configurations. These comprehensive findings underscore the substantial influence of the impeller blade on pressure, velocity magnitudes radial, axial, tangential shear stress, average pressure within the system.

Numerical analysis of buried pipelines under seismic slope instability

Damage to buried pipes under seismic landslide actions has been reported in many post-earthquake reconnaissance. The landslide-pipe problem in the technical literature has been often investigated using simplified analytical methods. However, the analytical methods ignore the real mechanism of pipe response under natural dynamic slope instability. The dynamic slope instability is significantly influenced by its lateral boundary interface (LBI) characteristics. In this study, slope-pipe interaction (SPI) under seismic loading, focusing on the effect of LBI properties, is evaluated by continuum numerical simulation using the SANISAND constitutive model in FLAC3D. The results show that the geometry of the failure mass varies from 2D to 3D by increasing the stiffness at the slope boundaries (from smooth to hard) and the maximum pipe deformation decreases by around 40%. Moreover, the response components of maximum axial stress, bending moment, and shear stress of the pipe occur at the end sections of the buried pipe and near the boundaries of the landslide zone. However, the maximum pipe deflection occurs in the middle section of the pipe. The results of shear force-shear displacement curves demonstrate that the soil-pipe interaction stiffness is variable along the pipe length and can be estimated by a hyperbolic equation.

A posteriori study on wall modeling in large eddy simulation using a nonlocal data-driven approach

The feasibility of wall modeling in large eddy simulation (LES) using convolutional neural network (CNN) is investigated by embedding a data-driven wall model developed using CNN into the actual simulation. The training dataset for the data-driven wall model is provided by the direct numerical simulation of turbulent channel flow at Reτ=400. The data in the inner layer, excluding y+≤10, are used in the training process. The inputs of the CNN wall model are the velocity components, and the outputs of the wall model are the streamwise and spanwise components of the wall shear stress. An a priori test has already been carried out in our previous study to assess the potential of CNN in establishing a wall model, and the results have shown the reasonable accuracy of the CNN model in predicting the wall shear stress. In this study, the focus is on the a posteriori test, and the performance of the CNN wall model is investigated in the actual LES under various conditions. Initially, the model is used in a simulation with the same specifications as those used for obtaining the training dataset, and the effect of the wall-normal distance of the CNN model inputs is investigated. Then, the model is tested for coarser grid sizes and higher Reynolds number flows to check its generalizability. The performance of the model is also compared with one of the commonly used existing wall models, called ordinary differential equation (ODE)-based wall model. The results show that the CNN wall model has better accuracy in predicting the wall shear stress in the a posteriori test compared to the ODE-based wall model. Moreover, it is able to predict the flow statistics with reasonable accuracy for the wall-modeled LES under various conditions different from those of the training dataset.

Acoustic emission and electromagnetic radiation of rock under combined compression-shear loading

This paper reports the use of inclined sandstone specimen in the uniaxial compression tests to achieve the combined compression and shear loading (CCSL) state. A series of CCSL tests are performed on the inclined specimens with an inclination angle of 0°, 3°, 5°, and 7°. Acoustic emission (AE) and electromagnetic radiation (EMR) during the failure process are monitored. The results show that the additional shear stress component caused by the inclined shape configuration of tested specimens has significant effects on the failure mode and mechanical properties. The increase of inclination causes the failure mode to transit from the axial splitting to shear failure, and decreases the peak strength and elastic modulus of the inclined specimens. The specimen inclination lowers the damage prior to the peak stress, but stimulates the damage development after the peak stress. EMR has better correlation with stress drop than AE. The 65% of the peak strength is defined as the stress threshold of rock failure. The inclination angle lowers stress threshold of rock failure and accelerates failure initiation

Mechanical Analysis and Simulation of Wood Textile Composites

Wood Textile Composites (WTCs) represent a new and innovative class of materials in the field of natural fiber composites. Consisting of fabrics made from willow wood strips (Salix americana) and polypropylene (PP), this material appears to be particularly suitable for structural applications in lightweight construction. Since the threads of the fabric are significantly oversized compared to classic carbon or glass rovings, fundamental knowledge of the mechanical properties of the material is required. The aim of this study was to investigate whether WTCs exhibit classic behavior in terms of fiber composite theory and to classify them in relation to comparable composite materials. It was shown that WTCs meet all the necessary conditions for fiber-reinforced composites in tensile, bending, and compression tests and can be classified as natural-fiber-reinforced polypropylene composites. In addition, it was investigated whether delamination between the fiber and matrix can be simulated by using experimentally determined mechanical data as input. Using finite element analysis (FEA), it was shown that the shear stress components of a stress tensor in the area of the interface between the fiber and matrix are responsible for delamination in the composite material. It was also shown that the resistance to shear stress depends on the geometric conditions of the reinforcing fabric.

The effect of highly elongated micro-contacts on rolling contact stress field

This study presents a boundary element model to analyze contact pressure distribution and the subsurface stress field in rough rolling elements. The methodology is demonstrated by the stress field of the backup roll and work roll in the cold/temper rolling process and characterized by a parallel orientation of roughness lay with rolling direction, resulting in a cluster of highly elongated elliptical micro-contacts. The numerical outcomes revealed that the amplitude of in-plane and out-of-plane shear stress components are strongly dependent on the ellipticity of the micro-contacts. Comparison with experimental results shows that the crack plane is oriented transversal to the rolling direction and the crack initiation location corresponds to the location where the shear stress components are maximum.

In situ measurement of three-dimensional intergranular stress localizations and grain yielding under elastoplastic axial-torsional loading

The three-dimensional grain-averaged response of solid bar samples under non-proportional (NP) elastoplastic axial-torsional loading was investigated using in situ high energy diffraction microscopy (HEDM) and companion crystal plasticity finite element (CPFE) modeling. Important stress metrics including applied shear (σθZ) and axial (σZZ) stress tensor components, stress and stress deviator tensor invariants (I1, J2, and J3), von Mises equivalent stress (σVMgrain), maximum resolved shear stress (mRSS), stress triaxiality (η), and lode angle parameter (θ‾) values were tracked for ∼ 300 grains under two different loading conditions: (1) Torsion-dominated loading (low NP) and (2) Tension-torsion loading (high NP) in equiatomic NiCoCr, a representative multicomponent face-centered cubic (FCC) superalloy. Overall, significant stress localizations existed within both samples as evidenced by the radial dependence of grain-resolved σθZ, σVMgrain , and J2; by comparison, I1, J3, η, and θ‾ metrics did not show discernible trends within the volume. These stress localizations reveal a complex interplay between axial and shear stress components (e.g., stress coupling) resulting in grain yielding near the sample surface largely driven by shear stress, whereas internal grain yielding was largely accommodated by axial stress. Grain-resolved stress localization trends were described well by the CPFE model, although some discrepancies in magnitude occurred, particularly for volumetric stress metrics (I1 and η) due to initial type II residual stress distributions. The superposition of initial residual stress states onto CPFE grain-resolved data significantly improved model accuracy for η. This suggests that residual stresses more strongly influence the simulation of volumetric rather than deviatoric (yield) stress metrics.

Quantitative shear stress imaging in high throughput in-line Si wafer inspection

Defect inspection methods are basic steps in electronic chip manufacturing and power device production processes in the semiconductor industry. Based on stress induced photoelastic anisotropy, in-line polarized stress imaging (PSI) and quantitative determination of the local shear stress component are possible in single crystalline silicon wafers. Hereafter, such polarization stress images are compared and validated quantitatively with results of finite element simulations based on different 3D elastic models. Uniquely, high resolution at high throughputs is achieved for 300 mm wafer size in the present quantitative stress imaging study.

Axisymmetric torsion of rigid disc parallel to penny shaped crack at the interface of layer bonded by two orthotropic half spaces

Two half-spaces and a layer of finite height of linear elastic orthotropic medium containing a penny-shaped crack and a circular rigid disk are considered such that the crack surface is parallel to the rigid disk, where both are situated at two different interfaces of layer and half-space and perpendicular to the axis of symmetry. The rigid disk is agitated by an axisymmetric torsion. The equations of equilibrium are solved with the use of Hankel’s integral transform method. Then, the tangential displacement components and shear stress components are determined. With the help of mixed-type boundary conditions, a system of integral equations is derived. After that, by utilizing a trial solution, these integral equations are transformed into a pair of simultaneous Fredholm integral equations of the second kind. The numerical results of the Fredholm integral equations are evaluated by converting it into a system of algebraic equations to interpret the physical quantity, like the stress intensity factor (SIF), which is demonstrated graphically. The obtained results are compared to the previously published article to ensure its accuracy.

New Approach of Normal and Shear Stress Components for Multiple Curvilinear Holes Which Weakened a Flexible Plate

In this article, a thin infinite flexible plate weakened by multiple curvilinear holes is considered. The strength shapes are mapped outside a unit circle with the assistance of particular conformal mapping under certain conditions. The mathematical model that governs the rounded forces of the current physical problem is the boundary value problem of elastic media. This study is applicable to many phenomena throughout nature, like tunnels, caves, and excavations in soil or rock. The Cauchy method for complex variables is used to get the closed forms of Gaursat functions and change the problem to a second-type integrodifferential equation with a Cauchy kernel, which is used for a large area of the contact problems. Then, the normal and shear stress components that act on the model are derived. Afterward, some of the physical applications are studied, and different stress components at specific values in each application are calculated and plotted using Maple 2023.