Elastic Formulation Research Articles

Regionalization of Climate Elasticity Preserves Dooge's Complementary Relationship

AbstractClimate elasticity of streamflow represents a nondimensional measure of the sensitivity of streamflow to climatic factors. Estimation of such elasticities from observational records has become an important alternative to scenario‐based methods of evaluating streamflow sensitivity to climate. Nearly all previous elasticity studies have used a definition of elasticity known as arc elasticity, which measures changes in streamflow about mean values of streamflow and climate. Using observational records in the western U.S., our findings reveal that elasticity definitions based on power law models lead to both regional and basin specific estimates of elasticity which are physically more realistic than estimates based on arc elasticity. Evaluating the ability of arc and power law elasticity estimators in reproducing Dooge's complementary relationship (DCR) between potential evapotranspiration and precipitation elasticities reveal that power law elasticities estimated from at‐site, panel, and hierarchical statistical models reproduce DCR, whereas corresponding estimators based on arc elasticity cannot reproduce DCR. Importantly, our regional elasticity formulations using panel and hierarchical formulations led to estimates of both regional and basin specific estimates of elasticities, enabling and contrasting streamflow sensitivity to climate across both basins and regions.

Improved Topical Delivery of Sorafenib-Meglumine Antimoniate: Elastic Nano-Liposomal Formulation for the Treatment of Cutaneous Leishmaniasis

Background: Cutaneous leishmaniasis (CL) is a widespread parasitic disease with significant health and social impacts. Current systemic treatments have severe side effects, highlighting the need for effective topical alternatives.Objective: This study aimed to develop and evaluate the effectiveness of sorafenib (SF) and meglumine antimoniate (MM) dual-loaded transferosomes (TRs) for topical treatment of CL.Methods: TRs were prepared using a thin film hydration method and incorporated into a carbopol gel. The formulations were characterized for vesicle size, zeta potential, entrapment efficiency, and deformability index. In vivo and ex vivo skin permeation studies were conducted, along with anti-leishmanial efficacy testing on an intramacrophage amastigote model using BALB/c mice.Results: The TRs exhibited a vesicle size of 186.1 ± 65.89 nm, zeta potential of -27.9 mV, and entrapment efficiency of 71.7 ± 3.9% for MM and 78.6 ± 4.2% for SF. Ex vivo studies showed enhanced skin permeation with cumulative permeation of 327.6 ± 29.4 µg/cm² for MM and 291.6 ± 28.3 µg/cm² for SF. In vivo results indicated a significant reduction in lesion size (0.1 ± 0.12 mm) and parasite load (2.7 ± 0.3 log scale).Conclusion: The MM-SF dual-loaded TRs demonstrated effective topical delivery and therapeutic potential for CL, providing a promising alternative to systemic treatments.

A Comprehensive Model to Compute Roll Gap Profile for a Rolling Mill Including Bearing and Mill Housing Deflections

Abstract Obtaining the desired strip profile is of paramount importance in cold rolling. The final strip profile depends on several parameters of the plants including crown and taper profiles and segmented backup roll positions. This paper aims to present a newly developed combined finite element–analytical model of the rolling mill able to model not only the roll cluster but also the mill housing and backup bearings deformations. This makes the proposed model especially suitable to simulate cluster-type rolling mills. The model exploits a modified elastic foundation formulation for a precise computation of the contact forces between rolls, taking into account rolls’ crown and taper. A reduced stiffness matrix approach is implemented for accurate mill housing modeling. A full 3D finite element model of the rolling mill and experimental data are used for the validation of the presented model. By exploiting the validated model, the paper shows that the deformation of supporting bearings and mill housing plays a crucial role in the strip profile calculation and should be included in any cluster-type roll mill model.

Introducing tunnel kinematic constraints into an elastic continuum formulation of tunnel–soil–pipeline interaction

This paper presents a new formulation of the problem of tunnelling effects on pipelines, which incorporates the tunnel kinematic constraints in the tunnel–soil–pipeline interaction analysis. This ‘constrained continuum formulation’ can be considered an extension of the original ‘two-stage’ elastic-continuum method, which traditionally neglects the mutual influence of the pipeline on the tunnel and vice versa. The new approach retains the advantage of using the greenfield condition as an input, but it allows closed-form consideration of the stiffening effect of the tunnel on the soil domain. The paper details the formulation and then provides an investigation into the effect of varying fixities: namely, both as ‘wished-in-place’ tunnels and as evolving, advancing constraints, with the tunnel ‘construction’. Normalised solutions are presented and compared with the previous formulation. In general, the addition of tunnel constraints leads to a stiffer soil action, and thus higher bending stressing in the pipeline; modelling of the tunnel kinematic constraint is conservative. The question of when such analysis is desired is discussed, concluding with a simple-to-use inequality suitable for design.

A continuum damage mechanics model for fatigue degradation and failure of short fiber reinforced composites

The present study is concerned with the development and implementation of a continuum damage mechanics model describing the degradation and failure of short fiber reinforced composites subjected to static and fatigue loads. Towards a numerically efficient formulation, the model is defined entirely on the macroscopic scale, although being motivated by microstructural considerations. Based on experimental observations, an anisotropic linear elastic formulation is adopted as a base formulation. The elastic model is coupled with a damage formulation assuming that damage is driven by the approach of a state point in strain space to a Tsai-Hill type failure envelope. The model is implemented as a user defined subroutine into a finite element formulation. As a numerically highly efficient alternative, an enhanced Wöhler-Miner type post-processing procedure is proposed. Both numerical approaches are found in good agreement with experimental data obtained in coupon experiments as well as in experiments on specimens with more complex geometry and loading conditions.

Influence of a change in activity regime on femoral bone architecture and failure behaviour.

The incidence and morbidity of femoral fractures increases drastically with age. Femoral architecture and associated fracture risk are strongly influenced by loading during physical activities and it has been shown that the rate of loss of bone mineral density is significantly lower for active individuals than inactive. The objective of this work is to evaluate the impact of a cessation of some physical activities on elderly femoral structure and fracture behaviour. The authors previously established a biofidelic finite element model of the femur considered as a structure optimised to loading associated with daily activities. The same structural optimisation algorithm was used here to quantify the changes in bone architecture following cessation of stair climbing and sit-to-stand. Side fall fracture simulations were run on the adapted bone structures using a damage elasticity formulation. Total cortical and trabecular bone volume and failure load reduced in all cases of activity cessation. Bone loss distribution was strongly heterogeneous, with some locations even showing increased bone volume. This work suggests that maintaining the physical activities involved in the daily routine of a young healthy adult would help reduce the risk of femoral fracture in the elderly population by preventing bone loss.

High-precision modelling of deformation of sandwich structures under bilateral symmetric and oblique-symmetric loading

The analysis and assessment of the stress-strain state (STS) of multilayer plates with orthotropic layers under the action of stationary transverse and tangential loads is an urgent task. It includes calculations on the strength and deformability of various homogeneous and multi-layered plates with layers of constant thickness but arbitrary structure according to the thickness of the plate. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexural components, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate. This significantly simplifies its modeling. For unflexural and exclusively flexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic, transverse-isotropic and orthotropic layers are constructed in an elastic formulation. That models takes full account deformation of transverse shear and transverse compression at transverse and the tangential loading of a plate. The model is continuous, that is, the number of equations and the order of differentiation of the calculation system of equations does not depend on the number of layers in the slab. This order of differentiation and the number of calculation equations can depend only on the order of iterative approximation of the model. A way of precise satisfaction of all defining correlations of the layers of material under keeping the conditions of their contact is found, while in the known continual models the dependence between the cross normal stress and cross deformation is only integral.The results of the analytical solution of the problem of deformation of a rectangular plate under boundary conditions of the Navier type under the action of transverse and tangential loads are given. By solving the test problems of the deformation of two-layer plates with transversally isotropic layers and three-layer plates with orthotropic layers and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, an assessment of the accuracy of the proposed refined models is given. The limits of admissible parameters of elastic characteristics of transversely isotropic and orthotropic plates for application of the offered models are established.

Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.

Effect of forward kinematics on the motion-trajectory oscillations and inertia forces

In vehicle-motion scenarios, non-negligible centrifugal-force component that cannot be balanced by the gravity force may develop, leading to increase in wheel-contact forces. This inertia-force component, which depends on vehicle forward velocity and oscillation amplitude, can arise when the vehicle negotiates straight road or railroad track. An approach based on the Frenet geometry and recorded motion trajectory (RMT) is introduced in this investigation to examine forward-kinematics effect on the motion geometry and inertia forces and to obtain general expression of the ratio between the amplitudes of the centrifugal and gravity forces. While the two three-dimensional vectors of applied and constraint forces define Frenet osculating plane of mass-center trajectories; for non-centroidal points, the osculating plane is defined by two vectors; the resultant of the applied and constraint forces and an inertia-force vector that depends on the angular velocity and angular acceleration. The concepts of instantaneous motion plane (IMP) and instantaneous zero-force axis (IZFA) are used to convert three-dimensional Cartesian forces into two-dimensional Frenet forces, based on the Frenet-Newton equations. Because contact forces can be formulated using two fundamentally different approaches; constraint contact formulation (CCF) and elastic contact formulation (ECF), the effect of the contact formulation on the IMP and IZFA orientation is examined. The results obtained, using railroad suspended wheelset example, demonstrate the following: (1) limitations of roller rigs that do not properly account for forward-velocity effect on the inertia forces; (2) difference between IMP and intermediate wheelset plane defined by wheelset lateral axis and global vertical axis; (3) non-negligible inertia centrifugal-force component in the absence of physical curvature; and (4) need to distinguish between physical and motion geometries; the latter is one that directly enters into the definition of the inertia forces. Trajectories with oscillation frequency independent of the oscillation amplitude for a given forward velocity are also identified.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

The relevance of analytical formulations predicting stiffener tripping

In search for the governing failure mode of externally pressurized ring-stiffened cylinders the focus is, due to its loading and slender geometry, drawn to elastic buckling. However, the hydrostatic pressure loading and imperfections cause nonlinear behaviour, and excessive deformation results in plastic failure, denoted by collapse. This implies that a true bifurcation point is often hardly observed, and this raises the question whether it is worthwhile to spend much effort to determine the elastic buckling pressures other than to ensure they are well above the collapse pressure? This paper focuses on the elastic buckling of ring frames, generally referred to as frame tripping. Some latest attempts in improving the accuracy of the analytical formulations will be questioned. Firstly, this study will compare non-linear buckling with linear-elastic material behaviour to elastic buckling by means of finite element analysis. Secondly, the non-linear elastic-plastic effects will be included too. Doing so, the effect of the geometrical non-linearity and material elastic-plastic non-linearity will be visible separately and will give an indication of the relevance of frame tripping results predicted by analytical elastic buckling formulations. Literature offers a variety on analytical formulations, each with an underlying idea to remove undesired assumptions that impair results. Even recently comprehensive formulations are published to establish an accurate value of the elastic tripping pressures. This paper shows whether those methods are still relevant in modern pressure hull design or not.

Emergent mesoscale correlations in active solids with noisy chiral dynamics.

We present the linear response theory for an elastic solid composed of active Brownian particles with intrinsic individual chirality, deriving both a normal mode formulation and a continuum elastic formulation. Using this theory, we compute analytically the velocity correlations and energy spectra under different conditions, showing an excellent agreement with simulations. We generate the corresponding phase diagram, identifying chiral and achiral disordered regimes (for high chirality or noise levels), as well as chiral and achiral states with mesoscopic-range order (for low chirality and noise). The chiral ordered states display mesoscopic spatial correlations and oscillating time correlations, but no wave propagation. In the high chirality regime, we find a peak in the elastic energy spectrum that leads to a non-monotonic behavior with increasing noise strength that is consistent with the emergence of the 'hammering state' recently identified in chiral glasses. Finally, we show numerically that our theory, despite its linear response nature, can be applied beyond the idealized homogeneous solid assumed in our derivations. Indeed, by increasing the level of activity, we show that it remains a good approximation of the system dynamics until just below the melting transition. In addition, we show that there is still an excellent agreement between our analytical results and simulations when we extend our results to heterogeneous solids composed of mixtures of active particles with different intrinsic chirality and noise levels. The derived linear response theory is therefore robust and applicable to a broad range of real-world active systems. Our work provides a thorough analytical and numerical description of the emergent states in a densely packed system of chiral self-propelled Brownian disks, thus allowing a detailed understanding of the phases and dynamics identified in a minimal chiral active system.

Breast simulation pipeline: From medical imaging to patient-specific simulations

BackgroundBreast-conserving surgery is the most acceptable operation for breast cancer removal from an invasive and psychological point of view. Before the surgical procedure, a preoperative MRI is performed in the prone configuration, while the surgery is achieved in the supine position. This leads to a considerable movement of the breast, including the tumor, between the two poses, complicating the surgeon's task. MethodsIn this work, a simulation pipeline allowing the computation of patient-specific geometry and the prediction of personalized breast material properties was put forward. Through image segmentation, a finite element model including the subject-specific geometry is established. By first computing an undeformed state of the breast, the geometrico-material model is calibrated by surface acquisition in the intra-operative stance. FindingsUsing an elastic corotational formulation, the patient-specific mechanical properties of the breast and skin were identified to obtain the best estimates of the supine configuration. The final results are a shape-fitting closest point residual of 4.00 mm for the mechanical parameters Ebreast=0.32 kPa and Eskin=22.72 kPa, congruent with the current state-of-the-art. The Covariance Matrix Adaptation Evolution Strategy optimizer converges on average between 5 to 30 min depending on the initial parameters, reaching a simulation speed of 20 s. To our knowledge, our model offers one of the best compromises between accuracy and speed. InterpretationSatisfactory results were obtained for the estimation of breast deformation from preoperative to intra-operative configuration. Furthermore, we have demonstrated the clinical feasibility of such applications using a simulation framework that aims at the smallest disturbance of the actual surgical pipeline.

Analytical solution for axial load transfer of floating piles

This paper presents an analytical solution for the analysis of axially-loaded compressible floating piles embedded in a homogeneous elastic soil layer of finite thickness. The mathematical model is based on a Tajimi-type elastic continuum formulation, that allows deriving an accurate closed-form expression for the frictional stresses developing at the soil-pile interface. The pile and its underlying soil column are considered as one-dimensional axially deforming rods, and solution of their governing equations leads to closed-form expressions for pile and soil displacements and stresses. The proposed model provides results of similar accuracy to boundary integral solutions for the special case of floating piles in elastic half-space, and to numerical analyses for the general problem of floating piles embedded in a layer of finite thickness. As such, the model is used to prepare charts for the settlement analysis of floating piles in practice, based on a parametric analysis that unveils the sensitivity of the axial response of floating piles to the different problem parameters.

Delve deep: Seismic revolution in acquisition and model building for exploration

Sparse-node acquisition is the latest approach to improve the earth model through the inclusion of ultra-long offsets and low-frequency generating sources. The new designs of node data collections with simultaneous shooting can be deployed on a regional basis, covering thousands of square miles in a cost-effective manner. In complex geologic settings such as salt environments, salt interpretation has a direct impact on subsalt imaging. However, salt interpretation can be quite time intensive and challenging. Full-waveform inversion (FWI), as a data-driven optimization algorithm with full-wavefield modeling, has become an essential tool for earth model building. However, its use in salt tectonic contexts, especially with streamer data collection, is limited because the frequencies acquired are not sufficiently low and the offsets are not sufficiently long. The recent advancement of the objective function enables the application of an FWI workflow, coupled with sparse-node acquisition in the deepwater settings of the Gulf of Mexico, to refine the salt geometry, correct the background velocity error in subsalt, and uncover a basin's structural framework. The increased frequency content of the FWI-updated fields enables us to extract the FWI-derived reflectivity (FDR), which is an image created by the FWI nonlinear iterative process with the full wavefield. Additionally, the classic extraction of amplitude-versus-angle/amplitude-variation-with-offset parameters employs the elastic Zoeppritz formulation to obtain rock-physics volumes and explain the amplitude on the prestack image gathers, which are typically generated by Kirchhoff or reverse time migration (RTM) depth migration. The next step is to substitute these migrations with FWI and output not only the poststack FDR but use it in a prestack mode to aid amplitude analysis at an early stage of model building.

A lode-dependent plasticity model for high-strength structural steel

In this study, a new plasticity model was developed to characterize the mechanical behavior of high-strength structural steel under multiaxial stress states. This model utilizes a Lode-dependent yield criterion in conjunction with a non-associated flow rule and isotropic hardening law to determine the evolution of the plastic flow stress. Further, a detailed numerical implementation algorithm for this plasticity model is presented, including the integration algorithm of the constitutive equations using an implicit elastic predictor-return mapping method and formulation of the consistent tangent modulus. To validate the proposed plasticity model, six types of Q690 high-strength steel specimens were tested under monotonic loading. These specimens can be used to obtain the elastoplastic response of the material under uniaxial tension, compression, pure shear, tensile shear, and plane strain state, respectively. Corresponding numerical simulations were performed for each specimen using ABAQUS/Standard with the proposed model being introduced into the analysis via the UMAT user subroutine. Then, the prediction performance of the proposed plasticity model was investigated by comparing the load-displacement curves of each test specimen obtained from both experiments and numerical simulations. It is shown that the proposed plasticity model can accurately predict the mechanical response of the high-strength steel under different stress states and enables to visualize the key stress state parameters and material strength variation at the failure region of each specimen, which confirms the promising prospect of this new model in practical applications.

Effects of Crosswind on Pantograph–Catenary Wear Using Nonlinear Multibody System Dynamic Algorithms

In this study, a multibody system (MBS) computational framework is developed to determine the exact location of the contact point and wear prediction resulting from the pantograph–catenary interaction. The railroad vehicle models in the MBS computational framework comprise rigid-body railroad vehicles, rigid-body pantograph systems, and flexible catenary systems. To avoid incremental rotation, the nonlinear finite element absolute nodal coordinate formulation is used to model a flexible catenary system in the MBS computational framework. To avoid co-simulation processes, the rigid-body railroad vehicle and the pantograph and flexible catenary systems were integrated into the MBS algorithms. The pantograph–catenary interaction is modeled using an elastic contact formulation developed to include the effect of pantograph–catenary separation and sliding contact. The proposed MBS approach evaluates the location of the contact point, contact force, and normal wear rate (NWR) from the mechanical and electrical contributions. This investigation considers the vibration caused by a crosswind scenario and determines the numerical result in the case of a steady crosswind scenario. The steady crosswind scenario contains the advantage of pantograph–catenary aerodynamic design, and the vibration of the catenary system remains significant after the excitation of a steady crosswind. In the case of a steady crosswind, the higher value of the steady crosswind effect significantly increases the mean contact force and the NWR from the mechanical contribution. After crosswind load disturbances, the mean contact force decreases, but the standard deviation of the contact force increases. Therefore, the NWR from the electrical contribution increases significantly. However, the total NWR increases with the crosswind velocity.

Strain, disfigurement and flexure tensors in continuum kinematics

In the differential geometric formulation of nonlinear elasticity, the strain tensor can intrinsically be regarded as a measure of change of the metric of the Riemannian manifold that embodies the continuum undergoing a deformation. Here, this classical concept is revisited and complemented with two newly defined higher order deformation measures, namely the disfigurement and flexure tensors, which respectively measure the change of the Levi-Civita connexion and the Riemann curvature of the continuum. The strain, disfigurement and flexure tensors can, thereby, roughly indicate the alteration of the geometry of a deforming body in the zeroth, first and second orders, respectively. In this light, one may predict that the disfigurement and flexure tensors should be related to the first and second covariant derivatives of the strain tensor. While this is true, the exact strain-disfigurement-flexure relations, as derived in this paper, are found to be involved and non-trivial, offering a deeper understanding of the deformation. In our discussions, distance-preserving, geodesic-preserving and curvature-preserving deformations are studied as three special cases to help better interpret the above-mentioned deformation measures. Some related results have also been found as by-products that might be of current interest or prove useful in future investigations.

A study on dynamic characteristics of the flutter for three-layer plates and shells flown around by a gas flow

The nonlinear flutter of viscoelastic three-layer plates and shallow shells with a structure asymmetrical in thickness, flown around by a supersonic gas flow is studied in this paper. Mathematical models of problems on the flutter of viscoelastic three-layer plates, cylindrical panels, and shells with a structure asymmetrical in thickness, flown around by a supersonic gas flow are developed. The flutter of viscoelastic three-layer plates is studied in linear and nonlinear formulations. The critical flutter velocities of elongated plates are compared with the results obtained in previously published studies, where the solutions were obtained in an elastic formulation. The flutter of viscoelastic three-layer plates, cylindrical panels with a rigid filler that resists transverse shear, flown from the outside by a supersonic flow, was studied. It is shown that an increase in the geometric parameter characterizing the flexural rigidity of the bearing layers of three-layer structures leads to an increase in the flutter velocity by 25–40%.

ВИКОРИСТАННЯ СИМЕТРІЇ ДЛЯ РОЗВ’ЯЗУВАННЯ ЗАДАЧ ДЕФОРМУВАННЯ ПЛИТ НА ЖОРСТКІЙ ОСНОВІ

The optimization of the design scheme of deformation of a rectangular multilayer plate with transversally isotropic layers resting on a rigid (non-deformable) foundation is proposed. The essence of optimization is to consider such a design diagram of the plate, in which the stress-strain state (SSS) of plate would be fully described by only one component, namely the unflexural component of symmetrical SSS relative to the middle surface of plate which is bilaterally symmetrically loaded. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is suggested to consider the design diagram of the plate, which is formed by supplementing it with a symmetric one about the contact surface of the foundation. In this case, the plate will be symmetrically loaded with respect to the middle surface of the plate, and the thickness of the plate will double. On the middle surface, the conditions of sliding contact of the upper and lower parts of the symmetrical plate, i.e. sliding contact of the plate with foundation are fulfilled. To model absolutely rigid contact at the border with the foundation, an additional thin layer of high rigidity ("non-deformable layer") is introduced into the supplemented plate. This does not change the essence of the calculation model of plate. The SSS of plate will be unflexural, which significantly simplifies its modeling. A two-dimensional model of deformation of multilayer rectangular plates on a rigid foundation with isotropic and transversally-isotropic layers is constructed in an elastic formulation for a unflexural SSS, with a high degree of iterative approximation, but three-dimensional by the nature display of the SSS. This model sufficiently takes into account transverse shear deformations and of transverse compression of the plate under transverse loading. The model is continuous, that is, the number of equations and the order of differentiation of the solving system of equations does not depend on the number of layers in the plate. This order of differentiation and the number of solving equations can depend only on the order of iterative approximation of the model. The derivation of the solving differential equations in the generalized forces and displacement functions is given, as well as the boundary conditions are obtained by the variational Lagrange method. The results of the analytical solution of the problem of deformation of homogeneous rectangular plate in sliding contact with a rigid foundation whith Navier-type boundary conditions under the action of a transverse sinusoidal load are given. A comparison of the calculation results with the exact three-dimensional solution ones was made. Keywords: multilayered plate, rigid foundation, transverse shear, transverse compression, continual model