Plane Stress Assumption Research Articles

A Cauchy like problem in plane elasticity

Let \(\Omega \) be a bounded domain in the plane, representing an elastic body. Let \(\Gamma_0 \) be a portion of the boundary \(\Gamma\) of \(\Omega \), \(\Gamma_0 \) being assumed to be paralled to the \(x\)-axis. It is proposed to determine the stress field in \(\Omega \) from the displacements and surface stresses given on \(\Gamma_0 \). Under the assumption of plane stress, it is shown that \(u_x + u_y\) is a harmonic function. An Airy stress function is introduced, from which the stress field is computed.

Numerical Analysis of the Rates of Displacements to Prescribe Constant Proportional Logarithmic Strains in Stretched Sheets

The end displacements that permit to prescribe proportional logarithmic strains in stretched sheets during biaxial loading are analyzed here using a finite element procedure. A cruciform specimen of the sheet which in-plane dimensions are normalized with respect to a parameter “l” is considered. The in-plane dimensions of the whole specimen are 8lx8l and the length of each arm is “2l”. The central region of the specimen is square with a side “2l” and a thickness “h/3” while the thickness of the whole sheet is “h”. This thickness is assumed much smaller than the in-plane dimensions of the sheet. The material of the sheet is considered incompressible, elastic-plastic and rate insensitive. Assuming that the behavior of the sheet can be described by generalized plane-stress assumptions, a large strain-plane-stress finite-element program is developed with a finite-strain version of J2 flow theory of plasticity as constitutive law. It is shown that for balanced biaxial state as well as for plane-strain state, the ratios of the increments of displacements at the boundaries of the sheet may be adjusted during the loading histories and the strains measured in the central zone such that x/l ≤0, 2.

Thermoelastic response of a fin exhibiting elliptic thickness profile: An analytical solution

A thermoelastic analytical solution of a variable thickness cooling fin problem is presented. A variable thickness annular fin mounted on a hot rotating rigid shaft is considered. The thickness of the fin is assumed to vary radially in a continuously variable nonlinear elliptic form. An energy equation that accounts for the conduction, convective heat loss from peripheral and edge surfaces, thickness variation and rotation is adopted. The thermoelastic equation is obtained under formal assumptions of plane stress and small strains. For given heat and centrifugal loads the temperature distribution in the fin and the corresponding state of stress are obtained by means of the analytical solutions of energy and thermoelastic equations, respectively.

Identification of elasto-plastic characteristics by means of air-bending test

A new identification method of material elasto-plastic characteristics by means of simple testing is proposed. The outcome of the procedure is the true-stress versus true-strain curve. The result is proposed in analytical form, according to Swift's law. The method requires acquisition of actual force-penetration values, carried out during a standard air-bending test. One of the main advantage of this method is that it allows an immediate identification of the material behaviour just in-process, that is to say that material characteristics can be accounted differently for each sheet subjected to bending; this is a fundamental preliminary step to achieve a bending adaptive control of sheets. Input requirements for identification include the geometry of tools and sheet, and the friction coefficient between sheet and die which needs to be estimated previously. In the proposed method, at a first stage the bending moment–curvature curve for the sheet is obtained by the numerical computation of force-penetration data through a multi-joint model of sheet during air-bending. Then, a simplified bending model, characterized by linear strain distribution across the thickness and plane-stress assumption, is employed to calculate the stress–strain curve.

Identification of welding residual stresses in rectangular plates using vibration responses(*)

A novel hybrid numerical/experimental identification procedure for the assessment of welding-induced residual stresses in rectangular plates is proposed and evaluated. This procedure explores the influence of the stress state on the dynamic responses of structural components, according to the so-named stress-stiffening effect. The technique consists in using a set of experimental natural frequencies of the welded plate and a mathematical model relating the residual stresses to the natural frequencies to formulate an optimization problem. The cost function represents the differences between the experimental and model-predicted dynamic responses and the design variables are interpreted as parameters of the mathematical model describing the stress distribution over the plate. A parameterized stress model suitable to the case of welding residual stresses is presented in terms of a differential equation that relates an Airy's stress function to the plastic strains resulting from the welding process. From this stress function, the stress components σx, σy and τxy (assuming plane stress state) are computed. Genetic Algorithms are used to solve the numerical optimization problem. To demonstrate the feasibility of the method, it is used for the assessment of residual stresses generated by TIG (GTAW) welding of a thin rectangular steel plate, for which experimentally measured natural frequencies and numerically computed residual stress distributions are available in the literature.(*) A preliminary version of this article has been presented in the 17th International Congress of Mechanical Engineering, promoted by Brazilian Society of Mechanical Sciences and Engineering, São Paulo, Brazil, 2003.

On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems

Closed form solutions to functionally graded rotating solid shaft and rotating solid disk problems are obtained under generalized plane strain and plane stress assumptions, respectively. The nonhomogeneity in the material arises from the fact that the modulus of elasticity of the material varies radially according to two different continuously nonlinear forms: exponential and parabolic. Both forms contain two material parameters and lead to finite values of the modulus of elasticity at the center. Analytical expressions for the stresses at the center are determined. These limiting expressions indicate that at the center of shaft/disk: (i) the stresses are finite, (ii) the radial and the circumferential stress components are equal, and (iii) the values of the stresses are independent of the variation of the modulus of elasticity. It is also shown mathematically that the nonhomogeneous solutions presented here reduce to those of homogeneous ones by an appropriate choice of the material parameters describing the variation of the modulus of elasticity.

Effect of Stress State on Mode II Stable Crack Extension

The effect of assumption of plane state of stress on the predictability of experimental results observed during the mode II stable crack growth (SCG) through 8 mm thick compact tension specimens (CTS) of a workhardening aluminum alloy (D16AT) has been studied. Experimental results include load-sliding displacement diagram, extent of SCG, crack front geometry and fracture surface fractographs. The experimental results show that the crack extends in its own plane, the fracture surface is flat, smooth and free of any shear lip. The crack front geometry, which is straight initially, remains mostly so throughout the SCG. Theoretical investigations have been done using an elastic-plastic finite element scheme and the COA/COD criterion as the criterion governing the growth. Finite element results, assuming plane stress and plane strain conditions separately, on the load-sliding displacement diagrams, J-resistance curve, plastic zones and variation of equivalent stress and strain along the crack-line ahead of the crack tip are also presented. The resistance curve is a straight-line and the magnitudes of equivalent stress and strain increases as the crack extension proceeds. In general, the predictions based on the assumption of plane state of stress are closer to the experimental results.

Crack propagation analyses with CTOA and cohesive model: Comparison and experimental validation

Two numerical models, namely an R-curve approach based on the crack tip opening angle (CTOA) and a cohesive model, are compared regarding their ability to predict ductile crack extension in thin aluminium sheets, which can be simulated under the assumption of plane stress. The experimental database is presented, the measuring techniques for the various quantities (optically and with clip gauges) are shown and the identification and validation of the respective model parameters are explained. A general concept for their identification is then derived for the case of thin walled structures under Mode I conditions. In order to investigate the performance of the models under different constraint conditions and the transferability of their parameters, C( T) specimens are used for parameter identification and M( T) specimens for validation. It is shown that for both models a single set of parameters describes the mechanical behaviour of both types of specimens. Cross-checking the two models, the crack tip opening angle is determined from the cohesive model calculations and compared with the experimental values.

Three dimensional stress distribution in FRP-to-concrete bond test specimens

One of the key factors affecting the behaviour and reliability of reinforced concrete structures strengthened with externally bonded fibre reinforced polymer (FRP) composites is the bond behaviour between the external composite and the concrete substrate. Extensive research has been carried out on external plate-to-concrete bond strength. This includes experimental studies using various test set-ups, theoretical work using fracture mechanics analysis and finite element analysis and the development of empirical bond strength models. In almost all the theoretical analyses to date, the plane stress assumption has been adopted although the actual stress distribution is likely to be 3D. This paper presents a finite element analysis on the stress distributions in a typical shear test set-up for FRP-to-concrete bond strength. Results show that the stress distribution is significantly different from plane stress assumption primarily because the difference between the width of the test FRP plate and that of the concrete block. It is also shown that the stress distribution is 3D even if the FRP plate has the same width as the concrete block because of the difference of the Poisson’s ratio between the constituent materials.

An h-adaptive modified element-free Galerkin method

Abstract In this work an h -adaptive Modified Element-Free Galerkin (MEFG) method is investigated. The proposed error estimator is based on a recovery by equilibrium of nodal patches where a recovered stress field is obtained by a moving least square approximation. The procedure generates a smooth recovered stress field that is not only more accurate then the approximate solution but also free of spurious oscillations, normally seen in EFG methods at regions with high gradient stresses or discontinuities. The MEFG method combines conventional EFG with extended partition of unity finite element (EPUFE) methods in order to create global shape functions that allow a direct imposition of the essential boundary conditions. The re-meshing of the integration mesh is based on the homogeneous error distribution criterion and upon a given prescribed admissible error. Some examples are presented, considering a plane stress assumption, which shows the performance of the proposed methodology.

Error Analysis on Finite Element Modeling of Involute Spur Gears

Finite element analysis can incorporate two-dimensional (2D) modeling if the geometry, load, and boundary conditions meet the requirements. For many applications, a wide range of problems are solved in 2D, due to the efficiency and costs of computation. However, care has to be taken to avoid modeling errors from significantly influencing the result. When the application area is nonlinear, such as when modeling contact problems or fracture analysis, etc, the 2D assumption must be used cautiously. In this paper, a large number of 2D and three-dimensional (3D) gear models were investigated using finite element analysis. The models included contact analysis between teeth in mesh, a gear body (disk), and teeth with and without a crack at the tooth root. The model results were compared using parameters such as the torsional (mesh) stiffness, tooth stresses and the stress intensity factors that are obtained under assumptions of plane stress, plane strain, and 3D analysis. The models considered variations of face width of the gear from 5 mm to 300 mm. This research shows that caution must be used especially where 2D assumptions are used in the modeling of solid gears.

Evaluation of two finite element formulations for a rapid 3D stress analysis of sandwich structures

For efficiently simulating the impact behavior of sandwich structures made from composite face sheets and a lightweight core a rapid and accurate 3D stress analysis is essential. For that reason, a three-layered finite element formulation based on plane stress assumptions was recently developed by Kärger et al. [Kärger, WA, Rolfes R, Rohwer K. A three-layered sandwich element with improved transverse shear stiffness and stress based on FSDT. Comput Struct, submitted for publication]. It has turned out, however, that under concentrated out-of-plane loads this element formulation lacks appropriate accuracy of stress results. Therefore, an improved finite element formulation is developed, which accounts for the full 3D stress state. In a post-processing routine, the transverse stresses are improved by using the Extended 2D Method, which was developed by Rolfes and Rohwer [Rolfes R, Rohwer K. Improved transverse shear stresses in composite finite elements based on first order shear deformation theory. Int J Numer Meth Eng 1997;40:51–60] and extended to a three-layered sandwich structure by Kärger et al. Both the finite element formulation by Kärger et al. and the new formulation presented in the present article use pure displacement approaches and require only C 0-continuity conditions, which simplifies integration into existing FE codes and allows combined application with other finite elements. Two examples demonstrate the accuracy and applicability of the two elements.

Stress distributions in elastic-plastic rotating disks with elliptical thickness profiles using Tresca and von Mises criteria

Analytical and numerical solutions for the elastic-plastic stress distribution in rotating variable thickness solid and annular disks are obtained under plane stress assumption. The thickness of the disk is assumed to vary radially in elliptic form which represents a wide range of continuously variable nonlinear cross-sectional profiles. Tresca's yield criterion and its associated flow rule are used to obtain analytical solutions for a linear hardening material. A computational model is developed to obtain solutions using the von Mises yield criterion, deformation theory of plasticity and a Swift-type hardening law. Both linear and nonlinear hardening materials are considered in solutions obtained by using von Mises criterion. The stresses, displacement and plastic strains are calculated for solid and annular disks rotating at different speeds and the results are presented in graphical forms.

Efficient and accurate multilayer solid-shell element: non-linear materials at finite strain

We present in this paper an efficient and accurate low-order solid-shell element formulation for analyses of large deformable multilayer shell structures with non-linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out-of-plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle, the in-plane and out-of-plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non-linear 3-D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane-stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well-defined lumped mass matrix, and provides double-side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non-linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out-of-plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out-of-plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non-linear composite multilayer shell structures. Copyright © 2005 John Wiley & Sons, Ltd.

Displacement Field and Strain Distribution in a Rotating Annular Disk

The displacement field and strain distribution in a thin rotating disk with constant thickness and density are found based on Mises’ yield criterion and its associated flow rule. The material of the disk is elastic-perfectly plastic and the assumption of plane stress is adopted. The solution is illustrated by an example.

Tresca?s yield criterion and linearly hardening rotating solid disks having hyperbolic profiles

An analytical solution for the stress distribution in rotating hyperbolic solid disk is obtained under plane stress assumption. The analysis is based on Tresca’s yield criterion, its associated flow rule and linear strain hardening material behavior. It is shown that the deformation behavior of the hyperbolic solid disk is different from that of the constant thickness disk. The plastic core consists of three different plastic regions with different mathematical forms of the yield criterion. Accordingly, three different stages of elastic-plastic deformation can be distinguished. The lower and upper bounds of the limit angular velocities for each stage are determined. It is also shown mathematically that in the limiting case the hyperbolic disk solution reduces to the solution of rotating uniform thickness solid disk. Die Trescasche Fliesbedingung und rotierende Vollscheiben mit hyperbolischem Profil und linearer Verfestigung

Dispersion relations for cylindrical bending motions of polarized piezoceramic bimorph plates with two facing edges free

Dispersion relations of thin parallel bimorph plates with polarized piezoceramics are obtained by using Hamilton’s principle. The coupled variational equations for piezoceramic bimorph plates are derived with the thin plate theory and the extension of electric potential in the thickness direction, which are valid for low frequency range. More specifically, coupled differential equations as well as dispersion relations subject to homogeneous natural-type boundary conditions on the two facing side-edges are derived for the cylindrical bending motions of both fully electroded and unelectroded bimorph plates of which surfaces are free from applied traction for both cases; the derivations are made through the expansions of mechanical displacements in the thickness co-ordinate with plane stress assumptions at major surfaces in the manner of Mindlin and of electric potential with vanishing second order components of electric potentials at major surfaces in the manner of Tiersten. Relations between the deflection gradient of fully electroded bimorph plate and the induced electric current are obtained. The complexity due to the additional inclusion of differential equations for the electric potential components may be alleviated through the reductions of the coupled differential equations. As an illustrative example, dispersion relations for the aforementioned four cases of bimorph plates composed of PZT5 are obtained. The dispersion curves are depicted and compared each other, and some differences and similarities are discussed.

Stress Analysis of Holes in Thick Plates

A review of the Stress Concentration Factors (SCFs) obtained from normal and oblique holes in thick flat plates loaded in uniaxial tension has been conducted. The review focuses on values from the plate surface and discusses the ramifications of making a plane stress assumption.

Particle debonding using different yield criteria

Abstract Effects of plastic anisotropy in relation to debonding of rigid inclusions embedded in an elastic–viscoplastic metal are studied. Full finite strain analyses are carried out for plane cells assuming plane stress or plane strain. The overall stress strain response is calculated, when the cell is subjected to a fixed biaxial stress state. Four phenomenological anisotropic yield criteria are considered, namely Hill [Hill, R., 1948. Proc. Roy. Soc. London Ser. A 193, 281–297], Barlat and Lian [Barlat, F., Lian, J., 1989. Int. J. Plasticity 5, 51–66], Barlat et al. [Barlat, F., Lege, D.J., Brem, J.C., 1991. Int. J. Plasticity 7, 693–712; Barlat, F., et al., 2003. Int. J. Plasticity 19, 1297–1319], or the von Mises isotropic yield surface. Also a non-normality flow rule is adopted in some of the studies. Significant effects of plastic anisotropy are seen on the plane stress cell, due to the initial extent and shape of the particular yield function considered. The required overall straining of the cell for debonding initiation is related to the extent of the yield surfaces, since a high yield stress promotes debonding. Additionally, the maximum overall stress level for the cell is lower for the Hill [Hill, R., 1948. Proc. Roy. Soc. London Ser. A 193, 281–297] and Barlat et al. [Barlat, F., Lege, D.J., Brem, J.C., 1991. Int. J. Plasticity 7, 693–712] materials than that predicted by the other three yield functions. In all cases analyzed the non-normality flow rule hastens the particle-matrix debonding. Keeping all material parameters fixed, the material response of the plane strain cell is considerably affected, due to debonding at a much reduced overall plastic strain compared to the corresponding plane stress cell.

Direct stress–strain representation for coated woven fabrics

An understanding of the complex behaviour of coated woven fabrics is vital for the design of state-of-the-art fabric structures. Fabric behaviour is typically defined using elastic constants based on plane stress assumptions. This paper considers two new methods of representing fabric response: (i) use of spline functions to define response surfaces, (ii) use of stress–strain mean and difference functions (proposed by Day [IASS symposium proceedings: shells, membranes and space frames 2 (1986) 17]. Both techniques provide direct correlation between stresses and strains, eliminating the assumption of plane stress. Extensive biaxial fabric testing is proposed to assess the validity of these approaches and extend their use.