Reissner Model Research Articles

Geometrically exact 3D beam theory with embedded strong discontinuities for modeling of localized failure in bending

In this work, we propose the elastoplastic model of a three-dimensional (3D) geometrically exact beam, which can capture localized bending deformation leading to a plastic hinge. This is a follow-up work to Tojaga et al. (2023), where we only studied the fracture of fiber-like beam structures that break in mode I or mode II, here extended to handle beam-like behavior with bending failure and the main difficulty of the non-vectorial character of large 3D rotations. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model, which leads to the multiplicative decomposition of the rotation tensor. We shown how to replace this with the corresponding additive decomposition of the rotation vector derivatives, to be performed in the material description in the tangent space of SO(3) manifold in the initial configuration. The plasticity model for which we give a more detailed implementation employs a Rankin-like multi-surface plasticity criteria, where each moment vector component (i.e. torsion or bending) is considered separately. The non-local variational formulation is proposed to deal with the softening phenomena characteristic of localized bending failure, where the fracture energy is introduced as the main parameter for softening. The discrete approximation is built in terms of the embedded-discontinuity finite element method (ED-FEM), which introduces a jump in rotation vector that can be handled at the level of a particular beam element. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best-approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and moments) from local (plastic curvature) variables. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Reduced model for fracture of geometrically exact planar beam: Non‐local variational formulation, ED‐FEM approximation and operator split solution

SummaryIn this work, we present a novel reduced model for computing the solution to a bending fracture problem of the geometrically exact beam. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model. The non‐local variational formulation is proposed to deal with the softening phenomena characteristic of fracture. The fracture energy is introduced as the main fracture parameter, rather than (in general) ill‐defined characteristic length used for previous non‐local models. The discrete approximation is built in terms of the embedded‐discontinuity finite element method (ED‐FEM), which needs no initial crack for optimal performance nor detailed crack kinematics description, such as in the X‐FEM or the phase‐field approach. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best‐approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and stress) from local (internal) variables including the crack opening. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Application of Bezier curves for description of structure shape in optimization of adhesive joints

The problem of topological optimization of a symmetrical double-shear adhesive joint has been solved. The suggested mathematical model of a joint with variable thickness generalizes the classic Holland–Reissner model. The shape of the doubler is described by means of the Bezier curve. Seeking parameters in the optimization problem are coordinates of reference points of the Bezier curve. Both joint length and doubler cross-section area can be considered an objective function. The restriction is applied on stress in adhesive film and doubler. The direct problem of finding the joint stress state at given geometric parameters was solved using the finite difference method. The genetic algorithm was used to solve the optimization problem. In order to improve the convergence of the genetic algorithm, the island model of evolution is suggested, which ensures quick evolution selection and stability of obtained results. The model problem is solved.

Consideration Upon Boundary Conditions in the Goland–Reissner Model

This paper analyzed a simple joint with the Goland–Reissner (G-R) theory. The study’s primary purpose is to define specific boundary conditions that depend on several factors to minimize the use of complex finite element analysis in the well-known applications. The problem depends on the length of the adhesions area and the magnitude of axial load P. The joint was first considered rigid to evaluate the stresses at its extremes, after which the theory of G-R was applied, considering the cases of equal and different adhesions. A mathematical approach was carried out starting from the equilibrium of the infinitesimal element. The results were finally compared with the Finite Element Method using FEM code.

The Effect of Collagen Viscoelastoplasiticty on Reissner’s Membrane Displacement: A Graphic Analysis

Background: The aim of this study is to analyze the effect of collagen viscoelastoplasticity on the bulge displacement of Reissner’s membrane that is observed in endolymphatic hydrops and Meniere’s disease.Methods: Viscoelastoplastic load–deformation characteristics for Reissner’s membrane were based on a reported collagen polymer model of the cochleo-saccular membranes. The projected bulge displacements of Reissner’s model membrane at key distention points were quantified trigonometrically and plotted graphically.Results: Initial deformation is characterized by a membrane laxity with substantial stretch at low tension with projected bulge displacement of Reissner’s membrane approaching 30%. Intermediate deformation is characterized by a linear membrane stiffness with projected bulge displacement of Reissner’s membrane in the range of 30-40%. Terminal deformation is characterized by reduced stiffness with a disproportionate increase in membrane stretch with projected bulge displacement of Reissner’s membrane reaching a critical value of 50%, indicating a hemi-circular profile with imminent risk of rupture.Conclusion: This collagen model of membrane viscoelastoplasticity demonstrates that at low pressure significant degrees of bulge displacement up to 30% can occur that may be reversible. The narrower 30-40% range of membrane displacement is one of the increasing deformity but without risk of rupture. Greater displacements approaching 50% indicate that the membrane is reaching a critical hemi-circular configuration with impending rupture.

Nanodeformations of microcapsules: comparing the effects of cross-linking and nanoparticles.

The mechanical properties of proteinaceous and composite microcapsules loaded with oil were measured by SFM and evaluated using the Reissner model. Comparison of the obtained results reveals significantly higher Young’s moduli of protein capsules due to intermolecular crosslinking. In contrast, conformational restrictions in composite microcapsules inhibit protein crosslinking leading to the reduction of their elasticity.

Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin–Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an expression of the shear correction factor variable (αn) in terms of the thickness (h) and the radius of curvature (R). The main advantage of the proposed model is that it has the possibility to analyze thin, medium and thick tunnel ventilation shafts. As a result, two comparisons were made: the first one, between the new model and the Mindlin–Reissner model without constitutive curvature with the shear correction factor αn=5/6 as a constant, and the other, between the new model and the tridimensional numerical models (solids and shells) obtained by finite element method for different slenderness ratios (h/R). The limitation of the proposed model is that it is to be formulated for a general linear-elastic and axial-symmetrical state with continuous distribution of the mass.

Applicability ranges for four approaches to determination of bending stiffness of multilayer plates

A linear static problem of bending of a multilayer plate with homogeneous isotropic layers is considered. The deflection is assumed to have harmonic shape in the tangential directions. For calculation of the bending stiffness we introduce two dimensionless parameters: a small thickness parameter equal to the ratio of the thickness to the deformation wavelength in the tangential directions, and a large inhomogeneity parameter equal to the ratio of the maximum and minimum Young’s moduli of the layers. The plane of these parameters is split into four regions, in which different models are available for calculating the bending stiffness. The first of these is the Kirchhoff - Love model based on the hypothesis of straight non-deformable normal. In the second region, the Timoshenko - Reissner model with a shear parameter is used, calculated by an asymptotic formula of the second order of accuracy. The third region is based on assumptions on inextensible normal fiber and large heterogeneity. Finally, in the fourth region, compression of the normal is taken into account, and an approximate formula for bending stiffness is proposed for a three-layer plate. Error estimation of the models is carried out on test examples by comparison with the exact numerical solution of the three-dimensional problem of the elasticity theory. The possibility of suitability of the bending stiffness for calculating the plate eigenfrequencies is discussed.

Comparative study of two geometrically non-linear beam models with different strain measures

In this work, two geometrically non-linear beam models which use different strain measures are investigated in problems with large rotations. The first is the Simo–Reissner model. The second is the Green–Lagrange model. The novel aspects of this work are: (1) the 1-D formulation of the second beam model which uses the Green–Lagrange strain tensor components so that is comparable to the Simo–Reissner beam model, (2) the demonstration of the numerical behaviour of this 1-D model after incorporating an invariant update procedure described on the rotation manifold SO(3), and (3) the comparison of the different strain measures of the two beam models through the use of three numerical examples.

Theoretical calculation of transmission loss of noise panels

In the paper basic theoretical models of transmission loss calculation are described. Here only single homogeneous isotropic panels are analysed. Three impedance values of one rigid and two flexible body models of panel are described and analysed. As flexible models the thin plate (Kirchhoff-Love plate model) i thick (Mindlin – Reissner model) are analysed.

On the decoupled biharmonic airy stress function for the square-end adhesive layer and sandwich structure core

To accurately model the stress transfer through a square ended sandwich structure core or a square ended adhesive layer, a Decoupled Biharmonic Airy stress function solution is derived. This solution satisfies the zero stress natural boundary conditions at the traction free surfaces and the matched displacement essential boundary conditions at the top and bottom material interfaces. This model asymptotically satisfies the biharmonic equation on the Airy stress function that arises from simultaneous imposition of point equilibrium and displacement field compatibility. For an example case, the stress field is investigated for an idealized state of pure shear. In this case, the shear and normal stress fields are found to be finite and differentiable throughout the domain. This includes finite valued through-thickness normal stresses at the sharp corners of an elastic medium. The resulting stress components are compared with the Goland–Reissner model and the Closed Form Higher Order (CFHO) model and it is found that the Goland–Reissner model and the CFHO model significantly underestimate the peak stress values as compared to the Decoupled Biharmonic model.

An elastic plate bending equation of second-order accuracy

A study is carried out of a thin plate of constant thickness made of linearly elastic material which is transversally isotropic and heterogeneous in the thickness direction. Asymptotic expansions in powers of the relative plate thickness are constructed, and the bending equation of second-order accuracy (the SA model) is delivered. The results of the SA model are compared with the Kirchhoff–Love classical model and with the Timoshenko–Reissner (TR) model, as well as with the exact solution. To this end, some problems for a functionally gradient plate bending, and for a multi-layer plate bending and free vibration are solved and analysed. The range of plate heterogeneity, for which the error of the approximate models is small, is established. The TR model and the SA model are proved to yields results close to each other and the exact results for a very broad range of heterogeneity. That is why the generalized TR model for one-layered homogeneous transversely isotropic plate is proposed. Parameters of this model are chosen so that the results are close to the exact results and the results by the SA model. For the Navier boundary conditions, the analytical solution of 3D problems for a rectangular heterogeneous plate is constructed.

Two-dimensional model of a plate made of an anisotropic inhomogeneous material

We present an asymptotic derivation of the two-dimensional equations of equilibrium of a thin elastic inhomogeneous plate manufactured of an anisotropic material of general form with 21 moduli of elasticity. We also consider simplified models obtained under special assumptions on the moduli. We use test examples to illustrate the error estimate of the proposed model and discuss its scope. The model is compared with the classical Kirchhoff–Love and Timoshenko–Reissner models.

Generalized Timoshenko‐Reissner models for beams and plates, strongly heterogeneous in the thickness direction

The paper is concerned with a thin beam and a thin plate made of a transversally isotropic linearly elastic material heterogeneous in the thickness direction. The ordinary Kirchhoff–Love and Timoshenko–Reissner models are known to be unacceptable for a strongly heterogeneous material or for a multilayered material with large ratio between the Young moduli of layers. A generalized Timoshenko–Reissner model is proposed for bending and free vibrations of such beams/plates. A multilayered beam/plate is reduced to one‐layer one with the equivalent elastic parameters. The range of application of such a model is very wide.

Generalized Timoshenko–Reissner model for a multilayer plate

A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.

Through-thickness stress distribution in the adhesive joint for the multilayer composite material

The adhesive joint Goland and Reissner model was generalized for the arbitrary number of the adherend layers. This model was used for the stress state analysis in the layered composite rod with metal edging.

Buckling of an axially compressed transversely isotropic cylindrical shell with a weakly supported curvilinear edge

The critical load and the buckling modes of a transversely isotropic circular cylindrical shell under axial compression are found. It is assumed that the curvilinear shell edge is free or weakly supported. In these cases, the buckling mode can be localized near this edge and the critical load can be lower than in the case of clamped edges. The transverse shear modulus is assumed to be small, so the solution is based on the Timoshenko–Reissner (TR) model. The deformations of the edge element are described using five general coordinates. Therefore, 25 = 32 possible combinations of boundary conditions are considered, depending on the restraint conditions imposed on these coordinates. In 15 cases, there is a chance of buckling near the shell edge and the corresponding behavior of the functions λ(q, g) is investigated. The role of the fifth boundary condition in the TR model, which does not exist in the Kirchhoff–Love (KL) model is studied. It is shown that, if the boundary condition H = 0 holds and g→0, then the results based on the TR and KL models coincide. If the restraint φ2 = 0 is imposed and g→0, then the TR model produces new results as compared with the KL model.

Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

Analytical particular solutions of the polyharmonic multiquadrics are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.

A Theoretical Analysis on the Thermal Buckling Behavior of Fully Clamped Sandwich Panels with Truss Cores

This article presents a theoretical analysis on the thermal buckling behavior of sandwich panels with truss cores under fully clamped boundary conditions, subjected to uniform temperature rise. The Reissner model is developed by ignoring the flexural rigidity of the core and considering the shear stiffness of the sandwich panel is only contributed by truss cores. By using double Fourier expansions to the virtual deformation mode, the critical temperature of sandwich panels is obtained. Theoretically predicted critical temperatures are in good agreement with those from FEM. The effect of boundary conditions and structure parameters of the sandwich panel are also discussed.

A new experimental approach for the estimation of bending moments in adhesively bonded single lap joints

The state of stresses in the adhesive layer may be described by a parameter known as edge moment factor. This parameter is also defined as normalized bending moment, or bending moment factor. The main goal of the present work is to determine the bending moment factor using the Digital Image Correlation method. For this purpose, 9 aluminum-epoxy single-lap-joint specimens with different adhesive thicknesses and ratios of segment of outer adherend length to the half of the overlap length were investigated. All specimens were tested under tensile loading. Deflections of single lap-joint specimens were estimated by the Digital Image Correlation method. In addition, Hart-Smith, Goland and Reissner and modified Goland and Reissner models were investigated in order to compare theoretical prediction with experimental data. The improvement to the Goland and Reissner model proposed by Tsai–Morton was more suitable to describe the mechanical behavior of investigated single-lap joints. Results show that the alternative methodology proposed here is a good way for estimating bending moment.