Semi-inverse Solutions Research Articles

On the explicit dynamics implementation and validation of partitioned rate-type constitutive relation for dampers

We propose a simple 3D nonlinear rate-type constitutive relation with just three parameters for elastomeric dampers capable of emulating double-shear data in frequency and strain sweep mode. The nonlinear relation derived using two scalar functions, namely Helmholtz free energy and the rate of dissipation function [1], result in a partitioned constitutive relation: Cauchy stress tensor is described in terms of the left Cauchy-Green stretch tensor and a tensor describing instantaneous elastic response, and the evolution equation for the instantaneous elastic response is derived by maximizing the rate of dissipation function. A compressible version of this model is implemented in Abaqus® The constitutive relation and the implementation are validated by comparing with known semi-inverse solutions obtained by specializing rate-type relation to special deformations. Material parameters are obtained for dampers made of synthetic and natural rubber by fitting the predictions of the model with that of double shear data. By using the same parameters, three dimensional simulations of a cylindrical damper subjected to sinusoidal force akin to that of an unbalanced machinery predict different hysteresis curves in compression/tension and shear. To that end, we show that the non-linear 3 parameter model is one of the simple models that can be formulated within the thermodynamic framework which can predict the inherent non-linearity of the damper operating under combined harmonic loading situations. A robust implementation (VUMAT) of the developed model is included with the article. We believe that this can aid engineers in tailoring dampers for its controlled dissipation across multiple modes of deformation as well as academicians in adding more value by building and implementing sophisticated models within this framework.

Semi-Inverse Solution of a Pure Beam Bending Problem in Gradient Elasticity Theory: The Absence of Scale Effects

The semi-inverse solutions of pure beam bending problems within the three-dimensional formulation of gradient elasticity theory as exact tests for the problem of estimating the efficient bending stiffness of so-called scale-dependent thin beams and plates due to the necessity of modeling sensing devices are presented. It is shown that the solutions within the gradient elasticity theory give classic beam bending stiffnesses and demonstrate the invalidity of the widespread results and estimates obtained in the past 15 years during study of scale effects within the gradient beam theories, according to which the relative bending stiffness grows by a hyperbolic law with decreasing thickness.

Revisiting bending theories of elastic gradient beams

In this paper, we discuss a problem of Bernoulli–Euler beam models development in the frame of the strain gradient and distortion gradient elasticity theories. Contradiction between some known size-dependent gradient beam models and analytical and numerical three-dimensional solutions found for the beam bending problems is shown. In particular, it is shown that in semi-inverse analytical solutions for a beam pure bending problem, the inverse squared dependence of normalized bending stiffness on the beam thickness could arise only due to wrong definition of the resultant bending moment and improper formulation of the boundary conditions on the top and bottom surfaces of the beam. In the correct semi-inverse solutions, finite values of normalized bending stiffness for ultra-thin gradient beams arise. The obtained results are also confirmed on the basis of 3D FE simulations realized for a pure and cantilever beam bending problems in the frame of simplified strain gradient elasticity theory. As a result, it is shown that the only correct approach for Bernoulli–Euler beam models development in gradient theories should ensures the fulfillment of boundary conditions on the top and bottom surfaces of the beam and corresponds to a model that assumed a uniaxial stress state of the beam. The consistent variational algorithm for the correct gradient beam models derivation is proposed.

On solutions of Saint-Venant’s problem for elastic dipolar bodies with voids

This study is dedicated to the Saint-Venant’s problem in the context of the theory of porous dipolar bodies. We consider a right cylinder consisting of an inhomogeneous and anisotropic material. In the equilibrium equations of this problem, the axial variable is regarded as a parameter. The main result describes a class of semi-inverse solutions to the Saint-Venant’s problem in terms of some generalized plane strain problems.

On the determination of semi-inverse solutions of nonlinear Cauchy elasticity: The not so simple case of anti-plane shear

We provide a systematic and complete analysis of the overdetermined problem that one obtains while considering the balance equations of unconstrained isotropic nonlinear Cauchy elastic bodies undergoing anti-plane shear deformations.

A semi-inverse problem of flows of fluids with pressure-dependent viscosities

A flow is said to be controllable if such a flow can be engendered by the application of appropriate surface tractions. The question, in which fluids are such flows possible leads to the inverse problem of determining constitutive relations, as such flows in general will not be possible in all materials. Thus the determination of the appropriate constitutive class in which certain types of deformations are possible is an inverse problem. A very special subclass of controllable flows is defined through semi-inverse methods. Such semi-inverse solutions are not possible in all bodies but only in special bodies and thus yet constitute an inverse problem. Here, we use a semi-inverse method to consider the flow of fluids whose viscosity depends on the pressure, glowing down an inclined plane, and a flow between two parallel plates. We illustrate that we have an inverse problem with regard to choices of constitutive relations by showing that such flows are only possible in fluids with special viscosity–pressure relationships. We also show, with the help of the simple flow problem that boundary layers can develop in virtue of the viscosity depending on the pressure, a rather novel feature.

Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.

On semi-inverse solutions for the time-dependent flows of a second-grade fluid

This paper deals with analytical solutions for the time-dependent equations arising in a second-grade fluid. The solutions have been developed by assuming certain forms of the stream function. Expressions for velocity components are obtained for flows in plane polar, axisymmetric cylindrical, and axisymmetric spherical polar coordinates. The obtained solutions are compared with existing results.

Semi-inverse solutions leading to two-dimensional boundary value problems in nonlinear elasticity theory

Semi-inverse solutions leading to two-dimensional boundary value problems in nonlinear elasticity theory

Saint-Venant's problem and semi-inverse solutions in linear viscoelasticity

This paper gives a treatment of Saint-Venant's problem for viscoelastic cylinders by reformulating the quasi-static equilibrium equations as a formal integro-differential operator over the cross section of the cylinder, with the axial variable playing the role of a parameter. Then, the conditions are established that the solution of Saint-Venant's problem may be treated as a generalized plane strain problem. Further, two classes of semi-inverse solutions to Saint-Venant's problem are described. These classes are used in order to obtain a solution for the relaxed Saint-Venant's problem.

Corrigendum ?Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity?

Corrigendum ?Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity?

Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity

Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross-sectional resultants of force and moment. Using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite-dimensional manifold. In particular, the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations. Moreover, the set of semi-inverse solutions can be analyzed locally.

A Relation between Semi-Inverse and Saint-Venant Solutions for Prisms

It is shown that in an infinite prism the linearization at a natural state of a family of Ericksen’s semi-inverse solutions is a combination of elementary St.-Venant solutions, namely, extension bending and torsion. Moreover, the span of these St. Venant solutions is precisely the linear manifold of solutions having locally uniformly bounded strain energy. This implies that any solution of a linearized problem in a finite prism continuable to a solution in the infinite prism in a manner that its energy on any portion of fixed length remains bounded, is an elementary St.-Venant solution.

Semi-inverse solutions of a non-newtonian fluid

Solutions for the equations of motion of an incompressible second grade fluid are obtained by employing semi-inverse methods in which we assume certain geometrical or kinematical properties of the fields. Specifically the problems studied in viscous fluids by Jeffery, Hamel and Görtler and Wieghardt, etc. are considered in a second grade fluid and the results for stream lines, velocities and pressure distribution are compared in the two cases.

Reduced equilibrium equations for perfectly elastic materials

Sufficient conditions are given on the coordinate systems which enable reduced equilibrium equations to be derived for perfectly elastic materials involving deformations which depend in an essential way only on two of the three coordinates. Reduced equilibrium equations given previously for plane and axially symmetric deformations are special cases of the equations given here. These equations considerably reduce the calculations involved in investigating possible solutions of finite elasticity, either exact semi-inverse solutions or approximate perturbation solutions. Moreover a formula for the pressure function appearing in the reduced equilibrium equations is given which relates to the corresponding pressure function associated with the inverse deformation. This formula is similar to one given previously for fully three dimensional deformations.

The Role of Saint Venant’s Solutions in Rod and Beam Theories

Within small displacement linear elasticity, Saint Venant derived semi-inverse solutions for bending and torsion of rods and beams. The resulting linear expressions for bending moments and torque in terms of curvatures and torsion are often taken to be those required by elastica theory in describing general configurations of rods and beams. It is here shown, by using a perturbation procedure based on an appropriate scaling of axial variations, that Saint Venant’s solutions combine naturally with elastica theory to describe configurations involuing large rotations and displacements. Also it is shown that the classical solution for beam flexure is essentially a bending solution with varying curvature. The contribution involving the bending function is, for general beam loading, only part of the first-order correction.

An asymptotic analysis of large deflections and rotations of elastic rods

The distortions within beams and rods undergoing large displacements and rotations are here derived from three-dimensional elasticity by an asymptotic procedure. This procedure, based on the premise that strains vary more gradually along the rod than in transverse directions, takes full account of the shape of the cross section, the traction conditions on the lateral boundary and of any material anisotropy. The manner in which it generates a hierarchy of sets of rod equations is outlined. In the fundamental set, the distortions over each cross section are the anti-clastic curvature and warping associated with the St.-Venant semi-inverse solutions for bending and torsion, suitably corrected to allow for large rotations. The corresponding equations governing the rod configuration are Kirchhoff's equations, with bending and torsional rigidities computed from the St.-Venant distortions. The procedure gives some three-dimensional substance to elastica theory, relating the constitutive assumptions to three-dimensional elasticity. It gives also a logical procedure for obtaining higher order corrections to the theory, and shows how St-Venant's hypothesis concerning details of end loading arises naturally.