Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

Elastic theory of unconstrained non-Euclidean plates

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.     

Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.     

Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements

Earlier it was shown in [1, 2] that the equations of classical nonlinear elasticity constructed for the case of small strains and arbitrary displacements are ill posed, because their use in specific problems may result in the appearance of “spurious” bifurcation points. A detailed analysis of these equations and the construction, in their stead, of consistent equations of geometrically nonlinear theory of elasticity can be found in [3]. Certain steps in this direction were also made in [4, 5]. In [3], it was also stated that the methods and applied program packages (APPs) based on the use of the classical relations of nonlinear elasticity require some revision and correction. In the present paper, this conclusion is justified and confirmed by numerical finite-element solutions of several three-dimensional geometrically nonlinear deformation problems and linearized problems on the stability of equilibrium of a rectilinear beam. These solutions were obtained by using two APPs developed by the authors and the well-known APP “ANSYS.” It is shown that the classical equations of the geometrically nonlinear theory of elasticity, which underly the first of the developed APP and the well-known APP “ANSYS,” often lead to overestimated buckling loads for structural members as compared with the consistent equations proposed in [1–3].     

Asymptotic construction of Reissner-like composite plate theory with accurate strain recovery

The focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry. The development starts with formulation of the three-dimensional (3-D), anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two-dimensional (2-D) variables. The variational asymptotic method is then used to rigorously split this 3-D problem into a linear one-dimensional normal-line analysis and a nonlinear 2-D plate analysis accounting for classical as well as transverse shear deformation. The normal-line analysis provides a constitutive law between the generalized, 2-D strains and stress resultants as well as recovering relations to approximately but accurately express the 3-D displacement, strain and stress fields in terms of plate variables calculated in the plate analysis. It is known that more than one theory may exist that is asymptotically correct to a given order. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like” plate theory. Although it is not possible in general to construct an asymptotically correct Reissner-like composite plate theory, an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of the Reissner-like formulation. Numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory. The present theory has excellent agreement with the previous theory and exact results.     

Linear and Nonlinear Aerodynamic Theory of Interaction between Flexible Long Structure and Wind

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Dynamic rheological properties of a fumed silica grease

The thermo-rheological characteristics of a fumed silica lubricating grease in linear and nonlinear oscillatory experiments have been investigated. The material rheological behavior represents a soft solid being thermo-rheologically complex. There is an abnormal temperature dependency in the range of ??10 to 10 °C which is related to the phase transition of the base oil. The dynamic moduli data in linear viscoelastic envelop (LVE) have been modeled using mode-coupling theory (MCT) in the whole temperature range. Two main relaxation mechanisms can be identified through linear and nonlinear viscoelastic properties related to interaction of the primary particle and its neighbor particles as well as a slow relaxation process which represents the escape of this particle from its “cage”. Finally, it is demonstrated that the dominant yielding process in large amplitude oscillatory experiments can be explained based on either particle cage rupture (consistent with MCT framework) or particle “hopping” out of its cage proposed in soft glassy rheology (SGR) model. It will be discussed that the governing mechanism depends on the applied frequency.     

Recasting theory of elasticity with micro-finite elements

In the classical theory of elasticity,a body is initially modeled as a homogeneous and dense assemblage of constituent "material particles".The kernel concept of elastic deformation is the displacement of the particle that associates the current configuration with the reference one.In this paper,we exploit an alternative constituent "micro-finite element",and use the stretch of the element as the essential quality to recast the theory of elasticity.It should be realized that such a treatment means that the elastic body can be modeled as a finite covering of elements and consequently characterized by a manifold.The recasting of the elasticity theory becomes more feasible for dealing with defects and topological evolution.     

CFD–DEM simulation of particle deposition characteristics of pleated air filter media based on porous media model

In this study, the three-dimensional physical model of pleated air filtration media was simplified to porous media model, and the calculation parameters of porous media were obtained based on experimental data. The model of V-shaped pleated air filter media is constructed, the height of the media pleat is 50 mm and the pleat thickness is 4 mm, the pleat angle is 3.7°. The Hertz-Mindlin contact model was modified by Johnson Kendall Roberts (JKR) adhesion contact model. The deposition process of particles in media was simulated based on computational fluid dynamics (CFD) theory and discrete element method (DEM). Results show that the CFD–DEM coupling method can be effectively applied to the macro research of pleated air filter media. The particles will form dust layer and dendrite structure on the fiber surface, and the dust layer will affect the subsequent air flow organization, and the dendrite structure will eventually form a “particle wall”. The formation of the “particle wall” will prevent the particles from moving further in the fluid domain, which makes area of pleated angle become the “low efficiency” part about the particle deposition. Compared with area of pleated angle, the particles are concentrated in the opening area and the middle area of the pleated to agglomerate and deposit.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by com-bining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momen-tum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and ana-lyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.     

A version of applied theory of thick shells

A version of an applied theory of shells of large thickness based on the introduction of force and kinematic hypotheses completing and extending the set of Love-Kirchhoff and Timoshenko-Reissner hypotheses is discussed. The complete system of equations including the elasticity relations, the geometric relations (displacements and strains), and the equilibrium equations is written out. The obtained system of equations is verified in several special cases. It is noted that the error of this theory does not exceed the squared thickness-to-radius ratio compared with unity.     

Converse problem of elasticity theory for an anisotropic medium

To prevent stress concentration it is of considerable interest that the body contour be found whose portions show no preference to brittle failure or plastic deformation. Such contours are called by us “equally rigid.” Two-dimensional problems are considered for finding the “equally rigid” form of the hole in an anisotropic medium. The lack of stress concentration on the coutour of the hole is the criterion which decides whether or not the hole is “equally rigid.” For an isotropic medium this converse problem of elasticity theory was solved in [1].     

Vibrations of structurally orthotropic laminated shells under thermal power loading

On the basis of the linearized version of equations obtained in a geometrically nonlinear statement and describing the nonaxisymmetric strain of nonshallow sandwich structure orthotropic shells under thermal power loading, the Rayleigh–Ritz method with polynomial approximation of displacements and shear strains is used to solve the problem of small free vibrations of axisymmetrically thermally preloaded freely supported three-layer conical shell. The causes of dynamical fracture of the shell under study are revealed.     

Structural rigidity optimization with frictionless unilateral contact

The problem of maximization the global rigidity (measured by the compliance) of an elastic structure with frictionless unilateral contact is considered in the framework of topology optimization. The frictionless unilateral contact is introduced in the continuous formulation of the elastic problem (under the assumption of small strains and small displacements) in the regularized form of an interface with an asymmetric behavior law relating the normal component of the stress vector transmitted through the contact surface to the normal displacement (in the case of contact with a rigid foundation) or the jump of normal displacement (in the case of internal contact of two surfaces of the elastic medium). Using the concept of homogeneous thermodynamical potentials, we extend a convergent and numerically efficient optimization algorithm introduced in the framework of linear elasticity to this nonlinear case of an elastic structure with unilateral contact. Numerical examples in two-dimensional elasticity are presented.     

Development of the “binary interaction” theory for entangled polydisperse linear polymers

Based on the theoretical principles previously described in the literature, the development of the “naïve” binary interaction model is detailed in this paper. The new theory is effectively a sweeping generalization of the “Double Reptation” model. The “switch function” has been shown to be an essential feature of any constraint release model for Doi–Edwards type molecular models that invoke the concept of a discrete slip-link tube and is used in our formulation. Using the assumption of a constant entanglement density, a slip link linear density evolution equation is derived to rigorously count matrix entanglements. This function has no counterpart in the conventional Doi–Edwards theory, or its derivatives, and is absolutely required to properly generalize the “Double Reptation” model so that nonlinear flows can be modeled. The binary interaction polydispersity model is complex mathematically but can be rigorously and justifiably simplified by suppressing the tube coordinate dependence using a boundary layer analysis. The simplification process can be continued to the continuum level to create a hierarchy of approximate binary interaction models, thereby making large-scale numerical simulations of complex flows viable, indeed straightforward.