A large deformation analysis for nonlinear elastic problems by hypoelastic theory

In this paper, large deformation problems for nonlinear elasticity are studied on the basis of hypoelastic theory. The expression of Cauchy elasticity is derived in the form of hypoclasticity. This makes it possible to solve large deformation for nonlinear elastic problems by the hypoelastic method. The variational principle of hypoclasticity and the Ritz method are used to obtain the numerical solution of a rectangular rubber membrane under uniaxial stretch.     

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.     

Deceleration and deformation of a liquid drop in a gas stream

An analytic method is developed for calculating the nonstationary motion and spreading of two-dimensional and axisymmetric liquid drops in a gas stream. The method is based on an expansion of the Navier-Stokes equations in a small parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–69, March–April, 1981.     

On large elastic deformation of prestressed right circular annular cylinders

We formulate and study inflation, extension and twisting of prestressed cylindrical shells that are isotropic in the stress free configuration. We establish that if the prestresses vary only radially in the annular cylinder then a deformation field of the form , θ=Θ+ΩZ, z=λZ is possible in annular cylinders made of any incompressible material and find sufficient conditions for the deformation to be possible when made of compressible materials. When the material is capable of undergoing large elastic deformations and has a non-linear constitutive relation, for the cases studied here, there is up to 26 percent variation in the boundary loads required to engender a given boundary displacement between the prestressed and stress free annular cylinders. On the other hand, the difference in the realized deformation field is only marginal (less than 2 percent). These are unlike the case wherein the material obeys Hooke's law and undergoes small deformations. This study has some relevance to the deformation of blood vessels.     

Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates

A combined experimental and analytical investigation has been performed to understand the mechanical behavior of two amorphous polymers—polycarbonate and poly(methyl methacrylate)—at strain rates ranging from 10⁻⁴ to 10⁴ s⁻¹. This range in strain rates was achieved in uniaxial tension and compression tests using a dynamic mechanical analyzer (DMA), a servo-hydraulic testing machine, and an aluminum split-Hopkinson pressure bar. DMA tension tests were used to characterize the viscoelastic behavior of these materials, with focus on the rate-dependent shift of material transition temperatures. Uniaxial compression tests on the servo-hydraulic machine (10⁻⁴ to 1 s⁻¹) and the split-Hopkinson pressure bar (10³ to 10⁴ s⁻¹) were used to characterize the rate-dependent yield and post-yield behavior. Both materials were observed to exhibit increased rate sensitivity of yield under the same strain rate/temperature conditions as the β-transition of the viscoelastic behavior. A physically based constitutive model for large strain deformation of thermoplastics was then extended to encompass high-rate conditions. The model accounts for the contributions of different molecular motions which become operational and important in different frequency regimes. The new features enable the model to not only capture the transition in the yield behavior, but also accurately predict the post-yield, large strain behavior over a wide range of temperatures and strain rates.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

On the calculation of some definite integrals in antenna theory

Summary The mutual radiation resistance of two identical linear antennae is usually calculated by using the e.m.f. method. However, in a recent paper it has been demonstrated by the author that sometimes a simple expression for the mutual radiation resistance of two antennae, derived on the basis of the Poynting vector method, may be used with advantage. In most cases, however, the results obtained by using this modified Poynting vector method contain integrals which are difficult to express by known functions. By equating the expression for the mutual radiation resistance derived on the basis of the Poynting vector method with that obtained by the e.m.f. method we may therefore obtain a useful mathematical formula expressing a definite integral by known functions. This heuristic procedure for generating formulae for some definite integrals has been used in several simple cases of pairs of antennae. Only an outline of the rather bulky calculations has been given here, details being reserved for separate papers on some new antenna formulae. Although the paper is mainly concerned with demonstrating the above mentioned method for generating integral formulae, the compilation of formulae for mutual radiation resistance of antennae, mostly not published before, should be of interest for those working in antenna theory.     

Diffusion of a fluid through an elastic solid undergoing large deformation

This paper is concerned with the modeling of slow diffusion of a fluid into a swelling solid undergoing large deformation. Both the stress in the solid as well as the diffusion rates are predicted. The approach presented here, based on the balance laws of a single continuum with mass diffusion, overcomes the difficulties inherent in the theory of mixtures in specifying boundary conditions. A “natural” boundary condition based upon the continuity of the chemical potential is derived by the use of a variational approach, based on maximizing the rate of dissipation. It is shown that, in the absence of inertial effects, the differential equations resulting from the use of mixture theory can be recast into a form that is identical to the equations obtained in our approach. The boundary value problem of the steady flow of a solvent through a gum rubber membrane is solved and the results show excellent agreement with the experimental data of Paul and Ebra-Lima (J. Appl. Polym. Sci. 14 (1970) 2201) for a variety of solvents.     

On the modelling of anisotropic elastic and inelastic material behaviour at large deformation

The purpose of this work is the formulation and discussion of an approach to the modelling of anisotropic elastic and inelastic material behaviour at large deformation. This is done in the framework of a thermodynamic, internal-variable-based formulation for such a behaviour. In particular, the formulation pursued here is based on a model for plastic or inelastic deformation as a transformation of local reference configuration for each material element. This represents a slight generalization of its modelling as an elastic material isomorphism pursued in earlier work, allowing one in particular to incorporate the effects of isotropic continuum damage directly into the formulation. As for the remaining deformation- and stress-like internal variables of the formulation, these are modelled in a fashion formally analogous to so-called structure tensors. On this basis, it is shown in particular that, while neither the Mandel nor back stress is generally so, the stress measure thermodynamically conjugate to the plastic “velocity gradient”, containing the difference of these two stress measures, is always symmetric with respect to the Euclidean metric, i.e., even in the case of classical or induced anisotropic elastic or inelastic material behaviour. Further, in the context of the assumption that the intermediate configuration is materially uniform, it is shown that the stress measure thermodynamically conjugate to the plastic velocity gradient is directly related to the Eshelby stress. Finally, the approach is applied to the formulation of metal plasticity with isotropic kinematic hardening.     

Phenomena of a liquid drop falling to a liquid surface

Two kinds of phenomenon have been observed when a liquid drop falls to a surface of the same liquid. The first, which can nearly always be observed, involves splash and some degree of penetration and cleavage and the conditions for this occurrence are identified. The experimental observations are compared with previous computational results. The second kind of colliding phenomena can be observed only by chance in an ordinary falling drop experiment and appears to be random. It includes the two phenomena investigated in this paper: the floating drop and the rolling drop.     

On superposed small deformations on a large deformation of an elastic Cosserat surface

Within the scope of the theory of a Cosserat surface, this paper is concerned with small deformations superposed on a large deformation in elastic shells and plates together with some related aspects of the subject. Special attention is given to problems of stability and vibrations of initially stressed isotropic plates.

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Résumé Dans le cadre de la théorie d'une surface de Cosserat, le présent rapport étudic, pour des plaques et coques élastiques, de petites déformations superposées à une grande déformation, ainsi que certains aspects connexes du sujet. Une attention particulière est accordée aux problèmes de stabilité et de vibrations de plaques isotropiques initialement sous tension.

     

Application of color structured light pattern to measurement of large out-of-plane deformation

Measurement of out-of-plane deformation is significant to understanding of the deflection mechanisms of the plate and tube structures.In this study,a new surface contouring technique with color structured light is applied to measure the out-of-plane deformation of structures with one-shot projection.Through color fringe recognizing,decoding and triangulation processing for the captured images corresponding to each deformation state,the feasibility of the method is testified by the measurement of elastic deflections of a flexible square plate,showing good agreement with those from the calibrated displacement driver.The plastic deformation of two alloy aluminum rectangular tubes is measured to show the technique application to surface topographic evaluation of the buckling structures with large displacements.     

A theory for large deformation and damage of interpenetrating polymer networks

Elastomers and gels can be formed by interpenetrating two polymer networks on a molecular scale. This paper develops a theory to characterize the large deformation and damage of interpenetrating polymer networks. The theory integrates an interpenetrating network model with the network alteration theory. The interpenetration of one network stretches polymer chains in the other network and reduces its chain density, significantly affecting the initial modulus, stiffening and damage properties of the resultant elastomers and gels. Double-network hydrogels, a special type of interpenetrating polymer network, have demonstrated intriguing mechanical properties including high fracture toughness, Mullins effects, and necking instability. These properties have been qualitatively attributed to the damage of polymer networks. Using the theory, we quantitatively illustrate how the interplay between polymer-chain stiffening and damage-induced softening can cause the Mullins effect and necking instability. The theory is further implemented into a finite-element model to simulate the initiation and propagation of necking instability in double-network hydrogels. The theoretical and numerical results are compared with experimental data from multiple cyclic compressive and tensile tests.     

A theory of coupled diffusion and large deformation in polymeric gels

A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.     

Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers

A self-consistent model for semi-crystalline polymers is proposed to study their constitutive behavior, texture and morphology evolution during large plastic deformation. The material is considered as an aggregate of composite inclusions, each representing a stack of crystalline lamellae with their adjacent amorphous layers. The deformation within the inclusions is volume-averaged over the phases. The interlamellar shear is modeled as an additional slip system with a slip direction depending on the inclusion's stress. Hardening of the amorphous phase due to molecular orientation and, eventually, coarse slip, is introduced via Arruda-Boyce hardening law for the corresponding plastic resistance. The morphology evolution is accounted for through the change of shape of the inclusions under the applied deformation gradient. The overall behavior is obtained via a viscoplastic tangent self-consistent scheme. The model is applied to high density polyethylene (HDPE). The stress-strain response, texture and morphology changes are simulated under different modes of straining and compared to experimental data as well as to the predictions of other models.     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.