On Obtaining Solutions in Nonlinear Viscoelasticity

1968 ◽  
Vol 35 (1) ◽  
pp. 129-133 ◽  
Author(s):  
R. M. Christensen
Keyword(s):  
Boundary Value Problems ◽  
Constitutive Relation ◽  
Nonlinear Viscoelasticity ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Static Solution ◽  
Fading Memory ◽  
Viscoelastic Cylinder ◽  
Nonlinear Stress

A derivation is given of a particular form of the isothermal nonlinear stress constitutive relation for materials with fading memory. This particular derivation results in a form for the constitutive relations that is well suited for application in solving boundary-value problems. The resulting forms are applied to obtain the exact quasi-static solution for the torsion of a right circular isotropic viscoelastic cylinder. Several effects due to the nonlinearity of the problem are discussed.

scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.

Verification of Plasticity Theory with Isotropic Hardening and Additive Decomposition of Left Deformation Tensor Using Digital Image Correlation System

2015 ◽  
Vol 240 ◽  
pp. 61-66 ◽  
Author(s):  
Marcin Gajewski ◽  
Cezary Ajdukiewicz ◽  
Andrzej Piotrowski
Keyword(s):  
Digital Image Correlation ◽  
Boundary Value Problems ◽  
Digital Image ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Experimental Tests ◽  
Image Correlation ◽  
Deformation Tensor ◽  
Isotropic Hardening ◽  
Complex Boundary

The development of measurement methods, and in particular digital image correlation (DIC) systems, which are designed to measure of entire displacements and deformations fields, opens up new areas of research. In general, the materials constitutive relations are formulated in such a way that material parameters could be determined with relatively simple experimental tests carried out on samples with uniform (approximately) stress and strain fields. Then it is possible to apply them to complex boundary value problems formulated e.g. in the small or large deformation theories. The application of DIC allows to verify the accuracy of their predictions by comparing the results of the experiment with solutions to boundary value problems obtained using the finite element method (FEM).

scholarly journals Implicit constitutive relations for nonlinear magnetoelastic bodies

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140959 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Boundary Value Problems ◽  
Large Deformations ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Small Displacement ◽  
Elastic Bodies ◽  
Classical Models ◽  
Linearized Model ◽  
Implicit Constitutive Relations

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

scholarly journals On the boundary conditions formulation in the problems of synthesis of woven 3d materials

2021 ◽  
pp. 114-121
Author(s):  
Евгений Валерьевич Мурашкин
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Stress Tensor ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Concrete Structures ◽  
Differential Constraints ◽  
Complete Formulation

В статье обсуждаются формулировки определяющих дифференциальных ограничений на поверхности наращивания на случай моделирования процессов формирования 3D материала, характеризующегося дополнительными характерными директорами (направлениями выкладки волокон в тканых материалах, арматуры в бетонных конструкциях). Выведена общая форма тензорного соотношения на поверхности наращивания, при учете дополнительных выделенных направлений. Определить набор совместных рациональных инвариантов тензора напряжений и характерных директоров. Дана инвариантно-полная формулировка определяющих соотношений на поверхности наращивания. Полученные результаты могут быть использованы для постановки и решения краевых задач, моделирующих процессы синтеза тканых 3D материалов. The article discusses the formulation of the defining differential constraints on the buildup surface in the case of modeling the processes of forming a 3D material characterized by additional characteristic directors (directions of laying fibers in woven materials, reinforcement in concrete structures). The general form of the tensor relation on the growing surface is derived, taking into account the additional selected directions. Determine the set of joint rational invariants of the stress tensor and characteristic directors. An invariant-complete formulation of the constitutive relations on the surface of the build-up is given. The results obtained can be used to formulate and solve boundary value problems that simulate the processes of synthesis of woven 3D materials.

scholarly journals About a new approach to calculating spiral clamps

2019 ◽  
pp. 59-67
Author(s):  
A N Danilin ◽  
S I Zhavoronok ◽  
L N Rabinsky
Keyword(s):  
Boundary Value Problems ◽  
Mutual Influence ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Design Parameters ◽  
Elastic Cylindrical Shell ◽  
Force Transfer ◽  
The Matrix ◽  
Generalized Displacements ◽  
Anisotropic Elastic

The bearing capacity of spiral clamps, which are mounted on wires (cables) for their tension, connection, repair, etc., is studied. The design of spiral clamps is formed from stretched spirals that are wound onto conductors with an interference fit, which makes it possible to obtain tensile connections practically inseparable. The general problem of the interaction of spiral clamps and overhead line conductor layers is formulated. Different asymptotic solutions are given for initial and boundary value problems, and the design parameters of spiral clamps are determined to provide their carrying capacity. A wire layer is represented by the energy approach as an equivalent anisotropic elastic cylindrical shell, and wire construction as a whole is considered as a system of cylindrical shells inserted each other and interacting by forces of pressure and friction. The equivalence of the elastic properties of the shell to the properties of the wire layer is established using energy averaging. The constitutive relations obtained using the Castigliano theorem relate the generalized displacements and the corresponding forces. The matrix in these ratios is a stiffness matrix or flexibility matrix of a spiral wire structure. Such approach allows variety of interaction problems for spiral clamps with conductor layers to be solved, and the force transfer mechanism to be investigated from common positions. Static equations are written from the equilibrium of the elementary shell ring. It is considered that the length of the clamp is so great that the mutual influence of its ends can be neglected; the clamp is modeled as semi-infinite shell. This model allows the different initial and boundary value problems to be formulated, depending on the boundary conditions and clamp mounting methods on a conductor.

scholarly journals On a new class of electro-elastic bodies. II. Boundary value problems

2013 ◽  
Vol 469 (2155) ◽  
pp. 20130106 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Electric Field ◽  
Boundary Value Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Electric Displacement ◽  
Cylindrical Tube ◽  
The Body ◽  
Elastic Bodies ◽  
Nonlinear Relation ◽  
Elastic Slab

In part I of this two-part paper, a new theoretical framework was presented to describe the response of electro-elastic bodies. The constitutive theory that was developed consists of two implicit constitutive relations: one that relates the stress, stretch and the electric field, and the other that relates the stress, the electric field and the electric displacement field. In part II, several boundary value problems are studied within the context of such a construct. The governing equations allow for nonlinear coupling between the electric and stress fields. We consider boundary value problems wherein both homogeneous and inhomogeneous deformations are considered, with the body subject to an electric field. First, the extension and the shear of an electro-elastic slab subject to an electric field are studied. This is followed by a study of the problem of a thin circular plate and a long cylindrical tube, both subject to an inhomogeneous deformation and an electric field. In all the boundary value problems considered, the relationships between the stress and the linearized strain are nonlinear, in addition to the nonlinear relation to the electric field. It is emphasized that the theories that are currently available are incapable of modelling such nonlinear relations.

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