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Applications of implicit constitutive theory for describing the elastic response of rocks and concrete

Bustamante R., Rajagopal K. R.

Archives of Mechanics

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2022

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Vol. 74, nr 6

513--547

EN

An implicit constitutive relation is proposed for elastic bodies, when the gradient of the displacement is assumed to be very small, and as a result the strains are small. The resulting constitutive relation is a non-linear relationship between the linearized strain and the stress. The model is used to fit data for rock and concrete. Some boundary value problems are studied within the context of homogeneous deformations, and also a problem with inhomogeneous deformations is analyzed, namely the inflation of a circular annulus. The predictions of this new implicit constitutive relation are compared with the predictions of the constitutive equations for linearized elastic bodies.

2

Computation and experimental comparison of the deformation behavior of pantographic structures with different micro-geometry under shear and torsion

Yang Hua, Müller Wolfgang H.

Journal of Theoretical and Applied Mechanics

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2019

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Vol. 57 nr 2

421--434

EN

Additive manufacturing methods, commonly known as 3D printing, allow more sophisticated designs to be created. Willingly designed substructures incorporated into the solid open up new possibilities for uncommon macroscopic deformation behavior. Such a man-made structure is also referred to as a metamaterial. A detailed simulation of a polymer-based metamaterial is challenging but accurately possible by means of the theory of elasticity. In this study, we present experimental investigations of a metamaterial composed of pantographic substructures of different internal geometry. The pantographic structures show an unexpected type of deformation, which can be modeled via elasticity with the aid of direct numerical simulation by using the Finite Element (FE) method. In other words, a detailed mesh is generated involving the substructure. The metamaterial is additively manufactured out of a common polymer showing nonlinear elastic deformation and, therefore, hyperelastic material models are used. Specifically, analytical solutions of a circular cylinder are examined and compared in the case of extension and torsion in order to comprehend the effects of the coefficients inherent to the energy function of the hyperelastic model. Finally, FE computations of pantographic structures are performed and compared with the experimental measurements.

3

Milestones of Direct Variational Calculus and its Analysisfrom the 17th Century until today and beyond - Mathematics meets Mechanics - with restriction to linear elasticity

Stein Erwin

Computer Assisted Methods in Engineering and Science

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2018

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Vol. 25, no. 4

141--225

EN

This treatise collects and reflects the major developments of direct (discrete) variational calculussince the end of the 17th century until about 1990, with restriction to classical linear elastome-chanics, such as 1D-beam theory, 2D-plane stress analysis and 3D-problems, governed by the 2nd order elliptic Lamé-Navier partial differential equations.The extension of the historical review to non-linear elasticity, or even more, to inelastic deformations would need an equal number of pages and, therefore, should be published separately.A comprehensive treatment of modern computational methods in mechanics can be found inthe Encyclopedia of Computational Mechanics.The purpose of the treatise is to derive the essential variants of numerical methods and algorithmsfor discretized weak forms or functionals in a systematic and comparable way, predominantly usingmatrix calculus, because partial integrations and transforming volume into boundary integrals with Gauss’s theorem yields simple and vivid representations. The matrix D of 1st partial derivativesis replaced by the matrix N of direction cosines at the boundary with the same order of non-zero entries in the matrix.

4

Nonlinear composites structural analysis with viscoelastic fibers

Sinaceur Amal, Sahli Ahmed, Sahli Sara

Advances in Science and Technology. Research Journal

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2018

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Vol. 12, no 2

158--169

EN

This paper presents a formulation for material and geometrical nonlinear analysis of composite materials by immersion of truss finite elements into triangular 2D solid ones using a novel formulation of the finite element method based on positions. This positional formulation uses the shape functions to approximate some quantities defined in the Nonlinear Theory of Elasticity and proposes to describe the specific strain energy and the potential of the external loads as function of nodal positions which are set from a deformation function. Because the nodal positions have current values in each node, this method naturally considers the geometric nonlinearities while the nonlinear relationships between stress and strain may be considered by a pure nonlinear elastic theory called hyperelasticity which allows to obtain linearised constitutive laws in its variational form. This formulation should be able to include both viscoelastic and active behavior, as well as to allow the consideration of nonlinear relations between stresses and deformations. It is common to adopt hyperelastic constitutive laws. Few are the works that use the strategy of approaching the problem such as fibers immersed in a matrix. The immersion of fibers in the matrix makes it possible to include both a viscoelastic behavior in a simple and direct way. The examples are simple cases, some of them even with analytical solutions, mainly for validation purposes of the presented formulations. By modeling a structure, the examples show the potentialities of the concepts and proposed formulations.

5

Antiplane deformation of a bimaterial with thin interfacial nonlinear elastic inclusion

Sulym H., Piskozub Y., Polanski J.

Acta Mechanica et Automatica

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2018

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Vol. 12, no. 3

190--195

EN

The problem of longitudinal shear of bimaterial with thin nonlinear elastic inclusion at the interface of matrix materials is considered. Solution of the problem is constructed using the boundary value problem of combining analytical functions and jump functions method. The model of the thin inclusion with nonlinear resilient parameters is built. Solution of the problem is reduced to a system of singular integral equations with variable coefficients. The convergent iterative method for solving such a system is offered for various nonlinear strain models, including Ramberg-Osgood law. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analysed for the tensely-deformed matrix under loading a uniformly distributed shear stresses and for a balanced system of the concentrated forces.

6

Euler–Bernoulli type beam theory for elastic bodies with nonlinear response in the small strain range

Janečka A., Průša V., Rajagopal K. R.

Archives of Mechanics

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2016

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Vol. 68, nr 1

3--25

EN

The response of many new metallic alloys as well as ordinary materials such as concrete is elastic and nonlinear even in the small strain range. Thus, using the classical linearized theory to determine the response of bodies could lead to a miscalculation of the stresses corresponding to the given strains, even in the small strain regime. As stresses can determine the failure of structural members, such miscalculation could be critical. We investigate the quantitative impact of the material nonlinearity in the Euler–Bernoulli type beam theory. The governing equations for the deflection are found to be nonlinear integro-differential equations, and the equations are solved numerically using a variant of the spectral collocation method. The deflection and the spatial stress distribution in the beam have been computed for two sets of models, namely the standard linearized model and some recent nonlinear models used in the literature to fit experimental data. The predictions concerning the deflection and the spatial stress distribution based on the standard linearized model and the nonlinear models are considerably different.

7

Computational challenges in the simulation of nonlinear electroelasticity

Steinmann P., Khoi Vu D.

Computer Assisted Methods in Engineering and Science

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2012

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Vol. 19, no. 3

199-213

EN

Nonlinear electroelasticity is not a new problem, its theory involving nonlinear deformation and nonlinear material behavior has been well established. However, the numerical simulation of nonlinear electroelas- ticity is until now still far from satisfactory, especially when the interaction between electric fields and matter cannot be considered as confined in the finite space occupied by the matter. It is understood that under the application of an electric field, the deformation of an elastic body is governed not always by what happens inside the material body but in many cases also by the environment surrounding it. This is notably true in the case of electronic electroactive polymers, the materials that emerge today as a lead- ing candidate in developing artificial muscles. In this work, we present a numerical analysis of nonlinear electroelasticity by assuming large deformation, nonlinear polarization and by paying attention to the contribution of the free space surrounding the bodies of interest.

8

A note on tensile instabilities and loss of ellipticity for a fiber-reinforced nonlinearly elastic solid

Merodio J., Neff P.

Archives of Mechanics

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2006

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Vol. 58, nr 3

293-303

EN

In this paper we examine the loss of ellipticity and the associated failure of fiber-reinforced compressible nonlinearly elastic solids under deformations leading to fiber extension. In particular, the analysis concerns a material model that consists of an isotropic base material augmented by a reinforcement depending on the fiber direction and referred to as a reinforcing model. We examine a reinforcement that introduces additional stiffness under simple shear deformations in the fiber direction. In previous contributions it was shown for this material that loss of ellipticity under uniaxial tensile loading in the fiber direction requires a non-convex reinforcing model. Here we generalize this result and show that loss of ellipticity under plane deformations not associated with uniaxial loading in the fiber direction but also creating fiber extension may occur for convex reinforcing models.

9

Numerical simulation of atomic positions in quantum dot by means of molecular statics

Dłużewski P., Traczykowski P.

Archives of Mechanics

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2003

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Vol. 55, nr 5-6

393-406

EN

Deformation of a crystal structure is considered here in terms of constitutive modelling based upon both the atomistic and continuum approaches. Atomistic calculations are made by using the Stillinger-Weber potential for the GaAs and CdTe structures. The stress-strain behaviour of the best-known anisotropic hyperelastic models are compared with the behaviour of the atomistic one in the uniaxial deformation test.

10

Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 2, The strongly curved shells

Hamdouni A., Millet O.

Archives of Mechanics

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2003

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Vol. 55, nr 2

177-219

EN

In the first part of this paper we have deduced a classification of asymptotic shallow shell models with respect to the level of applied forces, from the non-linear three-dimensional elasticity. We have used a constructive approach based on a dimensional analysis of the non-linear three-dimensional equilibrium equations, which naturally makes appear dimensionless numbers characterizing the applied forces (... and ...) and the geometry of the shell (... and C). To limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness ... of the shell, considered as the perturbation parameter. In the first part, we have studied the case of shallow shells corresponding to C=.... In the second part of this paper, we will study the case of strongly curved shells for which C=.... The classification that we obtain is then more complex. It depends not only on the force levels, but also on the existence of inextensional displacements which keep invariant the metric of the middle surface of the shell.

11

Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 1, The shallow shells

Hamdouni A., Millet O.

Archives of Mechanics

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2003

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Vol. 55, nr 2

135-175

EN

The purpose of this paper is to construct a classification of asymptotic shell models (inferred from the non linear three-dimensional elasticity) with respect to the applied forces and to the geometrical data. To do this, we use a constructive approach based on a dimensional analysis of the nonlinear three-dimensional equilibrium equations, which naturally gives rise to the appearance of dimensionless numbers characterizing the applied forces and the geometry of the shell. In order to limit our study to one-scale problems, these dimensionless numbers are expressed in terms of to the relative thickness ... of the shell, which is considered as the perturbation parameter. This leads, on the one hand, to distinguish shallow shells from strongly curved shells which have a different asymptotic behaviour, and on the other hand, to fix the applied force level. For each of these two classes of shells, using the usual asymptotic method, we propose a complete classification of two-dimensional shell models based on decreasing force levels, from severe to low. In the first part of this paper, we present the classification for shallow shells. We obtain successively the nonlinear membrane model, another membrane model, Koiter's non linear shallow shell model, and the linear Novozhilov-Donnell one, respectively for severe, high, moderate and low forces.

12

Design methods of machine elements of non-classical materials (in examples)

Dudziak M., Mielniczuk J.

Zeszyty Naukowe Politechniki Poznańskiej. Maszyny Robocze i Transport

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2002

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Nr 54

83-97

EN

The paper describes important methods of designing of construction characteristics: material characteristics, geometrical, stress - strain, and rheological properties. They can be helpful in designing machine elements of non-classical materials. The examples present designing taking into account the theory of nonlinear elasticity, limit load capacity and the theory of deformation of high-molecular materials.

13

A nonlinear model of a turbine blade by asymptotic analysis

Rodriguez J. M.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 1

101-113

EN

In this paper we obtain a limit model for a turbine blade fixed to a 3D solid. This model is a three-dimensional linear elasticity problem in the 3D part of the piece (the rotor) and a two-dimensional problem (the nonlinear shallow shell equations) in the 2D part (the turbine blade), with junction conditions in the part of the turbine blade fixed to the rotor. To obtain this model, we perform an asymptotic analysis, starting with the nonlinear three-dimensional elasticity equations on all the pieces and taking as a small parameter the thickness of the blade.