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Tuesday, November 10, 2020 | History

2 edition of Second gradient of strain in linear elasticity and its applications. found in the catalog.

Second gradient of strain in linear elasticity and its applications.

Seiichi Iwata

# Second gradient of strain in linear elasticity and its applications.

Published in [New York?] .
Written in English

Subjects:
• Elastic solids.,
• Particles (Nuclear physics),
• Difference equations.

• Classifications
LC ClassificationsQA935 .I9
The Physical Object
Pagination70 l.
Number of Pages70
ID Numbers
Open LibraryOL5346937M
LC Control Number72213456

In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. Without a doubt, the simplest approach to elasticity is linear-elasticity. This is a property that means that the relationship between stress and strain in the material is linear. Before a certain strain level, (sometimes small, sometimes pretty big) materials tend to “start” their strain-stress behavior win a linear .   After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. III. Non-Linear Elastic Constitutive Equations. From a specific strain energy function, we could derive the linear elastic coefficients by differentiating the strain energy with respect to the small strain coefficients. We could dervie both isotropic and anisotropic elastic coefficients from the strain .

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### Second gradient of strain in linear elasticity and its applications. by Seiichi Iwata Download PDF EPUB FB2

In this paper there is formulated a linear theory of deformation of an elastic solid in which the potential energy-density is a function of the strain and its first and second gradients.

This is a theory in which cohesive force and surface-tension are by: First and second strain gradient elasticity Lattice and continuum models Micro-structural effects abstract Mindlin, in his celebrated papers of Arch.

Rat. Mech. Anal. 16, 51–78, and Int. SolidsStruct. 1, –, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of. Second, the higher order elastic moduli, c i, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading.

These two effects are independent from each other and allow for separated identification of the corresponding material by: A linear elastic second gradient orthotropic two-dimensional solid that is invariant under $$90^{\circ }$$ rotation and for mirror transformation is considered.

A second strain gradient elasticity theory is proposed based on first and second gradients of the strain tensor. Such a theory is an extension of first strain gradient elasticity with double stresses. In particular, the strain energy depends on the strain tensor and on the first and second gradient terms of by: On first strain-gradient theories in linear elasticity on the two portions of S that intersect at C.

Equations (H), without the acceleration term, are linear forms of Toupin’s results [l, II. Second, the higher order elastic moduli, ci, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading.

These two effects are independent from each other and allow for separated identification of the corresponding material parameters. The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement.

The existence of the mixed, fifth-rank coupling term significantly complicates the problem. order elasticity constants required in the Toupin-Mindlin strain gradient theory. The method has been applied to a matrix-inclusion composite, showing that higher-order terms become more important as the stiffness contrast between inclusion and matrix increases.

⁄Corresponding author: [email protected] 1Email: [email protected] energy function of the (classical) strain and of its first gradient, which leads to the generation of symmetric stress fields (see Askes et al. [6] for historical details about the latter formulations and its applications).

Formulations in the boundary element method based on the strain gradient elasticity were pioneered by Polyzos et al. [7]. Elasticity Theory, Applications, and Numerics. Elasticity Theory and Applications, Second Edition, Revised. In strain gradient elasticity, the strain energy density of an elastic continuum is considered to be a function of the components of the strain tensor and its gradients.

Mindlin in his contribution on linear elasticity with microstructure (Mindlin, ) derived three forms for the formulation of strain gradient elasticity where, of course, all.

of continuum elasticity. First, and contrarily to classical elasticity, wave propagation in hexagonal (chiral or achiral) lattices becomes anisotropic as the frequency increases.

Second, since strain-gradient elasticity is dispersive, group and energy velocities have to be treated as di erent quantities. (Aifantis, ). This gradient elasticity model has been shown to eliminate strain singularities from dislocation lines and crack tips (Altan and Aifantis, ; Ru and Aifantis, ).

Even though this model could be formally obtained as a special case of the earlier gradient elasticity theories of the s, its.

We study the statics of some trusses, i.e. networks of nodes linked by linear springs. The trusses are designed in such a way that a few number of floppy modes are present and remain when considering the homogenized limit of the truss.

We then obtain linear elastic materials with exotic mechanical interactions which cannot be described in the classical framework of Cauchy stress theory.

Cordero NM, Forest S, Busso EP () Second strain gradient elasticity of nano-objects. J Mech Phys Solids – Google Scholar d’Agostino MV, Giorgio I, Greco L, Madeo A, Boisse P () Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite.

The second result can be derived by substituting the formula for displacement into the elastic stress-strain equations and simplifying. Point force in an infinite solid. The displacements and stresses induced by a point force acting at the origin of a large (infinite) elastic solid with Young’s modulus E and Poisson’s ratio are generated by.

Book Now. Elasticity Theory, Applications, and Numerics Next / / Elasticity Theory, Applications, and Numerics; zuvoq Elasticity Theory, Applications, and Numerics.

Elasticity Theory and Applications, Second Edition, Revised. The need for a third displacement gradient or, equivalently, a second strain gradient theory, and the sig-nificance of the cohesion parameter b 0 were recently re-examined for elastic fluids by Forest et al.

() and for isotropic solids by Cordero () and Ojaghnezhad and Shodja (). Mindlin's third displacement gradient/second strain. Advances in Engineering Plasticity and its Applications. Book models that can handle metallurgical reactions and the development of microstructure in the sharp rates and gradients of strain rate and temperature observed in extrusion.

Large strain elastic-viscoplastic torsion of cylindrical solid bars of glassy polymers is investigated. Elasticity: Theory and Applications reviews the theory and applications of elasticity.

The book is divided into three parts. The first part is concerned with the kinematics of continuous media; the second part focuses on the analysis of stress; and the third part considers the theory of elasticity and its applications to engineering problems.

Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski.

Using a suitable decomposition of the strain-gradient, it is found a necessary and. A linear elastic material is a material that exhibits a linear relationship between the components of the stress tensor and the components of the strain tensor.

A linear elastic material constitutive law, under the assumption of small deformation, is fully represented by a linear map between the stress matrix and the infinitesimal strain matrix.

Among the most common examples of material strain tensors used in nonlinear elasticity is the Seth-Hill family4 [] Er(U) = 1 2r(U 2r 1): r2Rnf0g logU: r= 0 () e e1 e1=2 e 1 e0 ~e1=2 Figure 1: Scale functions er;~erassociated with the strain tensors Er and Eer= 1 2.

Various gradient (nano)shell models can be easily formulated by simply replacing the classical constitutive equations of classical elasticity with the constitutive equation of the implicit gradient elasticity and its approximate versions of strain gradient and stress gradient counterparts.

In the literature, different proposals for a strain gradient plasticity theory exist. So there is still a debate on the formulation of strain gradient plasticity models used for predicting size effects in the plastic deformation of materials.

Three such formulations from the literature are discussed in this work. The deformation gradient tensor (,) = ⊗ is related to both the reference and current configuration, as seen by the unit vectors and, therefore it is a two-point tensor.

Due to the assumption of continuity of (,), has the inverse = −, where is the spatial deformation gradientby the implicit function theorem, the Jacobian determinant (,) must be nonsingular, i.e. (,) = (,) ≠. In this paper there is formulated a linear theory of deformation of an elastic solid in which the potential energy-density is a function of the strain and its first and second gradients.

We obtain elasticity equations of higher (in the general case, infinite) order than the equations of the classical theory. In contrast to the numerous known versions of the nonclassical theory (Cosserat, nonsymmetric, microstructure, micropolar, multipolar, and gradient), which also result in higher-order equations and contain elasticity relations for traditional and couple stresses with a.

Lattice models for the second-order strain-gradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of second-gradient models, we suggest a new.

Liang, X., Hu, S., Shen, S.: A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications. Compos. Struct.– () CrossRef Google Scholar. The general linear strain gradient elasticity theory has been strain gradient elasticity theory and its applications.

is a function of the strain and its first and second gradients. This. We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case.

Virtual elements are very close to mimetic finite differences (see, for linear elasticity, [L. Beira͂o da Veiga, M2AN The straight segment is the linear region where Hooke’s law is obeyed.

The slope of the straight region is $\frac{1}{k}$. For larger forces, the graph is curved but the deformation is still elastic—ΔL will return to zero if the force is removed.

Still greater forces permanently deform the object until it. Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear.

Regarding model validation and applications, thorough analyses of stretching, shearing and vibration phenomena of complex triangular lattices homogenized by the simplified second strain gradient elasticity model reveal the strong size dependency of lattice structures and hence provide pivotal information for practical applications of materials.

Request PDF | Centrosymmetric equilibrium of nested spherical inhomogeneities in first strain gradient elasticity | The first strain gradient linear elasticity theory (FSGLET) is invoked to study. Its five-part treatment covers functions of a complex variable, the basic equations of two-dimensional elasticity, plane and half-plane problems, regions with circular boundaries, and regions with curvilinear boundaries.

Worked examples and sets of problems appear throughout the text. edition. 26 s: 3. Gao, XL, Park, S. Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem.

Int J Solids Struct ; –   () A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior. International Journal of Solids and Structures() Questioning size effects as predicted by strain gradient plasticity. Applications include muscle, arteries, the heart, and embryonic tissues.

Strain-Energy Density Function; Linear Elastic Material; Boundary Value Problems. “This book can be used as a textbook for a course in nonlinear elasticity, as a reference book for a course in biomechanics, or as a reference book for researchers trying to learn.Review of Stress, Linear Strain and Elastic Stress-Strain Relations 39 11 1 1 12 1 2 13 1 3 21 2 1 22 2 2 23 2 3 31 3 1 32 3 2 33 3 3 ()() ().

n nn nn nn nn nn nn nn nn nn σσ σ σ σ σ σ σσ σ =+ + + + + ++ + () Note that in each parenthesis, there is a sum over the second index of σ and the index of second n. This sum can be.Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.

The law is named after 17th-century British physicist.