Incompatibility Tensor Research Articles

Equivalence of formulations for problems in elasticity theory in terms of stresses

The problem of finding the number of independent compatibility equations in strains in n-dimensional Euclidean space is discussed. The number in question coincides with the number of independent Beltrami-Michell compatibility equations in terms of stresses used when formulating the problem in elasticity theory and also with the number of components of the incompatibility tensor and of the Riemann-Christoffel tensor. For n = 3, two counterexamples are presented that show the impossibility to transfer three “diagonal” or three “nondiagonal” Beltrami-Michell equations from the domain of an elastic solid to its boundary. In this case, the formulation of the problem becomes nonequivalent to either classical or new formulating of the problem in terms of stresses.

208 不適合度テンソルを用いた内部応力場を伴なう金属微視組織の形成シミュレーション(組織形成)

208 不適合度テンソルを用いた内部応力場を伴なう金属微視組織の形成シミュレーション(組織形成)

A Dislocation-Crystal Plasticity Simulation on Large Deformation Considering Geometrically Necessary Dislocation Density and Incompatibility (1st Report, Definitions of GN Crystal Defects and Multiscale Modeling)

Plastic deformation and work-hardening of a crystal are caused by dislocation motions and dislocation accumulations, respectively. Recently, studies of crystal plasticity with the dislocation information have actively been done by many researchers. In this paper, both densities of isolated dislocation and dislocation pair corresponding respectively to the conventional GN and SS dislocation densities are uniformly defined as geometrical quantities, i.e., GN dislocation density and GN incompatibility tensors by extending Kröner's dislocation density and incompatibility tensors so that they are suitable for a crystal plasticity framework. Furthermore, we newly introduce a term of dynamic recovery that occurs in stage III of work-hardening of a single crystal into the expression of dislocation density. A new model of dislocation-crystal plasticity coupling deformation field with dislocation field is developed by introducing these dislocation densities into a hardening modulus matrix of crystal plasticity through the Bailey-Hirsch relation.

On the numerical treatment of initial strains in biological soft tissues

In this paper, different methodologies to enforce initial stresses or strains in finite strain problems are compared. Since our main interest relies on the simulation of living tissues, an orthotropic hyperelastic constitutive model has been used to describe their passive material behaviour. Different methods are presented and discussed. Firstly, the initial strain distribution is obtained after deformation from a previously assumed to be known stress-free state using an appropriate finite element approach. This approach usually involves important mesh distortions. The second method consists on imposing the initial strain field from the definition of an initial incompatible ‘deformation gradient’ field obtained from experimental data. This incompatible tensor field can be imposed in two ways, depending on the origin of the experimental tests. In some cases as ligaments, the experiment is carried out from the stress-free configuration, while in blood vessels the starting point is usually the load-free configuration with residual stresses. So the strain energy function would remain the same for the whole simulation or redefined from the new origin of the experiment. Some validation and realistic examples are presented to show the performance of the strategies and to quantify the errors appearing in each of them. Copyright © 2006 John Wiley & Sons, Ltd.

Remarques sur les contraintes résiduelles

Residual stresses which are currently observed in solid bodies can result from non-compatible initial strains. Theses stresses can then be determined, in general, from the incompatibility tensor associated to the initial strains tensor. However, even if the incompatibility tensor is zero, residual stresses may exist, when the solid is not simply-connected or when discontinuity surfaces are present. Several examples are provided. To cite this article: P. Bérest, G. Vouille, C. R. Mecanique 331 (2003).

応力関数法に関する考察

The discussion on solution procedures with stress components or stress functions is very few in comparison with the displacement method. In this paper some considerations of the stress function method in solving problems of elastostatics are given theoretically. A new stress function is introduced from the existing analogous relationship between stress function tensor and strain tensor through incompatibility tensor and compatibility equations in terms of strain components. Also, the relationship between this new stress function and the conventional stress function(Maxwell's stress function and Morera's stress function) is made clear. Fundamental equations which are indispensable in solving three-dimensional homogeneous isotropic elastic problems are derived in three case with use of each stress function. The effectiveness of our solution method with new stress function is shown in comparison with the expressions of displacement and stress components with the Galerkin vector. Finally, explicit forms on Fourier series solution of a simple supported thick plate subjected to a sinusoidal load are given.

A strain-gradient thermodynamic theory of plasticity based on dislocation density and incompatibility tensors

In this work, we discuss a thermodynamic theory of plasticity for self-organization of collective dislocations in FCC metals. The theory is described by geometrical tensor quantities of crystal defect fields such as dislocation density tensor, representing net mobile dislocation density and geometrically necessary boundaries, and the incompatibility tensor representing immobile dislocation density. Conservation laws for the two kinds of dislocation density are formulated with dislocation products and interactions terms. Based on the second law of thermodynamics, we drive basic constitutive equations for the dislocation flux, production and interaction terms of dislocations. We also derive a set of reaction-diffusion equations for the dislocation density tensor and incompatibility tensor which describes the vein and persistent slip band (PSB) ladder structures. These equations are analyzed using linear stability and bifurcation analysis. An intrinsic mesoscopic length scale is determined which provides an estimate for the wavelength of the PSBs. The basic aspects of the model are motivated and substantiated by analyzing the stress fields of various possible dislocation configurations using discrete dislocation dynamics.

Theory of Small-Angle X-Ray Scattering Caused by Incompatibilities in Elastic Anisotropic Media and the Application to Dislocations in H.C.P. Crystals

So, far, the Small-Angle X-Ray Scattering (SAXS) amplitude caused by internal stress sources which are described by an incompatibility field was calculated for isotropic and cubic crystal symmetry by KrOner and Sceger. In this work this problem has been solved for arbitrary crystal symmetry within the framework of the linear anisotropic elasticity theory. Using the unique decomposition of the elastic strain tensor into the deformational and incompatible parts as ansatz, the SAXS amplitude has been obtained by Fourier transformation of the basic equations of the internal stress theory. The integration problem can be reduced to the Fourier transformation of the incompatibility tensor only. A calculation of the volume dilatation field can be avoided in this way. The calculated SAXS amplitude is applied to dislocation loops in h.c.p. crystals. It is shown which information about these dislocation loops is available by increasing SAXS intensity contours.

Solution of the incompatibility problem in linear three‐dimensional anisotropic media for an isotropic incompatibility tensor

AbstractA solution of the incompatibility problem in three‐dimensional anisotropic elasticity is derived for the case that the incompatibility tensor has isotropic symmetry. To that end an infinitely extended linear elastic anisotropic medium is assumed. Under these conditions the internal stresses are obtained by pure differentiations from the corresponding fourth‐order stress function tensor. This result is then used to express the internal stress tensor as a convolution of the incompatibility tensor and the elastic Green's function tensor.

Analysis of Anisotropic Elasticity by Means of Internal Stress in a Reference Isotropic Elastic Body

AbstractA method of anisotropic elasticity problems is proposed which employs superposition of the analytic solution for an isotropic elastic body and the suitable internal stress determined by the boundary condition. Employing the incompatibility tensor and Kröner's stress functions, we can establish a simple formulation for the anisotropic elasticity. The difference of the stress fields in an isotropic material and an anisotropic material can be considered as an internal stress field caused by the incompatibility which can be evaluated from the solution of the isotropic material by use of its curl curl operations for the incompatibility tensor. Though it is difficult to get the analytic solution for general cases, some special cases with Airy's stress function are analyzed as simple examples. Since the internal stress can be expressed by the dislocations distributed continuously in the isotropic body, we can gain further physical insight into the anisotropy.

The internal mechanical state of solids with defects

In general, inelastic deformation is accompanied by internal mechanical stresses which also persist in the absence of external loads. A fundamental quantity for determining the state is the stress function tensor (or tensor potential), a field which can be calculated in terms of the incompatibility tensor, also a field. This tensor is not given a priori, but follows from the physics of the problem. Differential geometry is a most important and elegant tool to deal with the internal mechanical state, both on the geocometric and on the static side (the concept of mutually dual strain state and stress state). If stress-free strain is the cause of the internal stresses, then Riemannian geometry is adequate. More general geometries describe defects, namely Riemann-Carlan geometry describes dislocations in the form of Cartan's torsion of the strain space, and nonmetric (affine) geometry the point defects vacancy, self-interstitial and shear fault in the form of nonmctricity (a tensor field) of the strain space. The specific response to dislocations is torque stress which arises as Cartan's torsion of the stress space, whereas that to point defects is moment stress without torque entering as the nonmetricity of the stress space. The fundamental duality between strain space and stress space gives the theory a particular symmetry. Physical realizations of such (material) spaces are the alfine point structures, e.g. the Bravais lattices.

Dislocation-point defect dynamics in two-dimensional anisotropic liquids: New aspects of melting

The interaction of dislocations and vacancies in a locally correlated liquid is studied within a second quantized linear response theory. One- and two-particle dislocation Green's functions are calculated in RPA, using Zubarev's method. The elementary excitation spectrum and elastic response coefficient are discussed. Non-conservative and mass-conserving processes are differentiated, and the anisotropy of the melting statistics is found to influence the solidification temperature of the liquid. Quantum effects on the melting of Bose systems are discussed. An explicit relation between the elastic response coefficient and Hooke's tensor, based on Kröner's theory, valid in three dimensions, is given in an appendix, as well as its reduction to the two-dimensional case. A comparison with a recent microscopic derivation of this linear response law taking also dynamic properties of the incompatibility tensor into account is presented on a qualitative level.

Determination op the geometric characteristics of shells of revolution satisfying finite strain compatibility conditions: PMM vol. 42, n≗2, 1978. pp. 327–332

Taking account of shear strain, formulas to determine the Lamé coefficients, and the principal radii of curvature of coordinates of points of the middle surface of an arbitrary shell of revolution after it has been loaded by an axisymmetric load, are obtained in a quadratic approximation. The Lamé co-efficients and the radii of curvature of the deformed shell satisfy Codazzi —Gauss conditions and their associated compatibility equations for finite strain. It is shown that the finite strain compatibility equations obtained in this paper from the condition that the strain incompatibility tensor [1] of a three-dimensional solid equals zero, are satisfied identically if the small strain components satisfy the appropriate linear equations of strain compati-bility of the shell middle surface.

On the Force Exerted upon Continuously Distributed Dislocations in a Cosserat Continuum

AbstractThe force acting on distributed dislocations in a Cosserat continuum is studied and the expressions for the force and force density are given in tensorial form. The increase in the potential energy of the material body, caused by a displacement of distributed dislocations in a certain region of the body, is obtained. The force acting on those dislocations is deduced by differentiating it with respect to the distance through which the dislocations are displaced. The expression for the density of forces is given as follows: equation image where sjni is the tensor of dislocation density, ∑ij the symmetric part of stresses, Mjni the couple‐stress tensor, and ηij the incompatibility tensor.

On Inherent Stresses in an Elastic Solid

An analysis of inherent stresses in an elastic solid is described assuming inherent strain to be a stress-source. General solutions for two- and three-dimensional cases are obtained and applied in some examples. Theory of inherent stresses has been established using the concept of incompatibility of strain and the results are expressed in terms of "incompatibility tensor" in most cases. In this paper, stress functions including the incompatibility of inherent strain are integrated by parts and stress formulas are presented in terms of "inherent strain". In the inherent stress problems in engineering, the distribution of inherent strain is assumed to be known. The results in this paper may be conveniently applied to practical cases.

The basic equations of the dynamics of the continuous distribution of dislocations III. special problems

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell equations are derived in a series of papers [I, II, III]. The analogy of the elastic and electromagnetic fields is analyzed. In part [III] some special problems are discussed, such as the density of the forces acting on the dislocations, the energy dissipation during the movement of dislocations, which is expressed by an equation analogical to Ohm's law. The equations derived in the previous parts in four-dimensional symbolics are considered in the three-dimensional differential and integral form. It is found that in special cases the relations become the known ones of elastodynamics, hydrodynamics and the static theory of the continuous distribution of dislocations. It is found that Kroner's method of integrating the equations of the dislocation field by means of so-called incompatibility tensors is analogical to the integration of the Maxwell equations by means of Hertz vectors. The analogy between the elastic dislocation field and the electromagnetic field is discussed in detail.

三次元弾性論におけるCastiglianoの定理について

Abstract Southwell showed the relation between the stress functions-Maxwell's and Morera's-and the so-called“equations of compatibility”. His treatment of the surface integral, however, does not appear to the author to be very clear. Moreover, the consideration of the “dislocation”in an elastic body occupying a multiply-connected region, is lacking. These deficiencies are corrected in this paper. Maxwell's and Morera's stress functions are united to form a “stress functions tensor”(3). Then the “incompatibility tensor”is derived as the variational coefficient of the “stress energy” with respect to the “stress functions tensor.” Incidently, the surface-inregral appears, and is transformed into the sum of the porducts of line-ire egrals and the remaining surface-integral. The variation of the “stress-energy” can be written, finally in the form 28) from which the conditions of compatibilit y-both in the small and in the large-are derived, the latter being coincident with Volterra's theory.(9)