On the generalized virial theorem and Eshelby tensors

The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule.     

On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.     

On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity

This article presents some aspects of the formulation of finite strain elastoplasticity based on the multiplicative decomposition of the deformation gradient. A “canonical” structure of multiplicative elastoplasticity is discussed characterized by a geometrical setting relative to the intermediate configuration in terms of mixed-variant tensors and the exploitation of fundamental dissipation principles. The symmetric fourth-order elastoplastic moduli (so-called ‘Prandtl-Reuss-Tensors’ of the associative theory) appear as a consequence of the assumed metric-dependence of the flow criterion function in a characteristic structure which seems to be typical for large strain multiplicative elastoplasticity. Particular representations of “Prandtl-Reuss tensors” are outlined for isotropic response as well as for decoupled volumetric-isochoric stress response.     

On configurational compatibility and multiscale energy momentum tensors

In this work the continuum theory of defects has been revised through the development of kinematic defect potentials. These defect potentials and their corresponding variational principles provide a basis for constructing a new class of conservation laws associated with the compatibility conditions of continua. These conservation laws represent configurational compatibility conditions which are independent of the constitutive behavior of the continuum. They lead to the development of a new concept termed configurational compatibility, dual to the concept of configurational force. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by the integral is defect-free. It is shown that the Peach-Koehler force can be recovered as one of these invariant integrals. Based on the proposed defect potentials and their corresponding defect energies, two-field multiscale mixed variational principles can be employed to construct multiscale energy momentum tensors. An application is outlined in the form of a mode III elasto-plastic crack problem for which the new configurational quantities are calculated.     

An eigenvalue problem for tensors used in mechanics and the number of independent Saint-Venant strain compatibility conditions

A number of questions concerning the eigenvalue problem for a tensor \(\mathop A\limits_ \approx\) ∈ ?₄(Ω) with special symmetries are considered; here Ω is a domain of a four-dimensional (three-dimensional) Riemannian space. It is proved that a nonsingular fourth-rank tensor has no more than six (three) independent components in the case of a four-dimensional (three-dimensional) Riemannian space. It is shown that the number of independent Saint-Venant strain compatibility conditions is less than six.     

On tortuosity and areosity tensors for porous media

An average streamwise channel velocity is proposed as a more accurate representation of the actual intrapore velocity than the intrinsic phase average velocity. A relationship is derived between the average streamwise channel velocity and the interstitial velocity and superficial velocity. New definitions of tortuosity and areosity as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. The definitions produce results which conform with several other published results and are applicable to anisotropic media in general.     

On the representation of anisotropic elastic materials by symmetric irreducible tensors

In this paper we will derive a general framework for nonlinear anisotropic elastic materials. The method developed is based on concepts from the theory of symmetric irreducible tensors and group representation theory. Different expansions of constitutive functions with respect to specific basis functions will be introduced. It will be shown that these basis functions satisfy certain orthogonality relations which allow to establish an effective procedure for parameter identification. Finally the treatment of various symmetry groups will be discussed. Received July 12, 1999     

On the generalized Barnett–Lothe tensors for monoclinic piezoelectric materials

The three generalized Barnett–Lothe tensors L, S and H, appearing frequently in the investigations of the two-dimensional deformations of anisotropic piezoelectric materials, may be expressed in terms of the material constants. In this paper, the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x₃ = 0 are constructed based on the extended Stroh formalism. Then the three generalized Barnett–Lothe tensors are calculated from these eigenvectors and are expressed explicitly in terms of the elastic stiffness instead of the reduced elastic compliance. The special case of transversely isotropic piezoelectric materials is also presented.     

On the derivation of the acoustic matrix for isotropic linearly elastic shells

In this paper, we obtain the modes and velocities of acceleration waves on a thin hyperelastic shell in terms of the second fundamental form, which represents the geometrical properties of the shell, and of seven elastic moduli derived from the velocities in a plate of the same material. Some examples are studied, and approximations obtained in the case of a shallow shell.

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Sommario In questo lavoro si ottengono i modi e le velocità delle onde di accelerazione in una volta sottile iperelastica, con riferimento alla seconda forma fondamentale che rappresenta le proprietà geometriche della volta e a sette moduli elastici derivati dalle velocità in una piastra dello stesso materiale. Si studiano alcuni esempi e si presentano soluzioni approssimate nel caso di una volta ribassata.

     

On the calculation of predeformation-dependent dynamic modulus tensors in finite nonlinear viscoelasticity

To simulate the frequency-dependent behaviour of nonlinear viscoelastic structures under loadings which consist of a finite predeformation in combination with a superimposed harmonic deformation with small amplitude, frequency-domain formulations of the constitutive models are needed. For this purpose, a recently developed approach of finite viscoelasticity is considered and the corresponding dynamic modulus tensors are derived. The constitutive equations are geometrically linearized in the neighbourhood of the predeformation and evaluated in the frequency-domain. This procedure is applicable to arbitrary constitutive models and can be used to derive their frequency-domain formulations for finite element implementations as proposed by Morman and Nagtegaal [Morman, K.N., Nagtegaal, J.C., 1983. Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. International Journal for Numerical Methods in Engineering, 19, 1079–1103].     

On computation of the compliance and stiffness contribution tensors of non ellipsoidal inhomogeneities

We discuss computation of compliance/stiffness contribution tensors of an inhomogeneity of non-ellipsoidal shape and the difficulties that may be encountered in this context. These issues are illustrated on the example of one specific shape factor: concavity–convexity of the inhomogeneity shape. It is found, in particular, that the effect of the concavity factor depends on the specific constraint imposed. If the inhomogeneity volume is kept constant, then its compliance contribution rapidly increases with increasing concavity; on the computational side it leads to difficulties related to unusually high requirements to the accuracy of the computed volume averages quantities. If, on the other hand, the characteristic dimensions (for example, the distance between the farthest points) are kept constant, then the dependence of the compliance contribution on the concavity factor is almost linear.     

On the existence and the asymptotic stability of solutions for linearly viscoelastic solids

Communicated by C. Dafermos     

On the connection between the principal components of the stress and permeability tensors of porous media

The dependence of the flow properties of rocks on the conditions under which they are loaded is one of the least studied questions in the general problem of the connection between the physical properties of porous systems and the loads acting on them. The experimental [l] and theoretical [2, 3] studies so far made have dealt primarily with the dependence of the permeability of oil and gas collector rocks on the hydrostatic compression to which they are subjected. In the present paper, a connection is established between the parameters of a nonuniform anisotropic load and the permeability tensor of rock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 173–177, January–February, 1984.     

On the explosion of linearly distributed charge with curvilinear shape on the ground surface

We use the momentum version proposed by M. A. Lavrent'yev [1, 2] to treat the two-dimensional problem of the explosion of a linearly distributed charge of curvilinear shape on the ground surface. The problem of the explosion of a straight charge was solved for the first time in [2] in this version. The ground is assumed to be an ideal incompressible liquid at velocities exceeding some critical velocity which remains constant along the crater; beyond this boundary, the medium is fixed. The potential of the velocity is assumed to be constant on the charge and vanishing on the ground surface.     

On the constraint and scaling methods to derive the equations of linearly elastic rods

The theory of linearly elastic rods may be obtained from three-dimensional elasticity either by the method of internal constraints or by the scaling method. Both methods have been applied to obtain linear plate and shell equations ([1], [2]–[5]); the relationships between the two methods are discussed in [6]. For rods, a version of the constraint method has been developed in [7], whereas a scaling method has been presented in [12]. In this paper a direct comparison is made between the mechanical basis and analytical results of the constraint and the scaling methods, and it is shown how the scaling method yields the same Kirchhoff hypothesis that forms the starting point of the constraint method.

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Sommario La teoria delle travi elastiche lineari puó essere ottenuta a partire dalla teoria tridimensionale dell'elasticitá tanto con il metodo dei vincoli interni che con il metodo di riscalamento. Entrambi i metodi sono giá stati utilizzati per ottenere le equazioni delle piastre e dei gusci lineari ([1], [2]–[5]); in quel contesto, le relazioni tra i due metodi sono state discusse in [6]. Una versione del metodo dei vincoli appropriata al caso delle travi é stata sviluppata in [7], mentre i metodi di riscalamento per le travi si trovano esposti in [12]. Scopo di questo lavoro é compiere un paragone diretto tra i fondamenti meccanici e le risultanze analitiche, rispettivamente, del metodo dei vincoli e di quello di riscalamento, mostrando come il metodo di riscalamento imponga di accogliere proprio quelle ipotesi all aKirchhoff sulle quali si basa il metodo dei vincoli interni.