A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings

In this study, we first demonstrate that the J-integral in classical linear elasticity becomes path-dependent when the solid is subjected to combined electrical, chemical and mechanical loadings. We then construct an electro-chemo-mechanical J-integral that is path-independent under such combined multiple driving forces. Further, we show that this electro-chemo-mechanical J-integral represents the rate at which the grand potential releases per unit crack growth. As an example, the path-independent nature of the electro-chemo-mechanical J-integral is demonstrated by solving the problem of a thin elastic film delaminated from a thick elastic substrate.     

An assessment of the finite termination property of the defect correction method

This paper investigates the use of defect correction procedures for the solution of finite volume approximations to systems of conservation laws. Particular emphasis is laid on the order of accuracy obtained after a fixed finite number of iterations. It is shown that a high order of accuracy may be achieved after only one defect correction iteration, involving two inversions of a stable lower-order-accurate operator. However, this result is found to be critically dependent on the consistency of the lower-order operator, a property which does not always hold for conservative finite volume discretizations. Through numerical experiments, the lack of consistency of these schemes is found to inhibit severely the finite termination property of the defect correction process. Results are presented for linear advection, Poisson's equation, and the Euler equations.     

Nonlocal integral elasticity: 2D finite element based solutions

A finite element based method, theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle, is numerically implemented for solving 2D nonlocal elastic problems. The key idea of the method, known as nonlocal finite element method (NL-FEM), relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of (cross-stiffness) element’s matrices able to interpret the (nonlocality) effects induced in the element itself by the other elements in the mesh. An Eringen-type nonlocal elastic model is assumed with a constitutive stress–strain law of convolutive-type which governs the nonlocal material behaviour. Computational issues, as the construction of the nonlocal element and global stiffness matrices, are treated in detail. Few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness.     

Uniqueness theorems for the entropy in any differential body of complexity one

By using in an essential way a certain condition of mutual physical equivalence between admissible response functions for the heat flux, in a previous paper uniqueness theorems were proved for the response functions of the internal energy and of the equilibrium stress, in connection with differential bodies of complexity 1. It was then pointed out that the equality expressing the vanishing of the static internal dissipation uniquely determines the rate of entropy variation in terms of the rate of the internal energy variation and of the equilibrium stress. This paper shows, in a threefold manner, that the last result also holds if one does not impose the condition of physical equivalence. The first proof uses the assumption that the response functions are Euclidean invariant. The second proof uses (i) the weaker assumption of Galilean invariance and (ii) a greater degree of smoothness of the response function for the internal energy. Both of these proofs use an axiom postulating the possibility of putting the body in contact with a vacuum. The third proof of the uniqueness property for the entropy is independent of the isolation axiom and uses the assumptions of the second proof. Whereas any of the first two proofs is a consequence of the uniqueness theorem for the internal energy-proved here by using the afore-mentioned axiom-the third proof does not depend on this theorem. Rather, disregarding the above isolation axiom, it implies that uniqueness of the entropy is compatible with non-uniqueness of both the stress and internal energy.

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Sommario In un precedente articolo, utilizzando in maniera essenziale una certa condizione di mutua equivalenza fisica tra ammissibili funzioni costitutive per il flusso di calore, si dimostrano teoremi di unicità per le funzioni costitutive di energia interna e di stress statico in corpi di tipo differenziale e complessità 1. Inoltre si osserva che, di conseguenza, la uguaglianza esprimente l'annullarsi della dissipazione interna statica permette di determinare univocamente la velocità di variazione dell'entropia in termini della velocità di variazione dell'energia interna e dello stress statico. Nel presente lavoro si dimostra, in triplice maniera, che l'ultimo risultato vale anche se non si impone la accennata condizione di equivalenza fisica. Nella prima dimostrazione si usa l'ipotesi di invarianza Euclidea per le funzioni costitutive. Nella seconda si usa (i) la più debole ipotesi di invarianza Galileiana e (ii) un maggiore grado di regolarità per la funzione costitutiva dell'energia interna. In entrambe le dimostrazioni si usa un assioma che postula la possibilità di porre in contatto il corpo con il vuoto. La terza dimostrazione della proprietà di unicità per l'entropia, che usa le medesime ipotesi della seconda, è indipendente da tale assioma. Mentre nelle prime due dimostrazioni si richiama il teorema il teorema di unicità per l'energia interna-qui dimostrato con l'assioma di isolamento-la terza dimostrazione non dipende da tale teorema. Invece, se non si assume l'assioma di isolamento con il vuoto, l'unicità dell'entropia risulta compatibile con certe non-unicità per stress ed energia interna.

     

The variational structure of a nonlinear theory for spatial lattices

A theory for spatial lattices is presented in a variational setting and conditions restricting stable deformations are discussed. In particular, new results on the second variation of the energy are established and used to generate pointwise necessary conditions for locally energy-minimizing configurations.

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Sommario Viene presentata una teoria per i reticoli spaziali in un ambito variazionale e sono inoltre discusse condizioni che limitano deformazioni stabili. In particolare vengono stabiliti nuovi risultati sulla variazione seconda dell'energia. Tali risultati vengono usati per stabilire condizioni necessarie puntuali per configurazioni che minimizzano l'energia localmente.

     

The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material

In this paper, the interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of the air inside the crack was considered. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the electric permittivity of the air inside the crack and the gradient parameter of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials

The solutions of a limited-permeable crack (case I) or two collinear limited-permeable cracks (case II) in piezoelectric/piezomagnetic materials subjected to a uniform tension loading were investigated in this paper using the generalized Almansi’s theorem. At the same time, the electric permittivity and the magnetic permeability of air in crack were firstly considered. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were jumps of displacements across crack surfaces, not the dislocation density functions or the complex variable functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among electric displacement intensity factors, magnetic flux intensity factors and stress intensity factors at crack tips.     

The Space of Inextensional Displacements for a Partially Clamped Linearly Elastic Shell with an Elliptic Middle Surface

We consider a linearly elastic shell with an “elliptic” middle surface, clamped along a portion of its lateral face and subjected to body forces. Under weak regularity assumptions on the middle surface, we prove that the space of linearized inextensional displacements is reduced to zero, by using unique continuation results. Consequently, when the thickness of the shell goes to zero, the limit of the average with respect to the thickness of the three-dimensional displacement vector solves the “generalized membrane” shell model, according to the terminology introduced by P.G. Ciarlet and the first author. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Generalized Membrane Problem for Linearly Elastic Shells with Hyperbolic or Parabolic-Middle Surface

We consider a linearly elastic shell with a hyperbolic, or parabolic, middle surface, clamped along a large enough portion of its lateral face and subjected to body forces. We then show that the two-dimensional limit model found by the asymptotic analysis from the three-dimensional shell problem is the “generalized membrane” shell model, according to the terminology introduced by P.G. Ciarlet and V. Lods. We also identify the functional space where this model is well posed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Energy release rate and path-independent integral in dynamic fracture of magneto-electro-thermo-elastic solids

The energy release rate and associated energy flux integral in dynamic fracture of magneto-electro-thermo-elastic solids are formulated with the inclusion of multi-field fully coupled effects based on fundamental principles of thermodynamics. The difference between the global and local dynamic contour integrals is caused by unsteady state, mechanical body force, electricity conduction and thermal effect as the closed contour including crack faces is chosen. This formulation successfully captures the crack-tip singularity of coupled fields, offers the right expression for the crack driving force, and resolves the controversial issue on magneto-electro-thermo-elastic fracture criterion. Especially, for steady-state crack propagation in a magneto-electro-elastic solid, the path-independent dynamic contour integral is determined from the asymptotic near-tip field solution based on the Stroh-type formalism and the resulting dynamic energy release rate has an odd dependence on the dynamic magnetic induction intensity factor and the dynamic electric displacement intensity factor.     

A Property of Equations of Rigid/Plastic Material Obeying a Voce–Type Hardening Law

Using a simple example, the rotation of a rigid cone in rigid/plastic hardening material, the paper shows a qualitative difference between the solutions for two groups of hardening laws. The first group includes hardening laws with no saturation stress. In this case the solution under sticking conditions exists at any rotation angle of the cone up to infinity. The second group includes hardening laws with a saturation stress. For such laws the solution exists up to a finite value of the rotation angle. Once this angle has been reached, the solution breaks down. At the beginning of the process the behavior of the solution for both groups of hardening materials is similar. However, at the final stage the behavior of rigid/plastic hardening materials of the second group is similar to the behavior of rigid perfectly plastic materials. A specific hardening law with a saturation stress is applied to illustrate the general solution and the restrictions imposed by this law, and a priori specified interfacial law (sticking) on existence of the solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Small-Displacement Waves in Prestressed Bodies with Isotropic Incremental Elasticity Tensor

We consider a body at rest in a prestressed configurationwhich responds elastically to small incremental displacements fromthe incremental elasticity tensor is supposed isotropic. On the basis of the paper [1] we characterize the conditions for the propagation of longitudinal, transverse, and oblique small-displacement waves superimposed toFormulae for the propagation speeds of these waves are written in terms of the prestress components and Lamparameters. The amplitudes of longitudinal and transverse waves are eigenvectors for the prestress.     

Basic solution of a mode-I limited-permeable crack in functionally graded piezoelectric materials

In this paper, the basic solution of a mode-I crack in functionally graded piezoelectric materials was investigated by using the generalized Almansi’s theorem. In the analysis, the electric permittivity of air inside the crack were considered. To make the analysis tractable, it was assumed that the shear modulus, piezoelectric constants and dielectric constants vary exponentially with coordinate parallel to the crack. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the effects of the electric boundary conditions on the electric displacement fields near the crack tips can not be ignored. Simultaneously, the solution of the present paper will revert to a closed form one when the functionally graded parameter equals to zero.     

No-tension solids in the presence of thermal expansion: An explicit solution

In this paper the constitutive equation of no-tension materials is generalized in order to account for thermal dilatation. Subsequently, the explicit solution to the equilibrium problem of a circular ring subjected to two uniform radial pressures, p ₁ and p ₂, acting respectively on the inner and outer boundary and a temperature distribution depending linearly on the radius, is calculated. It is proven that if pressures p ₁ and p ₂ and the temperatures V ₁ and V ₂ of the inner and outer boundary, respectively, satisfy certain inequalities, then the well-known stress field corresponding to a linear elastic material is the solution for the non-linear elastic material under consideration. On the contrary, if these inequalities are not satisfied, the equilibrated stress field, negative semi-definite, is explicitly calculated and the corresponding displacement and fractures determined. In particular the dependence of the cracked regions on the temperatures V ₁ and V ₂ is analysed.

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Sommario In questo lavoro si generalizza l'equazione costitutiva dei materiali non resistenti a trazione, per tenere conto delle dilatazioni termiche. Successivamente si calcola la soluzione esplicita del problema di equilibrio di una corona circolare costituita da un materiale non resistente a trazione, soggetta a due pressioni radiali uniformi, p ₁ e p ₂, agenti rispettivamente sul bordo interno e sul bordo esterno e a una distribuzione di temperatura variabile linearmente col raggio. Si dimostra che se le pressini p ₁ e p ₂ e le temperature V ₁ e V ₂ rispettivamente del bordo interno ed esterno soddisfano alcune diseguaglianze, allora il noto campo di sforzo corrispondente ad un materiale elastico lineare è anche la soluzione per il materiale elastico non lineare qui considerato. Al contrario, se queste diseguaglianze non sono soddisfatte, lo stato di sforzo equilibrato e semidefinito negativo è calcolato esplicitamente e sono determinati i corrispondenti spostamenti e fratture. In particolare, viene analizzata la dipendenza della regione fratturata dalle temperature V ₁ e V ₂.

     

Eshelby integral formulas in gradient elasticity

Eshelby integral formulas play a fundamental role in mechanics of composite materials, because they provide an efficient tool for determining the average properties of dispersion-filled materials. For example, their use in the framework of the self-consistent averaging method actually gives a final and quite precise solution to the problem of determining effective physical and mechanical properties of filled composites up to large relative contents of inclusions and almost all relations between the phase characteristics of the composite. In the present paper, we generalize the Eshelby integral formulas to the gradient theory of elasticity. This provides the possibility for using efficient methods for estimating the average characteristics of micro and nano-structured materials in the framework of gradient theories, which permit taking the scale effects into account correctly, and hence find wider and wider applications in describing the mechanical and physical processes.     

Basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials

The basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials was studied in this paper using the generalized Almansi’s theorem. To make the analysis tractable, it was assumed that the shear modulus varies exponentially along the horizontal axis parallel to the crack. The problem was formulated through a Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in anisotropic material plane by using the non-local theory

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.     

Two instructive instabilities in non-linear elasticity: Biaxially loaded membrane,and rubber balloons

Elastic bodies buckle under compressive loads, i.e. solutions become non-unique, they bifurcate and the body becomes unstable. Similar phenomena occur in tension as is evidenced here by the symmetric biaxial loading of a square membrane. Symmetry breaking removes the non-uniqueness. Under non-symmetric loading the load-deformation curves become non-monotone, consequently a hysteresis occurs which is the reflection of a fold-type catastrophy. This instructive instability was discovered by Kearsley [1]. Here we investigate it more fully and present some additional aspects.Balloons have non-monotone pressure-radius relations which suggest non-trivial stability properties. A stability analysis is presented for two interconnected balloons. In this we follow — and expand on — the analyses presented by Dreyer et al. [2] and Kitsche et al. [3].General Invited Lecture presented at AIMETA '95 — 12th Congress of the Italian Association of Theoretical and Applied Mechanics, Napoli, 3–6 October, 1995.     

Determination of mechanical properties by instrumented indentation

The paper reviews the current state of the depth-sensing indentation (sometimes called nanoindentation), where the information on material behaviour and properties is obtained from the indenter load and depth, measured continuously during loading and unloading. It is shown how the contact parameters and principal characteristics are determined using pointed or spherical indenters. Indentation tests can be used for the measurement of hardness and elastic modulus, and also of the yield stress and for the construction of stress–strain diagrams, for the determination of the work of indentation and its components. Most devices use monotonic loading and unloading, but some also enable measurement under a small harmonic signal added to the basic monotonously increasing load. This makes possible continuous measurement of contact stiffness and the study of dynamic properties and the determination of properties of coatings. One section is devoted to the measurement on viscoelastic-plastic materials, where the delayed deforming must be considered during the measurement as well as in data evaluation. Instrumented indentation can also be used for the study of creep under high temperatures. The paper also discusses the errors arising in depth-sensing measurements and informs briefly about some other possibilities of the method.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.