Duality in constitutive formulation of finite-strain elastoplasticity based on F=FeFp and F=FpFe decompositions

A constitutive theory for large elastic–plastic deformations is presented by employing F=F^pF^e decomposition of the total deformation gradient. A duality in constitutive formulation based on this and the well-known Lee's decomposition F=F_eF_p is established for isotropic polycrystalline and single crystal plasticity.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

‘Les fleurs du mal’ – an adaptive wavelet method of arbitrary lines I: convection–diffusion problems« Les fleurs du MAL » – une méthode d'ondelettes adaptive de lignes arbitraires I : problèmes de convection–diffusion

Baudelaire's ‘les fleurs du mal’ refers to various new developments (‘les fleurs’) of the method ofarbitrarylines (mal), since it was first published (in C. R. Acad. Sci. Paris, Sér. I, in 1991). Here we revisit the basic mal (semi-discretization) methodology for stationary convection–diffusion problems and develop an adaptive, wavelet-based solver that is capable of capturing the thin layers that arise in such problems. We show the efficacy and high accuracy of the wavelet-mal solver by applying it to a challenging 2D problem involving both boundary and interior layers. To cite this article: X. Ren, L.S. Xanthis, C. R. Mecanique 332 (2004).     

Electrostatic stress analysis of an anisotropic piezoelectric half-plane under surface electromechanical loading

The generalized stress components on an anisotropic piezoelectric half-plane boundary under surface electromechanical loading are investigated. It is found that the behaviors of generalized stress components are related to matrices Γ and Ω, which have the same form as those for the purely elastostatic problem. Matrices Γ and Ω contain all the electro-mechanical coupling phenomena of the generalized stress components. All elements of matrices Γ and Ω are expressed explicitly in terms of generalized elastic stiffness for monoclinic piezoelectric materials with the plane of symmetry at x₃ = 0 and for transversely isotropic piezoelectric materials in which the coupled effects between the mechanical (electrical) deformations induced by electrical (mechanical) loadings are studied analytically. A numerical example of the electro-mechanical coupling behavior for PZT-4 is also given.     

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.     

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which X_a⁰ and X_c⁰ play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector X_a:=(YQ_a^t,YQ_a⁰)^t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime $\text{M}{}^{\mathrm{n+1}}$ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Then, constructed is a Poincaré group ISO_o(n,1) on space X:=X_a+X_b, of which X_b reflects the kinematic hardening rule in the model. We also find that the space (Q_a^t,q₀^a) is a Robertson–Walker spacetime, which is conformal to X_a through a factor Y, and conformal to X_c:=(ρQ_a^t,ρQ_a⁰)^t through a factor ρ as given by ρ(q₀^a)=Y(q₀^a)/[1−2ρ₀Q_a⁰(0)+2ρ₀Y(q₀^a)Q_a⁰(q₀^a)]. In the conformal spacetime the internal symmetry is a conformal group.     

An approach to characterization of the agricultural self-propelled machines stability

This paper presents an analytical algorithm with appropriate software specified for the approximation of the allowed critical slope of the solid flat terrain that guarantees static and/or dynamic stability of the specified self-propelled agricultural machines and their aggregates. This algorithm assumes machine as a rigid body, having 3 or 4 contact points (defined by wheels or crawlers), under uniform motion at different constant velocities and radii of curvature trajectories. Using this algorithm, based on the principles of theoretical mechanics combined with 3D analytical geometry, the computer program SSPM (stability of the self-propelled agricultural machines) has been coded. This software is intended to facilitate the analysis, comparison and optimization of different configurations of self-propelled agricultural machines in operation on horizontal and sloped flat terrains at constant velocities and radii of trajectory with respect to their static and dynamic stability. It calculates critical pitch and roll angles of the self-propelled machine and the maximum allowed slope of the flat terrain under the given conditions. The algorithm and the appropriate SSPM software were experimentally verified using the platform and low-scale tractor model. Average difference between calculated and experimental critical values of roll and pitch angles were about 4°     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

Simulations numériques de différentes méthodes d'éléments finis pour les problèmes de contact avec frottementNumerical implementation of different finite element methods for contact problems with friction

In this Note, we propose an efficient numerical treatment for solving contact problems with friction between deformable bodies. The discretized normal and tangential constraints at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers. Several numerical studies corresponding to this choice are achieved in the PLAST2 code. To cite this article: L. Baillet, T. Sassi, C. R. Mecanique 331 (2003).     

Modified kinematic hardening rule for multiaxial ratcheting prediction

A modified kinematic hardening rule is proposed in which one biaxial loading dependent parameter δ′ connecting the radial evanescence term [(α:n)ndp] in the Burlet–Cailletaud model with the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced into the framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted by Hassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quite well in all loading paths. The simulations of initial nonlinear part in ratcheting curves can be improved greatly while the evolutional parameter δ′ related to plastic strain accumulation is added into the modified model.     

On the use of universal relations in the modeling of transversely isotropic materials

A material is of coaxial type   if the Cauchy stress tensor T$\mathit{T}$ and the strain tensor B$\mathit{B}$ are coaxial for all deformations. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific  B$\mathit{B}$ we have that TB=BT$\mathit{TB}=\mathit{BT}$. We denote these materials B$\mathit{B}$-coaxial. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. We discuss the impact of B$\mathit{B}$-coaxial materials in the modeling of soft tissues. We conclude that B$\mathit{B}$-coaxial materials are a strong evidence that in real world materials two anisotropic invariants are always necessary to model in a meaningful and correct way single fiber reinforced materials.     

The effect of inextensibility on elastic surface waves

It is shown that when the complications associated with material anisotropy are absent a simple exact analysis can be given of the effect of unidirectional inextensibility on the propagation of surface waves in a semi-infinite elastic body. Provided that the direction of inextensibility e is not orthogonal to either m or m Λ n (m being the outward unit normal to the traction-free boundary of the body and n the wave normal), a unique surface wave exists with displacement everywhere orthogonal to e. The surface-wave solution is assembled from inhomogeneous plane waves in the usual manner, but a novel feature is the presence of a degenerate wave producing no displacement yet perturbing sinusoidally the tension in the inextensible fibres. When the aforementioned provisos are not met the surface wave either degenerates continuously into a shear wave (when $\text{(}\text{m}\text{Λ}\text{n}\text{)·}\text{e}\text{= 0,}\text{m}\text{·}\text{e}\text{≠ 0)}$, ceases to exist (when $\text{m}\text{·}\text{e}\text{= 0,}\text{n}\text{·}\text{e}\text{≠ 0)}$, or merges smoothly into a Rayleigh wave (when $\text{(}\text{e}\text{=±}\text{m}\text{Λ}\text{n}$, the inextensibility constraint then being inoperative).     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

Large bending behavior of creased paperboard. I. Experimental investigations

The large bending behavior of a creased paperboard is studied in the range of rotation θ ? [0°, 180°] – new results, apparently not reported previously in literature – with the aim to point out some crucial aspect involved in an adaptive robotic manipulation of the industrial cartons.The loading tests show a great variability of the mechanical behavior, depending dramatically on the crease indentation depth (also for the specimens obtained from the same carton): (a) when the damage induced during the crease formation is relatively small, the bending response is unusually complex: the moment constitutive function, m_L(θ), presents (up to) two peaks followed by unstable branches; (b) for greater indentation, the m_L(θ) is monotone.In the unloading case the response m_U(θ) is always monotone and is practically independent of the formation conditions of the crease. These behaviors can be easily described analytically using (piecewise) third degree splines.In a companion paper, the erection of a typical carton corner with unstable constitutive behavior is fully analyzed to detect the possible criticalities.     

A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

Homogénéisation de coques minces piézoélectriques perforées

In this Note we establish the limit constitutive law of a piezoelectric material with periodically perforated microstructures and whose reference configuration is a thin shell with fixed thickness. The justification of the new associated model (we show that the limit global and local problems are more complicated than the intial one) is obtained using the periodic unfolding method introduced by Cioranescu, Damlamian and Griso. To cite this article: M. Ghergu et al., C. R. Mecanique 333 (2005).     

Existence of Time-Periodic Solutions for a Magnetoelastic System in Bounded Domains

We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class C ². The mathematical model includes a nonlinear mechanical dissipation like ρ(u′)=|u′| ^p u′ and a periodic forcing function of period T. We prove the existence of T-periodic weak solutions when p∈[3,4] (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that p≥2.