An application of Curie’s principle to elastoplastic dynamics

This paper deals with the stability and the dynamics of a harmonically excited elastic–perfectly plastic unsymmetrical oscillator. Stability of the periodic orbits is analytically investigated with a perturbation approach. The occurrence of ratcheting effect is discussed for this system, and is related to the loss of symmetry of the periodic orbit in the phase space. Curie’s principle of symmetry is numerically verified for the symmetrical system with positive damping. Therefore, the observation of ratcheting phenomenon is necessarily associated to a breaking of symmetry in the constitutive behaviour, or in the forcing term. However, the generalized version of Curie’s principle has to be considered when a negative damping is introduced.     

A thermodynamic approach with internal variables using Lagrange formalism. Part I: General framework

We present some reflections on the application of the Lagrangian formalism for continuous media locally uniform subjected to internal irreversible evolutions. The Lagrangian density, defined as the time derivative of a non-equilibrium thermodynamic potential, [Thermodynamics of Relaxation Processes using Internal variables within a Lagrange-formalism. P. Germain’s Anniversary Volume 2000. Contiuum Thermomechanics: the Art and Science of Modeling Matter’s Behaviour, 2000], contains all the symmetry properties of the system. The generalised Lagrange co-ordinates correspond to the state and internal variables of the time derivative of the generalised Gibbs potential. The latter being used within the framework of the De Donder’s method, must also account for the memory effect of the physical medium.This first part is devoted to the thermodynamic framework called the distribution of non-linear relaxations approach (DNLR) developed by C. Cunat on the basis of the generalised Gibbs’ relation.     

Periodicity and chaos in a flexible crank-rocker mechanism

In this paper we introduce a crank-rocker mechanism at which the rocker is flexible. Using Hamilton’s principle we obtain the governing equations of motion for the elastic mode of the rocker. By applying the Bubnov–Galerkin global averaging method, we reduce the governing equations of motion to an ordinary differential equation which is Duffing’s oscillator with time varying coefficients. Through the application of Banach’s fixed-point theorem we predict the periodic solutions. Then we study the geometrical features of the motion near the 1 : 1, 1 : 2 and 2 : 1 commensurabilities. It is also shown that homoclinic and heteroclinic orbits can exist for the system.     

Analysis of temperature and stress field in Al alloy’s twin wire welding

By analyzing the shape of twin wire welding’s arcs and the track of droplets’ transition, the phenomenon that the twin wire welding’s fore arc and rear arc all deflect to the middle of the two arcs is found. Based on this the double ellipsoid heat source model is amended, and a heat source model which can apply to calculate the twin wire welding’s temperature field is put forward. This model is testified by actual experiment of temperature sampling. By comparing the temperature field of twin wire welding and single wire welding, the results show that twin wire welding has slender weld pool the end part of which is ellipsoid, and its HAZ is narrower than that of single wire welding. So, twin wire welding can not only reduce the Al alloy generating hot crack, but can also weaken the “overaging” softened phenomenon of heat treated strengthening Al alloy. In the end, the evolving rules of 2219 Al alloy’s longitudinal and transverse stress when welded with twin wire welding are analyzed.     

Initiation and stable growth of penny-shaped crack in aging viscoelastic transversally isotropic material

Considered is the long-term cracking of an aging transversally isotropic material containing a Mode I penny-shaped crack under remotely applied tensile stress. The aging material properties are described by the Boltzmann–Volterra’s linear theory for integral operators with non-difference kernels. It applied to wood, concrete, some polymers and rocks. Only the symmetric case is considered where the crack lies in the plane of isotropy. The modified Leonov–Panasyuk–Dugdale’s crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion. Volterra’s principle is applied to derive the equations of subcritical crack growth. Numerical calculations are made for subcritical crack growth for the specific example of transversally isotropic material simulating the behavior of reinforced concrete.     

Non-uniform, axisymmetric misfit strain： in thin films bonded on plate substrates/substrate systems： the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a ‘‘non-local’’ part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

On multiscale significance of Rice’s normality structure

The normality structure proposed by [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455.] provides a minimal framework of multiscale thermodynamics. As shown in this paper, Rice’s multiscale thermodynamic formalism is exactly consistent with Ziegler’s essential notion [Ziegler, H., 1977. An Introduction to Thermomechanics, North-Holland, Amsterdam.] that the entire constitutive response is determined by the knowledge of two scalar potential functions: an energy function and a dissipation function. In Rice’s multiscale thermodynamic formulation, the variational equation relating macroscale and microscale thermodynamic fluxes and forces plays a central role and ensures the equality between microscale and macroscale dissipation rate. The variational equation can be further reformulated into a principle of maximum equivalent dissipation. Based on the variation equation, the transformation from microscale to macroscale is characterized by two linear transformations with the same corresponding matrix.     

Simulation on projectile with high rotating speed penetrating into the moving vehicular door

The numerical program LS-DYNA, is used to simulate the process of the projectile with high rotating speed penetrating into the moving vehicular door. Because of the moving of the vehicular door, the projectile will turn, and the ballistic trajectory will migrate. The paper provides a method to calculate the projectile’s angle of turning’s curve. In the process of the penetration, the projectile’s moving speed is 300 m/s, rotating speed is 0, 3600 n/s and 6370 n/s. The vehicular door’s moving speed is 0, 40 m/s and 80 m/s. The projectile is the semi-sphere nose projectile whose diameter is 7.62 mm; the vehicular door’s thickness is 2 mm. The material model is the JOHN-COOK material model that can characterize strain, strain rate hardening and thermal softening effects. Through comparing with the results by simulation to study the effects of the projectile’s final velocity, the angle of rotation and the ballistic trajectory’s migration with different projectile’s rotating speeds and the vehicular door’s moving speeds.     

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.     

Missing boundary data recovering for the Helmholtz problem

This Note is dedicated to the numerical treatment of the ill-posed Cauchy–Helmholtz problem. Resorting to the domain decomposition tools, these missing boundary data are rephrased through an ‘interfacial’ equation. This equation is solved via a preconditioned Richardson algorithm with dynamic relaxation. The efficiency of the proposed method is illustrated by some numerical experiments. To cite this article: R. Ben Fatma et al., C. R. Mecanique 335 (2007).     

Crack trajectory instability: Propagation in inhomogeneous medium

The problem of crack trajectory stabilization in composite material is investigated. The equation for a crack path is found from the variational principle. It is considered as a path along which the extreme amount of energy is generated during the destruction. This statement corresponds to the variational problem analogous to the Lagrange–d’Alembert principle of classic mechanics and to the Fermat principle in optic and acoustic. For a crack path in inhomogeneous medium, nonlinear differential equation is obtained. Stability of the crack propagation in the inhomogeneous medium is considered. In particular, a 2D crack propagating in a composite material is considered. The path of propagation is assumed to cross layers or fibres. For layered and piece-wise continuous composites, the resulting governing equation corresponds to different kind of Duffing’s equation. The bifurcations of the trajectory and instability of crack path are investigated numerically. Conditions of crack trajectory stabilization are found. Properties of the materials that stabilized the crack trajectory are found.     

On the determination of the Young’s modulus of thin films using indentation tests

The main difficulty with the characterization of thin coatings using depth-sensing indentation tests is related to the determination of the contributions of the substrate and the film to the measured properties. In this study, three-dimensional numerical simulations of the Vickers hardness test are used in order to examine the influence of the elastic and plastic properties of the substrate and the film on the composite’s Young’s modulus results. The hardness of the film is equal to or higher than the substrate hardness. A study of the stress distributions and the indentation geometry of composites, film/substrate, was performed, taking into account the relative mechanical properties of the film and substrate. In addition, stress evolution during indentation was studied, in order to quantify the critical indentation depth under which the substrate is not elastically deformed. The accurate evaluation of the Young’s modulus of the films using weight functions is also examined: some of these have previously been proposed and one was introduced for this study. Two different fitting procedures were used to compare the results obtained from eight fictive film/substrate combinations using six weight functions. The first procedure, commonly used, considers the substrate’s modulus as a known parameter in the fitting process. In the second, the film and the substrate’s modulus are considered as unknown variables that are calculated simultaneously during the fitting process. The validity of the conclusions obtained using the fictive materials was checked by applying the weight functions to four real composites.     

On the use of Laplace transform inversion for reconstruction of relaxation spectra

We discuss the use of Fourier transforms to construct approximate inversion formulae for the Laplace transform and apply this technique to recover the relaxation spectrum from “data” for the relaxation modulus. We show that regularization by a Gaussian, as proposed by Davies and Anderssen [A.R. Davies, R.S. Anderssen, Sampling localization in determining the relaxation spectrum, J. Non-Newt. Fluid Mech. 73 (1997) 163–179] yields reasonable results even for data with significant noise. We also show that, in principle, other choices of regularization allow the relaxation spectrum to be reconstructed from data which are taken in any interval of time or frequency, however short and wherever located. We construct formulae which would do this, based on polynomial approximation of a function in an exponentially weighted space. This algorithm, however, turns out not to be practical, and we elucidate the reasons for that.     

Unsteady Motion of a Maxwellian Fluid Droplet in a Maxwellian Medium under the Action of Monotonic and Periodic Forces

Unsteady motion of a Maxwellian fluid droplet, which arises in a quiescent Maxwellian medium under the action of monotonic and periodic forces, is considered. In the initial period of time smaller than the relaxation time, the droplet is affected by elastic forces on the part of the fluid; moreover, the droplet itself is a viscoelastic material. A solution of the problem in the first approximation is found. The dependence of the amplitude of droplet velocity and the shift of the phase of oscillations on the relaxation time of the external and internal media and also on the frequency of oscillations of the driving force is analyzed. The passage to the limit in terms of density and viscosity of the internal medium is performed.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 55–65, July–August, 2005.     

The consistent Mindlin’s thermopiezoelectric equations and the principle of virtual work

The thermal field vector, an overlooked concept of importance, is introduced for the consistency in Mindlin’s thermopiezoelectric equations. The gradient of the temperature increment is defined as the thermal field vector; an analogous to the electric and magnetic field vectors (i.e., the gradients of the electric and magnetic potentials). The thermal field vector is used to state the universal gradient equations, and then, the principle of virtual work.     

An extension of the J-integral to piezoelectric materials

This paper provides an analysis of the crack propagation criterion for a thin piezoelectric plate with a symmetry of order six. On the basis of Gol’denveizer’s asymptotic integration method or Destuynder’s unidirectional zoom technique, we obtain an extension of the purely mechanical J-integral to piezoelectric materials, with a dependence of the gradient of energy of the plate only on zeroth order terms of asymptotic expansions.     

Full field methods and residual stress analysis in orthotropic material. I Linear approach

This work analyzes the problem of residual stress determination in an orthotropic material using the hole drilling technique combined with non-contact, full field optical methods. Due to the complex behavior of the material, first a solution algorithm for the isotropic case is analyzed, then the procedure is extended to solving the more complex problem. In the first part of the work, the simplified Smith–Schajer solution to the through-hole problem for an orthotropic material is analyzed, showing that the same linear least square approach used in the isotropic case applies to a large set of orthotropic materials; based on this analysis a simple residual stress measurement algorithm is developed using either analytical or numerically estimated calibration coefficients.In the second part of the work, the general solution is discussed: since in this case the simplified Smith–Schajer solution cannot be used, the Lekhnitskii’s analysis of the through-hole plate in tension is introduced and extended to handle residual stresses. On this basis a solution algorithm using the nonlinear fit of the theoretical displacement field capable of treating all the orthotropic materials at the cost of a more complex numerical procedure is proposed. The performances of both algorithms are tested against numerically generated noisy fields and experimental ones and show a good reliability and accuracy.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

Numerical simulation of debonding of adhesively bonded joint

This paper describes a numerical method to simulate the debonding of adhesively bonded joints. Assuming that the adhesive thickness and the adhesive Young’s modulus are small with respect to the characteristic length of the joint and to the Young’s modulus of the adherents, a simplified model is derived in the case of large displacements using the asymptotic expansion technique. Then, the problem of the crack growth is stated, in the case of a stable growth, as the search of the local minima of the total energy of the joint, sum of the mechanical energy and the Griffith’s fracture energy. This is made using the Newton’s method. To this end, the expressions of the first and second derivatives of the mechanical energy with respect to a crack front displacement are derived analytically. Finally, numerical examples are presented, highlighting the unstable character of the crack growth at initiation.     

Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green’s functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green’s functions are analytically degenerated to the special cases such as half-space, surface and full-space Green’s functions. Some typical numerical examples are also given to show the general features of the bi-material Green’s functions.