On the dynamic behavior of an elastoplastic oscillator

Summary We describe an experimental research on the behavior of an elastoplastic oscillator with one degree of freedom and we point out some differences with respect to the results of studies on a bilinear isteretical model. Therefore a mechanical scheme of an elastoplastic oscillator endowed with a continuous skeleton intrinsec curve is studied and the results of the theory agree with the experimental ones.

---

Sommario Si descrivono le esperienze condotte su un oscillatore elastoplastico a un grado di libertà. Si mettono in evidenza alcune differenze del comportamento reale con le risultanze teoriche, queste vengono spiegate attraverso lo studio di un modello teorico elastoplastico con caratteristica intrinseca curvilinea anzichè bilineare.

---

---

The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza e Tecnica delle Costruzioni del Politecnico di Milano.     

Two-dimensional bilinear oscillator: group-preserving scheme and steady-state motion under harmonic loading

We study the dynamic behavior of a single-mass, two-degree-of-freedom bilinear oscillator, whose restoring force obeys a bilinear elastoplastic law. For the purpose of calculation we develop a group-preserving scheme utilizing the internal symmetry group of the model in the plastic phase so as to satisfy the yield condition exactly at each time step. Then the oscillator is subjected to biaxial harmonic excitations, and the responses are displayed. Based on circular motion assumption and harmonic balance method, the closed-form formulae to estimate the steady-state responses are derived, which are then compared with the results calculated by the group-preserving scheme. The accuracy of the formulae is thus confirmed. Simple formulae to select minimum driving force amplitudes or maximum yield forces which cause the bilinear oscillator steadily oscillating in the plastic phase are derived. For the benefit of engineers’ use we also derive closed-form formulae to decide the force ratios required to sustain the user-specified dissipation per cycle.     

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

An improved formulation for domain stress evaluation by boundary element method in elastoplastic problems

An integral expression for domain stresses in the elastoplastic boundary element method is presented. Thus it is not necessary to evaluate the Cauchy. principal value in domain integral. This method shows some improvement in the accuracy and efficiency for the numerical process. A numerical result of axisymmetric elastoplastic problem has been worked out and a comparison with analytical result is given.This work had been supported by National Science Foundation of China.     

Stress-strain state and durability of mechanically inhomogeneous welds under low-cycle loading

Relations are proposed for the determination of the stress-strain state, strength, and life of butt welds with mild and hard interlayers under cyclic elastoplastic tension-compression. The accumulation of cyclic and quasistatic damages is determined with allowance for the redistribution of the cyclic elastoplastic strains and hardness of the stress state due to changes in the cyclic properties of separate regions of welds. The theoretical distribution of cyclic strains and the durability of welds under cyclic elastoplastic loading are supported by experimental data __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 29–38, February 2008.     

A FINITE ELEMENT—MATHEMATICAL PROGRAMMING METHOD FOR ELASTOPLASTIC PROBLEMS BASED ON THE PRINCIPLE OF VIRTUAL WORK

By expanding the yielding function according to Taylor series and neglecting the high order terms, the elastoplastic constitutive equation is written in a linear complementary form. Based on this linear complementary form and the principle of virtual work, a finite element-complementary method is derived for elastoplastic problem. This method is available for materials which satisfy either associated or nonassociated flow rule. In addition, the existence and uniqueness of solution for the method are also discussed and some useful conclusions are given. The project is supported by the National Natural Science Foundation of China     

Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. ^†Partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the first author was a Gibbs Instructor at Yale University. ^‡Partially supported by an MSRI Postdoctoral Fellowship, and by NSF grant DMS-9801664 (9996350).     

Analysis of elastoplastic deformation in metal-matrix composites with particulate reinforcements

In this paper, elastoplastic stress-strain behavior during tensile deformation of an aluminum alloy matrix composite containing alumina circular and non-circular particles is analyzed. In terms of cell models in conjunction with continuum plasticity theory, various periodic arrays of particles are assumed in a three-dimensional finite element simulation. The geometrical effects of particle volume fraction, shape, aspect ratio, array and distribution, as well as non-circular particle orientation on the overall elastoplastic stress-strain behavior are examined in view to design optimum microstructures of the composites. The project supported by the National Natural Science Foundation of China and the State Education Commission of China     

Finite amplitude,periodic motion of a body supported by arbitrary isotropic,elastic shear mountings

The undamped, finite amplitude, periodic motion of a load supported symmetrically by arbitrary isotropic, elastic shear mountings is investigated. Conditions on the shear response function sufficient to guarantee periodic motions for finite shearing with arbitrary initial data are provided. Some general results applicable for all simple shearing oscillators in the class are derived and illustrated graphically. The mechanical response of the general nonlinear shearing oscillator is compared with the response of a certain linear oscillator of comparable design. As consequence, certain static and dynamic aspects of the motion of an arbitrary nonlinear oscillator supported by shear springs are compared with those of a simple, linear oscillator for which the response is well-known and readily determined for the same initial data. The effect of a finite static shear deformation on the frequency equation for superimposed, small amplitude vibrations of the load is examined. The general analysis is applied to a class of hyperelastic biological tissues; and the frequency relation for finite amplitude oscillations of a load supported by soft tissue is derived. The finite amplitude oscillatory shearing of a general isotropic elastic continuum is described; and three universal relations connecting the stress and the oscillatory shearing deformation for every isotropic elastic material are presented.     

A self-consistent analysis for coupled elastoplastic damage problems

Based on irreversible thermodynamics and internal state variable theory, the volume-averaged Clausius-Duhem inequality is presented. In contrast to former investigations on damage-elastoplasticity, our evalustions are founded on the volume-averaged field equations of the analyzed elements and the self-consistent method. Hence, our results not only include the influence of void shapes but also consider the interaction among voids. Further, previous work about coupled elastoplastic damage problems only takes into account small initial void volume fractions. Our work, however, will be able to deal with elastoplastic damage problems with larger initial void volume fractions. Project supported by the National Natural Science Foundation of China (19762002) and Natural Science Foundation of Jiangxi in China     

The crack tip zone shielding effect for ductile particle reinforced brittle materials

The crack tip zone shielding effect for the ductile particle reinforced brittle materials is analyzed by using a micromechanics constitutive theory. The theory is developed here to determine the elastoplastic constitutive behavior of the composite. The elastoplastic particles, with isotropic or kinematical hardening, are uniformly dispersed in the brittle elastic matrix. The method proposed is based on the Mori-Tanaka's concept of average stress in the composite. The macroscopic yielding condition and the incremental stress strain relation of the composite during plastic deformation are explicity given in terms of the macroscopioc applied stress and the microstructural parameters of the composite such as the volume fraction and yield stress of ductile particles, elastic constants of the two phases, etc. Finally, the contribution of the plastic deformation in the particles near a crack tip to the toughening of the composite is evaluated. The project supported by National Natural Science Foundation of China     

Thermal stress measurement of quartz oscillator module packaging

The thermal stress of the quartz oscillator module packaging is investigated using digital-image correlation method (DICM), and the experimental results are given. Under the quartz oscillator module packaging, the quartz oscillator and the Fe−Sn−Cu alloy frame are joined together with the electroconductive adhesive (PI), and the electroconductive adhesive needs to be cured twice at 150°C and 275°C respectively. As the quartz oscillator and the Fe−Sn−Cu alloy frame have a distinct difference in both thermal expansion coefficients and mechanical properties and in process of packaging temperature rises or drops, the thermal stress is yielded easily. While temperature drops, the normal stress at the quartz oscillator edge is a tensile stress, which can make the quartz oscillator fracture. Project supported by the Natural Science Foundation of Tianjin (013605211).     

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modern structural engineering. This paper introduces the boundary element method (BEM) into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic media with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media. The project supported by the Basic Research Foundation of Tsinghua University, the National Foundation for Excellent Doctoral Thesis (200025) and the National Natural Science Foundation of China (19902007). The English text was polished by Keren Wang.     

The variational principle of structural analysis for damage-coupled elastoplastic creep problem

Based on the concept of the effective stress in continuum damage mechanics, this paper studies the damage-coupled problem in elastoplastic creep. By using the parametric variational principle developed from optimal control theory, a numerical principle for the problem discussed has been established, the proof for which is also given. It is found that the principle proposed has a normalized form and can easily be put into application by computer.This project is supported by the National Natural Science Foundation of China     

Geometric Singular Perturbations for Multiple Turning Points: Invariant Manifolds and Exchange Lemmas

In this paper, we consider singularly perturbed systems with multiple sets of turning points. Two types of related results contributing to geometric singular perturbation theory are obtained. The first result establishes a new class of invariant manifolds. The second result consists of several exchange lemmas that characterize the evolution of an invariant manifold passing through the vicinity of the slow manifold.Partially supported by NSF Grants DMS-0071931 and DMS-0406998.     

Elastoplastic model describing cyclic creeping behaviour of soft clays

The elastoplastic model in the deviatoric stress space is constructed to describe the cyclic undrained creeping behaviour of soft clays under cyclic stress by using Mises yield criterion and the concept of a field of hardening moduli. Furthermore, the effect of model parameters on the deformation is studied, and a method is given to determine quantitatively model parameters from results of cyclic triaxial tests of the saturated soft clay. The project supported by National Natural Science Foundation of China and Tianjin Twenty-one Century Youth Science Foundation     

A new hybrid stress element method for the analysis of elastoplastic solids

Based on a modified Hellinger/Reissner variational principle which includes the equivalent stress, equivalent plastic strain and non-conforming displacement increments as independent variables, a quadrilateral isoparametric hybrid stress element for the analysis of elastoplastic problem is proposed. By this formulation, the yield criterion and flow rule are satisfied in an average sense and greater accuracy can be obtained by using non-conforming displacement. A numerical example is presented to show that the present model has high accuracy and computational effectiveness.This project is supported by the Natural Science Foundation of the State Education Commission.     

A novel semi-inverse solution method for elastoplastic torsion of heat treated rods

Torsion rods are a primary component of many power transmission and other mechanical systems. The behavior of these rods under elastoplastic torsion is of major concern for designers. Different methods have so far been proposed which deal with the elastoplastic torsion of rods, most of which assume constant yield stress. This assumption produces rough and inaccurate results when the rods are heat treated, since in the process of heat treatment the form of yield stress distribution across the rod cross section changes. We propose a new method for calculating elastoplastic torsion of rods of simply connected cross section which is based on heat treatment observations. In our method the full plastic stress function is obtained by using the semi-inverse method. Elastoplastic stress function is obtained by generalizing the idea of the membrane analogy and using a piecewise continuous stress function. Since the proposed form of yield stress distribution can not be handled by the current Finite Element packages, we produce a computer package with a 3D graphical interface capable of calculating and displaying the 3D elastoplastic stress function, shear stress contours, and torque-angle of rotation per unit length. We show that our method produces excellent agreement for several known cross sections in comparison to methods which assume constant yield stress.     

Asymptotic response of a two DOF elastoplastic system under harmonic excitation. Internal resonance case

The study of a two DOF elastoplastic system is formulated in a suitable phase space, velocity and force, in which an originally multi-valued restoring force is represented by a proper function. The asymptotic response can thus be studied using the Poincaré map concept and avoiding approximate analytical techniques. On account of the peculiarity of this hysteretic system, which has a well-defined yielding point, its dynamics is studied in a reduced dimension phase space using an efficient numerical algorithm. It is shown that the asymptotic response is always periodic with the period of the driven frequency and is always stable. Thus the response of the oscillator is described by its frequency response curves at various intensities of the excitation. The results presented refer to a system with two linear frequencies in a ratio of 1 : 3. The response is highly complex with numerous peaks corresponding to higher harmonics. The effect of coupling in conditions of internal resonance is a strong modification of the frequency response curves and of the oscillation shape of the structure.     

Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.