P. Fotiu

Modal analysis of elastic-plastic plate vibrations by integral equations

Abstract A direct boundary element method for the vibration problems of thne elastic-plastic plates is presented. Dynamic fundamental solutions of a suitably shaped finite domain are used in modal form. The series Greens functions are separated into a quasistatic and a dynamic part. Often the series of the quasistatic part can be written in a faster converging form than the equivalent modal series. Analytical integration in the vicinity of the singularity is performed on the closed form fundamental solutions of the infinite domain, and only the non-singular differences from the actual Greens functions are represented in series form. This paper gives a general formulation of this method for Kirchhoff plates on an arbitrary elastic foundation. After integration, the resulting algebraic equations are arranged in a form most convenient for a time-stepping analysis of inelastic response. This rearrangement has to be performed only once, if the time step is kept constant. Constitutive equations are integrated by an implicit backward Euler scheme for plane stress. Applications are shown for impacted circular plates on several different foundations.

The propagation of spherical waves in rate-sensitive elastic-plastic materials

An integral equation method for the analysis of elastic-plastic wave propagation is presented. The elastic-plastic solution is thereby found as the superposition of the corresponding elastic result with waves produced by dynamically induced plastic strains. The solutions are represented in the form of integrals with elastodynamic Greens functions as integration kernels. The spherically symmetric problem of a dynamically loaded spherical cavity is considered and the corresponding Greens functions for this geometry are derived in closed form. Time convolution is carried out analytically over a prescribed time step and the spatial integration is performed by Gaussian quadrature. If the wave travels within each time step just the distance of one spatial element the evaluation of the integrals leads to a tridiagonal system of algebraic equations. Numerical results are compared to some known analytical solutions, proving the accuracy of the method. Computations are carried out for rate sensitive power law hardening-thermal softening materials.

Nonlinear flexural vibrations of layered plates

Abstract By means of a higher-order plate-bending theory developed by Whitney and Pagano along with Bergers hypothesis, large amplitude forced vibrations of moderately thick laminated specially orthotropic plates are investigated. The theory includes shear deformation and rotatory inertia in the same manner as Mindlins theory for isotropic homogeneous plates. The in-plane forces due to large deflections are assumed to be constant within the plate domain. Considering time-harmonic forcing a Kantorovich-Galerkin procedure provides the formulation of this problem in the lower band of the frequency domain. In the case of laminates made of isotropic layers an analogy to thin homogeneous plates is given, which is complete in the case of polygonal planforms and hard hinged supports. Furthermore, the plate deflection is determined by the solution of two (second-order) Helmholtz Klein Gordon boundary value problems. Inserting these results into a proper domain integral leads to Bergers normal force. This problem-oriented strategy renders the nonlinear frequency response functions of deflection of the undamped layered plate.

Dynamic Plasticity: Structural Drift and Modal Projections

A plastic source method is developed with structural applications as an alternative to the well-established incremental stiffness formulation of physically nonlinear problems. A combined model of rate-independent plasticity and damage is considered. Total solution is separated into two responses due to external loads and due to the material nonlinearities, which can be considered as defects in the structure. Modal decomposition is applied, where the quasistatic response due to external loads as well as defects is used in closed form whenever possible (e.g. for beam-like structures), which gives better accuracy compared to a series expansion of the total (quasistatic plus dynamic) solution. Defects act as sources of eigenstresses in the structure. Their intensity is determined in a time-stepping manner by means of the constitutive law. FFT is applied to responses in bending vibrations for inspection of changes in the frequency contents due to nonlinear material behaviour.

Material Science- and Numerical Aspects in the Dynamics of Damaging Structures

Commonly, macroscopically inelastic behavior is a result of changes of the materials micro-structure. In modern engineering science, frequent use is made of multi-component or multiphase materials, which exhibit a variety of different inelastic mechanisms on the microlevel. It is important to have knowledge of these mechanisms in order to give a thorough description of the macroscopic behavior. In a macroscopic continuum formulation non-uniformities on the microscale can only be considered as averaged quantities referred to a certain reference volume. Consequently, additional variables have to be introduced accounting for the microstructural state. These variables are called internal (sometimes hidden) variables. Since any rearrangements of the material microstructure are connected with energy dissipation, a change of the internal variables indicates a dissipative process. Therefore, variations of the microstate are at least partly irreversible in the thermodynamic sense. This corresponds to the conception of the microstructure being altered by formation and spreading of microdefects. In real materials such microdefects are identified as dislocations, microcracks and -voids, etc. There exists a vast field of literature concerning the formulation of inelastic behavior due to mechanisms on the microscale. In case of plastic flow first attempts to describe this phenomenon based on slip systems of single crystals are due to Bishop and Hill [3.5–1], [3.5–2] later contributions are given by Hill [3.5–3], [3.5–4], Lin [3.5–5], [3.5–6], Budiansky and Wu [3.5–7] and Havner [3.5–8] among others. Dislocation theories of plasticity and viscoplasticity have been introduced by numerous authors [3.5–9] — [3.5–12].

Analysis of viscoplastic sandwich beams using influence functions

ABSTRACT A method of analyzing inelastic structures is presented, in which inelastic strains are interpreted as sources of eigenstresses in the linear elastic structure. Stresses due to inelastic strains are calculated by means of influence functions. The method is applied to viscoplastic sandwich beams.

BEM analysis of grain boundary sliding in polycrystals

Abstract The paper presents a boundary element formulation of the stress analysis of an elastic polycrystal with viscous grain boundaries. The microstructure of the polycrystal is modelled as a periodic arrangement of hexagonal grains. Due to symmetry conditions the analysis can be restricted to a representative unit cell with special boundary conditions. These boundary conditions are derived in detail for an overall loading of the polycrystal by normal strains as well as by shear strains. Effective moduli are presented for a completely relaxed grain assemblage and they are compared to the outcome of several micromechanical theories. It is found that these theories cannot give an accurate description of the macroscopic behaviour of the crystal. Finally, an analysis of the stress relaxation of a crystal with nonlinear viscous grain boundaries is performed and the influence of the rate-sensitivity on the response is studied.

A modified generalized midpoint rule for the integration of rate-dependent thermo-elastic-plastic constitutive equations

The integration algorithm presented here is an extension of the widely used generalized midpoint rule. A simple but very effective method is derived to optimize the location of the collocation point (where plastic consistency is enforced) in order to achieve high accuracy for virtually unlimited sizes of the time step. These optimal locations are in the interval [Δt2, Δt], which automatically guarantees unconditional stability. The optimal weighting parameter θ is estimated from two explicit formulas. Hence, there is practically no increase in computational expense compared to applications of the conventional generalized midpoint rule. Furthermore, the method features a special formulation of plastic consistency, called a plastic predictor, which minimizes the necessary iterations at Gauss-point level. Numerical examples demonstrate the efficiency and accuracy of the algorithm for rate-dependent and rate-independent plasticity including combined kinematic and isotropic hardening, as well as thermal softening.

Dynamic Response of Elasto-Viscoplastic and Deteriorating Beams and Plates

A plastic source method as an alternative to the incremental stiffness formulation of physically nonlinear problems is generalized to include a continuum model of damage during cyclic plasticity. Solution of the associated linear elastic time-in-variant system is separated into a quasistatic and a dynamic portion, respectively. Forcing action is by the external dynamic loadings and by sources of plastic strain and accumulated damage in the yielding zones. Modal expansion technique is applied to the dynamic part of the solution. Intensity or the plastic sources is determined in a subroutine of the special constitutive law in a time stepping manner.

A Micromechanical Study of Grain Boundary Sliding by Boundary Elements

Grain boundary sliding is a significant deformation mechanism of metals at high temperatures and low stress levels. In such an environment grain boundary sliding plays an important part in the creep behavior of metals. The relative motion of the boundaries creates a non-uniform flow field inside each grain with stress concentrations at the grain junctions.