Institute of Applied Mechanics

Abstract A three‐field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this frame‐work, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piece‐wise constant functions and be L 2 ‐ortho‐gonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for non‐linear analysis.

Abstract In recent papers the author has shown that when determining optimum parameters for an absorber which minimizes the vibration response of a complex system, the latter may be treated as an equivalent single degree‐of‐freedom system if its natural frequencies are well separated. Emphasis was on minimizing the displacement response when the excitation was a harmonic force. In the present paper simple expressions for optimum absorber parameters are derived for undamped one degree‐of‐freedom main systems for harmonic and white noise random excitations with force and frame acceleration as input and minimization of various response parameters. These expressions can be used to obtain optimum parameters for absorbers attached to complex systems provided that optimization is with respect to an absolute, rather than a relative, quantity. The requirement that the natural frequencies should be well separated is investigated numerically for the different cases. The effect of damping in the main system on optimum absorber parameters is investigated also.

Abstract A class of ‘assumed strain’ mixed finite element methods for fully non‐linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part , formulated in the context of a three‐field variational formulation. The resulting class of mixed methods provides a possible extension to the non‐linear regime of well‐known incompatible mode formulations. In addition, this class of methods includes non‐linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2‐D, 3‐D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.

Abstract An updated version of the semi‐discretization method is presented for periodic systems with a single discrete time delay. The delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined. Stability charts are constructed for the damped and delayed Mathieu equation for different time‐period/time‐delay ratios. The convergence of the method is investigated by examples. Stability charts are constructed for 1 and 2 degree of freedom milling models. The codes of the algorithm are also attached in the appendix. Copyright © 2004 John Wiley & Sons, Ltd.

Abstract Rate‐independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non ‐ smooth fashion are considered in detail. It is shown that the standard Kuhn‐Tucker optimality conditions lead to the only computationally useful characterization of plastic loading. On the computational side, an unconditionally convergent return mapping algorithm is developed which places no restrictions (aside from convexity) on the functional forms of the yield condition, flow rule and hardening law. The proposed general purpose procedure is amenable to exact linearization leading to a closed‐form expression of the so‐called consistent (algorithmic) tangent moduli. For viscoplasticity, a closed ‐ form algorithm is developed based on the rate‐independent solution. The methodology is applied to structural elements in which the elastic domain possesses a non‐smooth boundary. Numerical simulations are presented that illustrate the excellent performance of the algorithm.

Channel flow with friction Reynolds number Reτ as high as 80 000 is treated by large-eddy simulation at a moderate cost, using the subgrid-scale model designed for detached-eddy simulations. It includes wall modeling, and was not adjusted for this flow. The grid count scales with the logarithm of the Reynolds number. Three independent codes are in fair agreement with each other. Reynolds-number variations and grid refinement cause trades between viscous, modeled, and resolved shear stresses. The skin-friction coefficient is too low, on the order of 15%. The velocity profiles contain a “modeled” logarithmic layer near the wall and some suggest a “resolved” logarithmic layer farther up, but the two layers have a mismatch of several units in U+.

Abstract In the classical problem a damped one degree‐of‐freedom absorber system is attached to a main system, which has one degree of freedom and is undamped. The optimum values of absorber stiffness and damping, which will minimize the resonant response of the main mass, are well known. In this paper the effect on these optimum conditions of light damping in the main system is studied. The authors show that optimum parameters for absorbers, which are attached to beams and plates, can be obtained simply and accurately from those for an equivalent one degree‐of‐freedom main system. This depends upon the concept of an effective mass for the elastic body and the representation of its response by the single relevant mode. It will be shown in a later paper that for more complex elastic bodies such as cylindrical shells, for which the natural frequencies are more closely spaced, these simple concepts do not predict accurately optimum absorber parameters.

Abstract In this paper, a formulation is presented for the finite element treatment of multibody, large deformation frictional contact problems. The term multibody is used to mean that when two bodies mechanically contact, both may be deformable. A novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first , before deriving the corresponding finite element equations This feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general. In particular, the approach yields a characterization of the frictional constraint (assuming a Coulomb law) suitable for arbitrary discretizations in either two or three dimensions. A geometric framework is constructed within which both frictionless and frictional response are naturally described, making subsequent finite element discretization a straightforward substitution of finite‐dimensional solution spaces for their continuum counterparts. To our knowledge, this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature. The development includes exact linearization of the statement of virtual work, which enables optimal convergence properties for Newton‐Raphson solution strategies, and which appears to be highly desirable (if not essential) for the general robustness of implicit finite element techniques. Since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur, the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non‐linear behaviour. This assertion is substantiated through presentation of a variety of examples in both two and three dimensions.

Abstract We show that, for rigid body dynamics, the mid‐point rule formulated in body co‐ordinates exactly conserves energy and the norm of the angular momentum for incremental force‐free motions, but fails to conserve the direction of the angular momentum vector. Further, we show that the mid‐point rule formulated in the spatial representation is, in general, physically and geometrically meaningless. An alternative algorithm is developed which exactly preserves energy, and the total spatial angular momentum in incremental force‐free motions. The implicit version of this algorithm is unconditionally stable and second order accurate. The explicit version conserves exactly angular momentum in incremental force‐free motions. Numerical simulations are presented which illustrate the excellent performance of the proposed procedure, even for incremental rotations over 65 degrees. The procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells.

Abstract Time finite element methods are developed for the equations of structural dynamics. The approach employs the time‐discontinuous Galerkin method and incorporates stabilizing terms having least‐squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time‐discontinuous Galerkin and least‐squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics. These results show that, for particular interpolations, the time finite element method exhibits improved accuracy and stability.

Crack deflection and the subsequent growth of delamination cracks can be a potent source of energy dissipation during the fracture of layered ceramics. In this study, multilayered ceramics that consist of silicon nitride (Si 3 N 4 ) layers separated by boron nitride/silicon nitride (BN/Si 3 N 4 ) interphases have been manufactured and tested. Flexural tests reveal that the crack path is dependent on the composition of the interphase between the Si 3 N 4 layers. Experimental measurements of interfacial fracture resistance and frictional sliding resistance show that both quantities increase as the Si 3 N 4 content in the interphase increases. However, contrary to existing theories, high energy‐absorption capacity has not been realized in materials that exhibit crack deflection but also have moderately high interfacial fracture resistance. Significant energy absorption has been measured only in materials with very low interfacial fracture resistance values. A method of predicting the critical value of the interfacial fracture resistance necessary to ensure a high energy‐absorption capacity is presented.

Abstract The classical normal mode method of determining response is extremely useful for practical calculations, but depends upon the damping matrix being orthogonal with respect to the modal vectors. Approximations that allow the method to be used when this condition is not satisfied have been suggested; the simplest approach is to neglect off‐diagonal terms in the triple matrix product formed from the damping and modal matrices. In this paper the errors in response caused by this approximation are determined for several simple structures for a wide range of damping parameters and different types of excitation. Based on these results a criterion, relating modal damping and natural frequencies, is formulated; if this is satisfied, the errors in response caused by this diagonalization procedure are within acceptable limits.

Abstract Optimum parameters are determined for absorbers, which, when attached to one mass of a main system with two degrees of freedom, minimize the harmonic response of that mass. Comparison is made with the absorber parameters that are determined by treating the main system as an equivalent one degree‐of‐freedom system and using classical results. Close agreement is obtained if the ratio of the two natural frequencies of the main system is reasonably large. This is in agreement with the author's recent work on optimum absorber parameters which minimize the response of elastic bodies. The extension of the method to multi degree‐of‐freedom main systems is outlined. The conditions for which different values of these parameters are predicted when the response is minimized over narrow and broad frequency bands are determined.

Abstract A class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward‐Euler return maps as a particular case. More importantly, it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary Helmholz free energy function (B‐stability). Since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm, the requirement of B‐stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity. The analysis that follows depends critically on the assumption of convexity. In particular, the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation. The proposed algorithms obey the discrete counterpart of this classical principle.

Abstract In a companion paper 1 the authors show that the parameters of an absorber which will minimize the resonant response of a simple elastic body can be determined from known results by treating the body as an equivalent single degree‐of‐freedom system. In this paper cylindrical shells are considered as examples of dynamically complex structures, for which the ratio of the natural frequencies of adjacent modes tends towards unity. It is shown that as dynamic complexity increases optimum absorber parameters for the reduction of resonant response deviate increasingly from those for an equivalent single degree‐of‐freedom system. Absorbers can be used also to reduce the random response of structures. Simple expressions for optimum parameters are given for an undamped main system, which has one degree of freedom and is subjected to white noise excitation. Optimum absorber parameters for beams, plates and cylindrical shells show similar qualitative behaviour for random and harmonic response with the concept of an equivalent single degree‐of‐freedom system being applicable only for the simpler structures.

This paper examines various methods for defining effective linear systems for the earthquake response analysis of simple hysteretic structures. A broad class of approximate methods is considered including: harmonic equivalent linearization, resonant amplitude matching, dynamic mass, constant critical damping, geometric stiffness, geometric energy, stationary random equivalent linearization, and average period and damping. A technique for estimating the accuracy of different approximate methods is presented. A new linearization scheme that may be applied to both degrading and nondegrading hysteretic systems is proposed.

Abstract In the present work, rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. Concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail. Both internal and external constraints are dealt with on an equal footing. The present approach thus circumvents the use of rotational variables throughout the whole time discretization. After the discretization has been completed a size‐reduction of the discrete system is performed by eliminating the constraint forces. In the wake of the size‐reduction potential conditioning problems are eliminated. The newly proposed methodology facilitates the design of energy–momentum methods for multibody dynamics. The numerical examples deal with a gyro top, cylindrical and planar pairs and a six‐body linkage. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract In Parts I to V of the present work, the formulation and finite element implementation of a non‐linear stress resultant shell model are considered in detail. This paper is concerned with the extension of these results to incorporate completely general non‐linear dynamic response. Of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and, for the Hamiltonian case, the total energy. A main goal of this paper is the design of non‐linear time‐stepping algorithms, and the construction of finite element interpolations, which preserve exactly these fundamental constants of motion. It is shown that only a very special class of algorithms, namely a formulation of the mid‐point rule in conservation form , exactly preserves the total linear and angular momentum. For the Hamiltonian case, a somewhat surprising result is proved: regardless of the degree of non‐linearity in the stored‐energy function, a generalized mid‐point rule algorithm always exists which exactly conserves energy The conservation properties of a time‐stepping algorithm need not, and in general will not, be preserved by the spatial discretization. Precise conditions which ensure preservation of these conservation properties are derived. A number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology.

Abstract This paper presents the results of a numerical investigation in which the maximum response of six hysteretic systems is calculated for an ensemble of twelve earthquakes. Inelastic response spectra are constructed for a range of response ductility. An effective linear period and damping are calculated for each system and ductility by determining those parameters which minimize an RMS response spectrum error. Conclusions are presented concerning the effects of deterioration, stiffness degradation, cracking and ductility on the effective linear system parameters.

Dyes incorporated in molecular sieve crystals can reveal information about the orientation of guests in these crystals through polarization microscopy (see Figure), Raman scattering, and second‐harmonic generation measurements. Pyroelectric studies of the dye molecule para ‐nitro‐aniline inserted in the molecular sieve A1P0 4 ‐5 are described that show which of two rival models explaining the ordering of guests in molecular sieves actually holds. magnified image