Fictitious Material Research Articles

Locking materials and the topology of optimal shapes

We study a problem of structural optimisation using the fictitious material approach. This is connected with the equilibrium of locking materials, which can be approximated by strongly non linear materials. A finite element simulation allows us to experiment with some conjectures about the topology of the optimal solutions.

Finite element analysis of textile composite preform stamping

The forming or draping of a textile composite preform may result in large changes in the fibrous microstructure of the preform. This change in the local fiber orientation leads to significant changes in the fabric permeability as well as the mechanical properties of the ensuing composite structure. Therefore, this change in orientation of the tows of the preform needs to be known accurately to calculate the various effective properties of the composite. A new finite element approach for stamping analysis of a plain-weave textile composite preform has been developed. This model is simple, efficient and can be used in the existing finite element codes. The model represents the preform as a mesh of 3-D truss elements and 3-D shell elements. The truss elements model the tows, which are allowed to both scissor and slide relative to one another. The shell elements represent a fictitious material that accounts for inter-tow friction and fiber angle jamming. The model takes into account large strains and large deformations. In-plane uniaxial tension tests have been performed on plain-weave specimens for determining the constitutive law of the transforming medium and to show the inter-tow sliding. Application of the model is demonstrated by simulating the stamping of a preform by a spherical punch. The results from the simulation show good correlation with results from the experiments.

Dynamics of elastic-plastic shear frames with secondary structures: shake table and numerical studies

Results from experimental and numerical studies of earthquake-excited small-scale primary–secondary structures are presented. The primary structure considered is a plane three-storey shear frame with a fundamental frequency of 5.5 Hz. The columns of the first floor are built with soft aluminium and they are stressed beyond its linear range of behaviour. After each test the elastic–plastic columns are replaced by a new set of undeformed virgin aluminium bars. The elastic–plastic shear frame is tested with and without an attached secondary structure. The secondary structure is modelled as an elastic SDOF oscillator, and its natural frequency is tuned to the fundamental frequency of the shear frame. Alternatively, the oscillator is mounted on the horizontal beam of the second and third floor. The base excitation of the structural model is characterized by a broad band random process with constant spectral density in a frequency range between 3 and 30 Hz. In the numerical study, the digital recorded acceleration of the base excites the mechanical model of the investigated structures. Numerical outcomes assuming fictitious unlimited elastic material behaviour of the shear frame are set in contrast to results from experiments and computational simulations where the measured non-linear force displacement relation of the elastic–plastic floor is approximated by a piecewise linear curve. The effect of elastic–plastic materials on the dynamic interaction between primary and secondary structure is shown and the difference to unlimited elastic material behaviour is worked out in detail. Copyright © 2001 John Wiley & Sons, Ltd.

Determination of constitutive properties of thin metallic films on substrates by spherical indentation using neural networks

The indentation test has been developed into a popular method for investigating mechanical properties of thin films. However, there exist only some empirical or semi-analytical methods for determining the hardness and Young’s modulus of a film from pyramidal indentation of the film on a substrate, where the deformation of film and substrate is subjected to be elastic–plastic. The aim of the present paper is to show how constitutive properties and material parameters may be determined by using a depth-load trajectory which is related to a fictitious bulk film material. This bulk film material is supposed to possess the same mechanical properties as the real film. It is assumed that the film and the substrate exhibit elastic–plastic material properties with nonlinear isotropic and kinematic hardening. The determination of the depth-load trajectory of the bulk film is a so-called inverse problem. This problem is solved in the present paper using both the depth-load trajectory of the pure substrate and the depth-load trajectory of the film deposited on this substrate. For this, use is made of the method of neural networks. Having established the bulk film depth-load trajectory, the set of material parameters entering in the constitutive laws may be determined by using e.g. the method proposed by Huber and Tsakmakis (Huber, N., Tsakmakis, Ch., 1999. Determination of constitutive properties from spherical indentation data using neural networks. Part II: plasticity with nonlinear isotropic and kinematic hardening. J. Mech. Phys. Solids 47, 1589–1607).

GENERAL PRINCIPLES OF MODELLING PHYSICAL-MECHANICAL PROPERTIES OF CONGLOMERATES/BENDRIEJI KONGLOMERATŲ FIZIKINIŲ-MECHANINIŲ SAVYBIŲ MODELIAVIMO PRINCIPAI

A basic mathematical analogue for modelling grading of a mineral part of conglomerates is presented (1). The given method, in comparison with the ones used at present, has a number of advantages: a) it allows us to define instantly the possibility of getting the planned grading of a mineral part of the conglomerates out of mineral materials in possession: b) it enables us to get the optimal mineral part of the conglomerate, according to the property of conglomerate that is desired, for example, the cheapest conglomerate or the largest density of conglomerate; c) it enables us to get and analyse fictitious (not corresponding to the Standard) mineral material composition. The general mathematical analogue (2)–(4) for modelling mineral part composition of conglomerate and prognosis of its physical-mechanical properties is based on basic mathematical analogue (1). For both mathematical analogues the following symbols are used: c j—weight multiplier of component j (for example, the price of one mass unit of component j); xj —quantity of component (mineral material) j in parts of the unit: m—number of components: aij —quantity of component j in percent, that pass through sieve i; b min, i bmax, i —limited quantities of a mineral part of conglomerate in percent, that can be passed through sieve i n—number of sieves dvj —coefficient of additional inequality v; corresponding component j: hv —limit value of additional inequality v; s—number of additional inequalities; gwj —coefficient of physical-mechanical equality wcorresponding component j; Pw —absolute term of physical-mechanical equality w; t—number of the physical-mechanical equalities. The general mathematical analogue (2)-(4) is multi- criterion. The final solution can be chosen by including coefficients of influence or by a person who takes part in modelling.

NUMERICAL SOLUTION OF SINGLE-PHASE STEFAN PROBLEM USING A FICTITIOUS MATERIAL

A method to solve the single-phase Stefan problem with external heat flux imposed on the phase-change surface is investigated. The method is based on the fixed-grid enthalpy formulation, and uses a fictitious material with vanishingly small specific heat and anisotropic thermal conductivity. The viability of the method is demonstrated on the one-dimensional laser ablation problem with a uniform beam and the two-dimensional problem with a Gaussian beam.

Time harmonic eddy currents in non‐linear media

One way to deal with non‐linearity in the case of time harmonic excitation of the electric and magnetic fields in ferromagnetic media is to introduce an effective material to model the ferromagnetic region. This fictitious material is constant throughout a period but inhomogeneous and it takes into account the non‐linear relationship between the field quantities. Several methods to create an effective magnetization curve are presented and different finite element formulations are applied to these. Some of these methods are known from the literature for 1D and 2D problems but not for 3D ones and they have only be used for a magnetic vector potential formulation so far. In the present paper they are extended for general vector and scalar potential formulations. Further possible ways to introduce such an effective material to take into account saturation effects are shown. All these different methods are investigated on a non‐linear 3D time harmonic eddy current problem using complex formalism.

Displacement estimates of pipe elbows prior to plastic collapse loads

The non-differentiability of the plastic dissipation calculated through the Von-Mises yield function leads to convergence difficulties when using mathematical programming to solve Köiter's kinematical formulation of the classical limit analysis. This problem is avoided replacing the plastic dissipation by the strain energy of a fictitious viscoelastoplastic material with a nearly infinite Young modulus. Classical limit analysis can only give information about the limit load multiplier and the plastic collapse mechanism. Based on Zarka's method and using the finite element method and mathematical programming, it is possible to obtain not only the limit load but also an estimate of the elastoplastic displacements. This is very useful because construction codes usually impose limits on the electroplastic displacements. Some pipeline systems are examined using a 1-dimensional shell type finite element to illustrate the procedure. The results obtained are compared with simplified analytical solutions and with alternative numerical results using 2-dimensional shell elements and realistic materials.

A shallow crack assessment scheme for generalised material behaviour in bending

An engineering J-integral estimation procedure has been developed from data generated using finite element computations. The study has been focused on shallow edge cracked geometries with crack depths up to 10% of the specimen depth and subjected to severe localised plastic deformation under pure bending loading. The material behaviour considered is of a generalised nature—one that consists of a perfectly plastic plateau preceding the work hardening region, which is represented by power law. J-estimation equations were developed from curve-fitting a large number of J versus remote strain curves obtained from finite element analyses. The validity of the proposed scheme was confirmed by comparison with finite element solutions on specimens with fictitious and real material curves. The current work, together with a previously presented scheme for tension loading, hence provides a comprehensive treatment of defect assessment of shallow cracked geometries under severe localised plastic deformation.

Perfectly matched layers for acoustic waves in viscous media: Applications to underwater acoustics

Berenger’s perfectly matched layer (PML) has recently been proved to exist for elastodynamic equations [Chew and Liu, Schlumberger-Doll-Res. Rep. (1995)]. This fictitious material absorbs all waves with an arbitrary incident angle and an arbitrary frequency without giving rise to any reflections. Therefore, when used as an absorbing boundary condition (ABC) at the computational edge in a finite-difference method, the PML provides orders of magnitude higher absorption than other existing ABCs. In this work, the PML is further extended as an ABC for a finite-difference simulation of acoustic waves in viscous media. For such attenuative media, an additional term involving the time-integrated wavefield is introduced to account for the coupling between the attenuation from the PML and the normal viscous attenuation. This ABC is highly effective in absorbing outgoing waves at the computational edge even when a dipping interface intersects the outer boundary. This new material ABC is ideal for parallelization on multiprocessor computers. The algorithm is validated by analytical and numerical solutions. Various two- and three-dimensional numerical simulations will be shown to demonstrate the applications in underwater acoustics and other areas. [Work supported by a Presidential Early Career Award for Scientists and Engineers through EPA and by Sandia National Laboratories.]

Ultrasonic 2-D SH crack detection in anisotropic solids

A 2-D SH flaw scattering model in anisotropic media is developed in this paper. The situation to be simulated is the scattering of waves generated by a probe incident upon an infinite strip-like flaw between two different homogeneous anisotropic solids. The scattering problem is solved by combining a 2-D SH probe model with a 2-D SH scattering model for an incoming plane wave. The probe model is based upon the assumption that the traction is known at the probe interface and the flaw is modeled by the so called spring boundary conditions. The solution is obtained by means of Fourier methods. In addition to the displacement field, a probe signal response which is directly proportional to the voltage output in a pulse-echo experiment is calculated. Numerical examples are presented both in the frequency and the time domain for the total displacement field and in the time domain for the probe signal response for two fictitious material configurations. An example where the flaw is a surface-breaking crack is also given.

PERFECTLY MATCHED LAYERS FOR ELASTODYNAMICS: A NEW ABSORBING BOUNDARY CONDITION

The use of perfectly matched layers (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, we will first prove that a fictitious elastodynamic material half-space exists that will absorb an incident wave for all angles and all frequencies. Moreover, the wave is attenuative in the second half-space. As a consequence, layers of such material could be designed at the edge of a computer simulation region to absorb outgoing waves. Since this is a material ABC, only one set of computer codes is needed to simulate an open region. Hence, it is easy to parallelize such codes on multiprocessor computers. For instance, it is easy to program massively parallel computers on the SIMD (single instruction multiple data) mode for such codes. We will show two- and three-dimensional computer simulations of the PML for the linearized equations of elastodynamics. Comparison with Liao’s ABC will be given.

Numerical modeling of ablation phenomena as two-phase Stefan problems

A fixed domain numerical scheme for ablation problems based on the enthalpy formulation and with temperature as the main dependent variable is presented. A fictitious material which occupies the region where the material has been removed is introduced, and its material properties are: null specific heat and any value for the conductivity, preferably that of the lower temperature phase. The ablating surface is transformed on an internal interface and the thermal load there is represented as a concentrated internal heat source on the interface. The resulting two-phase Stefan problem is discretized with the scheme presented in a series of previous papers. Finally, several numerical examples are presented.

A shallow crack assessment scheme for generalised material behaviour in tension

A previously proposed estimation procedure for J-strain relationships in power law materials has been further augmented so that it can be used for general engineering applications to real material stress-strain curves. The extension covers two important aspects of material curve representation. First the effects of including the material constant α of the power law on J are included—this inclusion facilitates a more accurate curve fitting of real material curves. Secondly, a set of equations is presented to account for the presence of a perfectly-plastic yield plateau, in addition to the strain hardening portion. The validity of the augmented procedure was confirmed by comparison with finite element solutions on specimens with fictitious and real material curves.

Effect of coatings with smoothly changing transverse rigidities on the stiffness of encapsulated anisotropic fiber composites

The beneficial role of anisotropic coatings with radially dependent transverse stiffness on the improvement of the whole stiffness of the fiber composite is presented in this paper. Such a typical composite is studied as being, macroscopically, a transversely isotropic material by arranging the variation of the effective moduli in its respective representative volume element. This model is formed by a transversely isotropic cylindrical fiber, a transversely isotropic cylindrical annulus as coating of the fiber and an annulus of isotropic matrix covering the encapsulated fiber. The material surrounding the matrix, called the equivalent composite, corresponds to a fictitious material equivalent of the matrix together with the rest of the fibers included in the real composite and therefore attains effective moduli, which average the actual mechanical properties of the composite. Solutions are established in a closed form for the longitudinal and transverse elastic moduli, E Lc and E Tc, of the composite, and for the shear modulus, G Tc, for various composite models. The study of the variation of these mechanical properties in terms of the percentage amount of the matrix reveals interesting information concerning the overall stiffness of the composite, which is influenced by the extent, as well as by the mechanical properties of the coatings.

Optimal design and relaxation for reinforced plates subject to random transverse loads

We consider a Kirchhoff plate subject to a random transverse load. We reinforce the plate with stiffeners. For a prescribed area fraction of stiffeners we seek their optimal layout so that on average, the plate is as stiff as possible with respect to the random load. We provide a mathematical formulation of this problem. To ensure the existence of a solution we relax the problem to include generalized designs. The relaxation procedure rests upon the derivation of new optimal lower bounds on the compliance energy for the effective elasticity of composites made from stiffeners on multiple scales. Our bounding method follows the program of Hashin and Shtrikman. However our method is novel as no fictitious comparison material is used in the derivation. This relaxation is new and is the extension to the two-dimensional setting of the relaxation given by Cheng and Olhoff ( Int. J. Solids Struct., 17 (1981) 305 23, 795–810) for one-dimensional plate problems, when the plate thickness is allowed to take two values. The relaxed formulation of the problem can be solved numerically.

Calculation of eddy current losses in nonlinear ferromagnetic materials

A complex analysis of the nonlinear diffusion problem in ferromagnetic materials under steady-state excitation is presented. The problem is solved by considering an equivalent fictitious material where the relative permeability is assumed to be constant in time but different from point to point and is related to the nonlinear B-H characteristic curve with the help of the stored magnetic co-energy density. Eddy current losses are calculated in a one-dimensional thick steel plate. A comparison made with results obtained from the classical step-by-step method shows a good agreement. >

Material failure by void coalescence in localized shear bands

For a void containing ductile material a continuum model of the loss of stress carrying capacity due to void coalescence is suggested. This model is included in a set of constitutive reactions for a ductile porous material, such that the process of void coalescence is assumed active, when the local void volume fraction has exceeded a critical value. Predictions of the model are illustrated by a number of simple analyses of shear band development in uniformly strained solids. Furthermore, the failure model is applied in a numerical analysis of strain localization and fracture by a void sheet mechanism in the matrix material between two larger voids. When coalescence has taken place in a small material region, this part of the material has no more stress-carrying capacity. In the continued computations such totally failed material is described as a fictitious elastic-plastic material with a very low yield stress.

Frictionless contact of elastic bodies by finite element method and mathematical programming technique

The work presents an analysis of the contact problems between elastic bodies without friction. Extended variational principles in the contact problems constitute the theoretical foundations of the analysis. In fact, kinematically admissible displacement fields are subjected to the non-interpenetration conditions so that the contact problem being equivalent to a minimization of the potential energy with contraints on the displacements may be reduced to a particular case of the mathematical programming techniques. Applications are limited to plane state of stress or strain cases where small deformations are considered. The geometrical space is discretized into finite elements, namely the 12 degree of freedom hybrid triangles or the 16 degrees of freedom isoparametric bi-linear quadrangle elements. A special procedure of linearization is adopted to use the quadratic programming algorithm already prepared for large scale problems. In these conditions, a computer program has been written in FORTRAN IV H and the results obtained from this present good agreement comparing to analytical solutions or other numerical solution in the literature. Finally, theoretical efforts have been performed to generalize the method by introducing a concept of fictitious locked material which represents the assumed contact space between the two bodies. A general and new condition of non-interpenetration is proposed for general cases of two elastic solids in contact.

Assessing probabilities of compound events in a judicial context

Subjects presented with fictitious criminal case material were found to overestimate the probabilities of compound as compared with component events. An analogy to bayesian cascaded inference behaviour was discussed. Results of functional measurement and correlational procedures suggested a three‐strategy model of compound probability estimation, applicable where there are two component events. According to the model, subjects base their choice of an information processing strategy on the larger and smaller component probability assessments and on a criterion value for each.