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In order to understand how hardening of the binder phase in particulate reinforced metal matrix composites (PRMMC) influences the effective strength, we present in this work a numerical framework consisting of the direct method (DM) and statistical models. Using this approach we created a large number of statistically equivalent representative volume element (SERVE) models to represent an exemplary PRMMC material WC-20 Wt.% Co and predicted its effective strengths using DM. After the global strength was calculated from each SERVE sample all derived data are interpreted by Bayesian network and diagnostic testing. By doing so the relationship between material strength and few selected characteristics have been clarified. The study shows the formulated approach as a novel means for investigating how the overall mechanical properties of random heterogeneous materials react to a certain constituent. Meanwhile, the study also demonstrates how statistical models, in particular the Bayesian network, can be used as a powerful supplement to micromechanical models for result analysis and knowledge discovery.

In this paper, a numerical technique is presented for evaluating the influence of particle size in particulate reinforced metal matrix composites (PRMMCs) on two critical material parameters, namely, the ultimate strength and endurance limit. To demonstrate this technique a representative PRMMC material, tungsten carbide–cobalt hard metal (WC–Co), was selected. As a typical random heterogeneous material, WC–Co demonstrates a noteworthy randomness in its composite structure. In order to take this characteristic into consideration, a novel numerical approach is adopted in this work, which is a conjunction between so-called direct methods (DMs) and statistical analyses. As the approach requires a large number of samples to be evaluated, numerical efficiency in solving optimization problems resulted from DM is a critical issue. For the purpose of comparison with the lower bound formulation, a numerical scheme formulated from the upper bound method is selected in this work. Overall, the study clarifies how the material strength of WC–Co with a fixed binder content is influenced by the reinforcement WC particle size and shows how the proposed approach can be used as a viable means for investigating the relationship between properties and structural characteristics of PRMMC materials.

This paper introduces a numerically-based methodology for determining the load-bearing capacity of particulate reinforced metal matrix composites (PRMMC) under both monotonic and cyclic loading. A multi-scale approach combining shakedown analysis with homogenization was used to evaluate the influence exerted by the composite structure on the global behavior of PRMMCs. The results were interpreted using statistical methods in order to take account of the randomness associated with the composite structure of the material. The general work flow of the approach can be summarized as follows: first, a large number of representative volume elements (PRMMC) were constructed from real material images using an in-house code. Next, lower bound limit and shakedown problems were solved via the interior-point method. Finally, results were converted to their corresponding macro quantities and evaluated statistically. With this approach we investigated a representative PRMMC structure, WC/Co, with two different binder contents and scrutinized the contribution of the composite structure over composite strength.

A numerical approach is presented to determine the load bearing capacity of structural elements made of heterogeneous materials subjected to variable loads. Melan’s lower-bound shakedown theorem is applied to representative volume elements. Combined with the homogenization technique, the material effective properties are determined through transformation from the mesoscopic to macroscopic admissible loading domains. For the numerical applications, finite element method and large-scale nonlinear optimization, based on an interior-point-algorithm, are used. The methodology is illustrated by the application to regular and random heterogeneous materials. This way, the proposed method provides a direct numerical approach to evaluate the macroscopic strength of heterogeneous structures as a useful tool for the design of structures.

The load-bearing capacity of cylindrical pressure vessels closed by Klopperboden torispherical drumheads with piping nozzle connections placed in the head's knuckle region is determined by using shakedown analysis. The pressure vessels under consideration are subjected to internal pressure, an axial force in the direction of the nozzle, in-plane bending moment at the nozzle, and temperature loading, all of which may vary independently. In particular, the interactions are investigated in several combinations of two and three of these loads, leading to two- and three-dimensional loading domains. The corresponding elastic and shakedown limits are computed based on Melan's statical shakedown theorem. The obtained results are compared to those taken from literature where available.

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Pade approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.

Predicting strengths and understanding how these values related to the underlying composite structure is essential for the design and application of particulate reinforced metal matrix composites (PRMMCs). In order to investigate how ultimate strength and endurance limit of an exemplary PRMMC material, WC-20 wt% Co, are related to other structural and mechanical characteristics, an integrated numerical approach consisting of direct methods (DM) and artificial neural network (ANN) is presented in this work. Using few features obtained from elastic and DM analyses as inputs, multiple regression and classification ANNs were established to predict global material strengths. With this approach, the study implied that the distribution pattern of the stress field, in particular the one pertained to the binder phase, has a nontrivial influence over global composite strengths.

Wave propagation in nonlinear elastic media with microstructure is studied. As an illustrative example, a 1D model of a layered composite material is considered. Geometrical nonlinearity is described by the Cauchy–Green strain tensor. For predicting physical nonlinearity the expression of the energy of deformation as a series expansion in powers of the strains is used. The effective wave equation is derived by the higher-order asymptotic homogenization method. An asymptotic solution of the nonlinear cell problem is obtained using series expansions in powers of the gradients of displacements. Analytical expressions for the effective moduli are presented. The balance between nonlinearity and dispersion results in formation of stationary nonlinear waves that are described explicitly in terms of elliptic functions. In the case of weak nonlinearity, an asymptotic solution is developed. A number of nonlinear phenomena are detected, such as generation of higher-order modes and localization. Numerical results are presented and practical significance of the nonlinear effects is discussed.

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.

A finite element model of gradient-enhanced damage viscoplasticity for dynamic analysis of thin-walled shell structures is proposed. To take void nucleation and growth into account, the free-energy function is enhanced by introducing a non-local damage term, which is defined in terms of a new non-local variable and a local damage parameter. Local constitutive laws considering viscoplastic behavior, isotropic hardening and isotropic ductile damage leading to softening are used. The performance of the proposed model is demonstrated through numerical simulations of several examples including shock-wave loaded plates.

We propose a numerical method for the computation of shakedown loads of engineering structures subjected to varying thermo-mechanical loading. The method is based on Melan’s lower bound shakedown theorem using the von Mises yield criterion. The resulting nonlinear convex optimization problem is presented in a generalized formulation and then solved by an interior-point algorithm, which is characterized by a problem-tailored solution strategy, particularly suitable for application to large-scale engineering structures. Theoretical and numerical issues of the algorithm are described. It’s efficiency is shown by application to thermo-mechanical problems from power plant engineering. The results are compared to those found in literature as well as to calculations with other optimization codes lancelot , ipopt and ipdca .

Based on the static theory of strain-gradient viscoplasticity proposed by Anand et al. (2005) , a one-dimensional dynamic analysis is derived for finite element computation of isotropic hardening materials. The kinetic energy is assumed to be composed of the conventional and internal kinetic energy. The internal energy is described phenomenologically in terms of the equivalent plastic strain in order to capture the heterogeneity of plastic flow. Herein the microscopic density is assumed to be related to the macroscopic one through a microscopic-inertia parameter. The macroscopic-force balance and microscopic-force balance including inertia effects are derived. The performance of the proposed formulation is illustrated through the numerical simulation of a one-dimensional dynamic problem. A parameter study to find the microscopic-inertia parameter is carried out. At last, suitable microscopic boundary conditions and dynamic effects are discussed through comparison with the conventional plasticity.

Cold drawing of bimetallic wires is an efficient technology for fabrication of filamentary composite materials. In the present paper an analytical approach to the plastic flow of a two-component mono-filamentary wire through a converging conical die is proposed. Developed solution is based on a number of simplified assumptions concerning the radial flow pattern, the perfectly plastic materials’ behavior, and the perfect bonding conditions between the components. The stress and the strain-rate fields are determined. Obtained results for the drawing stress show a good agreement with available experimental data. Examination of the interfacial stresses localized at the boundary between the components allows to identify the range of the input parameters in which the pure radial flow can exist.

The phenomenon of the load diffusion from a fibre to a surrounding matrix is analysed for the 2D case. We use an approximate analytical approach based on the asymptotic reduction of the governing biharmonic problem into two harmonic problems. The comparison of the obtained solutions with known results of other authors shows an acceptable accuracy of the proposed asymptotic simplifications. All solutions are obtained in closed analytical form. The case of perfect bonding between fibre and matrix for a single fibre and for a periodic system of fibres is firstly considered. Then we study the influence of the interface stiffness. In the case when only a single fibre is loaded, the influence of all other fibres is predicted by means of a three-phase model. The proposed approach gives a possibility to solve the problems for a broken fibre and for a broken matrix, as well as for partly debonded fibres. The important problem of infinite matrix cracks is also solved in the present paper. The obtained results can be used for the calculation of pull-out and push-out tests, as well as for the investigation of the fracture of composite materials.

An asymptotic approach for simulation of the imperfect interfacial bonding in composite materials is proposed. We introduce between the matrix and inclusions a flexible bond layer of a volume fraction c(3) and of a non-dimensional rigidity λ(3), derive a solution for such three-component structure, and then set c(3)→0, λ(3)→0. In the asymptotic limit depending on the ratio λ(3)/c(3) different degrees of the interface's response can be simulated. A problem of the axial shear of elastic fibre-reinforced composites with square and hexagonal arrays of cylindrical inclusions is considered. The performed analysis is based on the asymptotic homogenization method, the cell problem is solved using the underlying principles of the boundary shape perturbation technique. As a result, we obtain approximate analytical solutions for the effective shear modulus and for the stress field on micro level depending on the degree of the interfacial debonding. Developed solutions are valid for all values of the components’ volume fractions and properties. In particular, they work well in cases of rapid oscillations of local stresses (e.g., in the case of densely packed perfectly rigid inclusions), while many of other commonly used methods may face computational difficulties.

The thermo-mechanical behaviour of a fluid-filled metal foam is influenced by the characteristics of the solid skeleton and the pore-fluid. During the deformation process both constituents exchange momentum and energy. Based on the theory of porous media a fluid-filled foam is modeled as a binary mixture consisting of a metal skeleton and a pore-gas. The porous solid skeleton is assumed to behave thermo-elasto-plastically. The pore-gas is considered as an ideal gas. For both constituents different phase temperatures and thermo-mechanical coupling mechanisms are taken into account. The resulting system of differential equations is solved using the finite element method. An example demonstrates the applicability and validity of this approach.

In this paper, the theoretical background for the failure analysis of fibre-reinforced composites under variable repeated loads in the framework of direct methods is presented. It is based on a local shakedown analysis in a representative volume element of the composite and the use of averaging techniques to study the influence of each component (matrix, fibre and interface) on the macroscopic response of such composite.

An experimental study of hot spots occurrence in railway disc brakes is reported on. The aim of this study was to better classify and to explain the thermal gradients appearance on the surface of the disc. Thermographic measurements with an infrared camera have been carried out on the rubbing surface of brake discs on a full-scale test bench. Based on thermography, a classification of hot spots observed in disc brakes is proposed. A detailed investigation of the most damaging thermal gradients, called macroscopic hot spots (MHS) is given. From these experimental researches, a scenario of hot spots occurrence is suggested step by step. Influence of parameter such as pad stiffness and pad contact length on hot spots developments is studied. Observations give new highlights on the conditions of hot spots appearance. Discussion of the theoretical approaches compared to experimental observations is proposed.

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Pade approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.

The present paper reports on modeling, numerical simulation, and experimental investigation of plates subjected to impulsive loading. The kinematical hypothesis used for the theoretical description of the transient response includes transverse shear deformations, rotary inertia, and geometrical nonlinear effects. The material modeling accounts for elastic–plastic behavior, isotropic and kinematical hardening, and strain rate sensitivity. The numerical simulation of the transient inelastic vibrations is performed using isoparametric finite elements. Both the Chaboche and the Bodner–Partom viscoplastic constitutive laws are used to trace the evolution of the material characteristics in the framework of a layered shell model. The theoretical and numerical developments are checked by experimental investigations of thin steel plates subjected to shock waves. These experiments are performed in a shock tube with various impact periods and loading histories. The topics addressed in this report include (a) the correlation of experimental and simulated transient inelastic response using the Chaboche and Bodner–Partom models, (b) the sensitivity of the predicted structural response to variations of the material parameters identified on the basis of uniaxial tension tests, (c) the effect of the transverse shear stress distribution on the local evolution of the material behavior and on the global dynamic response, (d) the evolution of deflections, stresses, and plastic zones under blast loading conditions.

Purpose: In a meta-analysis of the literature we evaluated the present knowledge of the material properties of cortical and cancellous bone to answer the question whether the available data are sufficient to realize anisotropic finite element (FE)-models of the proximal femur. Material and method: All studies that met the following criteria were analyzed: Young's modulus, tensile, compressive and torsional strengths, Poisson’s ratio, the shear modulus and the viscoelastic properties had to be determined experimentally. The experiments had to be carried out in a moist environment and at room temperature with freshly removed and untreated human cadaverous femurs. All material properties had to be determined in defined load directions (axial, transverse) and should have been correlated to apparent density (g/cm3), reflecting the individually variable and age-dependent changes of bone material properties. Results: Differences in Young's modulus of cortical [cancellous] bone at a rate of between 33% (58%) (at low apparent density) and 62% (80%) (at high apparent density), are higher in the axial than in the transverse load direction. Similar results have been seen for the compressive strength of femoral bone. For the tensile and torsional strengths, Poisson’s ratio and the shear modulus, only ultimate values have been found without a correlation to apparent density. For the viscoelastic behaviour of bone only data of cortical bone and in axial load direction have been described up to now. Conclusions: Anisotropic FE-models of the femur could be realized for most part with the summarized material properties of bone if characterized by apparent density and load directions. Because several mechanical properties have not been correlated to these main criteria, further experimental investigations will be necessary in future.

A comparative study of vibrations of elasto-viscoplastic circular plates subjected to shock-wave impulses is presented. Results of a finite element analysis are related to experimental data. Two types of elasto-viscoplastic constitutive equations are used: the Chaboche and Bodner–Partom model. They are integrated using the trapezoidal rule whereas the equations of motion are solved by the central difference method. The problem of the modelling of boundary conditions and outside damping is discussed. A good correlation between numerical and experimental results and between two types of constitutive models is observed.

Results from the modelling of the initiation and the evolution of microvoids governed by triaxiality and effective plastic strain in heterogeneous materials are presented. In particular, the damage evolution in a microalloyed thermomechanically treated steel consisting of brittle hard phases embedded in a ductile matrix is studied. Results obtained by numerical simulations using the finite element method are compared with experimental investigations of a notched cylindrical tensile bar. For this, two kinds of continuum damage models are used: the model of Rousselier, which assumes the yield stress as function of damage, and the concept of effective strains and stresses proposed by Lemaitre.

Resume Knowles et Sternberg ont donne la definition de l'integrale de contour M , relative a la symetrie de dilatation dans le domaine elastique (loi de conservation). Les conditions d'independance du contour de cette integrale correspondent a des cas d'applications tres restrictifs. On propose d'elargir le champ d'application en definissant pour chaque type de transformation une integrale M modifiee independante du contour. La demonstration de cette loi de conservation associee a la transformation est abordee.

PII S0093-6413(99)00029-4 SHAKEDOWN ANALYSIS OF COMPOSITES D. Weichert, A. Hachemi and F. Schwabe Institute of General Mechanics, RWTH-Aachen Templergraben 64, 52056 Aachen, Germany (Received 16 December 1998," accepted for print 15 March 1999) Introduction One of the advantages of composite materials compared to conventional materials stems from the different mechanical characteristics of each component of a composite. This may be used to design materials for specific technological purposes, where in some sense controversial material properties are required. As example one may quote metal-matrix-composites (MMC's) exhibiting at the same time high hardness and fracture toughness [1]. To predict failure of this type of materials, it is important to understand the complex processes on the microstructural level leading to failure, including in particular inelastic effects, and to link them to the macroscopic material properties. In this paper, failure of metal-matrix-composites under variables loads due to accumulated plastic deformations is investigated. For this, shakedown-analysis [2] is carried out on the micro- level. Then, by use of the technique of homogenisation [3], the results are linked to the overall material properties on the macro-level. The methodology is analogous to that given by Suquet [3] for the case of limit analysis of heterogeneous media (see also [4, 5]). General considerations and definitions The considered composite material is composed of inclusions, embedded according to a regular pattern as shown in FIG. 1 in an elastic-perfectly plastic metal matrix. The macroscopic behaviour of this heterogeneous material is observed on the scale (Xl, x2) and the microscopic behaviour on the scale (Yl, Y2). 309

Here we have investigated under which conditions elastic-perfectly plastic, cracked bodies subjected to variable loads shake down. For this purpose, an extension of the static shakedown theorem (Melan’s theorem) is presented by using a crack analysis developed by Nguyen Quoc Son in the framework of the concept of generalized standard materials.

A generalization of Melan's shakedown theorem is presented taking into account geometrical eAects and plastic ductile damage. Numerical results illustrate the proposed method. # 1998 Elsevier Science Ltd. All rights reserved

A numerical method is proposed to predict the failure of mechanical structures. It is based on the generalization of the classical static shakedown theorem to damaged inelastic structures. Ductile plastic damage is taken into account by using the models of Lemaitre and Shichun-Hua. The obtained results are compared in the special case of limit analysis to those obtained by incremental methods.

An extension of Melan's shakedown theorem to a certain class of inelastic material with damage and kinematical hardening is presented. The developed numerical method is illustrated by two examples.

An analytical and numerical study of an extended J-type integral for inhomogeneous materials is presented for three-dimensional mode-I-crack configurations. This integral is developed by using the Eshelby's energy moment tensor and consists of two parts, a volume integral enclosing the crack front and a surface integral embracing any inhomogeneities. For numerical calculations, the 3D integral appears to be an attractive approach for obtaining pointwise values of the J-integral along a crack front. Numerical examples are presented for a center cracked body subjected to quasi-static loading.

In this paper a numerical model is presented which aims to simulate occurrence and evolution of local damage. Different failure criteria are proposed and incorporated into the finite element (FE) code CRACKAN. Particle cracking is suggested by critical normal stresses. Decohesion between hard phase (HP) and metal matrix (MM) is assumed to occur if the stresses normal to the interface reach a critical value. First, simple numerical examples are chosen to get a better understanding, how geometric and physical characteristics of the constituents influence the stress concentration in HP which is, in view of the chosen failure criteria, an indicator for the occurrence of these phenomena. Unit cell models are adopted in order to investigate the interaction behaviour between HP. It can be seen that the different microstructural characteristics are in competition and that the pattern of interaction at last determines the type of local damage. Finally, the damage behaviour in front of a main crack is modelled. During loading, HP will fail due to cleavage and affect redistribution of the microscale fields of stresses and strains.

An analytical expression of the J-integral for inhomogeneous materials is presented in a form suitable for the numerical analysis of arbitrary three-dimensional (3-D) mode-I crack configurations. This integral is given using the principle of virtual work and Eshelby's energy moment tensor. The virtual crack extension method was developed from finite element considerations and was based on the calculations of the released energy when a crack in a finite element model was extended by a small amount Δa. For numerical calculations, the 3-D analogue of the volume integral appears to be an attractive approach for obtaining pointwise values of the stress intensity factors along a crack front. The formulation is easily incorporated into a finite element program, but can also conveniently be used as part of a post-processing program, which uses stress and displacement data from a finite element analysis to calculate the stress intensity factors. Numerical examples are presented, using twenty node, isoparametric, quarter-point elements, for the compact tension specimen, homogeneous and inhomogeneous materials. Stress intensity factors are calculated for various moduli of the inclusion and distances of the crack from the interface. Numerical results are compared with results obtained by two-dimensional models.

A two-dimensional shakedown theory for shells under combined mechanical and thermal loading conditions is presented. Linear elastic-ideal plastic as well as linear elastic-limited linear hardening material behaviour is taken into account. Numerical examples of technical relevance illustrate the proposed method.

Using the tool of Convex Analysis, a method is proposed to determine upper bounds of the safety factor in the shakedown problem. As an example, the case of loads on a segment for spherical yield conditions is considered.

The quasistatical shakedown behaviour of elastic-plastic bodies is investigated under the assumption of small strains but finite displacements. A condition guaranteeing shakedown is obtained. Two methods for the treatment of particular problems are presented and one of them is illustrated by the example of a von Karman-type plate.

A finite element method for the shakedown-assessment of sheets under combined thermal and mechanical loads is presented. The generalized standard material model is adopted allowing for a certain class of kinematical hardening materials; it is realized by a simple two-surface yield condition. Sample calculations illustrate the method.