Modeling macroscopic extended continua with the aid of numerical homogenization schemes

Even in the range of small elastic deformations the behavior of foams is not well described by only two elastic constants. Usually the manufacturers give values of the material parameters depending on the loading conditions. This problem is investigated on a microscopic scale by a simple beam model and on the macroscopic scale by an extended continuum model. It has been found that this approach shows the size effect [J. Mater. Sci. 18 (1983) 2572] that cannot be described within the framework of the standard continuum mechanical setting. The existence of the size effect within this model can be explained by independent rotations which do not scale with the displacement field.While macroscopic material parameters are generally unknown for foams the macroscopic properties are derived from the microscale where the parameters are assumed to be known. After evaluation of the microscopic constitutive equations, which are also considered to be known, the quantities are mappped back to the macroscale by a homogenization procedure. This approach is known from literature as FE² model, see e.g. [V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials, PhD-thesis, Technical University of Eindhoven, 2002], [Int. J. Numer. Meth. Eng., 54 (2002) 1235] or [Arch. Appl. Mech., 72 (2002) 300]. It is shown that the Cosserat continuum and the FE² model are able to describe the same effects qualitatively.     

A nonlinear Timoshenko beam formulation based on the modified couple stress theory

This paper presents a nonlinear size-dependent Timoshenko beam model based on the modified couple stress theory, a non-classical continuum theory capable of capturing the size effects. The nonlinear behavior of the new model is due to the present of induced mid-plane stretching, a prevalent phenomenon in beams with two immovable supports. The Hamilton principle is employed to determine the governing partial differential equations as well as the boundary conditions. A hinged–hinged beam is chosen as an example to delineate the nonlinear size-dependent static and free-vibration behaviors of the derived formulation. The solution for the static bending is obtained numerically. The solution for the free-vibration is presented analytically utilizing the method of multiple scales, one of the perturbation techniques.     

Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM

It is verified that, under certain continuity conditions, the boundary integral equations (BIEs) of both displacement and displacement derivative can be expressed in a variety of regularized forms with at most weak singularities for any points within the domain or on its boundary. A series of algorithms embedded in the regularized formulations are presented to calculate the field variables within the domain or on its boundary. Detailed numerical results are given to check and compare the validities of the proposed algorithms and some practical effective algorithms are discovered. Due to the character of the at most weak singularities in the formulations, the algorithms require no special numerical treatments, but only the general Gauss quadrature to implement. To the end, the continuity requirements for the field variables and the validities of the algorithms are discussed.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

Stress, stress asymmetry and couple stress: from discrete particles to continuous fields

Explicit and closed expressions for the stress and couple-stress fields for discrete (classical) mechanical systems in terms of the constituents’ degrees of freedom and interactions are derived and compared to previous results. This is done by using an exact and general coarse graining formulation, which allows one to predetermine the resolution of the continuum fields. Since the full dynamics of the pertinent fields is considered, the results are not restricted to static states or quasi-static deformations; the latter comprise mere limiting cases, which are discussed as well. The fields automatically satisfy the equations of continuum mechanics. An explicit expression for the antisymmetric part of the stress field is presented; the question whether the latter vanishes, much like its nature when it does not, have been debated in the literature. Physical explanations of some of the obtained results are offered; in particular, an interpretation of the expression for the stress field provides an argument in favor of its uniqueness, yet another topic of debate in the literature. The formulation and results are valid for single realizations, and can of course be used in conjunction with ensemble averaging. Part of the paper is devoted to a biased discussion of the notion of coarse graining in general, in order to set the presented results in a certain perspective. Although the results can be applied to molecular (nanoscale included) and granular systems alike, the presentation and some simplifying assumptions (which can be easily relaxed) target granular systems. The results should be useful for the analysis of experimental and numerical findings as well as the development of constitutive relations.     

Analysis of particle-to-particle elastic interactions in syntactic foams

This work analyzes the elastic interaction between a pair of hollow particles embedded in a dissimilar medium subjected to remote uniaxial tensile loading. The Boussinesq-Papkovich stress function approach is combined with a multi-pole series expansion to derive a semi-analytical solution of the Navier-Cauchy equation and determine stress and displacement fields in the inclusions and the surrounding medium. Stress profiles, stress intensification factors, circumferential stress resultants in the particles, and strain energy storage are presented for a broad range of particle wall thicknesses, sizes, and relative inter-particle distances. Results specialized to glass-vinyl ester systems demonstrate that particle wall thicknesses and size distributions can be used to modulate the effects of particle-to-particle interaction on the overall mechanical behavior of syntactic foams.     

Stress and crack propagation in the surface layer of carburized stable austenitic alloys during cooling

While operating in furnaces for thermal-chemical treatment, the furnace equipment cast from stable austenitic steel is exposed to many unfavourable factors that contribute to the formation of cracks and deformations, ultimately resulting in withdrawal of this equipment from further use. This study discusses the issue of microstresses formation in the carburized, surface layer of cast steel caused, during temperature changes, by different coefficients of thermal expansion of the structural constituents – carbides and austenitic matrix, of this steel and impact of the stresses on the development of cracks running from the surface to the core of the material. To study this problem a model of carbides network at the austenite grain boundary, for the carburized Fe-Ni-Cr alloy, with an oxides layer on its surface has been developed. It was used in the simulation analysis of the influence of carbides network depth and oxides layer thickness on stresses generated on the casts surface and in the subsurface zone during rapid cooling.     

Analysis of stress localization in hydroturbine stator-scroll system

Based on 3D finite-element models, a procedure for analyzing a spatial stress-strain state in the stator-scroll assembly of hydroturbine units has been developed. It allows determination of stresses and their localization zones in structures that contain not only identical but also similar components. The output data visualization makes it possible to identify the places of stress localization in the system and in its components. __________ Translated from Problemy Prochnosti, No. 1, pp. 132–137, January–February, 2007. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2005, Kiev, Ukraine).     

Warping shear stresses in nonuniform torsion by BEM

 In this paper a boundary element method is developed for the nonuniform torsion of simply or multiply connected prismatic bars of arbitrary cross section. The bar is subjected to an arbitrarily distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint–Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a BEM approach. Both the warping and the torsion constants using only boundary discretization together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest. Received: 13 November 2001 / Accepted: 2 October 2002     

Residual and thermoelastoplastic stress distributions in a heat treated solid cylinder

An important consideration in design is the determination of residual stresses developed during heat treatment of steel. By selecting an appropriate heat treatment technique, one can control the level of residual stresses in the components. Obviously beneficial residual stresses in the material increase the life of components. On the other hand, the unexpected failure of components that is later attributed to detrimental residual stresses is not uncommon.

In this paper, residual stress distribution in a quenched long solid cylinder with temperature dependent properties is evaluated. Using three different speeds of cooling, the level of residual stresses for each speed was determined and compared with the others. From the stresses calculated, it was found that the tensile and tangential residual stresses are more sensitive to the speed of cooling than the radial one. For theoretical analysis a quasi-static uncoupled thermoelastoplastic analysis, based on incremental theory of plasticity, is developed and a numerical procedure for successive elastic solution approximation is formulated.     

Monotonic and Cyclic Deformations of Case‐Hardened Steels Including Residual Stress Effects

Abstract: Monotonic and cyclic deformations of case‐hardened steel specimens under axial loading were investigated experimentally and analytically. A finite element (FE) model for the case‐hardened specimens was constructed to study multiaxial stresses due to different plastic flow behaviour between the case and the core, as well as to evaluate residual stress relaxation and redistribution subsequent to cyclic loading. The multiaxial stress is shown to increase the effective stress on the surface, and, therefore, unfavourable to yielding or fatigue crack nucleation. The residual stresses are shown to relax or redistribute, even in the elastic‐behaving region, when any part of a case‐hardened specimen or component undergoes plastic deformation. Multi‐layer models were used to analyse and predict monotonic and cyclic deformation behaviours of the case‐hardened specimen based on the core and case material properties, and the results are compared with the experimental as well as FE model results. The predicted monotonic stress–strain curves were close to the experimental curves, but the predicted cyclic stress–strain curves were higher than the experimental curves.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

Numerical prediction of disorder effects in solid foams using a probabilistic homogenization scheme

The present study is concerned with a numerical prediction of uncertainties in the macroscopic mechanical properties of microheterogeneous materials with uncertain microstructure. As a model material, solid foams are employed. The stochastic information on the uncertainty is gained in multiple numerical homogenization analyses of small-scale testing volume elements. The local relative density, the cell size distribution, the cell geometry and the spatial orientation of the testing volume elements are assumed to form the set of the relevant stochastic variables. Selected microstructural cases are analyzed for their macroscopic material response. Based on the probability distributions for the stochastic variables defining the microstructures of the testing volume elements, the probability distributions for the mesoscopic material properties are obtained. For the numerical homogenization of the testing volume elements, an enhanced finite element technique is employed, where the components of the macroscopic deformation gradient are introduced as generalized degrees of freedom. Assuming periodic boundary conditions, the global degrees of freedom interact with the conventional displacement degrees of freedom of the discretized microstructure via special boundary coupling elements. The mesoscopic stresses are obtained in a rather efficient manner as the generalized reaction forces for the global degrees of freedom.     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

Thermal Stress Analysis in Particulate Composite Components

Abstract: Thermal stresses resulting from flow of hot and cold water were obtained in sinks manufactured with acrylic casting dispersions by both experimental and numerical analysis. The experimental work was performed in two poly(methyl methacrylate) (PMMA) -silica based particulate composites, compositions of which are different by the volume fraction of filler and the particle size. The thermal stresses in sinks were obtained using strain gauges. The alternate flow of cold and hot water using a mixed tap, results in the formation of thermal gradients, leading to the occurrence of thermal stresses . The experimental results were compared with the numerical results obtained with a commercial finite element software. For this analysis, some thermal parameters such as the convective heat transfer coefficient, the density, the thermal conductivity and the specific heat were experimentally obtained. In general, a good agreement between experimental and numerical results was observed. The results of this study, together with the results of fatigue and fracture tests, will be used in a tolerance analysis of defects in the products manufactured with these composites.     

Effect of stress path on pre-peak damage in rock induced by macro-compressive and -tensile stress fields

The importance of the stress path on pre-peak (micro-) damage in rock material is addressed. Cracks, induced by macro-compressive stresses and macro-tensile stress fields are studied systematically on thin slices of crinoidal limestone samples. The effect of the sequence of macro-compressive and tensile stress fields, on the presence of the cracks is quantified. Firstly, samples damaged by compressive stresses only or tensile stresses only are studied. Hereafter, as a first case, a sample damaged firstly by compressive stresses and secondly by tensile stresses is studied. As a second case, samples damaged firstly by a tensile stress field, followed by compressive stresses are studied and compared to the first case. In the discussion, also the recorded cumulative acoustic emission energy and the clustering of acoustic emission events are used. The differences of both cases are highlighted: in the second case, more damage is observed than in the first case.     

Energy release rate and mode-mixity of adhesive joint specimens

Fracture behaviour of adhesive joints under mixed mode loading is analysed by using the beam/adhesive-layer (b/a) model, in which, the adherends are beamlike and the adhesive is constrained to a thin flexible layer between the adherends. The adhesive layer deforms in peel (mode I), in shear (mode II) or in a combination of peel and shear (mixed mode). Macroscopically, the ends of the bonded part of the joints can be considered as crack tips. The energy release rate of a single-layer adhesive joint is then formulated as a function of the crack tip deformation and the mode-mixity is defined by the shear portion of the total energy release rate. The effects of transversal forces and the flexibility of the adhesive layer are included in the b/a-model, which can be applied to joints with short crack length as well as short bonding length. The commonly used end-loaded unsymmetric semi-infinite joints are examined and closed-form solutions are given. In comparison to the singular-field model in the context of linear elastic fracture mechanics, the b/a-model replaces the singularity at the crack tip with a stress concentration zone. It is shown that the b/a-model and the singular-field model yield fundamentally different mode-mixities for unsymmetric systems. The presented closed-form b/a-model solutions facilitates parametric studies of the influence of unbalance in loading, unsymmetry of the adherends, as well as the flexibility of the adhesive layer, on the mode mixity of an adhesive joint.     

Shear deformation effect in second-order analysis of frames subjected to variable axial loading

In this paper a boundary element method is developed for the second-order analysis of frames consisting of beams of arbitrary simply or multiply connected constant cross section, taking into account shear deformation effect. Each beam is subjected to an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the beam deflection, the axial displacement and to a stress function and solved employing a BEM approach. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress function using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

A two scale anisotropic damage model accounting for initial stresses in microcracked materials

In a recent study [15], we proposed a class of isotropic damage models which account for initial stresses. The present paper extends this approach to anisotropic damage due to growth of an arbitrarily penny-shaped microcracks system. The basic principle of the upscaling technique in the presence of initial stress is first recalled. Then, we derive a closed-form expression of the elastic energy potential corresponding to a system of arbitrarily oriented microcracks. It is shown that the coupling between initial stresses and damage is strongly dependent of the microcracks density and orientation. Predictions of the proposed model are illustrated through the investigation of the influence of initial stresses on the material response under non monotonous loading paths. Finally, by considering a particular distribution of microcracks orientation, described by a second order damage tensor, it is shown that the model is a generalization of the macroscopic damage model of Halm and Dragon [9], for which a physically-based interpretation is then proposed.