A computational homogenization approach for the yield design of periodic thin plates. Part I: Construction of the macroscopic strength criterion

The purpose of this paper is to propose numerical methods to determine the macroscopic bending strength criterion of periodically heterogeneous thin plates in the framework of yield design (or limit analysis) theory. The macroscopic strength criterion of the heterogeneous plate is obtained by solving an auxiliary yield design problem formulated on the unit cell, that is the elementary domain reproducing the plate strength properties by periodicity. In the present work, it is assumed that the plate thickness is small compared to the unit cell characteristic length, so that the unit cell can still be considered as a thin plate itself. Yield design static and kinematic approaches for solving the auxiliary problem are, therefore, formulated with a Love–Kirchhoff plate model. Finite elements consistent with this model are proposed to solve both approaches and it is shown that the corresponding optimization problems belong to the class of second-order cone programming (SOCP), for which very efficient solvers are available. Macroscopic strength criteria are computed for different type of heterogeneous plates (reinforced, perforated plates,…) by comparing the results of the static and the kinematic approaches. Information on the unit cell failure modes can also be obtained by representing the optimal failure mechanisms. In a companion paper, the so-obtained homogenized strength criteria will be used to compute ultimate loads of global plate structures.     

Strength homogenization of matrix-inclusion composites using the linear comparison composite approach

A homogenization procedure to estimate the macroscopic strength of nonlinear matrix-inclusion composites with different strength characteristics of the matrix and inclusions, respectively, is presented in this paper. The strength up-scaling is formulated within the framework of the yield design theory and the linear comparison composite (LCC) approach, introduced by Ponte Castaneda (2002) and extended to frictional models by Ortega et al. (2011), which estimates the macroscopic strength of composite materials in terms of an optimally chosen linear thermo–elastic comparison composite with a similar underlying microstructure. In the paper various combinations for the underlying material behavior for the individual phases of the composite are considered: The matrix phase can be a quasi frictional material characterized either by a Drucker–Prager-type (hyperbolic) or an elliptical strength criterion, which predicts a strength limit also in hydrostatic compression, while the inclusion phase either may represent empty pores, pore voids filled with a pore fluid, rigid inclusions, or solid inclusions, whose strength characteristics also maybe described by a Drucker–Prager-type or an elliptical strength criterion. For generating the homogenized strength criterion efficiently in such general cases of matrix-inclusion composites, a novel algorithm is proposed in the paper. The validation of the proposed strength homogenization procedure for selected combinations of strength characteristics of the matrix material and the inclusions is conducted by comparisons with experimental results and alternative existing strength homogenization models.     

Yield design of thin periodic plates by a homogenization technique and an application to masonry wallsCalcul à la rupture des plaques minces périodiques par une technique d'homogénéisation et une application aux murs en maçonnerie

A homogenization method for determining overall yield strength properties of thin periodic plates from their local strength properties is proposed within the framework of the yield design theory. The proposed method is applied to the determination of the in-plane and out of plane strength criterion for masonry described as a regular assemblage of infinitely resistant bricks separated by Coulomb interfaces. To cite this article: K. Sab, C. R. Mecanique 331 (2003).     

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.     

Optimum design of laminated composite foam-filled sandwich plates subjected to strength constraint

Optimum design of laminated composite sandwich plates with both continuous (core thickness) and discrete (layer group fiber angles and thicknesses) design variables subjected to strength constraint is studied via a two-level optimization technique. The strength of a sandwich plate is determined in a failure analysis using the Tsai–Wu failure criterion and the finite element method which is formulated on the basis of the layerwise linear displacement theory. In the first level optimization of the design process, the discrete design variables are temporarily treated as continuous variables and the corresponding minimum weight of the sandwich plate is evaluated subject to the strength constraint using a constrained multi-start global optimization method. In the second level optimization, the optimal solution obtained in the first level optimization is used in the branch and bound method for solving a discrete optimization problem to determine the optimal design parameters and the final weight of the plate. Failure test of laminated composite foam-filled sandwich plates with different lamination arrangements are performed to validate the proposed optimal design method. A number of examples of the design of laminated composite foam-filled sandwich plates are given to demonstrate the feasibility and applications of the proposed method.     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

Experimental and theoretical analysis of the deformation of transversely isotropic plates under creep conditions

This paper describes the results of calculations and experiments on the torsion of plates made of isotropic and transversely isotropic VT-20 and 1163T alloys with low resistance to creep strain in the direction perpendicular to the median surface. The numerical simulation results for plates of different thicknesses related to the class of rigid and flexible plates are compared using the pure bending theory and the finite element method. It is found that the curvature values are smaller in the case of deformation of a plate made of anisotropic material into a sign-variable saddle surface than in the case of a plate of isotropic material. The calculation in the assumption of pure bending provides an upper bound of the curvature difference in the deformation of plates made of transversely isotropic and isotropic materials.     

Upper bound limit analysis of active earth pressure with different fracture surface and nonlinear yield criterion

Conventional calculations of static and seismic active earth pressures of soils on a retaining wall are formulated assuming the soils obeying a linear Mohr–Coulomb yield criterion. However, experimental evidences show that the strength envelopes of almost all geomaterials are nonlinear in nature over a wide range of normal stresses. In this paper, the strength envelope of the backfill behind a retaining wall is considered to follow a nonlinear yield criterion. A simple method is proposed for calculating the static and seismic active earth pressures acting against a retaining wall using a nonlinear yield criterion. This method is based on the upper bound theorem of limit analysis. Both translational and rotational fracture surfaces are employed in the formulation for calculating active earth pressures. Quasi-static representation of earthquake effects using a seismic coefficient concept is adopted for seismic active earth pressure calculations. Instead of using directly the actual nonlinear yield criterion, a linear Mohr–Coulomb yield criterion, which is tangential to the nonlinear yield criterion, is used to formulate the active earth pressure problem as a classical nonlinear programming problem. A nonlinear sequential quadratic programming algorithm is used to search for the maximum solution. In order to assess the validity of the proposed method, values of active earth pressures for different values of seismic coefficients and nonlinear parameters in the yield criterion are calculated and compared with solutions obtained using an extended Rankine’s active earth pressure theory. For the case of static active earth pressure, the upper bound solutions using the present method with a translational fracture surface are equal to the extended Rankine’s theoretical solutions and are slightly smaller than those obtained using the present method with a rotational fracture surface. For the case of seismic active earth pressure, numerical results obtained using the present method with a rotational fracture surface is very close to the extended Rankine’s theoretical solutions. A study is conducted to investigate the effects of the parameters in the nonlinear yield criterion on the active earth pressures.     

确定复合材料宏观屈服准则的细观力学方法

运用细观力学中的均匀化方法，分析了含周期性微结构复合材料的宏观屈服准则，并对Hill-Tsai准则进行了修正。从基于复合材料细观结构的代表性胞元入手，运用塑性极限理论中的机动分析以及有限元方法，计算了细观结构的极限载荷域。通过宏细观尺度对应关系，得到复合材料的宏观屈服准则。     

STRENGTH PROPERTIES OF JOINTED ROCK MASSES BASED ON THE HOMOGENIZATION METHOD

This paper aims to determine the strength properties of jointed rock masses by means of the homogenization method.To reflect the microstructure of jointed rock masses,a representative element volume (R...     

Dynamic plastic behavior of circular plate usingunified yield criterion

Dynamic plastic behavior of a rigid-plastic simply supported circular plate undermoderate partial uniformly distributed impulsive load is complemented by using a unified yieldcriterion which consists of a generic of convex piecewise linear yield criteria. Upper bound andlower bound plastic responses of the plate under rectangular pulse are obtained; responsebehavior of the plate with respect to the Mises criterion is derived by a proximal manner. Theinconstant circumferential moment distribution and the non-linear flow velocity distribution in theperiods corresponding to the two motion phases are suggested in this paper. Static and kinematicadmissibility of the dynamic plastic solutions is discussed and two types of moment profiles forthe plates under intense dynamic load are supposed for studying in the future.     

A lower bound approach to the yield loci of porous materials

A lower bound approach is proposed for the first time to solve the macroscopic yield loci of porous materials. The results are then compared with Gurson's upper bound yield loci and those of the experiments. It is shown that the present analysis is much more in accordance with the experimental results than the Gurson's.     

Extensions of the Prager-Shield theory of optimal plastic design

The Prager-Shield associated displacement field method for optimal plastic design is extended to multi-component specific cost functions and multiple load conditions, and a lower bound theorem based on kinematic requirements only is introduced. Since any statically admissible stress field results in an upper bound, the proposed theorem provides a simple method for establishing bounds on the optimal cost. By a simple substitution of parameters into the general equations presented, the optimality criteria can be obtained for particular design problems. Examples of optimal fibre-reinforced plates are given.     

基于均匀化理论韧性复合材料塑性极限分析

运用细观力学中的均匀化方法分析了韧性复合材料的塑性极限承载能力.从反映复合材料细观结构的代表性胞元入手,将均匀化理论运用到塑性极限分析中,计算由理想刚塑性、Mises组分材料构成的复合材料的极限承载能力.运用机动极限方法和有限元技术,最终将上述问题归结为求解一组带等式约束的非线性数学规划问题,并采用一种无搜索直接迭代算法求解.为复合材料的强度分析提供了一个有效手段.     

Bounds and estimates for the effective yield surface of porous media with a uniform or a nonuniform distribution of voids

This paper aims at studying the effects of a nonuniform distribution of voids on the macroscopic yield response of porous media with a rigid-perfectly plastic matrix. For this purpose, a semi-analytical model, recently proposed by Bilger et al. [Bilger, N., Auslender, F., Bornert, M., Masson, R., 2002. New bounds and estimates for porous media with rigid perfectly plastic matrix. C. R. Mecanique 330, 127–132], is extended to more general situations where the local porosity can fluctuate. The microstructure is described by a generalized Hashin-type assemblage of hollow spheres and the distribution of the local porosity is obtained from a three-dimensional simulated microstructure. The matrix layer around the voids is discretized into concentric sub-layers so as to take better into account the plasticity gradient along the radial direction. Classical homogenization techniques then provide new self-consistent estimates and upper bounds for the macroscopic yield surface. These results are compared first to the predictions of the Gurson model and its extensions and then to numerical results derived from three-dimensional Fast Fourier Transform (FFT) calculations carried out with the same material porosity distribution. A good agreement is obtained with the three-dimensional FFT calculations and with Gurson–Tvergaard's predictions even for high triaxiality and without fitting any parameter. Nevertheless, when the heterogeneous distribution of voids tends to form clusters, the proposed model fails to capture the properties of the macroscopic yield surface for large triaxiality factors.     

Assessment of existing and introduction of a new and robust efficient definition of the representative volume element

Accurate numerical homogenization necessitates the thorough determination of the Representative Volume Element (RVE). There exists several seminal works on the notion of the RVE in homogenization, its definitions and methods of determination for efficient computation of composite effective properties. The objective of the current work is to assess the ability of numerical RVE determination methods to deliver accurate effective properties of composite materials. This paper demonstrates that common and well-established RVE determination methods, based on studying the convergence rate of the effective properties with respect to the volume element size, are invalid for the case of composites reinforced by randomly oriented fibers and yield erroneous estimates of their effective properties. Following the failure of traditional RVE determination methods, we proposed a new RVE determination criterion that is not based on the average property stability, but its statistical variations. Our new proposed criterion has been shown to be more accurate than other criteria in computing the effective properties of composites for aspect ratios up to 60. Moreover, the proposed criterion does not necessitate a convergence study over the volume element size, hence reducing considerably the RVE determination cost. Finally, our work questions the validity of many published works dealing with composites including heterogeneities of high aspect ratios.     

Perturbation methods for the analysis of the dynamic behavior of damaged plates

This paper presents an analytical method for the analysis of the dynamic behavior of damaged plates. The proposed approach allows the derivation of mode shapes and corresponding curvature modes for plates with various kinds of defects. Damage is modeled as a localized reduction in the plate thickness. Both point and line defects are considered to model notches or line cracks and delaminations in the plate. Small thickness reductions are considered so that the dynamic behavior of the damage plate can be analyzed through perturbations with respect to the undamaged modes. Results are presented to demonstrate the sensitivity of the curvature modes with respect to the considered low damage levels. Also, the curvature modes are used for the estimation of the strain energy of the plate and for the formulation of a damage index which can be used to provide damage location and extent information.     

Application of the gradient strength criterion and the boundary-element method to a plane stress-concentration problem

A numerical algorithm for strength analysis of plane structural elements with stress concentrators is developed using the gradient strength criterion and the boundary-element method. As the first calculation test, the brittle fracture of a plate with a circular hole in tension is evaluated. To verify the algorithm and to compare the results of analysis with experimental data available in the literature, we consider symmetric and asymmetric problems of fracture of glass plates with a narrow elliptic hole in tension and compression. For all the problems, the accuracy of numerical results is estimated by comparison with analytical solutions. In comparison with the classical criteria, the use of the gradient strength criterion leads to better agreement between theoretical estimates and experimental data. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 214–221, July–August, 1999.     

Hardness-packing density scaling relations for cohesive-frictional porous materials

Recent progress in instrumented nanoindentation makes it possible today to test in situ phase properties and structures of porous materials that cannot be recapitulated ex situ in bulk form. But it requires a rigorous indentation analysis to translate indentation data into meaningful mechanical properties. This paper reports the development and implementation of a multi-scale indentation analysis based on limit analysis, for the assessment of strength properties of cohesive-frictional porous materials from hardness measurements. Based on the separation-of-scale condition, we implement an elliptical strength criterion which results from the nonlinear homogenization of the strength properties of the constituents (cohesion and friction), the porosity and the microstructure, into a computational yield design approach to indentation analysis. We identify the resulting upper bound problem as a second-order conical optimization problem, for which advanced optimization algorithms became recently available. The upper bound yield design solutions are benchmarked against solutions from comprehensive elastoplastic contact mechanics finite element solutions and lower bound solutions. Furthermore, from a detailed parameter study based on intensive computational simulations, we identify characteristic hardness-packing density scaling relations for cohesive-frictional porous materials. These scaling relations which are developed for two pore-morphologies, a matrix-pore morphology and a polycrystal (perfect disordered) morphology, are most suitable for the reverse analysis of the strength parameters of cohesive-frictional solids from indentation hardness measurements.     

基于双剪应力屈服准则的受内压管道爆破压力分析

采用双剪应力屈服准则,对在内压作用下的无缺陷管道进行了塑性极限分析,得到了管道爆破压力的计算公式;并且将结果与基于Tresca、Mises、平均剪应力屈服准则得到的爆破压力进行了比较.研究结果表明:爆破压力随着管道材料的应变硬化指数的增大而减小,随着管道厚径比的增大而增大;此外,基于双剪应力屈服准则得到的管道爆破压力为爆破压力的上限,而基于Tresca屈服准则得到的爆破压力为管道爆破压力的下限.