Two-scale and full-scale analyses of elastoplastic honeycomb blocks subjected to flat-punch indentation

In this study, we perform comparative two-scale and full-scale analyses of elastoplastic regular hexagonal honeycomb blocks subjected to in-plane flat-punch indentation. Zigzag and armchair types of cell arrangements are considered for the blocks; relatively thick cell walls are assumed to focus on the effect of cell wall yielding rather than cell wall deflection. For the two-scale analysis, the fully implicit homogenization scheme developed by Asada and Ohno [Asada, T., Ohno, N., 2007. Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis. International Journal of Solids and Structures 44, 7261–7275] is rebuilt by introducing half unit cells to halve the analysis domains in unit cells. For the full-scale analysis, three cell sizes are considered. It is shown that the two-scale analysis provides no apparent dependence on the two types of cell arrangements because of their equivalence under in-plane loading, whereas the full-scale analysis reveals that the local deformation in honeycomb blocks is highly dependent on cell arrangement. Nevertheless, the two-scale analysis is found to be successful in predicting the indentation load-displacement relation and the macroscopic localization direction in honeycomb blocks.     

Global–local analysis of granular media in quasi-static equilibrium

A method of global–local analysis is developed for quasi-static equilibrium problems for granular media. The two-scale modeling based on mathematical homogenization theory enables us to formulate two separate boundary value problems in terms of macro- and microscales. The macroscale problem governs the equilibrium of a global structure composed of granular assemblies, while the microscale one is posed for the particulate nature of a local structure with the friction-contact mechanism between particles. The local structure is identified with a periodic representative volume element, or equivalently, a unit cell, over which averaging is performed. The mechanical behavior of unit cells is analyzed by a discrete numerical model, in which spring and friction devices connect rigid particles, whereas the continuum-based finite element method is used for the macroscopic one. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method for granular materials.     

考虑微结构连接性的双尺度结构自然频率拓扑优化

多孔材料因具有轻量化、高孔隙率和减振/散热等优良多物理特性,在航空航天等领域具有广阔应用前景。采用拓扑优化方法对含多种多孔材料的结构进行结构与材料微结构构型一体化设计,有助于获得具有优良力学性能的结构设计。然而,传统逆均匀化微结构设计方法无法确保不同多孔材料微结构之间的连接性,设计结果不具备可制造性。本文面向含多种多孔材料的双尺度结构基频最大化设计问题,考虑不同微结构之间的连接性,协同设计多孔材料的微结构构型及其在宏观尺度下的布局。采用均匀化方法计算多孔材料的宏观等效力学性能,通过对不同多孔材料微结构单胞的边界区域采用相同的拓扑描述确保双尺度优化过程中任意空间排布下不同微结构的连接性,并通过优化算法确定微结构间的连接形式及微结构拓扑。在宏观尺度,提出结合离散材料插值模型和RAMP插值模型RAMP (Rational Approximation of Material Properties)的多孔材料各向异性宏观等效刚度及质量插值模型,获得清晰的多孔材料宏观尺度布局并减轻优化过程中伪振动模态的影响。建立以双尺度结构基频最大化为目标,以材料用量为约束的优化列式,推导灵敏度表达式,并基于梯度优化算法求解双尺度结构拓扑优化问题。数值算例表明,采用本文优化方法能够有效确保基频最大化双尺度结构设计中不同多孔材料微结构之间的连接性,增强优化设计结果的可制造性。     

Applying the Explicit Time Central Difference Method for Numerical Simulation of the Dynamic Behavior of Elastoplastic Flexible Reinforced Plates

Based on a stepwise algorithm involving central finite differences for the approximation in time, a mathematical model is developed for elastoplastic deformation of cross-reinforced plates with isotropically hardening materials of components of the composition. The model allows obtaining the solution of elastoplastic problems at discrete points in time by an explicit scheme. The initial boundary value problem of the dynamic behavior of flexible plates reinforced in their own plane is formulated in the von Kármán approximation with allowance for their weakened resistance to the transverse shear. With a common approach, the resolving equations corresponding to two variants of the Timoshenko theory are obtained. An explicit “cross” scheme for numerical integration of the posed initial boundary value problem has been constructed. The scheme is consistent with the incremental algorithm used for simulating the elastoplastic behavior of a reinforced medium. Calculations of the dynamic behavior have been performed for elastoplastic cylindrical bending of differently reinforced fiberglass rectangular elongated plates. It is shown that the reinforcement structure significantly affects their elastoplastic dynamic behavior. It has been found that the classical theory of plates is as a rule unacceptable for carrying out the required calculations (except for very thin plates), and the first version of the Timoshenko theory yields reasonable results only in cases of relatively thin constructions reinforced by lowmodulus fibers. Proceeding from the results of the work, it is recommended to use the second variant of the Timoshenko theory (as a more accurate one) for calculations of the elastoplastic behavior of reinforced plates.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Dual finite element methods in homogenization for elastic–plastic fibrous composite material

The problem of homogenization for a periodic, elastic–perfectly plastic, fiber reinforced, composite material is considered. The overall mechanical behavior of the material is described using the anisotropic model of elastic–plastic body with kinematic hardening. The appropriate initial–boundary value problem, set for one repeatable cell of the composite, is solved in order to find the effective constitutive relations. The cell problem is solved using the finite element method formulated in two dual forms: in displacements and in stresses. Stress functions are used in the latter formulation.     

Transient thermoelastoplastic stress analysis for a hollow sphere using the incremental theory of plasticity

An analysis based on the incremental strain theory is formulated for solving the problem of an elastoplastic hollow sphere subjected to a transient temperature distribution. Thermal and material properties are assumed to be temperature dependent and the behaviour of the medium to be characterized by the Ramberg-Osgood stress-strain relation. A method of successive elastic solutions is used to obtain a numerical solution. An illustrative example shows that the effective stress is not a monotonie function of the radius, but is much dependent on the history, gradient, and distribution of the temperature in the hollow sphere. In addition, unloading in the plastically deformed region is confirmed from the detailed discussion on the distribution of strains. As a result, the analysis based on the total strain theory is not permissible for solving this kind of elastoplastic problems subjected to transient thermal loading. In the following analysis the problem is treated in a quasi-static sense and the inertia terms in the thermoelastoplastic equations are neglected.     

A new exact integration method for the Drucker–Prager elastoplastic model with linear isotropic hardening

This paper presents the exact stress solution of the non-associative Drucker–Prager elastoplastic model governed by linear isotropic hardening rule. The stress integration is performed under constant strain-rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory. Based on the time-continuous stress solution, a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex. Explicit expression for the algorithmically consistent tangent tensor is also derived. In addition, a fully analytical strain solution is also derived for the stress-driven case using constant stress-rate assumption. In order to get a deeper understanding of the features of these solutions, two numerical test examples are also presented.     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.     

Alternating minimization for data-driven computational elasticity from experimental data: kernel method for learning constitutive manifold

Data-driven computing in elasticity attempts to directly use experimental data on material, without constructing an empirical model of the constitutive relation, to predict an equilibrium state of a structure subjected to a specified external load. Provided that a data set comprising stress–strain pairs of material is available, a data-driven method using the kernel method and the regularized least-squares was developed to extract a manifold on which the points in the data set approximately lie (Kanno 2021, Jpn. J. Ind. Appl. Math.). From the perspective of physical experiments, stress field cannot be directly measured, while displacement and force fields are measurable. In this study, we extend the previous kernel method to the situation that pairs of displacement and force, instead of pairs of stress and strain, are available as an input data set. A new regularized least-squares problem is formulated in this problem setting, and an alternating minimization algorithm is proposed to solve the problem.     

Mathematical and numerical modeling of the non-associated plasticity of soils—Part 2: Finite element analysis

The method of the implicit standard material has allowed the formulation of a consistent mathematical model of the boundary value problem for the non-associated plasticity of soil. The mean accomplished steps are the achievement of the bipotential function, the recovering of the stress–strain relationship under a normality rule, introduction of the bifunctional and the proof of the solution existence. Here the mathematical model is discretized by the finite element method. First, the stress update scheme was formulated, the tangent matrix is explicitly derived and then the non-linear system is solved by the Newton–Raphson method where a new algorithm using a symmetrical tangent matrix is improved. This is in opposition to conventional non-associated plasticity, which uses a non-symmetric tangent matrix. Through the numerical examples we show the feasibility and the efficiency of the algorithm. It is also seen that we perform some studies of the numerical solutions, particularly the comparison between associated and non-associated limit load.     

A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation

In the context of the linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials, the uniqueness of solution of a natural initial, mixed boundary value problem is proved. The proof depends on an equation of energy balance formulated entirely in terms of temperature and velocity fields.     

气泡在液体中运动过程的数值模拟

本文用数值方法预测气泡在液体中的百定常运动。运用位标函数进行界面的隐含跟踪并且与有限体积法相结合构成一种可行的计算方法。本文方法允许在界面处存在很大的物性差，而且较容易将表面张力引入控制方程。我们对气液两相流中单个气泡的运动进行了计算，得到了与实验结果符合很好的数值结果。     

Homogenization of viscoelastic-matrix fibrous composites with square-lattice reinforcement

The present paper provides details on the application of asymptotic homogenization techniques to the analysis of viscoelastic-matrix fibrous composites with square-lattice reinforcement and their effective properties. The correspondence principle allows transformation of the governing boundary value problems to quasi-static ones. Thereafter, the homogenization procedure is used. To solve the cell problem, modified boundary shape perturbation procedure is proposed. The resulting Laplace transforms are inverted by the effective and accurate Gaver algorithm. The proposed approach, however, yields a computationally intense solution.     

A second-moment incremental formulation for the mean-field homogenization of elasto-plastic composites

In this paper, the incremental formulation for the mean-field homogenization (MFH) of elasto-plastic composites is enriched by including second statistical moments of per-phase strain increment fields, thus combining two advantages. The first one is to handle non-monotonic loading histories and the second is to better account for the heterogeneity of microscopic fields. The proposal is currently restricted to elasto-plasticity with J2 flow theory in each phase, under the small perturbation hypothesis. The formulation crucially exploits the return mapping algorithm for the J2 model, with its two steps: elastic predictor, and plastic corrections. It is shown that the second-moment measure of the average von Mises stress in each phase at the elastic predictor step plays a major role in the computation of both the average stress and the comparison tangent operator. The proposal is implemented for an extended Mori-Tanaka scheme. Predictions are compared to results provided by full-field, finite element computations of representative volume elements or unit cells, for various composite materials, with polymer or metal matrices. There are cases where the predictions of the proposed modeling improve significantly over those of a first-order incremental formulation.     

Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids

In this study, a general framework is developed to analyze microscopic bifurcation and post-bifurcation behavior of elastoplastic, periodic cellular solids. The framework is built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. We thus derive the eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes. The orthogonality allows the macroscopic increments to be independent of the eigenmodes, resulting in a simple procedure of the elastoplastic post-bifurcation analysis based on the notion of comparison solids. By use of this framework, then, bifurcation and post-bifurcation analysis are performed for cell aggregates of an elastoplastic honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it is shown that the eigenmode has the longitudinal component dominant to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It is further shown that under equi-biaxial compression, the flower-like buckling mode having occurred in a macroscopically stable state changes into an asymmetric, long-wave mode due to the sextuple bifurcation in a macroscopically unstable state, leading to the localization of microscopic buckling in deltaic areas.     

Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis

The objective of this contribution is to develop an elastic-plastic-damage constitutive model for crystal grain and to incorporate it with two-scale finite element analyses based on mathematical homogenization method, in order to characterize the macroscopic tensile strength of polycrystalline metals. More specifically, the constitutive model for single crystal is obtained by combining hyperelasticity, a rate-independent single crystal plasticity and a continuum damage model. The evolution equations, stress update algorithm and consistent tangent are derived within the framework of standard elastoplasticity at finite strain. By employing two-scale finite element analysis, the ductile behaviour of polycrystalline metals and corresponding tensile strength are evaluated. The importance of finite element formulation is examined by comparing performance of several finite elements and their convergence behaviour is assessed with mesh refinement. Finally, the grain size effect on yield and tensile strength is analysed in order to illustrate the versatility of the proposed two-scale model.     

Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains

In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.     

Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept

This paper presents boundary element method (BEM) formulations for usual and sensitivity problems in (small strain) elasto-plasticity using the concept of the local consistent tangent operator (CTO). “Usual” problems here refer to analysis of nonlinear problems in structural and solid continua, for which Simo and Taylor first proposed the use of the CTO within the context of the finite element method (FEM). A new implicit BEM scheme for such problems, using the CTO, is presented first. A formulation for sensitivity analysis follows. It is shown that the sensitivity of the strain increment, associated with an infinitesimal variation of some design parameter, solves a linear problem which is governed by the (converged value of the) same global CTO as the one that appears in the usual problem. Numerical results for both usual and sensitivity problems are shown for a one-dimensional example. They demonstrate the effectiveness of the present approach. In particular, accurate sensitivities with respect to material parameters (e.g., exponent of the power-type hardening law) are obtained even with few integration cells and for large load increments.