考虑损伤的修正梯度弹性理论

梯度弹性理论在描述材料微结构起主导作用的力学行为时具有显著优势,将其与损伤理论相结合,可在材料破坏研究中考虑微结构的影响.基于修正梯度弹性理论,将应变张量、应变梯度张量和损伤变量作为Helmholtz自由能函数的状态变量,并在自然状态附近对自由能函数作Taylor展开,进而由热力学基本定律,推导出修正梯度弹性损伤理论本构方程的一般形式.编制有限元程序,模拟土样损伤局部化带的发展演化过程.结果表明,修正梯度弹性损伤理论消除了网格依赖性;损伤局部化带不是与损伤同时发生,而是在损伤发展到一定程度后再逐渐显现出来.     

Homogenisation of random composites via the multiscale finite‐element method

The transition between the chosen microstructure and microvariables and the material properties on the macrolevel is always a sensitive point in the theory of homogenisation. In this talk we will observe the transfer of data between the scales based on the multiscale finite element method where in each Gauss point of the macromesh a micromesh is attached. For a given deformation gradient provided from the macroscale one calculates microfluctuations satisfying periodic boundary conditions and from those the effective first Piola‐Kirchhoff stress tensor for each Gauss point. The latter provides a possibility to calculate the elasticity tensor on the macrolevel. We study a microstructure containing elliptical cracks of random aspect ratio and orientation. The results based on such procedure show the dependence of the macrovariables on the crack ellipticity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal material orientation of nonlinear orthotropic materials

The problems of optimal material orientation are studied in the case of orthotropic elastic materials. It is assumed that the stress-strain relation (material behavior) is nonlinear and can be described by a transcendental relation including a logarithmic function. The orientation of the material is established from the condition that the elastic energy density attains its maximal (or minimal) value.Tartu University, Estonia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 335–346, May–June, 1999.     

Deformation of orthotropic composites with unidirectional ellipsoidal inclusions under matrix microdamages

In the present paper, a model of deformation of stochastic composites under microdamaging is developed for the case of orthotropic composite, when the microdamages are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by three-axial ellipsoidal inclusions with orthotropic symmetry of elastic properties. It is assumed that the loading process leads to accumulation of damages in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a composite with orthotropic components are taken as the basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. Based on the analytical and numerical approach, an algorithm for the determination of nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the matrix. Using the numerical solution, nonlinear stress-strain diagrams for the orthotropic composite in the case of biaxial extension are obtained. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 121–130, January–March, 2008.     

Characterizing digital microstructures by the Minkowski-based quadratic normal tensor

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures.     

Micromechanical creep model for pure copper

A Unit Cell made of copper is simulated and investigated under creep conditions within the framework of micromechanics. Geometrical 3D model of the copper microstructure is represented as a Unit Cell with grains of random crystallographical orientation and geometry. Such simulation enables algorithm of Voronoi tessellation. The stress-strain behavior of grains in the general case is anisotropic due to the ordered crystalline structure. The anisotropic model for a material with a cubic symmetry is implemented in Abaqus and used to assign behavior of grain interior in elastic and creep regions. Material parameters for elastic model are taken from elastic tests of single crystal copper [1]. Power law material parameters for creep model are taken from creep test performed for single crystal copper [2]. The model parameter ξ is validated numerically. Creep results are presented for the case of proportional loading during the primary and secondary creep. Statistical analysis of creep curves received for 55 different realizations of Unit Cell geometry is carried out. As a result confident interval and mathematical expectation of creep data are calculated. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of the stress-strain state of a ring-reinforced cylindrical section of an underground pipe

A method is presented for solving the problem of determining the stress-strain state of closed circular cylindrical shells in an elastic medium. The problem relates to the design of underground pipelines. The work of cylindrical shells is examined from the viewpoint of the theory of thin-walled three-dimensional systems, with allowance being made for the unilateral character of the interaction with the elastic medium. The stress-strain state of a cylindrical section of an underground pipe reinforced in the middle by a ring is investigated. It is shown that different factors influence the stress-strain state of the shell of the pipe.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 66–72, 1987.     

Anisotropy of elasticity of a composite with irregularly oriented anisometric filler particles

The effects of orientation and shape of filler particles on the elastic properties of composites have been analyzed. The elastic constants of a composite with irregularly oriented filler particles were calculated by using the method of orientational averaging of the properties of a representative structural element. The elastic constants of the structural element were found according to a known generalized Eshelby solution for a finite concentration of ellipsoidal inclusions. The diagrams of elasticity anisotropy for a transversely isotropic structural element and an orthotropic composite with irregularly oriented inclusions are presented. A quantitative estimate for the degree of anisotropy of elastic properties of composites is suggested. Data on the influence of shape anisometry of inclusions on the anisotropy coefficient of filled composites are also reported.     

Deformation of composites with arbitrarily oriented orthotropic fibers under matrix microdamages

In the present work, a model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of a composite with orthotropic inclusions, when microdefects are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by triaxial arbitrarily oriented ellipsoidal inclusions with orthotropic symmetry of the elastic properties. It is assumed that the process of loading leads to accumulation of damage in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules in the case of orthotropic components are taken as basic relations. The fracture criterion is specified as the limiting value of the intensity of average shear stresses acting in the intact part of the material. On the basis of the analytic and numerical approach, we propose an algorithm for the determination of nonlinear deformation properties of the investigated material. The nonlinearity of composite deformations is caused by the finiteness of deformations. By using the numerical solution, the nonlinear stress–strain diagrams are predicted and discussed for an orthotropic composite material for various cases of orientation of inclusions in the matrix.     

An adaptive integration scheme for heat conduction in additive manufacturing

Solidification dynamics are important for determining final microstructure in additively manufactured parts. Recently, researchers have adopted semi-analytical approaches for predicting heat conduction effects at length and time scales not accessible to complex multi-physics numerical models. The present work focuses on improving a semi-analytical heat conduction model for additive manufacturing by designing an adaptive integration technique. The proposed scheme considers material properties, process conditions, and the inherent physical behavior of the transient heat conduction around both stationary and moving heat sources. Both the adaptive integration scheme and a technique for calculating only the points within the melt pools are described in detail. The full algorithm is then implemented and compared against a simple Riemann sum integration scheme for a variety of cases that highlight process and material variations relevant to additive manufacturing. The new scheme is shown to have significant improvements in computational efficiency, solution accuracy, and usability.     

Some details of the description of a disordered condensed system using the theory of defect states of orientational order

Using the theory of defect states of orientational order, we describe a disordered condensed system as an elastic medium with linear topological singularities. We show that elastic stress fields produced by linear disclinations are Abelian. In the quasistationary linear approximation, we obtain expressions for linear dislocation and disclination tensor potentials. We show that using the theory of defect states of orientational order, we can describe the α and β relaxations in a supercooled liquid as relaxation processes in the respective disclination and dislocation subsystems.     

Stress-strain state in the matrix of stochastically reinforced composites

Conclusion We proposed a method of studying the concentration of stresses and strains in the matrix of composites with a stochastic structure in a three-dimensional formulation. The method is based on the use of tensor operators assigned at the inclusion-matrix interface and results from the theory of effective moduli of stochastically reinforced composites.Advantages of the proposed approach include its relative simplicity and clarity, as well as the fact that it can be used to analyze the three-dimensional stress and strain concentration in the matrix of composites with components having very different properties. However, for high values of the volume concentration of reinforcement c₁ > 0.6, it is necessary to use the results of exact solutions obtained, for example, within the framework of deterministic models. The correction that is introduced here is connected with the average of the stresses and strains over an inclusion. The character of their distribution over the interface remains the same as before.The numerical results obtained here show the significant effect of the relative dimensions of the inclusions on the effective elastic properties and the stress concentration in the matrix. Comparison of theoretical values of the longitudinal elastic modulus with experimental results can serve as grounds for validating the proposed variant of choosing the tensor L₀ in the determination of the corrected characteristics and stress-strain state of the matrix.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 396–402, May–June, 1989     

Nonlinear constitutive behavior of orthotropic materials

The problem of optimal material orientation is studied in the case of nonlinear elastic materials. Optimal orientations corresponding to extreme (maximal or minimal) energy density are obtained for orthotropic materials. The material behavior (the stress-strain relation) is simulated in a general form, which includes, as particular cases, different versions of the power law stress-strain relations. The optimality conditions are derived for the general cases. Local and global extremums are determined for particular cases.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Tartu University, Estonia. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 445–454, March–April, 2000.     

Elasticity of composites with irregularly oriented shape-anisotropic filler particles

A variant of determining the elastic characteristics of composites containing irregularly oriented shape-anisotropic filler particles of two types (short fibers and thin platelets) is considered. The effective elastic constants of the composites are calculated by using the method of orientational averaging of elastic characteristics of isolated transversely isotropic structural elements reinforced with unidirectionally oriented short fibers or coplanarly arranged thin platelets. The superposition of elastic properties of the irregularly oriented structural elements, with account of their orientational distribution in the composite material, is accepted. The calculation results are compared with experimental data for the effective elastic moduli of polymeric composites reinforced with short glass fibers and of polymeric nanocomposites containing the platelet-type particles of organically modified montmorillonite. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 285–300, May–June, 2006.     

A material model of nonlinear fractional viscoelasticity

Multiscale methods are frequently used in the design process of textile reinforced composites. In addition to the models for the local material structure it is necessary to formulate appropriate material models for the constituents. While experiments have shown that the reinforcing fibers can be assumed as linear elastic, the material behavior of the polymer matrix shows certain nonlinearities. These effects are mainly due to strain rate dependent material behavior. Fractional order models have been found to be appropriate to model this behavior. Based on experimental observations of Polypropylene a one-dimensional nonlinear fractional viscoelastic material model has been formulated. Its parameters can be determined from uniaxial, monotonic tensile tests at different strain rates, relaxation experiments and deformation controlled processes with intermediate holding times at different load levels. The presence of a process dependent function for the viscosity leads to constitutive equations which form nonlinear fractional differential equations. Since no analytical solution can be derived for these equations, a numerical handling has been developed. After all, the stress-strain curves obtained from a numerical analysis are compared to experimental results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of material parameters of heterogeneous viscoelastic curing polymers

Two goals characterize the present contribution: First, the development of a numerical approach for determining the properties of the material microstructure, and second, the shift of the focus of the inverse analysis from investigating a purely elastic material toward the parameter identification related to heterogeneous inelastic materials. As a rule, the constitutive laws in this case involve a greater number of material parameters the determination of which requires different kinds of tests. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of Statistically Similar RVEs for 3D Microstructures

This contribution presents a method to construct three-dimensional Statistically Similar RVEs (SSRVEs) for the simulation of dual phase steel (DP steel). Since the microstructure of DP steel strongly influences the overall material properties, it should be incorporated in numerical calculations. For this purpose the FE² method can be applied and for an efficient computation SSRVEs with a reduced complexity compared to the real microstructure have to be defined, which still represent the mechanical response of the material accurately. The construction method is based on the minimization of a least-square functional considering suitable statistical measures describing the inclusion morphology of a given real microstructure. The mechanical response of the SSRVEs is compared to the response of the real microstructure in virtual mechanical tests. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于变体积约束的阻尼材料微结构拓扑优化研究北大核心CSCD

阻尼复合结构的抑振性能取决于材料布局和阻尼材料特性.该文提出了一种变体积约束的阻尼材料微结构拓扑优化方法,旨在以最小的材料用量获得具有期望性能的阻尼材料微结构.基于均匀化方法,建立阻尼材料三维微结构有限元模型,得到阻尼材料的等效弹性矩阵.逆用Hashin-Shtrikman界限理论,估计对应于期望等效模量的阻尼材料体积分数限,并构建阻尼材料体积约束限的移动准则.将获得阻尼材料微结构期望性能的优化问题转化为体积约束下最大化等效模量的优化问题,建立阻尼材料微结构的拓扑优化模型.利用优化准则法更新设计变量,实现最小材料用量下的阻尼材料微结构最优拓扑设计.通过典型数值算例验证了该方法的可行性和有效性,并讨论了初始微构型、网格依赖性和弹性模量等对阻尼材料微结构的影响.