Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.

Displacement Field Variable Modeling Method for Heterogeneous Materials in Wind Power Blade Core Plates

In order to study the mechanical properties of the heterogeneous core plate of the wind turbine blade,a modeling method of the core plate based on displacement field variables is proposed.Firstly,the wind turbine blade core plate was modeled according to the theory of modeling heterogeneous material characteristics.Secondly,the three-point bending finite element model of the wind turbine blade core plate was solved by the display dynamic equation to obtain the deformation pattern and force-deformation relationship of the core plate.Finally,the three-point bending static test was conducted to compare with the finite element analysis.The test results show that:the damage form of the wind turbine blade core plate includes elasticity,yield,and failure stages.The main failure modes are plastic deformation,core material collapse,and panel-core delamination.The failure load measured by the test is 1.59 kN,which is basically consistent with the load-displacement result obtained by the simulation,with a difference of only 1.9%,which verifies the validity and reliability of the model.It provides data references for wind turbine blade structure design.

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.

Multi-Scale Analysis of Fretting Fatigue in Heterogeneous Materials Using Computational Homogenization

This paper deals with modeling of the phenomenon of fretting fatigue in heterogeneous materials using the multi-scale computational homogenization technique and finite element analysis(FEA).The heterogeneous material for the specimens consists of a single hole model(25% void/cell,16% void/cell and 10% void/cell)and a four-hole model(25%void/cell).Using a representative volume element(RVE),we try to produce the equivalent homogenized properties and work on a homogeneous specimen for the study of fretting fatigue.Next,the fretting fatigue contact problem is performed for 3 new cases of models that consist of a homogeneous and a heterogeneous part(single hole cell)in the contact area.The aim is to analyze the normal and shear stresses of these models and compare them with the results of the corresponding heterogeneous models based on the Direct Numerical Simulation(DNS)method.Finally,by comparing the computational time and%deviations,we draw conclusions about the reliability and effectiveness of the proposed method.

A THEORY OF EFFECTIVE THERMAL CONDUCTIVITY FOR MATRIX-INCLUSION-MICROCRACK THREE-PHASE HETEROGENEOUS MATERIALS BASED ON MICROMECHANICS

The effective thermal conductivity of matrix-inclusion-microcrack three-phase heterogeneous materials is investigated with a self-consistent micromechanical method (SCM) and a random microstructure finite element method(RMFEM). In the SCM, microcracks are assumed to be randomly distributed and penny-shaped and inclusions to be spherical, the crack effect is accounted for by introducing a crack density parameter, the effective thermal conductivity is derived which relates the macroscopic behavior to the crack density parameter. In the RMFEM, the highly irregular microstructure of the heterogeneous media is accurately described, the interaction among the matrix-inclusion-microcracks is exactly treated, the inclusion shape effect and crack size effect are considered. A Ni/ZrO2 particulate composite material containing randomly distributed, penny-shaped cracks is examined as an example. The main results obtained are: (1) the effective thermal conductivity is sensitive to the crack density and exhibits essentially a linear relationship with the density parameter: (2) the inclusion shape has a significant effect on the effective thermal conductivity and a polygon-shaped inclusion is more effective in increasing or decreasing the effective thermal conductivity than a sphere-shaped one; and (3) the SCM and RMFEM are compared and the two methods give the same effective property in the case in which the matrix thermal conductivity A, is greater than the inclusion one lambda(2). In the inverse case of lambda(1) < lambda(2), the two methods as the as the inclusion volume fraction and crack density are low and differ as they are high. A reasonable explanation for the agreement and deviation between the two methods in the case of lambda(1) < lambda(2) is made.

Review of heterogeneous material objects modeling in additive manufacturing

This review investigates the recent developments of heterogeneous objects modeling in additive manufacturing(AM),as well as general problems and widespread solutions to the modeling methods of heterogeneous objects.Prevalent heterogeneous object representations are generally categorized based on the different expression or data structure employed therein,and the state-of-the-art of process planning procedures for AM is reviewed via different vigorous solutions for part orientation,slicing methods,and path planning strategies.Finally,some evident problems and possible future directions of investigation are discussed.

A NEW APPROXIMATE SOLUTION TO THE MULTIPLE SCATTERING THEORY FOR THE ELASTIC PROPERTIES OF HETEROGENEOUS MATERIALS

The multiple scattering theory has been a powerful tool in determining the effective properties of heterogeneous materials. In this paper , a simple relationship between the scattering theory and the micromechanics theory based on the Eshelby principle has been suggested. According to the relationship, a new and simple approximate solution to the exact multiple scattering theory has been given in terms of Eshelby' s S-tensor. The solution easily shows those known results for isotropic composites with spherical inclusions and for tracnsversely isotropic composites, and first gives non-setf-consistent (average t-matrix) and symmetric self-consistent (effective medium or coherent potential) approximate results for isotropic composites with spheroidal inclusions.

Irregular Characteristics and Parameter Assessment of Machined Surface of Heterogeneous Materials

Surface roughness is one of the main indexes in qua li ty assessment of machined components. Surface generation by material removal pro cess depends on the machining process mechanism. The material removal mechanisms are different for machining common materials and heterogeneous materials. Machi ned surface profiles of conventional engineering materials are determined by the moving tracks of tool edges on workpiece surface, the roughness mainly depends on the cutting parameters and the geometrical shape of cutting tool. Heterogeneo us materials consist of two or more separate materials, their properties vary fr om one phase to another and change along with measurement direction. When he terogeneous materials are cut, a quantity of machining-conduced imperfections o ccurs in the machined surface, part of the surface profiles do not directly result from the cutting of tool edges but from the imperfections, the surface te xture may confuse or disappears. The imperfections distribute randomly and their shapes are irregular, the spacing of profile peaks and valleys is irregular and un-periodical, therefore, they cannot be distinguished by wavelength. The prof iles of machined surface of heterogeneous materials have dense, narrow and sharp peaks and valleys. The amplitude distribution of profile peak and valley is dis persed and unsymmetrical, and usually the profile has a positive skewness. Ten p oint height of irregularities or root-mean-square deviation of the profile is more appropriate parameter than maximum height or arithmetical mean deviation of the profile for describing the height characteristics of roughness, and statist ical method and random process method are used to describe the irregularity distribution of the profile.

Silicon-based optoelectronic heterogeneous integration for optical interconnection

The performance of optical interconnection has improved dramatically in recent years.Silicon-based optoelectronic heterogeneous integration is the key enabler to achieve high performance optical interconnection,which not only provides the optical gain which is absent from native Si substrates and enables complete photonic functionalities on chip,but also improves the system performance through advanced heterogeneous integrated packaging.This paper reviews recent progress of silicon-based optoelectronic heterogeneous integration in high performance optical interconnection.The research status,development trend and application of ultra-low loss optical waveguides,high-speed detectors,high-speed modulators,lasers and 2D,2.5D,3D and monolithic integration are focused on.

Complex fields in heterogeneous materials under shock:modeling, simulation and analysis

In this mini-review we summarize the progress of modeling, simulation and analysis of shock responses of heterogeneous materials in our group in recent years. The basic methodology is as below. We first decompose the problem into different scales. Construct/Choose a model according to the scale and main mechanisms working at that scale. Perform numerical simulations using the relatively mature schemes. The physical information is transferred between neighboring scales in such a way: The statistical information of results in smaller scale contributes to establishing the constitutive equation in larger one. Except for the microscopic Molecular Dynamics(MD) model, both the mesoscopic and macroscopic models can be further classified into two categories,solidic and fluidic models, respectively. The basic ideas and key techniques of the MD, material point method and discrete Boltzmann method are briefly reviewed. Among various schemes used in analyzing the complex fields and structures, the morphological analysis and the home-built software, GISO, are briefly introduced. New observations are summarized for scales from the larger to the smaller.

Artificial Photosynthesis(AP):From Molecular Catalysts to Heterogeneous Materials

The development of green and renewable energy sources is in high demand due to energy shortage and productivity development.Artificial photosynthesis(AP)is one of the most effective ways to address the energy shortage and the greenhouse effect by converting solar energy into hydrogen and other carbon-based high value-added products through the understanding of the mechanism,structural analysis,and functional simulation of natural photosynthesis.In this review,the development of AP from natural catalysts to artificial catalysts is described,and the processes of oxygen production,hydrogen production,and carbon fixation are sorted out to understand the properties and correlations of the core functional components in natural photosynthesis,to provide a better rational design and optimization for further development of advanced heterogeneous materials.

3-D NUMERICAL SIMULATION OF FRACTURE PROCESSES IN HETEROGENEOUS BRITTLE MATERIALS

By using the lattice model combined with finite element methods andstatistical techniques, a numerical approach is developed to establish mechanical models ofthree-dimensional heterogeneous brittle materials. A special numerical code is introduced, in whicha lattice model and statistical approaches are used to simulate the initial heterogeneity ofmaterial properties. The size of displacement-load step is adap-tively determined so that only fewelements would fail in each load step. When the tensile principal strain in an element exceeds theultimate strain of this element, the element is considered broken and its Young's modulus is set tobe very low. Some important behaviors of heterogeneous brittle materials are indicated using thiscode. Load-displacement curves and figures of three-dimensional fracture patterns are alsonumerically obtained, which are similar to those observed in laboratory tests.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ──DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.

Sub-Homogeneous Peridynamic Model for Fracture and Failure Analysis of Roadway Surrounding Rock

The surrounding rock of roadways exhibits intricate characteristics of discontinuity and heterogeneity.To address these complexities,this study employs non-local Peridynamics(PD)theory and reconstructs the kernel function to represent accurately the spatial decline of long-range force.Additionally,modifications to the traditional bondbased PD model are made.By considering the micro-structure of coal-rock materials within a uniform discrete model,heterogeneity characterized by bond random pre-breaking is introduced.This approach facilitates the proposal of a novel model capable of handling the random distribution characteristics of material heterogeneity,rendering the PD model suitable for analyzing the deformation and failure of heterogeneous layered coal-rock mass structures.The established numerical model and simulation method,termed the sub-homogeneous PD model,not only incorporates the support effect but also captures accurately the random heterogeneous micro-structure of roadway surrounding rock.The simulation results obtained using this model show good agreement with field measurements from the Fucun coal mine,effectively validating the model’s capability in accurately reproducing the deformation and failure mode of surrounding rock under bolt-supported(anchor cable).The proposed subhomogeneous PD model presents a valuable and effective simulation tool for studying the deformation and failure of roadway surrounding rock in coal mines,offering new insights and potential advancements.

A homogenization method for nonlinear inhomogeneous elastic materials

Background Fast simulation techniques are strongly favored in computer graphics,especially for the nonlinear inhomogeneous elastic materials.The homogenization theory is a perfect match to simulate inhomogeneous deformable objects with its coarse discretization,as it reveals how to extract information at a fine scale and to perform efficient computation with much less DOF.The existing homogenization method is not applicable for ubiquitous nonlinear materials with the limited input deformation displacements.Methods In this paper,we have proposed a homogenization method for the efficient simulation of nonlinear inhomogeneous elastic materials.Our approach allows for a faithful approximation of fine,heterogeneous nonlinear materials with very coarse discretization.Modal analysis provides the basis of a linear deformation space and modal derivatives extend the space to a nonlinear regime;based on this,we exploited modal derivatives as the input characteristic deformations for homogenization.We also present a simple elastic material model that is nonlinear and anisotropic to represent the homogenized materials.The nonlinearity of material deformations can be represented properly with this model.The material properties for the coarsened model were solved via a constrained optimization that minimizes the weighted sum of the strain energy deviations for all input deformation modes.An arbitrary number of bases can be used as inputs for homogenization,and greater weights are placed on the more important low-frequency modes.Results Based on the experimental results,this study illustrates that the homogenized material properties obtained from our method approximate the original nonlinear material behavior much better than the existing homogenization method with linear displacements,and saves orders of magnitude of computational time.Conclusions The proposed homogenization method for nonlinear inhomogeneous elastic materials is capable of capturing the nonlinear dynamics of the original dynamical system well.

Effect of insoluble materials on the volumetric behavior of rock salt

This paper focuses on the presence of nodules of insoluble materials within salt specimens,and their effect on the volumetric strain measurements and the dilatancy phenomenon.We analyzed experimental results of over 120 conventional triaxial compression tests,and found that in 20%of the cases,the volumetric strain measurements were atypical.We also noted that the natural variability of the specimens can lead to a non-negligible data scattering in the volumetric strain measurements when different specimens are subjected to the same test.This is expected given the small magnitude of those strains,but it occasionally implies that the corresponding specimens are not representative of the volumetric behavior of the studied rock.In order to understand these results,we numerically investigated salt specimens modeled as halite matrices with inclusions of impurities.Simulations of triaxial compression tests on these structures proved that such heterogeneities can induce dilatancy,and their presence can lead to the appearance of tensile zones which is physically translated into a micro-cracking activity.The modeling approach is validated as the patterns displayed in the numerical results are identical to that in the laboratory.It was then employed to explain the observed irregularities in experimental results.We studied the natural variability effect as well and proposed a methodology to overcome the issue of specimen representativity from both deviatoric and volumetric perspectives.

Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies

4-node, 8-node and 8（4）-node quadrilateral plane isoparametric elements are used for the solution of boundary value problems in linear isotropic Cosserat elasticity. The patch test is applied to validate the finite elements. Engineering problems of stress concentration around a circular hole in plane strain condition and mechanical behaviors of heterogeneous materials with rigid inclusions and pores are computed to test the accuracy and capability of these three types of finite elements.

Statistical Analyses of the Strengths of Particulate Reinforced Metal Matrix Composites(PRMMCs)Subjected to Multiple Tensile and Shear Stresses

For design and application of particulate reinforced metal matrix composites(PRMMCs),it is essential to predict the material strengths and understand how do they relate to constituents and microstructural features.To this end,a computational approach consists of the direct methods,homogenization,and statistical analyses is introduced in our previous studies.Since failure of PRMMC materials are often caused by time-varied combinations of tensile and shear stresses,the established approach is extended in the present work to take into account of these situations.In this paper,ultimate strengths and endurance limits of an exemplary PRMMC material,WC-Co,are predicted under three independently varied tensile and shear stresses.In order to cover the entire load space with least amount of weight factors,a new method for generating optimally distributed weight factors in an n dimensional space is formulated.Employing weight factors determined by this algorithm,direct method calculations were performed on many statistically equivalent representative volume elements(SERVE)samples.Through analyzing statistical characteristics associated with results the study suggests a simplified approach to estimate the material strength under superposed stresses without solving the difficult high dimensional shakedown problem.

Heterostructure engineering in electrode materials for sodium-ion batteries:Recent progress and perspectives

Sodium-ion batteries(SIBs)have stepped into the spotlight as a promising alternative to lithium-ion batteries for large-scale energy storage systems.However,SIB electrode materials,in general,have inferior performance than their lithium counterparts because Nat is larger and heavier than Lit.Heterostructure engineering is a promising strategy to overcome this intrinsic limitation and achieve practical SIBs.We provide a brief review of recent progress in heterostructure engineering of electrode materials and research on how the phase interface influences Nat storage and transport properties.Efficient strategies for the design and fabrication of heterostructures(in situ methods)are discussed,with a focus on the heterostructure formation mechanism.The heterostructure's influence on Nat storage and transport properties arises primarily from local distortions of the structure and chemomechanical coupling at the phase interface,which may accelerate ion/electron diffusion,create additional active sites,and bolster structural stability.Finally,we offer our perspectives on the existing challenges,knowledge gaps,and opportunities for the advancement of heterostructure engineering as a means to develop practical,highperformance sodium-ion batteries.

Numerical Simulation of the Mechanical Properties and Failure of Heterogeneous Elasto-Plastic Materials

A general numerical approach was developed to simulate the mechanical properties and the failure of heterogeneous elasto-plastic materials using statistical distributions of the material properties. An appropriate elastic-plastic constitutive relation is used to describe the material behavior and failure in each element, with a two-parameter Weibull distribution used to produce the initial heterogeneous material property variations. An adaptive incremental load-step is applied so that only one or a few elements （or integration points） change their status （i.e., from elastic to plastic, or from plastic to strain failure） within one load step. A failed element is then assigned a very small modulus to simulate the failure rather than removing it from the model, which keeps the continuity of the geometric mesh. The numerical results show that the model is suitable for simulating the effective mechanical properties and failure of heterogeneous materials with local elasto-plastic constitutive relations.