Modeling and simulation of relaxation in viscoelastic open cell materials and structures

Open cell materials with cubic anisotropy and structures made thereof are investigated with respect to their linear viscoelastic properties, in particular their relaxation behavior. The study is concerned with the prediction of the effective behavior which results from the isotropic bulk material properties as well as the cellular architecture. Finite Element Method simulations of three-dimensional structures are employed to predict the effective response to a wide range of loading modes in the time domain.For predicting the properties of the cellular materials and structures by the Finite Element Method different modeling strategies are employed. The first approach is a periodic unit cell method modeling an infinite medium by means of periodic boundary conditions. This way the entire effective linear viscoelastic constitutive behavior can be computed. However, it is not possible to capture effects as being attributed to traction free faces and load introduction in specimens or structures. A second approach follows to account for these effects by generating finite sample models to represent situations which occur in experimental testing. Finally, an analytical constitutive material law is developed to model linear viscoelasticity for cubic anisotropy in the time domain. It is implemented into the commercial Finite Element software ABAQUS/Standard and the material parameters are gained from the unit cell investigations. This enables the simulation of structures, parts, and components which consist or contain such cellular materials.     

Computational homogenization of cellular materials

In this work we propose to study the behavior of cellular materials using a second-order multi-scale computational homogenization approach. During the macroscopic loading, micro-buckling of thin components, such as cell walls or cell struts, can occur. Even if the behavior of the materials of which the micro-structure is made remains elliptic, the homogenized behavior can lose its ellipticity. In that case, a localization band is formed and propagates at the macro-scale. When the localization occurs, the assumption of local action in the standard approach, for which the stress state on a material point depends only on the strain state at that point, is no-longer suitable, which motivates the use of the second-order multi-scale computational homogenization scheme. At the macro-scale of this scheme, the discontinuous Galerkin method is chosen to solve the Mindlin strain gradient continuum. At the microscopic scale, the classical finite element resolutions of representative volume elements are considered. Since the meshes generated from cellular materials exhibit voids on the boundaries and are not conforming in general, the periodic boundary conditions are reformulated and are enforced by a polynomial interpolation method. With the presence of instability phenomena at both scales, the arc-length path following technique is adopted to solve both macroscopic and microscopic problems.     

On the uniaxial compression behavior of regular shaped cellular metals

The numerical simulation of random cellular metals is still connected to many unsolved problems due to their stochastic structure. Therefore, a periodic model of a cellular metal is developed for fundamental studies of the mechanical behavior and is numerically investigated under uniaxial compression. The influence of differing hardening behaviors and differing boundary conditions on the characteristics of the material is investigated. Recommendations for the numerical simulation are derived. In contrast to common models, experimental samples of the same geometry are easy to manufacture and the results of the experiments show good agreement with the finite element calculations. Based on the proposed concept of a unit cell with periodic boundary conditions, it is possible to derive constitutive equations of cellular materials under complex loading conditions.     

Meshless meso-modeling of masonry in the computational homogenization framework

In the present study a multi-scale computational strategy for the analysis of structures made-up of masonry material is presented. The structural macroscopic behavior is obtained making use of the Computational Homogenization (CH) technique based on the solution of the Boundary Value Problem (BVP) of a detailed Unit Cell (UC) chosen at the mesoscale and representative of the heterogeneous material. The attention is focused on those materials that can be regarded as an assembly of units interfaced by adhesive/cohesive joints. Therefore, the smallest UC is composed by the aggregate and the surrounding joints, the former assumed to behave elastically while the latter show an elastoplastic softening response. The governing equations at the macroscopic level are formulated in the framework of Finite Element Method (FEM) while the Meshless Method (MM) is adopted to solve the BVP at the mesoscopic level. The material tangent stiffness matrix is evaluated at both the mesoscale and macroscale levels for any quadrature point. Macroscopic localization of plastic bands is obtained performing a spectral analysis of the tangent stiffness matrix. Localized plastic bands are embedded into the quadrature points area of the macroscopic finite elements. In order to validate the proposed CH strategy, numerical examples relative to running bond masonry specimens are developed.     

An enhanced affine formulation and the corresponding numerical algorithms for the mean-field homogenization of elasto-viscoplastic composites

This paper deals with the prediction of the overall behavior of a class of two-phase elasto-viscoplastic composites, based on mean-field homogenization. For this, important improvements are made to the recently-proposed affine formulation. The latter theory linearizes the rate-dependent inelastic constitutive equations of each phase’s material and transforms them into fictitious linear thermo-elastic relations in the Laplace–Carson domain. The main contributions of the present work are threefold. Firstly, complete mathematical developments including a full treatment of internal variables are carried out, enabling the modeling of the response under unloading and cyclic histories. Secondly, robust and accurate computational algorithms are proposed. Thirdly, an extensive validation of the predictions against reference unit cell finite element results is conducted for a variety of materials and loadings. A good agreement between predictions and reference results is observed.     

轻质高强点阵材料及其力学性能研究进展

点阵材料是一种新型轻质高强材料, 同时具备形状控制、致动、能量吸收和传热等多种功能. 文章综述了点阵材料的拉伸主导型设计原则、点阵构型和制备工艺. 拉伸主导型点阵材料的比强度和比刚度明显强于一般胞元材料, 在低密度时质量效率更加突出. 根据材料的基本构型特征主要介绍了三维八角点阵以及夹层点阵材料, 比较分析了熔模铸造法和冲压折叠成型工艺的特点. 总结了研究点阵材料力学性能的理论方法和试验研究成果, 研究表明缺陷对点阵材料力学性能的影响明显小于一般胞元材料. 对点阵材料在形状控制与致动、传热和数值计算方面的应用研究成果进行了介绍. 文中归纳了作者近期在炭纤维点阵复合材料方面的工作, 给出了制备炭纤维隐身点阵格栅的探索性工作. 主要包括炭纤维点阵复合材料的三维编织工艺和二维点阵格栅的嵌锁工艺以及隐身点阵格栅反射率试验测试结果.      

Multifunctional design of two-dimensional cellular materials with tailored mesostructure

Two-dimensional cellular materials (prismatic honeycombs) provide a range of properties that make them suitable for multifunctional applications involving heat dissipation and structural performance. In this paper we present two-scale homogenization-based finite element scheme for convective heat transfer and structural characterization of 2-D cellular metals with uniform and graded cell sizes of various topologies as well as with mixed cell-topologies. For convective heat transfer analysis, the cells are modeled implicitly as temperature-dependent sinks modeling the out-of-plane fluid convection through the cells; the sink strength is determined via a micromechanics problem of heat transfer in a cell. For structural analysis, the cellular material is represented as a micropolar continuum with linear elastic constitutive equations obtained via micromechanics solution of a representative unit cell. The analyses are then used in conjunction with an optimization algorithm to design cellular materials with functionally tailored mesostructures. The analysis and design framework enables tailoring cellular materials with graded cell structures of a given topology as well as with cell structures that combine multiple topologies.     

Dynamic overshooting in 2D periodic materials with square voids caused by sudden flaw appearance

The dynamic response caused by the sudden appearance of a flaw in materials in which square voids are distributed in a doubly periodic manner is investigated. A broad range of relative densities corresponding to cellular materials as well as to almost solid ones is considered. In the former case the materials are characterized by either bending or stretch dominated behavior, and different layouts corresponding to both these cases are examined. The flaw models a break of an intervoid ligament in a material which is subjected to a far-field tensile loading. Both cases of a first flaw in undamaged material as well as the case of a new flaw appearance in an already damaged material are addressed.     

Investigation of stress-strain behavior of single walled carbon nanotube/rubber composites by a multi-scale finite element method

The excellent properties of carbon nanotubes have generated technological interests in the development of nanotube/rubber composites. This paper describes a finite element formulation that is appropriate for the numerical prediction of the mechanical behavior of rubber-like materials which are reinforced with single walled carbon nanotubes. The considered composite material consists of continuous aligned single walled carbon nanotubes which are uniformly distributed within the rubber material. It is assumed that the carbon nanotubes are imperfectly bonded with the matrix. Based on the micromechanical theory, the mechanical behavior of the composite may be predicted by utilizing a representative volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure. Therefore, non-linear spring-based line elements are employed to simulate the discrete geometrical structure and behavior of the single-walled carbon nanotube. On the other hand, the matrix is modeled as a continuum medium by utilizing solid elements. In order to describe its behavior an appropriate constitutive material model is adopted. Finally, the interfacial region is simulated via the use of special joint elements of variable stiffness which interconnect the two materials in a discrete manner. Using the proposed multi-scale model, the stress-strain behavior for various values of reinforcement volume fraction and interfacial stiffness is extracted. The influence of the single walled carbon nanotube addition within the rubber is clearly illustrated and discussed.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

Modeling of cementitious materials exposed to isothermal calcium leaching,considering process kinetics and advective water flow. Part 2: Numerical solution

The second part of the paper presents numerical solutions of the mathematical model of hydro-chemo-mechanical behavior of cementitious materials exposed to contact with deionized water of part 1. The model defines kinetics of the calcium leaching process instead of a direct application of a curve describing equilibrium between solid calcium in the material skeleton and the calcium dissolved in the pore solution. It further takes into account the advective flux of calcium ions. Both aspects are new as compared to previous models. The weak form of the governing equations of the model is derived first using the Galerkin method. Then, the equations are discretized in space with finite elements and in time domain with finite differences, and finally the procedures used for numerical solution of their discretized form are presented. Three numerical examples are solved to test the numerical solution procedure proposed and demonstrate its robustness for solution of 1D and 2D problems concerning fast and slow leaching of cement-based materials. The effect of various factors on the results concerning chemical degradation of structures made of cementitious materials is analyzed as well.     

Contactless optical extensometer for textile materials

In this paper we present a contactless extensometer. For some flexible materials, with great displacements and deformations, contact during measurement is not acceptable. In fact, contact measurement can modify the tensile behavior, as is the case for fibrous materials. Contactless extensometers usually have to print or glue some marks on the sample, which may cause problems during measurement. These extensometers typically use digital image processingto obtain deformation data. The principle used in this study uses the natural periodicity or surface patterns inherent in most textile materials without any image processing. During deformation the distance between two periods or pattern elements changes, allowing this method to measure the real-time modification of this in-plane distance. The extensometer consists of two parts: an optical device and a signal processing unit performing a Fourier analysis. Some results obtained during a tensile test on woven fabrics and non-wovens are presented here.     

锚固正交各向异性岩体的本构关系和破坏准则

从理论上对由系统锚杆加固的正交各向异性岩体,取出包含锚杆的表征单元.根据“等效材料”的概念,在原岩体的本构关系和破坏准则中计人锚杆刚度和强度的“贡献”,从而建立了相应锚固岩体的木构关系和破坏准则.然后列举算例,考察了表征单元的应力-应变关系和破坏强度随锚杆安置角度变化的各向异性表现.最后将计算与一个简单试验的结果作了对比,看到二者的吻合程度较好,因而初步地验证了所提力学模型的可靠性.     

Dynamic crushing and energy absorption of regular,irregular and functionally graded cellular structures

The in-plane dynamic crushing of two dimensional honeycombs with both regular hexagonal and irregular arrangements was investigated using detailed finite element models. The energy absorption of honeycombs made of a linear elastic-perfectly plastic material with constant and functionally graded density were estimated up to large crushing strains. Our numerical simulations showed three distinct crushing modes for honeycombs with a constant relative density: quasi-static, transition and dynamic. Moreover, irregular cellular structures showed to have energy absorption similar to their counterpart regular honeycombs of same relative density and mass. To study the dynamic crushing of functionally graded cellular structures, a density gradient in the direction of crushing was introduced in the computational models by a gradual change of the cell wall thickness. Decreasing the relative density in the direction of crushing was shown to enhance the energy absorption of honeycombs at early stages of crushing. The study provides new insight into the behavior of engineered and biological cellular materials, and could be used to develop novel energy absorbent structures.     

Nonlinear nonplanar vibration of a functionally graded box beam

A functionally graded material (FGM) is a type of material designed to change continuously within the solid. It can be designed for specific applications such as thermal barrier coatings, corrosion protection, biomedical materials, space/aerospace industries, automotive applications, compliant mechanisms etc. In these applications, many primary and secondary structural elements can be idealized as beams. So, the aim of the present work is to study the nonlinear nonplanar vibration of a clamped-free slender box beam made of a FGM. More specifically, the cross section consisting of two isotropic materials, connected by a FG layer, is considered. To correctly describe the dynamic characteristics of the system, the nonlinear integro-differential equations used in this work, which consider the flexural–flexural–torsional couplings that occur in the nonplanar motions of the beam, include both geometric and inertial nonlinearities. In addition, the Galerkin method is applied to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge–Kutta method. A detailed parametric analysis using several tools of nonlinear dynamics, unveils the complex dynamics of the FG beam in the main resonance region. The FG beam displays a complex nonlinear dynamic behavior with several coexisting planar and nonplanar solutions, leading to an intricate bifurcation scenario. Special attention is given to the symmetry breaking of beam dynamics and its influence on the bifurcations and instabilities. The results show that even small variations in cross section and material gradation have profound influence on the bifurcation diagrams and the dynamic behavior of the structure.     

Disintegration of axisymmetric body of revolution made of a material of complex chemical composition in a high-enthalpy airflow

We present a numerical solution of the problem of the disintegration in a high enthalpy airflow of an axisymmetric body of revolution made of a plastic of complex chemical composition. The system of partial differential equations for the laminar multicomponent boundary layer is solved by the pivotal method. The use of an experimental kinetic condition for the disintegration in place of the vapor pressure curve, which is usually used for single-component subliming materials, permits the use of a single technique for calculating both coking and noncoking materials of complex composition. Calculated is the composition of the products of combustion of materials consisting of the chemical elements, H, C, N, O. As an example the results are presented of the calculation of the disintegration of a sphere made from a material similar to textolite in composition.The author wishes to thank G. A. Tirskii for guidance in this study and E. A. Gershbeyn for discussions of the solution scheme.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Dual-Configuration Fiber Bragg Grating Sensor Technique to Measure Coefficients of Thermal Expansion and Hygroscopic Swelling

We propose a method based on the dual-configuration fiber Bragg grating (FBG) sensor to measure the coefficients of thermal expansion (CTE) and hygroscopic swelling (CHS) of polymeric materials. The Bragg wavelength shifts are documented in “two” small but different polymer-FBG assemblies while they are subjected to environmental loading conditions (temperature or moisture). The behavior of the infinite polymer/FBG assembly is reconstructed numerically from the data obtained from the two configurations. The coefficients, then, can be determined from the simple governing equation derived for the infinite assembly. The proposed method is implemented for an underfill material. The validity of measurements is corroborated by a commercially available tool, and the repeatability of measurements is verified by an experiment with a different configuration.     

Modeling the rubbing contact in honeycomb seals

Metallic honeycomb labyrinth seals are commonly used as sealing systems in gas turbine engines. Because of their capability to withstand high thermo-mechanical loads and oxidation, polycrystalline nickel-based superalloys, such as Hastelloy X and Haynes 214, are used as sealing material. In addition, these materials must exhibit a tolerance against rubbing between the rotating part and the stationary seal component. The tolerance of the sealing material against rubbing preserves the integrity of the rotating part. In this article, the rubbing behavior at the rotor–stator interface is considered numerically. A simulation model is incorporated into the commercial finite element code ABAQUS/explicit and is utilized to simulate a simplified rubbing process. A user-defined interaction routine between the contact surfaces accounts for the thermal and mechanical interfacial behavior. Furthermore, an elasto-plastic constitutive material law captures the extreme temperature conditions and the damage behavior of the alloys. To validate the model, representative quantities of the rubbing process are determined and compared with experimental data from the literature. The simulation results correctly reproduce the observations made on a test rig with a reference stainless steel material (AISI 304). A parametric study using the nickel-based superalloys reveals a clear dependency of the rubbing behavior on the sliding and incursion velocity. Compared to each other, the two superalloys studied exhibit a different rubbing behavior.     

Elastic moduli of laminated anisotropic composites

The present investigation is concerned with the development of a theoretical basis for determining the elastic moduli of laminated anisotropic materials within the framework of the theory of plates and shells, and the establishment of sound experimental procedures for the confirmation of the predicted results. A general theory is formulated whereby the properties of a laminated anisotropic composite can be predicted once the material properties, the thickness and the orientation of each unit ply are known. Treated in detail are the cross-ply and angle-ply laminates, these configurations being of increasing importance to designers and analysts of filament-wound materials. Laminated materials of this type may, depending on lamination parameters, exhibit coupling between in-plane strain and bending or twisting curvature which must be considered in the analysis and testing of such materials. Based on an understanding of the predicted mechanical behavior, an experimental program is designed, using glass-filament-reinforced resin cross-ply and angle-ply plates and cylindrical pressure vessels as test specimens, which confirms the validity of the theory and presents experimental data heretofore not available.