On stability of elastic domain during isothermal solid-solid phase transformation in a tube configuration

Under isothermal quasi-static stretching the phasetransition of a superelastic NiTi tube involves the formation(during loading) and vanishing(in unloading) of a high strain(martensite) domain.The two events are accompanied by arapid stress drop/rise due to the formation/vanishing of domain fronts.From a thermodynamic point of view,both areinstability phenomena that occur once the system reaches itscritical state.This paper investigates the stability of a shrinking cylindrical domain in a tube configuration during unloading.The energetics and thermodynamic driving force of thecylindrical domain are quantified by using an elastic inclusion model.It is demonstrated that the two domain fronts exhibit strong interaction when they come close to each other,which brings a peak in the total energy and a sign changein the thermodynamic driving force.It is proved that suchdomain front interaction plays an important role in controlling the stability of the domain and in the occurrence of stressjumps during domain vanishing.It is also shown that the process is governed by two nondimensional length scales(thenormalized tube length and normalized wall-thickness) andthat the length scale dependence of the critical domain lengthand stress jump for the domain vanishing can be quantifiedby the elastic inclusion model.     

NON-LOCAL MODELING ON MACROSCOPIC DOMAIN PATTERNS IN PHASE TRANSFORMATION OF NiTi TUBES

Recent experiments revealed many new phenomena of the macroscopic domain patterns in the stress-induced phase transformation of a superelastic polycrystalline NiTi tube during tensile loading. The new phenomena include deformation instability with the formation of a helical domain, domain topology transition from helix to cylinder, domain-front branching and loading-path dependence of domain patterns. In this paper, we model the polycrystal as an elastic continuum with nonconvex strain energy and adopt the non-local strain gradient energy to account for the energy of the diffusive domain front. We simulate the equilibrium domain patterns and their evolution in the tubes under tensile loading by a non-local Finite Element Method （FEM）. It is revealed that the observed loading-path dependence and topology transition of do- main patterns are due to the thermodynamic metastability of the tube system. The computation also shows that the tube-wall thickness has a significant effect on the domain patterns： with fixed material properties and interfacial energy density, a large tube-wall thickness leads to a long and slim helical domain and a severe branching of the cylindrical-domain front.     

Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations

This paper studies the effects of structural and material length scales on the equilibrium domain patterns in thin-walled long tube configurations during stress-induced elastic phase transition under displacement-controlled quasi-static isothermal stretching. A nonconvex and nonlocal continuum model is developed and implemented into a finite element code to simulate the domain formation and evolution during the phase transition. The morphology and evolution of the macro-domains in different tube geometries are investigated by both analytical (energy analysis) and numerical (nonlocal finite element) methods. Energy minimization is used as the principle to explain the experimentally observed macroscopic domain patterns in a NiTi polycrystal tube. It is found that the domain pattern, as the minimizer of the system energy, is governed by the relative values of the material length scale g, tube-wall thickness h and tube radius R through two nondimensional factors: h/R and g/R. Physically, h/R and g/R serve as the weighting factors of bending energy and domain-wall energy over the membrane energy in the minimization of the total energy of the tube system. Theoretical predictions of the effects of these length scales on the domain pattern are quantified and confirmed by the computational parametric study. They all agree qualitatively well with the available experimental observations.     

Scaling relationship on macroscopic helical domains in NiTi tubes

Macroscopic helix-shaped deformation domains were observed in NiTi polycrystalline shape memory alloy tubes during the stress-induced martensitic phase transition of the material under uniaxial quasi-static isothermal stretching. Further experiments showed that the shape and size of the helical domain not only varied with the external applied nominal strain but also depended on the tube geometry. In this paper, we analyze and quantify the free energy of the tube system in the presence of the helical domain by the procedures of the elementary mechanics method. We prove that the shape of the helix is determined by the competition between the domain front energy and the elastic-misfit bending strain energy of the tube system. The former favors a short helical domain and promotes the domain merging into a cylindrical domain, while the latter favors a long slim helical domain. Based on the principle of minimization of free energy, the equilibrium shape of the helical domain is predicted and its dependence on the material properties, the tube geometry and the applied strain is expressed by a power-law scaling relationship. Clear physical understandings of the experimentally observed helical domain patterns are obtained and the results agree well quantitatively with the available experimental data.     

On equilibrium domains in superelastic NiTi tubes — helix versus cylinder

Macroscopic cylinder- and helix-shaped deformation domains were observed in NiTi polycrystalline shape memory alloy tubes during the phase transition under uniaxial quasi-static isothermal stretching. Further experiments showed that the occurrence of cylinder or helix domain and its subsequent isothermal evolution strongly depend on the domain volume, tube wall-thickness and loading history. This paper studies the energetics of the cylindrical and helical domains using an elastic inclusion model. It is demonstrated that the total misfit strain energy of an equilibrium domain in tube essentially depends on two nondimensional length scales: the normalized effective domain length and the normalized wall-thickness. Based on such understanding, we quantify the length scale dependence in the energy of the equilibrium cylindrical and helical domains. The energetic preference of each type of domain is predicted and the critical condition for the helix → cylinder domain transition is established.     

Microstructural modeling of crack nucleation and propagation in high strength martensitic steels

A dislocation-density based multiple-slip crystalline plasticity formulation, a dislocation-density grain boundary (GB) interaction scheme, and an overlapping fracture method were used to investigate crack nucleation and propagation in martensitic steel with retained austenite for both quasi-static and dynamic loading conditions. The formulation accounts for variant morphologies, orientation relationships, and retained austenite that are uniquely inherent to lath martensitic microstructures. The interrelated effects of dislocation-density evolution ahead of crack front and the variant distribution of martensitic blocks on crack nucleation and propagation are investigated. It is shown that dislocation-density generation ahead of crack front can induce dislocation-density accumulations and plastic deformation that can blunt crack propagation. These predictions indicate that variant distribution of martensitic blocks can be optimized to mitigate and potentially inhibit material failure.     

正多边形基多胞薄壁管的吸能特性

多胞薄壁结构具有轻量化、高比吸能的特点,在汽车、轮船、航空航天等领域得到了广泛的应用。已有研究表明结构的耐撞性与结构的拓扑方式及胞元数量密切相关。为了研究结构形状和拓扑优化对其吸能效果的影响,基于正多边形结构,通过内嵌多边形和外接圆管的方式设计了两类新型多胞薄壁结构,并对这两类多胞薄壁结构进行准静态和落锤冲击实验,利用高速相机记录结构的变形模式,并定量分析了结构的吸能特性。实验结果表明:除正三角形二级内嵌四边形所得结构在准静态加载实验后期出现了局部失稳现象外,其余结构在准静态和落锤冲击实验过程中均保持垂直受压,结构变形模式与吸能效果较好。通过比较两类结构的实验结果得出:不论是在准静态加载还是在落锤冲击的情况下,内嵌多边形结构的各项吸能指标都明显优于外接圆管的结构;同等质量的情况下,内嵌四边形结构的吸能效果明显优于内嵌三角形的结构。     

Crushability maps for structural polymeric foams in uniaxial loading under rigid confinement

A family of epoxy-based polymeric foams with various initial porosity levels was subjected to quasi-static uniaxial loading in rigid confinement (uniaxial strain) to investigate their crushability characteristics. Two issues were investigated. The first issue was the uniformity of deformation in a specimen as a function of porosity level by creating a grid of equally spaced thin stripes on the surface and by monitoring their pattern during the experiment. It was found that the higher the porosity of foam, the more non-uniform the deformation in the specimen. However, the localized non-uniform deformation did not affect the global stress-strain response, especially at large deformations. The second issue was the development of a new analysis tool, called “crushability map”. The purpose of the tool is to depict the evolution of porosity, bulk density and energy absorption as functions of applied strain, stress, and porosity. These maps can assist in characterizing the residual crushability or energy absorption capability of foams as a function of residual porosity. The maps can be used as a design tool for selection of suitable foams for a given application in conjunction with various design criteria.     

非贯通裂隙岩体损伤演化率相关性及变形特征

含非贯通裂隙岩体是自然界中岩体的主要赋存形式,其裂隙几何特征对岩体的强度及变形均产生显著影响。应变率对岩体的损伤演化及黏滞效应也具有显著的率相关性。首先,运用模型元件的方法,将非贯通裂隙岩体动态破坏过程视为具复合损伤、静态弹性特性、动态黏滞特性的非均质点组成,对黏弹性响应的Maxwell体进行改进,将细观损伤体与裂隙损伤演化的宏观损伤体根据等效应变假设并联组成宏细观复合损伤体,构建综合考虑岩体宏细观缺陷的动态损伤模型;其次,基于断裂力学及应变能理论,对岩体宏观裂隙动态扩展的能量机制进行分析,综合考虑初始裂隙应变能、裂隙动态损伤演化过程应变能、裂隙闭合应变能,得到裂隙岩体宏观动态损伤变量计算公式;最后,将模型计算结果与实验结果进行比较,模型计算结果与实验结果吻合较好,证明了模型的合理性,同时利用模型讨论了裂隙倾角、应变率、岩石性质对岩体变形特征的影响规律。     

Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials

Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress-strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress-strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.     

Dynamic and Quasi-Static Propagation of Compaction Waves in a Low-Density Epoxy Foam

We conducted dynamic and quasi-static compression experiments with low-density (ρ = 120 kg/m³) epoxy foam specimens. The specimens had a 10.0-mm-square cross-section and a length of 19.3 mm. Dynamic experiments were conducted with a modified split Hopkinson pressure bar (SHPB), and the quasi-static experiments were conducted with a hydraulic load frame device (MTS-810). In both cases, the specimens were loaded from one end at a constant velocity. Equally spaced grid lines were marked on the specimens to monitor the deformation history. Digital images taken at equally spaced time intervals gave the positions of each grid line. These images showed that a constant end-face velocity V produced a compaction wave front that traveled at a constant velocity C in both dynamic and quasi-static experiments. We described these results with a shockwave analysis that used a locking solid material model.     

The role of dislocation walls for nanoindentation to shallow depths

Dislocation events are seen as excursions or pop-in events in the load–displacement curve of nanoindentation experiments. Two nanoindenters have been used to examine the difference between quasi-static and dynamic loading during indentation. Yield excursions were present in the load–displacement curves of both the statically and dynamically loaded single crystal nickel samples. Only one major excursion occurred in each quasi-static indent, nominally loaded at 100 μN/s while staircase yielding was observed under dynamic loading indentation with a 45 Hz oscillation of 2 nm superimposed on a 60 μN/s loading rate. Thermal activation analysis is used to explain the arrest and reinitiation of the yielding with activation volumes being modeled. For nanoindentation experiments differences between quasi-static and dynamic loading are described by the models presented. It is proposed that insight into the plastic deformation mechanisms associated with such plastic instabilities will provide one of the keys to length scale effects necessary to understanding nanostructures.     

Topological evolution in dense granular materials: A complex networks perspective

One of the great challenges in the science of complex materials – materials capable of emergent behavior such as self-organized pattern formation – is deciphering their “inherent” structural design principles as they deform in response to external loads. We have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to: (i) uncover such design principles and (ii) identify suitable metrics that quantify the evolution of structure during deformation. Herein, we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks. Attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level, i.e., a particle and its immediate neighbors, as well as the macroscopic level. We explore network motifs and other topological properties at these multiple length scales, in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation, spatially and with respect to strain. Key processes or rearrangement events that cause loss of connectivity within the material domain, e.g. microbanding and force chain buckling, are investigated. Network statistics of these processes, previously shown to be major sources of energy dissipation and nonaffine deformation, are then tied to corresponding trends observed in the evolving macroscopic network. It is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity.     

含随机填充孔圆形蜂窝结构的面内冲击性能

研究多孔材料细观结构与宏观力学性能之间的关系, 建立具有固定相对密度的含随机固体填充孔的圆形蜂窝结构模型。在此模型的基础上具体讨论了不同孔洞填充比和冲击速度对圆形蜂窝结构变形模式、动态冲击平台应力以及能量吸收性能的影响。研究结果表明:填充孔在蜂窝变形过程中有局部牵制作用, 蜂窝材料变形模式仍为准静态模式、过渡模式、动态模式; 当变形模式为过渡模式或动态模式时, 结构的平台应力与速度的平方成线性关系, 存在明显的速度效应; 高速冲击下, 含固体填充孔的蜂窝结构单位质量吸收的能量高于规则蜂窝结构。研究结果可为蜂窝材料的研究和设计提供参考。     

Variational formulation and stability analysis of a three dimensional superelastic model for shape memory alloys

This paper presents a variational framework for the three-dimensional macroscopic modelling of superelastic shape memory alloys in an isothermal setting. Phase transformation is accounted through a unique second order tensorial internal variable, acting as the transformation strain. Postulating the total strain energy density as the sum of a free energy and a dissipated energy, the model depends on two material scalar functions of the norm of the transformation strain and a material scalar constant. Appropriate calibration of these material functions allows to render a wide range of constitutive behaviours including stress-softening and stress-hardening. The quasi-static evolution problem of a domain is formulated in terms of two physical principles based on the total energy of the system: a stability criterion, which selects the local minima of the total energy, and an energy balance condition, which ensures the consistency of the evolution of the total energy with respect to the external loadings. The local phase transformation laws in terms of Kuhn–Tucker relations are deduced from the first-order stability condition and the energy balance condition.The response of the model is illustrated with a numerical traction–torsion test performed on a thin-walled cylinder. Evolutions of homogeneous states are given for proportional and non-proportional loadings. Influence of the stress-hardening/softening properties on the evolution of the transformation domain is emphasized. Finally, in view of an identification process, the issue of stability of homogeneous states in a multi-dimensional setting is answered based on the study of second-order derivative of the total energy. Explicit necessary and sufficient conditions of stability are provided.     

What is behind the plastic strain rate?

The plastic strain rate plays a central role in macroscopic models on elasto-viscoplasticity. In order to discuss the concept behind this quantity, we propose, first, a kinetic toy model to describe the dynamics of sliding layers representative of plastic deformation of single crystalline metals. The dynamic variable is given by the distribution function of relative strains between adjacent layers, and the plastic strain rate emerges as the average hopping rate between energy wells. We demonstrate the behavior of this model under different deformations and how it captures the elastic-to-plastic transition. Second, the kinetic toy model is reduced to a closed evolution equation for the average of the relative strain, allowing us to make a direct link to macroscopic theories. It is shown that the constitutive relation for the plastic strain rate does not only depend on the stress, but also on the macroscopic applied deformation rate, contrary to common practice.     

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.     

Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method

The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.