Adhesion and friction of an elastic half-space in contact with a slightly wavy rigid surface

The paper deals with the estimation of the pressure distribution, the shape of contact and the friction force at the interface of a flat soft elastic solid moving on a rigid half-space with a slightly wavy surface. In this case an unsymmetrical contact is considered and justified with the adhesion hysteresis. For soft solids as rubber and polymers the friction originates mainly from two different contributions: the internal friction due to the viscoelastic properties of the bulk and the adhesive processes at the interface of the two solids. In the paper the authors focus on the latter contribution to friction. It is known, indeed, that for soft solids, as rubber, the adhesion hysteresis is, at least qualitatively, related to friction: the larger the adhesion hysteresis the larger the friction. Several mechanisms may govern the adhesion hysteresis, such as the interdigitation process between the polymer chains, the local small-scale viscoelasticity or the local elastic instabilities. In the paper the authors propose a model to link, from the continuum mechanics point of view, the friction to the adhesion hysteresis. A simple one-length scale roughness model is considered having a sinusoidal profile. For partial contact conditions the detached zone is taken to be a mode I propagating crack. Due to the adhesion hysteresis, the crack is affected by two different values of the strain energy release rate at the advancing and receding edges respectively. As a result, an unsymmetrical contact and a friction force arise. Additionally, the stability of the equilibrium configurations is discussed and the adherence force for jumping out of contact and the critical load for snapping into full contact are estimated.     

An extended strain energy density failure criterion by differentiating volumetric and distortional deformation

We extend Sih’s strain energy density criterion (Sih, 1974) for crack kinks and material failure by weighting differently the volumetric and distortional parts in the extended strain energy density factor. The work is inspired by the factor that failure by microscopic shearing governed by distortion and microscopic separation controlled by hydrostatic tension represent distinct deformation processes, and should be treated differently as we count their influences to material failure. With the weight parameter introduced to the extended strain energy density factor criterion, we explain satisfactorily several critical experiments which reported crack kink in samples subjected to mixed-mode loading. The extended strain energy density idea is also used to derive a generalized pressure-dependent yielding criterion, which supplies a theoretical basis for those novel strength criteria for materials like bulk metallic glasses. Corresponding methods to determine the two material parameters, the critical strain energy density factor and the weight parameter quantifying the relative contribution by distortion over volumetric deformation, are discussed.     

Clustering instability in adhesive contact between elastic solids via diffusive molecular bonds

Motivated by experimental observations that cell-cell and cell-matrix adhesion often involves formation of discrete patches of dense molecular bonds, we consider the plane strain problem of two elastic half-spaces, each covered with a layer of lipid membrane, joined together by mobile molecular bonds that diffuse along the interface under the combined action of a thin layer of glycocalyx repellers and an externally applied tensile stress. We show that, for a range of bond density values with or without the applied stress, the state of a uniform distribution of bonds is intrinsically unstable with respect to perturbations in bond density distribution. This instability is found to be primarily driven by elastic deformation energies in the bulk and the membrane. The change in free energy associated with a cosine perturbation in bond density distribution indicates that there exists a critical wavelength beyond which the perturbation becomes unstable and a fastest growing wavelength that tends to dominate as the instability grows. These length scales have typical values in the order of a micrometer, in agreement with the general characteristic size of bond clusters observed in cell adhesion.     

Interfacial welding of dynamic covalent network polymers

Dynamic covalent network (or covalent adaptable network) polymers can rearrange their macromolecular chain network by bond exchange reactions (BERs) where an active unit replaces a unit in an existing bond to form a new bond. Such macromolecular events, when they occur in large amounts, can attribute to unusual properties that are not seen in conventional covalent network polymers, such as shape reforming and surface welding; the latter further enables the important attributes of material malleability and powder-based reprocessing. In this paper, a multiscale modeling framework is developed to study the surface welding of thermally induced dynamic covalent network polymers. At the macromolecular network level, a lattice model is developed to describe the chain density evolution across the interface and its connection to bulk stress relaxation due to BERs. The chain density evolution rule is then fed into a continuum level interfacial model that takes into account surface roughness and applied pressure to predict the effective elastic modulus and interfacial fracture energy of welded polymers. The model yields particularly accessible results where the moduli and interfacial strength of the welded samples as a function of temperature and pressure can be predicted with four parameters, three of which can be measured directly. The model identifies the dependency of surface welding efficiency on the applied thermal and mechanical fields: the pressure will affect the real contact area under the consideration of surface roughness of dynamic covalent network polymers; the chain density increment on the real contact area of interface is only dependent on the welding time and temperature. The modeling approach shows good agreement with experiments and can be extended to other types of dynamic covalent network polymers using different stimuli for BERs, such as light and moisture etc.     

Predictive nonlinear constitutive relations in polymers through loss history

A model is presented that calculates the highly nonlinear mechanical properties of polymers as a function of temperature, strain and strain rate from their molecular structure. The model is based upon the premise that mechanical properties are a direct consequence of energy stored and energy dissipated during deformation of a material. This premise is transformed into a consistent set of structure–property relations for the equation of state and the engineering constitutive relations in a polymer by quantifying energy storage and loss at the molecular level of interactions between characteristic groups of atoms in a polymer. The constitutive relations are formulated as a set of analytical equations that predict properties directly in terms of a small set of structural parameters that can be calculated directly and independently from the chemical composition and morphology of a polymer.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

高分子材料平面应变拉伸变形局部化

将基于应变软化玻璃状高分子材料微观特征建立的ＢＰＡ８－链分子网络模型引入ＵｐｄａｔｉｎｇＬａｇｒａｎｇｅ有限元方法，建立了适于变形局部化分析的大变形弹塑性有限元驱动应力法．在此基础上，数值模拟了初始各向同性高分子材料平面应变拉伸变形局部化的传播过程．探讨了ＢＰＡ模型对具有加工硬化特性的结晶性高分子材料变形分析的适应性；分析了局部化传播过程中颈缩截面的非均匀应力三轴效应；最后，讨论了网格尺寸以及初始几何不均匀性对颈缩扩散以及应力三轴效应的影响     

重组竹顺纹冲击力学性能研究

重组竹是一种新型竹基复合材料,其力学性能优于落叶松等木材。为评价重组竹在动态加载下的顺纹抗冲击力学性能,以密度1.06 g/cm3、含水率8.52%、龄期3～5年的毛竹基重组竹为研究对象,通过准静态单轴压缩和循环加卸载以及动态加载实验,研究了重组竹加载变形过程、各项力学性能指标以及对应变率的敏感性。结果表明:重组竹顺纹压缩过程可以分为弹性变形和弹塑性变形阶段,破坏类型为延性破坏,其各项强度指标随应变率的提高而提高,动态增长因子与应变率之间呈现线性关系,斜率为0.0024;重组竹压缩过程中的应变比能与应变之间呈线性关系,且随应变率的增长而增大,证明其吸能能力随着应变率的增大而提高。实验结果证明,重组竹顺纹具有良好的抗冲击力学性能和显著的应变率效应。     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

A model for the viscoelastic behavior of polymers at finite strains

Summary A constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property. The model is based on the concept of transient networks, and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains. The breakage and reformation rates for adaptive links are assumed to depend on some strain energy density. The viscoelastic behavior is described by an integral constitutive equation, where the relaxation functions satisfy partial differential equations with coefficients depending on the strain history. Adjustable parameters of the model are found by fitting experimental data for a number of polymers in tension at strains up to 400 per cent. To validate the constitutive relations, we consider loading with different strain rates, determine adjustable parameters at one rate of strains, and compare prediction of the model with observations at another rate of strains. Fair agreement between experimental data and results of numerical simulation is demonstrated when the rates of strains differ by more than a decade. Received 1 July 1997; accepted for publication 7 October 1997     

Strain energy density prediction of fatigue crack growth from hole of aging aircraft structures

Fatigue crack growth rate depends not only on the load amplitude, but also on the morphology of crack path. The strain energy density theory has the ability to analyze crack growth rate. A strain energy density crack growth model is proposed. It can predict the lifetime of fatigue crack growth for mixed mode cracks while an equation for mode I crack is also obtained. The validity of the model is established with two cases: a center-crack panel and cracks emanating from the edge of a hole. The stress intensity factor expression for the former case is analytical while that of the latter is calculated numerically using finite elements. The results are compared with the testing data. Good agreement shows that the proposed model is useful.     

Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers

Viscoelasticity characterizes the most important mechanical behavior of elastomers. Understanding the viscoelasticity, especially finite strain viscoelasticity, of elastomers is the key for continuation of their dedicated use in industrial applications. In this work, we present a mechanistic and physics-based constitutive model to describe and design the finite strain viscoelastic behavior of elastomers. Mathematically, the viscoelasticity of elastomers has been decomposed into hyperelastic and viscous parts, which are attributed to the nonlinear deformation of the cross-linked polymer network and the diffusion of free chains, respectively. The hyperelastic deformation of a cross-linked polymer network is governed by the cross-linking density, the molecular weight of the polymer strands between cross-linkages, and the amount of entanglements between different chains, which we observe through large scale molecular dynamics (MD) simulations. Moreover, a recently developed non-affine network model (Davidson and Goulbourne, 2013) is confirmed in the current work to be able to capture these key physical mechanisms using MD simulation. The energy dissipation during a loading and unloading process of elastomers is governed by the diffusion of free chains, which can be understood through their reptation dynamics. The viscous stress can be formulated using the classical tube model (Doi and Edwards, 1986); however, it cannot be used to capture the energy dissipation during finite deformation. By considering the tube deformation during this process, as observed from the MD simulations, we propose a modified tube model to account for the finite deformation behavior of free chains. Combing the non-affine network model for hyperelasticity and modified tube model for viscosity, both understood by molecular simulations, we develop a mechanism-based constitutive model for finite strain viscoelasticity of elastomers. All the parameters in the proposed constitutive model have physical meanings, which are signatures of polymer chemistry, physics or dynamics. Therefore, parametric materials design concepts can be easily gleaned from the model, which is also demonstrated in this study. The finite strain viscoelasticity obtained from our simulations agrees qualitatively with experimental data on both un-vulcanized and vulcanized rubbers, which captures the effects of cross-linking density, the molecular weight of the polymer chain and the strain rate.     

On hyperelastic stress-strain law of F-actin bundles

Parallel F-actin bundles are a class of organized semiflexible polymers that play a critical role in cell mechanics, including cell adhesion, cell spreading, cell migration, mitosis and intracellular transport. Here we develop an analytical model of hyperelastic behaviors of an F-actin bundle by considering a wormlike chain confined in a harmonic potential. Closed form solutions are obtained for the axial stress—strain relation of an F-actin bundle under stretch.     

预延伸非晶聚合物材料各向异性变形局部化

给出一种可描述预延伸各向异性特性的背应力张量三维表达式，引入大变形弹塑性有限元驱动应力法，结合ＢＰＡ８ 链细观分子网络模型，模拟了预延伸各向异性非晶聚合物材料平面应变拉伸变形局部化力学行为．详细讨论了预延伸比（ＩｎｉｔｉａｌＤｒａｗｉｎｇＲａｔｉｏ；ＩＤＲ）和预延伸方向（ＩｎｉｔｉａｌＤｒａｗｉｎｇＤｉｒｅｃｔｉｏｎ；ＩＤＤ）对变形抗力、颈缩规律、剪切带方向以及试件中心部位链延伸比的影响．     

Constitutive equations in finite elasticity of rubbers

A constitutive model is derived for the elastic behavior of rubbers at arbitrary three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk material. With reference to the concept of constrained junctions, the chain ends are assumed to be located at some distances from appropriate junctions. These distances are not fixed, but are altered under deformation. An explicit expression is developed for the distribution function of vectors between junctions (an analog of the end-to-end distribution function for a flexible chain with fixed ends). An analytical formula is obtained for the strain energy density of a polymer network, when the ratio of the mean-square distance between the ends of a chain and appropriate junctions is small compared with the mean-square end-to-end distance of chains. Stress–strain relations are derived by using the laws of thermodynamics. The governing equations involve three adjustable parameters with transparent physical meaning. These parameters are found by fitting experimental data on plain and particle-reinforced elastomers. The model ensures good agreement between the observations at uniaxial tension and the results of numerical simulation, as well as an acceptable prediction of stresses at uniaxial compression, simple shear and pure shear, when its parameters are found by matching observations at uniaxial tensile tests.     

动态压缩下Zr基非晶合金失效释能机理

为研究Zr 基非晶合金动态压缩条件下的失效释能机理,采用力学试验机、霍普金森杆、高速摄影、差示扫描量热分析(differential scanning calorimetry, DSC)、扫描电镜(scanning electron microscope, SEM)等,得到了材料应力应变曲线、高速摄影图像、失效式样微观形貌及DSC 曲线,根据实验数据计算了材料的晶化激活能,并拟合了材料的JH-2(Johnson-Holmquist Ⅱ)模型,对材料动态失效过程进行有限元数值模拟。实验结果表明,压缩条件下材料为脆性断裂,断口处观察到典型的脉状纹样及液滴状结构,材料失效过程伴随着释能现象;数值模拟结果表明,材料裂纹局部的瞬时内能大于材料晶化激活能。动态压缩下材料的失效释能机理即为材料破碎释放储存的弹性势能,并导致材料局部晶化释能,释能强度与应变率成正相关。     

Propagation of compaction waves in metal foams exhibiting strain hardening

One-dimensional models for compaction of cellular materials exhibiting strain hardening are proposed for two different impact scenarios. The models reveal the characteristic features of deformation under the condition of decreasing velocity during the impact event. It was established that an unloading plastic wave of strong discontinuity propagates in the foam and it has a significant dynamic effect on the foam compaction and energy absorption. The proposed models are based on the actual experimentally derived stress strain curves. The compaction mechanism in three aluminium based foam materials, two of them with relatively low density – Alporas and Cymat with 9% and 9.3% relative density, respectively and a higher density Cymat foam with 21% relative density, is analysed. Numerical simulations were carried out to verify the proposed models.The predictions of the proposed models are compared with published analytical models of compaction of cellular materials which assume a predefined densification strain. It is shown that the approximation of a cellular material with significant strain hardening by the Rigid Perfectly-Plastic-Locking (RPPL) model can lead to an overestimation of the energy absorption capacity for the observed stroke due to the non-uniform strains along the compacted zone of the actual material in contrast to the predefined constant densification strain in the RPPL model. The assumption of a constant densification strain leads also to an overestimation of the maximum stress, which occurs under impact.     

On Computational Issues in Large Deformation Analysis of Rubber Bushings*

ABSTRACT

Accurate bushing analysis requires a locking free finite element formulation, an appropriate selection of the strain energy density function, and an adequate use of bulk modulus to assure numerical stability and accuracy. In this paper, the pressure projection finite element method is employed. The method projects displacement-calculated pressure onto a lower order pressure field, based on the Babuska-Brezzi condition, to avoid volumetric locking and pressure oscillation. Mooney-Rivlin and Cubic strain energy density functions are used to study the material effect on the predicted rubber behavior in tension-compression and shear deformation modes, and the need to use a higher order strain energy density function for bushing analysis is identified. The effect of bulk modulus on bonded rubber behavior in bushings with respect to bushing shape factor is studied, and the minimum allowable bulk modulus to impose incompressibility in bushing analysis is characterized. The load-deflection response of annular bushings subjected to axial, torsional, and radial deformations are analyzed and results are compared to linear approximations. An effort is made to demonstrate how a Mooney-Rivlin model cannot capture load-displacement nonlinearities in bushing axial and torsional deformations. Two- and three-dimensional results are compared and the applicability of two-dimensional analysis is discussed.