Inhomogeneous deformation of elastomer gels in equilibrium under saturated and unsaturated conditions

A variational method is employed to obtain governing equations and boundary conditions describing finite strain equilibrium configurations of elastomeric gels. Three situations are considered: a liquid saturated gel, an unsaturated gel, and a gel in equilibrium with a vapor of its own liquid. Surface tractions can lead to equilibrium transitions between these cases. The liquid saturated gel is regarded as immersed in a liquid bath. If this bath becomes depleted, then the gel is unsaturated. The degree of unsaturation - a measure of the amount of liquid that would restore a state of saturation - affects the subsequent mechanical behavior. If the unsaturated system is further allowed to condense or evaporate its liquid component at the gel surface, then a new state of equilibrium is achieved. The transition between the unsaturated case and the case of being in equilibrium with the vapor phase corresponds to the chemical potential variable of the gel changing its value from one that is determined by a volume constraint to the value of the chemical potential in the vapor phase. A finite element method is created on the basis of the variational method and demonstrated in the context of eversion, a deformation that imposes very large finite strains. Liquid migration within the gel is not modeled as our focus is on equilibrium states that occur after all such non-equilibrium processes come to rest.     

A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: Application to thermally responsive gels

An elastomeric gel is a cross-linked polymer network swollen with a solvent, and certain gels can undergo large reversible volume changes as they are cycled about a critical temperature. We have developed a continuum-level theory to describe the coupled mechanical deformation, fluid permeation, and heat transfer of such thermally responsive gels. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model based on a Flory–Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical–mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. We have numerically implemented our theory in a finite element program. We show that our theory is capable of simulating swelling, squeezing of fluid by applied mechanical forces, and thermally responsive swelling/de-swelling of such materials.     

Stress relaxation and creep of polymer gels in solvent under uniaxial and biaxial deformations

Stress relaxation and creep of polymer gels in solvent under various deformation modes such as uniaxial, strip-biaxial, and equibiaxial were theoretically investigated. The magnitudes of relaxed stress and the creep at equilibrium under each deformation mode were derived by a thermodynamic consideration of gel system. Combining a constitutive equation of gel with the equation of motion of polymer network, the stress and strain under each deformation mode have been formulated. The theory proposed here was applied to the rectangular gels under various deformations to calculate the stress relaxation and creep behavior of polymer networks in solvent. Two methods different in treatment of swelling under the constant strain or stress were employed for the calculation: one is based on the assumption that the swelling proceeds isotropically, and the other considers the anisotropic swelling process. The results obtained by the two methods mainly differ in the diffusion mode determining the swelling behavior. The possibility of undershoot of relative strain in load-free direction in the creep is also expected.     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

Fluid-pressure eigenstates and bifurcation in tension specimens under lateral pressure

An analysis is presented of an eigenstate that may be significant in deformation processes where part of the surface of a body is subjected to loading by uniform fluid pressure. The ‘fluid-pressure eigenstate’ is a configuration in which quasi-static incremental deformation is possible under surface traction-rates that are related to the instantaneous velocity field in a certain way, the fluid pressure being momentarily stationary. Deformation processes exist such that, given certain rate boundary-conditions, uniqueness of the incremental deformation is guaranteed at every instant up to a fluid-pressure eigenstate. For a cylindrical specimen, of arbitrary cross-section, of elastic/plastic or incompressible, finite elastic material it is shown that the first fluid-pressure eigenstate to be reached on a path of uniform stretching corresponds to the instant at which the ‘effective load’ reaches a maximum. No fluid-pressure eigenstates are reached in isotropic Cauchy-elastic solids under all-round fluid pressure loading provided the physically reasonable conditions that the instantaneous bulk and shear moduli remain positive are satisfied.     

pH敏感凝胶平衡溶胀的有限元研究

1943年Flory提出第一个凝胶溶胀理论以来,凝胶溶胀理论的发展极大地促进了凝胶科学的发展．在诸多凝胶溶胀理论中,Flory-Rehner凝胶溶胀理论是最为经典的理论之一,成功地预测了聚合物胶体的诸多溶胀变形特性．但该模型是基于一种最为简单的自由连接链模型,存在精度较差的局限．因此,本文采用能够表征网络缠结拓扑等微观结构影响的弹性自由能模型,即由Edwards-Vilgis提出的Slip-Link模型,构造自由能模型并形成pH敏感凝胶的平衡溶胀理论,并基于Abaqus有限元分析了微观结构及几何限制作用对pH敏感凝胶力学行为的影响．分析结果可为微流体控制阀的设计提供参考．     

A coupled theory of fluid permeation and large deformations for elastomeric materials

An elastomeric gel is a cross-linked polymer network swollen with a solvent (fluid). A continuum-mechanical theory to describe the various coupled aspects of fluid permeation and large deformations (e.g., swelling and squeezing) of elastomeric gels is formulated. The basic mechanical force balance laws and the balance law for the fluid content are reviewed, and the constitutive theory that we develop is consistent with modern treatments of continuum thermodynamics, and material frame-indifference. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model for the free energy based on a Flory-Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical-mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. As representative examples of application of the theory, we study (a) three-dimensional swelling-equilibrium of an elastomeric gel in an unconstrained, stress-free state; and (b) the following one-dimensional transient problems: (i) free-swelling of a gel; (ii) consolidation of an already swollen gel; and (iii) pressure-difference-driven diffusion of organic solvents across elastomeric membranes.     

Swelling and instability of a gel annulus

We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent.For equilibrium states,the free-energy function of the gel can be converted into a strain energy function,and as a result the gel can be treated as a compressible hyperelastic material.Asymptotic methods are used to study the inhomogeneous swelling in order to obtain the leading-order solution.Some analytical insights are then deduced.Because of the compressive hoop stress in this state,at some stage instability can occur,leading to wrinkles in the gel.An incremental deformation theory in nonlinear elasticity is used to conduct a linear bifurcation analysis for understanding such instability.More specifically,the critical loading for the onset of a wrinkled state is obtained.Detailed discussions on the behaviors of various physical quantities in this critical state are given.It is found that the critical mode number,while insensitive to the material parameters,is greatly influenced by the ratio of the inner and outer radii of the gel.Also,an interesting finding is that the critical swelling ratio is an increasing function of this geometrical parameter,which implies that a thin annulus is more likely to be unstable than a thick one.     

Development of a Novel Experimental Device to Investigate Swelling of Elastomers in Biodiesel Undergoing Multiaxial Large Deformation

The lifetime of an elastomeric product depends on the nature of mechanical loading and the environmental condition during the service. In this context, at least two important aspects contribute to the degradation of the elastomeric parts in service: diffusion of aggressive liquids leading to swelling and fluctuating multiaxial mechanical loading leading to fatigue failure. Moreover, the amount of swelling of elastomers in solvent is affected by the presence of mechanical loading. Hence, it is essential to understand the interactions between the two phenomena for durability analysis of the component. The present study investigates the swelling of elastomers due to diffusion of palm biodiesel in the presence of static multiaxial large deformation. For this purpose, new experimental device and specimen are developed. The device consists of a hollow diabolo elastomeric specimen attached to specially-designed circular metallic grips and plates such that immersion tests can be conducted while the specimens are simultaneously subjected to various mechanical loadings: simple tension, simple torsion and combined tension-torsion. Thus, diffusion of liquids takes place in the material which concurrently undergoes multiaxial large deformation. Two types of elastomers are investigated: Nitrile Rubber (NBR) and Polychloroprene Rubber (CR). The particular features of the device and specimen are discussed and perspectives for further improvement are drawn.     

Revisiting poromechanics in the context of microporous materials

Poromechanics offers a consistent theoretical framework for describing the mechanical response of porous solids, fully or partially saturated with a fluid phase. When dealing with fully saturated microporous materials, which exhibit pores of the nanometre size, aside from the fluid pressure acting on the pore walls additional effects due to adsorption and confinement of the fluid molecules in the smallest pores must be accounted for. From the mechanical point of view, these phenomena result into volumetric deformations of the porous solid: the so-called “swelling” phenomenon. The present work investigates how the poromechanical theory should be refined in order to describe adsorption and confinement induced swelling in microporous solids. Firstly, we report molecular simulation results that show that the pressure and density of the fluid in the smallest pores are responsible for the volumetric deformation of the material. Secondly, poromechanics is revisited in the context of a microporous material with a continuous pore size distribution. Accounting for the thermodynamic equilibrium of the fluid phase in the overall pore space, the new formulation introduces an apparent porosity and an interaction free energy. We use a prototype constitutive relation relating these two quantities to the Gibbs adsorption isotherm, and then calculate the induced deformation of the solid matrix. Agreement with experimental data found in the literature is observed. As an illustrating example, we show the predicted strains in the case of adsorption of methane on activated carbon.     

高应变率大变形下的聚丙烯/尼龙共混高聚物损伤型本构特性

对两种采用不同相容剂的聚丙烯（PP）和尼龙（PA）共混高聚物材料在大变形下的粘弹性力学行为进行研究,着重考察应变率效应和损伤的演化,从而分析不同的界面分子设计对共混体系材料宏观性能的作用。在准静态及冲击实验研究的基础上,基于ZWT非线性粘弹性模型,并结合了遗传算法,分别得到了能有效描述两种共混高聚物大变形阶段计及损伤的非线性粘弹性本构关系。两种材料在不同加载条件下表现出明显不一致的性能,原因在于其损伤演化的率相关性,且两种材料的大变形机制存在一定的差别,能用ZWT方程进行描述的范围也不一样。     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: II. Kink Band Stability and Maximally Dissipative Band Broadening

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can support stationary kinks in plane deformation if the reinforcing is sufficiently great. If the deformation on one side of the kink involves a load axis aligned with the fiber axis, then the more general plane deformation on the other side of the kink is characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. Here, the ellipticity status of the two correlated deformation states is shown to span all four possible ellipticity/nonellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be nonelliptic. Maximally dissipative quasi-static kink motion is examined and interpreted in terms of kink band broadening in on-axis loading. Such maximally dissipative kinks nucleate only in compression as weak kinks, with subsequent motion converting nonelliptic deformation to elliptic deformation. The associated fiber rotation involves three periods: an initial period of slow rotation, a secondary period of rapid rotation, and a final period of essentially constant orienation.     

Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

A network of polymers can imbibe a large quantity of a solvent and swell, resulting in a gel. The swelling process can be markedly influenced by a mechanical load and geometric constraint. When the network, solvent, and mechanical load equilibrate, inside the gel the chemical potential of the solvent is homogeneous, but the concentration of the solvent and the deformation of the network can be inhomogeneous. We use the chemical potential of the solvent and the deformation gradient of the network as the independent variables of the free-energy function, and show that the boundary value problem of the swollen gel is equivalent to that of a hyperelastic solid. We implement this approach in the finite-element package, ABAQUS, and analyze examples of swelling-induced deformation, contact, and bifurcation. Because commercial software like ABAQUS is widely available, this work may provide a powerful tool to study complex phenomena in gels.     

The effect of fiber recruitment on the swelling of a pressurized anisotropic non-linearly elastic tube

We study the effect of fiber recruitment on the mechanical response of a fiber reinforced non-linearly elastic tube that is both swollen and pressurized. Attention is restricted to cylindrically symmetric tube deformation. The constitutive model permits fibers to support tension, but not compression. While many combinations of pressure and swelling cause all of the fibers to be recruited for load support, both large swelling and large deswelling can give rise to fiber derecruitment at certain locations in the tube. This leads to less channel opening than would be the case if the fibers provided support while contracted. The transition between mechanically active and mechanically inactive fibers can be described in terms of the quasi-static motion of a fiber recruiting interface.     

Chemo-Mechanical Consolidation of Clays: Analytical Solutions for a Linearized One-Dimensional Problem

Consolidation (and swelling) of clayey soils caused by change in chemistry of pore fluid is addressed. Such phenomena are caused by changes in the concentration of various species in the solution and result primarily from a stress-independent deformation of individual clusters, and from a mechanical weakening or strengthening of the clay solid matrix in the presence of stress. Second, transport of chemicals that involves concentration gradients induces additional driving forces of osmotic consolidation due to semipermeable membrane nature of clay. In this paper an extension of Terzaghi's model of the mechanical consolidation to incorporate chemical loading of soil is proposed. A linearized model is used to solve analytically two one-dimensional problems of consolidation of a homogeneous layer simulating a landfill liner with drained or undrained boundaries. The numerical results show a strong dependence of distribution of pore pressure on the chemical load and chemically induced settlements of soil to be comparable to the mechanical ones.     

A continuum constitutive model for the mechanical behavior of woven fabrics

We propose a new approach for developing continuum models for the mechanical behavior of woven fabrics in planar deformation. We generate a physically motivated continuum model that can both simulate existing fabrics and predict the behavior of novel fabrics based on the properties of the yarns and the weave. The approach relies on the selection of a geometric model for the fabric weave, coupled with constitutive models for the yarn behaviors. The fabric structural configuration is related to the macroscopic deformation through an energy minimization method, and is used to calculate the internal forces carried by the yarn families. The macroscopic stresses are determined from the internal forces using equilibrium arguments. Using this approach, we develop a model for plain weave ballistic fabrics, such as Kevlar®, based on a pin-joined beam geometry. We implement this model into the finite element code ABAQUS and simulate fabrics under different modes of deformation. We present comparisons between model predictions and experimental findings for quasi-static modes of in-plane loading.     

Experimental investigation of the properties of polyelectrolyte gel

A polyelectrolyte gel, e.g. polyacrylamide gel has interesting thermo-mechanical properties. It swells in solution, increasing its volume by large factors and the swelling may be induced by a change of temperature, by the acetone content of the solution or by its acidity. The swelling can be reversed, i.e. the gel may be shrunk back to its original shape. The preparation of the gel is described here and quantitative details of the swelling or shrinking are reported.Particular emphasis is placed on the effect of a load on the properties of the gel. It is shown that a tensile load significantly affects the volume changes induced by changes of acidity of the solution.Some peculiar properties of the gel emerged during the investigation, such as anisotropy in thin gels and a decrease of the electrical resistance as an immediate consequence of both loading and unloading.     

简支饱和多孔弹性梁的非线性弯曲

基于饱和多孔介质理论和弹性梁的大挠度弯曲假设,在多孔弹性梁轴线不可伸长,孔隙流体仅沿轴向方向扩散的限制下,建立了微观不可压饱和多孔弹性梁大挠度拟静态响应的一维非线性数学模型.在此基础上,利用Galerkin截断法,分析了两端可渗透的简支多孔弹性梁在突加横向均布载荷作用下的非线性弯曲,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶随时间的响应曲线.数值结果表明:当载荷较小时,大挠度非线性与小挠度线性理论的结果相差很小,而当载荷较大时,非线性大挠度理论的结果小于相应线性小挠度理论的结果,并且这种差异随着载荷的增大而增大.同时,在载荷突加于梁上时,多孔弹性梁骨架起初不变形,孔隙流体压力等效力偶由零突增为非零,其值与外载荷保持平衡.随着时间的增加,梁的挠度增加,等效力偶逐渐减小为零,最终多孔梁骨架承担全部的外载荷.     

RESEARCH ON DYNAMIC MECHANICAL RESPONSE AND CONSTITUTIVE MODEL OF PORCINE LIVER

在交通事故中,腹部器官常因冲击载荷作用而受到伤害,严重时甚至危及生命.肝损伤是腹部损伤中最为常见的一种,致死率很高,了解肝脏的动态力学性能对于事故中肝脏的损伤评估及防护设计有着重要的意义.从新鲜的猪肝组织中取肝实质部分制作试样,利用英斯特朗材料试验机对其进行两种加载率（0.004 s^-1,0.04 s^-1）和两种加载方向（垂直肝脏表面和平行于肝脏表面）的准静态压缩试验,并压缩至破坏.利用改进的分离式霍普金森压杆（split Hopkinson pressure bar,SHPB）实验装置沿平行于肝脏表面方向进行三种高应变率（1 300 s^-1,2 400 s^-1,4 500 s^-1）的动态压缩试验.结果表明：所有应变率下的猪肝压缩应力应变曲线都呈非线性凹向上特征,初始阶段应力值很低,应变约30%后应力幅值显著增大;准静态压缩时,两种应变率（0.004 s^-1,0.04 s^-1）和两种加载方向下肝脏组织破坏应力和破坏应变等力学性能无显著不同,平均破坏应变为48%,平均破坏应力为0.45 MPa.高应变率下肝脏组织的流动应力明显高于准静态下的流动应力,表现出一定的率敏感性.采用Yeoh型超弹性本构模型描述猪肝组织准静态力学性能,基于黏超弹性模型理论,提出了一个能描述肝脏组织从低应变率到高应变率范围力学性能的率相关本构模型,该模型与实验结果有很好的一致性.