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.     

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.     

Biaxial Mechanical Evaluation of Planar Biological Materials

A fundamental goal in constitutive modeling is to predict the mechanical behavior of a material under a generalized loading state. To achieve this goal, rigorous experimentation involving all relevant deformations is necessary to obtain both the form and material constants of a strain-energy density function. For both natural biological tissues and tissue-derived soft biomaterials, there exist many physiological, surgical, and medical device applications where rigorous constitutive models are required. Since biological tissues are generally considered incompressible, planar biaxial testing allows for a two-dimensional stress-state that can be used to characterize fully their mechanical properties. Application of biaxial testing to biological tissues initially developed as an extension of the techniques developed for the investigation of rubber elasticity [43, 57]. However, whereas for rubber-like materials the continuum scale is that of large polymer molecules, it is at the fiber-level (∼1 μm) for soft biological tissues. This is underscored by the fact that the fibers that comprise biological tissues exhibit finite nonlinear stress-strain responses and undergo large strains and rotations, which together induce complex mechanical behaviors not easily accounted for in classic constitutive models. Accounting for these behaviors by careful experimental evaluation and formulation of a constitutive model continues to be a challenging area in biomechanics. The focus of this paper is to describe a history of the application of biaxial testing techniques to soft planar tissues, their relation to relevant modern biomechanical constitutive theories, and important future trends. This revised version was published online in July 2006 with corrections to the Cover Date.     

A constitutive theory for shape memory polymers. Part II: A linearized model for small deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modelling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

A constitutive theory for shape memory polymers. Part I: Large deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

Epoxy-based electroactive polymer gels

We describe a new type of electroactive polymer gel. In contrast to cross-linked polyacrylates, this system is based on triepoxides cross-linked with polyfunctional amines. The stoichiometry is selected to form a cross-linked gel with excess amino hydrogens. In acid solutions these become cationic and the ionic interactions cause the gel to swell. We describe the mechanical properties of the gels, their response to changes in pH and to electrical activation. The response time is in rough agreement with what would be expected for a diffusional process through a gel of this size (about 100 μm). Samples of smaller dimensions would be expected to respond more rapidly.     

Bending actuation in polyurethanes with a symmetrically graded distribution of one-way shape memory alloy wires

Structures are being actuated by embedding shape memory alloy (SMA) wires into compliant materials, such as polyurethane. To achieve bending actuation, these wires are placed in opposing wire configurations, where multiple wires are often employed to enhance the amplitude of the bending actuation response. In this investigation, a procedure has been developed for fabricating polyurethanes with a symmetrically graded distribution of SMA wires. The effects of grading the distribution of one-way SMA wires have been characterized using full-field displacement deformation measurements obtained with the digital image correlation (DIC) technique. These measurements have been used in a one-dimensional (1D) model of bending actuation to determine the “equivalent two-way shape memory effect (SME)” of the graded wire distribution. To utilize the 1D actuation model, the constitutive properties of the polyurethane structure predicted by rule-of-mixture formulations were reduced to account for the differences in strain between the SMA wires and the polyurethane matrix. The graded wire distribution was also found to significantly stiffen the polyurethane structure. The level of equivalent two-way SME therefore became limited by the maximum recovery stress of the SMA wires, with a maximum level that was approximately 75% less than previously measured levels in an opposing wire configuration. However, the bending actuation behavior was more symmetric, and the actuated bending deflections were similar to those observed when using more compliant materials. It was also predicted that the symmetrically graded wire distribution would exhibit a better balance between actuation amplitude and uniformity, which combined with the more symmetric actuation behavior makes the graded wire distribution potentially more desirable for achieving higher actuation frequencies with distributed actuation concepts in new applications, such as miniaturized double diaphragm pumping devices.     

形状记忆合金混杂复合材料薄壁梁主动变形模型

提出具有变形主动驱动作用的SMA纤维混杂复合材料单闭室薄壁截面梁的力-位移本构关系模型.基于变分渐近法导出具有SMA主动纤维的复合材料薄壁空心梁的二维截面刚度系数以及截面内力(矩)与位移(转角)关系方程,含SMA纤维层合板材料性能由混合率进行预测.基于Tanaka的SMA应力应变关系以及Lin的线性相变动力模型,导出了SMA诱发的轴力、扭矩与弯矩的数学表达式.由该文建立的具有拉伸-扭转-弯曲静变形耦合的一般公式出发,讨论周向均匀刚度配置以及周向反对称刚度配置特殊情形,并给出了简化的本构方程.在不考虑SMA纤维含量和温度变化的情况下,本文的模型可以退化为普通纤维复合材料单闭室薄壁截面梁的已有结果.通过数值计算揭示了SMA对弯曲-扭转静变形特性的作用规律,分析了SMA纤维含量、驱动温度和复合材料铺层角的影响.     

Separating viscoelasticity and poroelasticity of gels with diferent length and time scales

Viscoelasticity and poroelasticity commonly coexist as time-dependent behaviors in polymer gels.Engineering applications often require knowledge of both behaviors separated;however,few methods exist to decouple viscoelastic and poroelastic properties of gels.We propose a method capable of separating viscoelasticity and poroelasticity of gels in various mechanical tests.The viscoelastic characteristic time and the poroelastic difusivity of a gel define an intrinsic material length scale of the gel.The experimental setup gives a sample length scale,over which the solvent migrates in the gel.By setting the sample length to be much larger or smaller than the material length,the viscoelasticity and poroelasticity of the gel will dominate at diferent time scales in a test.Therefore,the viscoelastic and poroelastic properties of the gel can be probed separately at diferent time scales of the test.We further validate the method by finite-element models and stress-relaxation experiments.     

Multiaxial Mechanical Characterization of Interpenetrating Polymer Network Reinforced Acrylic Elastomer

The acrylic elastomer membrane VHB 4910 is a material widely used for applications as Dielectric Elastomer Actuators DEA. For suitable actuation performance however, it is necessary to pre-strain the very compliant membrane. This reduces the lifetime of DEA due to early failure of the tensioned membrane. Interpenetrating Polymer Network Reinforced Acrylic Elastomers (IPN) are produced by introducing a curable additive into the pre-strained acrylic elastomer membrane. While curing at elevated temperature, the additive forms a second polymeric network that supports part of the pre-strain in the acrylic membrane. This leads to a free standing material that combines the actuation performance of pre-strained VHB 4910 with an excellent long-term reliability. This work presents a detailed mechanical characterization of acrylic IPN membranes. To reduce the experimental effort required to characterize the nonlinear elastic behavior, we developed a unique specimen design that enables the assessment of uni- and biaxial stress states within one experiment. Slight changes in the material composition of IPN-membranes lead to substantial variations in their mechanical properties. The extraction of material behavior in different kinematic states within a single sample thus reduces the uncertainty on the determination of constitutive models. An extensive experimental campaign was carried out involving uniaxial and equibiaxial tension and relaxation. Image based local deformation measurements and iterative finite element calculations were applied to derive constitutive model parameters that describe the mechanical response in a wide range of planar strain and strain rate.     

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.     

Nonlinear multiscale modeling of polymer materials

In this study, a hyperelastic multiscale modeling technique is used to predict elastic properties of polycarbonate and polyimide polymer systems using a set of widely accepted atomistic force fields. The model incorporates molecular simulations and a nonlinear, continuum mechanics-based, constitutive formulation that incorporates the behavior of the polymer materials as predicted from molecular simulations. The predicted properties of the polymers using multiple force fields are compared to experimentally measured values. Both static and dynamic molecular simulations are performed using molecular mechanics energy minimizations and molecular dynamics simulation techniques, respectively. The results of this study indicate that static molecular simulation is a useful tool to predict the bulk-level nonlinear mechanical behavior of polymers for finite deformations. It is found that the AMBER force field yields the most accurate predicted mechanical and physical properties of the modeled polymer systems compared to the other force fields used in this study.     

Degeneration character of gels under cyclic loading and its constitutive relation

An interesting degeneration behavior of gel materials under cyclic load is found in experiment. In order to explain this phenomenon, a constitutive model is proposed to simulate the microstructure of gels. Further the corresponding constitutive relation and its simplified form that contains six parameters is proposed to describe the nonlinearity and degeneration of gels under cyclic loading and unloading. The data of compression experiments for two typical gels are simulated by computer. The theoretical results are consistent with experiments very well.     

A thermodynamic model of physical gels

Physical gels are characterized by dynamic cross-links that are constantly created and broken, changing its state between solid and liquid under influence of environmental factors. This restructuring ability of physical gels makes them an important class of materials with many applications, such as in drug delivery. In this article, we present a thermodynamic model for physical gels that considers both the elastic properties of the network and the transient nature of the cross-links. The cross-links’ reformation is captured through a connectivity tensor M at the microscopic level. The macroscopic quantities, such as the volume fraction of the monomer ?, number of monomers per cross-link s, and the number of cross-links per volume q, are defined by statistic averaging. A mean-field energy functional for the gel is constructed based on these variables. The equilibrium equations and the stress are obtained at the current state. We study the static thermodynamic properties of physical gels predicted by the model. We discuss the problems of un-constrained swelling and stress driven phase transitions of physical gels and describe the conditions under which these phenomena arise as functions of the bond activation energy E_a, polymer/solvent interaction parameter χ, and external stress p.     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

Separating viscoelasticity and poroelasticity of gels with diferent length and time scales简

Viscoelasticity and poroelasticity commonly coexist as time-dependent behaviors in polymer gels. Engineering applications often require knowledge of both behaviors separated; however, few methods exist to decouple viscoelastic and poroelastic properties of gels. We propose a method capable of separating viscoelasticity and poroelasticity of gels in various mechanical tests. The viscoelastic char- acteristic time and the poroelastic diffusivity of a gel define an intrinsic material length scale of the gel. The experimen- tal setup gives a sample length scale, over which the solvent migrates in the gel. By setting the sample length to be much larger or smaller than the material length, the viscoelasticity and poroelasticity of the gel will dominate at different time scales in a test. Therefore, the viscoelastic and poroelastic properties of the gel can be probed separately at different time scales of the test. We further validate the method by finite-element models and stress-relaxation experiments.     

Mechanics of inhomogeneous large deformation of photo-thermal sensitive hydrogels

A polymer network can imbibe copious amounts of solvent and swell, the resulting state is known as a gel. Depending on its constituents, a gel is able to deform under the influence of various external stimuli, such as temperature, pH-value and light. In this work, we investigate the photo-thermal mechanics of deformation of temperature sensitive hydrogels impregnated with light-absorbing nano-particles. The field theory of photo-thermal sensitive gels is developed by incorporating effects of photochemical heating into the thermodynamic theory of neutral and temperature sensitive hydrogels. This is achieved by considering the equilibrium thermodynamics of a swelling gel through a variational approach. The phase transition phenomenon of these gels, and the factors affecting their deformations, are studied. To facilitate the simulation of large inhomogeneous deformations subjected to geometrical constraints, a finite element model is developed using a user-defined subroutine in ABAQUS, and by modeling the gel as a hyperelastic material. This numerical approach is validated through case studies involving gels undergoing phase coexistence and buckling when exposed to irradiation of varying intensities, and as a microvalve in microfluidic application.     

The effect of large deformation and material nonlinearity on gel indentation

A gel, an aggregate of polymers with solvents, has dual attributes of solid and liquid as solvent migrates in and out of the polymer network. Indentation has recently been used to characterize the mechanical properties of gels. This paper evaluates the effects of large deformation and material nonlinearity on gel indentation through theoretical modeling and finite element analysis. It is found that large deformation significantly affects the interpretation of the experimental observations and the classical relation between indentation force and depth has limitations for large deformation. The material nonlinearity does not play a very important role on indentation experiment so that the poroelasticity is a good approximation. Based on these observations, this paper proposes an alternative approach to measure the mechanical properties of gels, namely, uniaxial compression experiment.     

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

聚氯乙烯弹性体静动态力学性能及本构模型

为揭示聚氯乙烯弹性体在静、动态载荷下的力学性能,采用万能材料试验机和改进的分离式霍普金森压杆实验装置获得了材料在应变率为0.001、0.01、0.1、1 510、2 260和3 000 s?1下的应力应变曲线,并以屈服强度为整形器优选参数,对比了紫铜、铜版纸和铅等3种整形器材料的整形效果。使用修正的ZWT非线性黏弹性本构模型描述聚氯乙烯弹性体在静、动态载荷下的力学性能。结果表明:聚氯乙烯弹性体在静态载荷下具有应变率效应和显著的超弹性特性,动态载荷下表现出较明显的应变率效应和较强的抗变形能力,且静动态载荷下的力学行为受应变历史影响较大。3种整形器材料中铜版纸的整形效果最好。修正后的ZWT非线性黏弹性本构模型能够得到统一参数的本构表达式,且各应变率下的拟合结果与实验结果具有较好的一致性。