Stress–strain relations for hydrogels under multiaxial deformation

Constitutive equations are derived for the elastic response of swollen elastomers and hydrogels under an arbitrary deformation with finite strains. An expression is developed for the free energy density of a polymer network based on the Flory concept of flexible chains with constrained junctions and solvent-dependent reference configuration. The importance of introduction of a reference configuration evolving under swelling is confirmed by the analysis of experimental data on nanocomposite hydrogels subjected to swelling and drying. Adjustable parameters in the stress–strain relations are found by fitting observations on swollen elastomers, chemical gels (linked by covalent bonds and sliding cross-links), and physical gels under uniaxial stretching, equi-biaxial tension, and pure shear. Good agreement is demonstrated between the observations and results of numerical simulation. A pronounced difference is revealed between the effect of solvent content on elastic moduli of chemical and physical gels.     

Polymer Networks with Slip-links: 2. Constitutive Equations for a Cross-linked Network

Stress–strain relations are derived for the mechanical response of elastomers at arbitrary three-dimensional deformations with finite strains. An elastomer is treated as an incompressible network of chains bridged by permanent (chemical cross-links and physical cross-links whose lifetime exceeds the characteristic time of deformation) and temporary (entanglements modeled as slip-links) junctions. Two types of chains are introduced in the network to distinguish between permanent and temporary nodes. Type-I chains have free ends, and their motion at the micro-level is constrained by a random number of slip-links. Type-II chains are Gaussian chains permanently connected to the network. Concentration of type-I chains is fixed, while the number of type-II chains per unit volume can change under deformation. The governing equations involve two (networks with constant concentrations of type-II chains) or three (networks where the content of type-II chains is affected by mechanical factors) material parameters. These parameters are found by fitting observations on rubbers, thermoplastic–elastomers, and thermoplastic-elastomer composites. Good agreement is demonstrated between the experimental data in uniaxial tensile tests and the results of numerical simulation at elongations up to 1,000%. It is shown that the adjustable parameters are affected by chemical composition and molecular architecture of polymers in a physically plausible way.     

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.     

Virtual rheological experiments on linear alkane chains confined between titanium walls

Molecular dynamics simulations are used to investigate the rheological behavior of confined linear alkane chains under shear flow conditions. The simulated fluids are contained between atomistic, pure titanium walls, coupled to an external bath to maintain the temperature at a constant value. Shear flow conditions are imparted by moving the metal walls in opposite directions, and the corresponding density, velocity, and temperature profiles of the systems are calculated. For the shear rates investigated, all linear alkane chains exhibit a tendency for slip at the walls, evidenced by density profiles divided into two distinct regions and small discontinuities in the velocity and temperature profiles at the wall/fluid interface. Compared to equilibrium, alkane chains show higher mean-square end-to-end distances, characterized by higher components along the shear flow direction. Received: 1 January 2000 Accepted: 7 November 2000     

Volume changes in hydrogels subjected to finite deformations

Constitutive equations are derived for the elastic response of hydrogels under an arbitrary deformation with finite strains. An expression is proposed for the free energy density of a hydrogel based on the Flory concept of a network of flexible chains with constrained junctions whose reference configuration differs from the initial configuration of a fully swollen gel. Adjustable parameters in the stress–strain relations are found by fitting observations on poly(acrylamide) and gellan hydrogels under uniaxial tension and compression. The effect of elongation ratio on osmotic Poisson's ratio is examined numerically.     

Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.      

Dilute solution rheology: experimental results and finitely extensible nonlinear elastic dumbbell theory

Experimental results for intrinsic viscosity and for intrinsic complex viscosity of polymer solutions were compared with the rheological predictions of the finitely extensible, nonlinear elastic (FENE) dumbbell theory of a dilute suspension. The FENE dumbbell adequately models the intrinsic viscosity of flexible polymers, but less successfully portrays the behavior in small amplitude oscillatory motion. Expressions for the high frequency asymptotic limit of the intrinsic complex viscosity of a FENE dumbbell suspension, and the mean-square end-to-end distance of FENE dumbbells in steady shear flow are given.     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

压缩-剪切复合应力下PMMA材料的静态力学特性研究

通过压缩具有一定倾斜角(0°,10°,15°,20°和25°)试件和双剪切模型试件,实现了单轴压缩、压缩-剪切复合应力以及纯剪切三种应力状态,得到PMMA(聚甲基丙烯酸甲酯)在相应应力状态下的应力-应变曲线,同时对不同应力状态下试件的破坏模式进行了分析。结果表明:在不同受力环境中材料的强度和破坏的机理不同;同单轴压缩状态下相比,材料在压缩-剪切复合应力状态下屈服极限、强度极限以及破坏应变均不同程度的增大,呈现明显的"剪切增强"现象。单轴压缩与压缩-剪切应力状态下试件的破坏模式均为在试件短对角面上出现明显的剪切屈服带,由应力分析得出试件剪应力在短对角面上达到最大,引起在此平面上分子链间滑动从而产生应变软化形成剪切屈服带;双剪切试件的破坏模式为与剪切面呈45°的斜面。     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.     

A micro-mechanical model for the response of filled elastomers at finite strains

Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.     

Polymer Networks with Slip-links: 1. Constitutive Equations for an Uncross-linked Network

Constitutive equations are derived for the mechanical response of polymers at three-dimensional deformations with finite strains. A polymer is treated as an incompressible network of flexible chains with free ends whose motion at the micro-level is constrained by a random number of slip-links. The slip-links move affinely with macro-deformation, whereas chains can slide with respect to slip-links. When a free end of a chain slides through a slip-link, the slip-link disappears. Stress–strain relations are developed by using the laws of thermodynamics. They involve only one material constant with a transparent physical meaning.     

The effect of annealing on the viscoplastic response of semicrystalline polymers at finite strains

A constitutive model is developed for the viscoplastic behavior of a semicrystalline polymer at finite strains. A solid polymer is treated as an equivalent heterogeneous network of chains bridged by permanent junctions (physical cross-links, entanglements and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. In the sub-yield region of deformations, junctions between chains in meso-domains slide with respect to their reference positions (which reflects sliding of nodes in the amorphous phase and fine slip of lamellar blocks). Above the yield point, this sliding process is accompanied by displacements of meso-domains in the ensemble with respect to each other (which reflects coarse slip and disintegration of lamellar blocks). To account for the orientation of lamellar blocks in the direction of maximal stresses and formation of micro-fibrils in the post-yield region of deformations (which is observed as strain-hardening of specimens) elastic moduli are assumed to depend on the principal invariants of the right Cauchy–Green tensor for the viscoplastic flow. Stress–strain relations for a semicrystalline polymer are derived by using the laws of thermodynamics. The constitutive equations are determined by six adjustable parameters that are found by matching observations in uniaxial tensile tests on injection-molded isotactic polypropylene at elongations up to 80%. Prior to testing, the specimens were annealed at various temperatures ranging from 110 to 163 °C. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. The effect of annealing temperature on the material parameters is studied in detail.     

The viscoelastic and viscoplastic behavior of low-density polyethylene

Three series of tensile tests with constant cross-head speeds (ranging from 5 to 200 mm/min), tensile relaxation tests (at strains from 0.03 to 0.09) and tensile creep tests (at stresses from 2.0 to 6.0 MPa) are performed on low-density polyethylene at room temperature. Constitutive equations are derived for the time-dependent response of semicrystalline polymers at isothermal deformation with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical cross-links and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. The viscoelastic behavior of a polymer is modelled as thermally-induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). The viscoplastic response reflects mutual displacement of meso-domains driven by macro-strains. Stress–strain relations for uniaxial deformation are developed by using the laws of thermodynamics. The governing equations involve five material constants that are found by fitting the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. It is shown that observations in conventional creep tests reflect not only the viscoelastic, but also the viscoplastic behavior of an ensemble of meso-regions.     

Relaxational interactions and viscoelasticity of polymer melts: Part II. Rheological properties in shear and elongational flows

The rheological equation of state derived in Part I on the basis of relaxation equations of chain dynamics is analyzed for the steady and transient shear and uniaxial elongational flows of monodisperse polymers. The effect of superslow relaxation processes associated with basic macro-molecular motions that occur on a characteristic scale essentially greater than the so-called distance between entanglements was investigated in these flows. It is shown that the relaxation times in the region of linear and non-linear viscoelasticity are self-consistent. Theoretical predictions are in good agreement with the experimental data for melts of nearly monodisperse flexible polymers.     

Conformational evolution under steady shear flow: a comparison between cyclic and linear block copolymers

Cyclic block copolymer is a special type of block copolymer having no free ends. Comparison of the dynamic behavior between cyclic and linear block copolymers enables an understanding of the role of chain ends in dynamics of the latter. In relation to this point, analysis was made on the conformational dynamics for a cyclic bead-spring type diblock copolymer chain, AoB, under the steady shear flow. Further comparison was made on the conformational behavior of the AoB chain and that of two symmetric linear triblock copolymer chains, A–B–A and B–A–B. For these chains, the mobility was set to be higher for the A segments than the B segments. Thus, for the AoB chain under the steady shear flow, the segments of the A block exhibit less orientational anisotropy than those of the B block. This orientational contrast is enhanced for the A–B–A chain partly because the constraint for the motion of the segments is less near the chain ends than near the chain center. Nevertheless, for the B–A–B chain, the segmental orientation over the A block becomes more anisotropic than that over the B block. Detailed analysis shows that this result is attributable to a high orientational correlation for the segments of two end B blocks, in particularly for those near the block junctions. The correlated B segments exert a tensile force on the A block thereby significantly enhancing the orientational anisotropy of the A segments.     

A unified multiaxial fatigue damage parameter

According to the critical plane principle, a unified multiaxial fatigue damage parameter is presented based on the varying behaviour of the strains of the critical plane. Both the parameters of the maximum shear strain amplitude and normal strain excursion between adjacent turning points of the maximum shear strain on the critical plane are considered in the multiaxial fatigue damage parameter presented. An equivalent strain amplitude is made with both parameters of the maximum shear strain amplitude and normal strain excursion by means of von Mises criterion. Thus a new multiaxial fatigue damage parameter proposed in this paper may be used under either proportional or nonproportional loading, and may also be reduced to a uniaxial form. It is used to predict multiaxial fatigue life and good agreement is demonstrated by experimental data. The project is supported by the National Doctoral Foundation of China and National Natural Science Foundation of China.     

Modelling the viscoplastic response of polyethylene in uniaxial loading–unloading tests

Two series of uniaxial cyclic tests are performed on low-density polyethylene at room temperature. In the first series of experiments, injection-molded specimens are stretched to several maximal strains ε_max in the region of sub-yield deformations with a constant cross-head speed, mm/min, and retracted down to the zero stress with the same strain rate. In the other series, loading–unloading tests are carried out with the maximal strain ε_max=0.10 and cross-head speeds ranging from 5 to 200 mm/min. A constitutive model is derived for the viscoplastic behavior of a semicrystalline polymer at small strains. A polymer is modelled as an equivalent network of chains bridged by permanent junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is treated as an ensemble of meso-regions connected by links (crystalline lamellae). Deformation of a specimen induces sliding of junctions with respect to their reference positions both at active loading and unloading (this process reflects sliding of junctions in amorphous regions and fine slip of crystalline lamellae). At retraction, sliding of junctions is accompanied by mutual displacements of meso-domains (that reflects coarse slip and fragmentation of lamellar blocks). The constitutive equations are determined by 5 adjustable parameters that are found by matching the experimental stress–strain curves.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.