An affine microsphere approach to modeling strain-induced crystallization in rubbery polymers

Upon stretching a natural rubber sample, polymer chains orient themselves in the direction of the applied load and form crystalline regions. When the sample is retracted, the original amorphous state of the network is restored. Due to crystallization, properties of rubber change considerably. The reinforcing effect of the crystallites stiffens the rubber and increases the crack growth resistance. It is of great importance to understand the mechanism leading to strain-induced crystallization. However, limited theoretical work has been done on the investigation of the associated kinetics. A key characteristic observed in the stress–strain diagram of crystallizing rubber is the hysteresis, which is entirely attributed to strain-induced crystallization. In this work, we propose a micromechanically motivated material model for strain-induced crystallization in rubbers. Our point of departure is constructing a micromechanical model for a single crystallizing polymer chain. Subsequently, a thermodynamically consistent evolution law describing the kinetics of crystallization on the chain level is proposed. This chain model is then incorporated into the affine microsphere model. Finally, the model is numerically implemented and its performance is compared to experimental data.     

Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature

A boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bi-material interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of complex weighting functions for both displacements and tractions, along with corresponding non-standard numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-insensitive solutions for a range of two-dimensional interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject. As a final application, the method is applied to study the scaling behavior of epoxy–metal butt joints.     

Simulation of rate dependent plasticity for polymers with asymmetric effects

Polycarbonate is an amorphous polymer which exhibits a pronounced strength-differential effect between compression and tension. Also strain rate and temperature influence the mechanical response of the polycarbonate. The concept of stress mode dependent weighting functions is used in the proposed model to simulate the asymmetric effects for different loading speeds. In this concept, an additive decomposition of the flow rule is assumed into a sum of weighted stress mode related quantities. The characterization of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the mode angle, such that stress mode dependent scalar weighting functions can be constructed. The resulting evolution equations are updated using a backward Euler scheme and the algorithmic tangent operator is derived for the finite element equilibrium iteration. The numerical implementation of the resulting set of constitutive equations is used in a finite element program for parameter identification. The proposed model is verified by showing a good agreement with the experimental data. After that the model is used to simulate the laser transmission welding process.     

Experimental determination of strain-induced anisotropy during nonproportional straining of an A1/Mg alloy at room temperature

A servohydraulic, computer controlled MTSn axial-torsion testing machine with a bi-axial clip-on extensometer is used to test thin-walled tubes of an A1/Mg alloy under strain control. A plastic offset strain of 10⁻⁴ determines the yield surfaces. Straining and yield surface probing is governed by a computer program which also controls digital data acquisition. Yield surfaces in stress and in strain space as well as the axial and shear stress-strain diagrams can be reconstructed from the digitally recorded data. The specimens were subjected to a strain path in the form of a regular 16-sided polygon which was followed on some specimens by a square path. The total inelastic strain path length can exceed 15% while the equivalent strain excursion is less than 2%. It is shown that yield surfaces measured on specimens withclose initial stress-strain diagrams are very consistent and that yield surface probing has an insignificant effect on subsequent yield surfaces. Yield surfaces are shown to translate, changein shape and size and to exhibit a cross effect. A post processor which includes a least square smoothing routine calculates the area and the centroid of each yield surface. The size increase is initially rapid but the rate of increase decreases as a saturation is approached. After strining for less than 1% in a fixed direction a characteristic yield surface shape is established. Yield surfaces obtained at the same point in strain space with identical prestrain direction of at least 1% but with increased amounts of accumulated plastic strain have the same shape but show an increase in size. The yield surfaces differ in shape and size when the same strain point is reached from different directions. The centroid of the yield surface in stress space moves almost in a circular path for a polygonal strain path. All stress space yield surfaces contain the origin but this is not the case for the surfaces in strain space.     

Mobility anisotropy and nonlinear effects in molecular viscoelasticity theory of linear polymers

Various methods of taking the mobility anisotropy into account in modeling of the behavior of polymer fluids in a simple shear flow are compared. Both the qualitative coincidence of the results of the analysis and the difficulties of quantitatively estimating the anisotropy microparameters on the basis of the available data concerning steady shear are noted. For axial tension the dependence of steady viscosity on tensile stress is considered. This dependence is determined by two independent scalar functions: the slip coefficient and the flow anisotropy coefficient.Barnaul. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–12, July–August, 1995.     

Saint-Venant problem for solids with helical anisotropy

We discuss the solution of Saint-Venant’s problem for solids with helical anisotropy. Here the governing relations of the theory of elasticity in terms of displacements were presented using the helical coordinate system. We proposed an approach to construct elementary Saint-Venant solutions using integration of ordinary differential equations with variable coefficients in the case of a circular cylinder with helical anisotropy. Elementary solutions correspond to problems of extension, of torsion, of pure bending and of bending of shear force. The solution of the problem is obtained using small parameter method for small values of twist angle and numerically for arbitrary values. Numeric results correspond to problems of extension–torsion. Dependencies of the stiffness matrix (in dimensionless form) on angle between the tangent to the helical coil and the axis of the cylinder corresponding to stiffness on stretching and torsion are illustrated graphically for different values of material and geometrical parameters.     

Limits on the resolution of correlation PIV iterative methods. Practical implementation and design of weighting functions

Elsewhere in this volume (Nogueira et al. (2005) Exp Fluids, in press), the conceptual background that explains the possibility of resolving wavelengths smaller than the size of the interrogation window, with no basic restrictions but sampling, has been explained. Here, a practical implementation of the concepts is performed. To achieve this resolution in iterative PIV processing, an appropriate weighting function can be used, as commented in that reference. Here, the constraints for the design of such weighting functions are presented and analysed. This opens a line of work on possible weighting functions to develop, since the weightings used in these iterative methods, like local field correction particle image velocimetry (LFC-PIV) (Nogueira et al. (1999) Exp Fluids 27(2):107–116), have not been optimised yet. As an example, different weighting functions are commented and tested both on synthetic and real images. The results on these new weightings indicate that the current ones can be improved and the optimisation criteria are open for further advancement.     

An improved algorithm for calculating relaxation time spectra from material functions of polymers with monodisperse and bimodal molar mass distributions

Using the basic concept of Emri and Tschoegl, the algorithm for calculating relaxation time spectra has been improved such that excellent results are provided in the difficult case of polymers with narrow molar mass distributions. These spectra can be compared with those calculated by nonlinear regularization (Weese 1992), which we regard as a very exact method, and show equally good results with even less mathematical effort. Examples of dense relaxation time spectra (up to eight points per decade) are given for nearly monodisperse polystyrene melts and for mixtures of these up to four components. The relaxation time spectra describe the dynamic mechanical experimental data in each case with an overall error of less than 3%. The filtering method used to avoid physically senseless oscillations has been proven to resolve the characteristic peaks contributed by monodisperse polymers accurately.     

Constitutive modeling of strain-induced softening in swollen elastomers

Under cyclic loading, elastomeric material exhibits strong inelastic responses such as stress-softening due to Mullins effect, hysteresis and permanent set. The corresponding inelastic responses are observed in both dry and swollen rubbers. Moreover, it is observed that inelastic responses depend strongly on the swelling level. For engineering applications involving the interaction and contact between rubber components and solvent, the understanding and consideration of swelling are essential pre-requisites for durability analysis. In this paper, a simple phenomenological model describing Mullins effect in swollen rubbers under cyclic loading is proposed. More precisely, the proposed model adopts the concept of evolution of soft domain microstructure with deformation originally proposed by Mullins and Tobin. The swollen rubbers are obtained by immersing dry ones in solvent until desired degrees of swelling are achieved. Subsequently, their mechanical responses, in particular Mullins effect, under cyclic loading are investigated. These experimental data are used to assess the efficiency of the proposed model. Results show that the model agrees qualitatively well with experiments. Furthermore, the model captures well the fundamental features of strain-induced softening.     

Rheological behavior of sewage sludge and strain-induced dewatering

In order to study the strain-induced water release in sewage sludge and its connection with rheological behavior, two types of rheological tests have been carried out. The rheology of sewage sludge samples stemming from a urban sewer was first characterized using a Couette cell system. In particular, the yielding behavior and the elastic modulus of the sludge has been considered under shear flow conditions. In these pure shear tests the reproducibility of the measurements was rather poor, limiting this study to low strains. Consequently, a second type of rheological tests, namely the squeeze test, which is more appropriate for these paste-like materials, has been considered. The rheological behavior along with the dewatering efficiency have been studied under the squeeze flow conditions. Surprisingly, it was found that, under certain conditions, the strain-induced water release mechanism became more effective when decreasing the squeeze speed. This was interpreted in terms of a competition between the paste flow and the water filtration through the porous media made up by the flocs.     

Anisotropic plasticity model coupled with Lode angle dependent strain-induced transformation kinetics law

A phenomenological macroscopic plasticity model is developed for steels that exhibit strain-induced austenite-to-martensite transformation. The model makes use of a stress-state dependent transformation kinetics law that accounts for both the effects of the stress triaxiality and the Lode angle on the rate of transformation. The macroscopic strain hardening is due to nonlinear kinematic hardening as well as isotropic hardening. The latter contribution is assumed to depend on the dislocation density as well as the current martensite volume fraction. The constitutive equations are embedded in the framework of finite strain isothermal rate-independent anisotropic plasticity. Experimental data for an anisotropic austenitic stainless steel 301LN is presented for uniaxial tension, uniaxial compression, transverse plane strain tension and pure shear. The model parameters are identified using a combined analytical–numerical approach. Numerical simulations are performed of all calibration experiments and excellent agreement is observed. Moreover, we make use of experimental data from ten combined tension and shear experiments to validate the proposed constitutive model. In addition, punch and notched tension tests are performed to evaluate the model performance in structural applications with heterogeneous stress and strain fields.     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V/V, volume average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Three-dimensional solutions for general anisotropy

The Stroh formalism is extended to provide a new class of three-dimensional solutions for the generally anisotropic elastic material that have polynomial dependence on x₃, but which have quite general form in x₁,x₂. The solutions are obtained by a sequence of partial integrations with respect to x₃, starting from Stroh's two-dimensional solution. At each stage, certain special functions have to be introduced in order to satisfy the equilibrium equation. The method provides a general analytical technique for the solution of the problem of the prismatic bar with tractions or displacements prescribed on its lateral surfaces. It also provides a particularly efficient solution for three-dimensional boundary-value problems for the half-space. The method is illustrated by the example of a half-space loaded by a linearly varying line force.     

Necking in glassy polymers: Effects of intrinsic anisotropy and structural evolution kinetics in their viscoplastic flow

Two recently proposed developments of the Glass–Rubber constitutive model for glassy polymers treat the viscoplastic deformation as intrinsically anisotropic, and incorporate the kinetics of structural evolution. These features enable the model to capture better the distinctive features of glassy polymers’ constitutive response: post-yield strain-softening and strain-hardening and effects of pre-existing molecular orientation. They have been combined to form a new variant of the model, and the consequences for necking have been explored. Uniaxial extension of prismatic bars was simulated using the finite element method, employing a numerical implementation of the new model, with material parameters of polystyrene. Strain localization predicted with the new model was found to be systematically retarded as compared to predictions with the original (intrinsically isotropic) version of the model, for the same conditions. In particular, the effect of frozen-in molecular orientation was examined. This was found to retard strain localization for stretching parallel to the orientation direction, for both models. But the localization predicted with the new model was always significantly less pronounced than with the original model. Indeed, for sufficiently high pre-orientation (e.g. a uniaxial stretch of 2.2), localization could be effectively prevented with the new model, under conditions when otherwise failure by necking is predicted. Such results can all be explained in terms of a linear stability analysis. They suggest that all previous simulations of necking in glassy polymers made using intrinsically isotropic representations of polymer viscoplasticity may have over-predicted the rate of strain localization.     

Elastoplastic formulation for friction with orthotropic anisotropy and rotational hardening

In various fields of engineering, it is important to clarify the frictional sliding behavior over a wide scale. In this study, we formulate an anisotropic friction model with the orthotropy and rotation of a sliding surface based on the elastoplastic theory. This model can also describe preliminary microscopic sliding and rate-dependent frictional response. Further, basic experimental results of anisotropic frictional sliding can be pertinently represented by the present model. We also employ this model with the finite element method and analyze typical frictional contact problems. We then demonstrate the effect of anisotropy parameters on the numerical results.     

Nucleation of micro-cracks for polycrystalline metal with anisotropy: micro-stress evaluation

Presented is the local stresses on the crystallographic plane as they are influenced by the metal fracture with anisotropy. It is based on the nucleation of micro-cracks and its unstable equilibrium in a polycrystal with texture. Crystallographic texture causes non-uniform distribution of the crack nucleus orientations owing to their preference to expand or open on certain crystallographic planes. This is the main cause of anisotropy of cleavage fracture stress of textured metal. “Oriented” micro-stresses in textured metal contribute to the anisotropic effect. In view of what has been said, accumulated plastic strains at fracture is analysed.