The contribution presents a new micro-mechanically based network model for the description of the elastic response of rubbery polymers at large strains and considers details of its numerical implementation. The approach models a rubber-like material based on a micro-structure that can be symbolized by a micro-sphere where the surface represents a continuous distribution of chain orientations in space. Core of the model is a new two-dimensional constitutive setting of the micro-mechanical response of a single polymer chain in a constrained environment defined by two micro-kinematic variables: the stretch of the chain and the contraction of the cross section of a micro-tube that contains the chain. The second key feature is a new non-affine micro-to-macro transition that defines the three-dimensional overall response of the polymer network based on a characteristic homogenization procedure of micro-variables defined on the micro-sphere of space orientations. It determines a stretch fluctuation field on the micro-sphere by a principle of minimum averaged free energy and links the two micro-kinematic variables in a non-affine format to the line-stretch and the area-stretch of the macro-continuum. Hence, the new model describes two superimposed contributions resulting from free chain motions and their topological constraints in an attractive dual geometric structure on both the micro- and the macro-level. Averaging operations on the micro-sphere are directly evaluated by an efficient numerical integration scheme. The overall model contains five effective material parameters obtained from the single chain statistics and properties of the network with clearly identifiable relationships to characteristic phenomena observed in stress–strain experiments. The approach advances features of the affine full network and the eight chain models by a substantial improvement of their modeling capacity. The excellent predictive performance is illustrated by comparative studies with previously developed network models and by fitting of various available experimental data of homogeneous and non-homogeneous tests.

A micro-macro approach to rubber-like materials—Part I: the non-affine micro-sphere model of rubber elasticity

Co-continuous structures can be regarded as the coexistence of at least two continuous structures within the same volume. Blends with co-continuous structures may combine the properties of both components in a favorable way, for example, mechanical moduli. This review article deals with the identification, characterization, and properties of co-continuous structures as well as with the development of co-continuous structures during the melt blending process. Co-continuous structures usually can be formed within a composition region about the phase inversion composition, which mainly depends on the viscosity ratio. On the other hand, co-continuous structures can be found independent of composition as intermediate stages during the initial state of morphology development and during phase inversion process in blends in which the component finally forming the dispersed phase forms the matrix in early mixing states. In addition, even at low volume fractions of one component, stable co-continuous morph...

Formation of Co-continuous Structures in Melt-Mixed Immiscible Polymer Blends

Continuous fibre reinforced thermoplastic composites are increasingly applied in aircraft structures. These high-performance thermoplastic matrices need processing at high temperatures; therefore, thermal residual stresses arise due to the mismatch in coefficients of thermal expansion between the fibres and the thermoplastic matrix. Since residual stresses are inherently present in virtually all composite materials and influence the properties of the composite structures significantly, it is of utmost importance that the residual thermal stresses are taken into account in both design and analytic modelling of composite structures. In order to understand the effects of residual stresses and find ways to decrease their magnitude or use them to our advantage, the factors responsible for the residual stress build-up need to be understood. This first part of a literature review discusses these factors, focusing on thermoplastic composites, and in particular the material properties of their constituents and processing parameters governing stress build-up.

Residual stresses in thermoplastic composites—A study of the literature—Part I: Formation of residual stresses

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 coupled theory of fluid permeation and large deformations for elastomeric materials

Recent progress toward understanding the rheology of dilute solutions of flexible polymers is reviewed, emphasizing experimental results from flows imaging single deoxyribonucleic acid (DNA) molecules and filament-stretching rheometry of dilute polystyrene Boger fluids, as well as Brownian dynamics (BD) simulations of these flows. The bead-spring and bead-rod models are presented, the range of their applicability discussed, and methods presented for inclusion of hydrodynamics interactions, excluded volume, and other physical effects within BD simulations. After reviewing and updating work in the linear viscoelastic regime, the primary focus shifts to the more complex nonlinear regime. While BD predictions of the conformations of 20 to 100 micron long DNA molecules in strong shear and extensional flows has been in good to excellent agreement with the corresponding experiments, predictions of the polystyrene dilute solution rheometry data have been hit or miss, with poorer results obtained for the higher mo...

The rheology of dilute solutions of flexible polymers: Progress and problems

Application of the methodology of Pade approximants to a Taylor expansion of the inverse Langevin function led to an accurate analytical expression The approximation, retaining a finite extendibility of the Langevin spring, enables a convenient analysis of experimental data and analytical manipulations of material models

A Padé approximant to the inverse Langevin function

Interfacial tension plays an important role in the manufacture of thermoplastic matrix composites and in the development and stabilization of blend morphologies The purpose of this work was to develop a rapid method to determine the interfacial tension The work entailed the microscopic tracking of shape changes of polystyrene fibers imbedded in poly(methyl methacrylate) The measured interfacial tension was found to agree with a value extrapolated from literature data to ±20% This method reduces the analysis time over traditional techniques, does not require complicated equipment, and provides reasonable precision

Estimation of Interfacial Tension Using Shape Evolution of Short Fibers

The mechanics of retraction of polymeric fibers imbedded in a polymeric matrix was analyzed. The shape evolution was described using a one-dimensional equation of equilibrium representing a balance between interfacial tension and viscous resistance. Microscopic tracking of polystyrene fibers in poly(methyl methacrylate) matrix was performed and compared with the data for the reverse material combination. The analysis of hydrodynamic and viscous factors was conducted using experimental results. It was found that the model provides an adequate description for the micromechanics of imbedded fibers. A characteristic time scale for the retraction was obtained.

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Analysis of a retraction mechanism for imbedded polymeric fibers

The interfacial tension between molten, high molecular weight samples of polycarbonate and poly(methyl methacrylate) was measured using the imbedded fiber retraction (IFR) technique. A value of 1.44 ± 0.16 dyn/cm was measured at 240 °C. This value was obtained on commercial resins with zero‐shear bulk viscosities and at a temperature typically encountered in the processing of blends and/or composites. Refinements to the IFR method were implemented which reduced the experimentation time. Consequently, measurements of the interfacial tension can now be routinely obtained in several hours for materials with zero‐shear bulk viscosities on the order of 104 Pa s.

Evaluation of the interfacial tension between high molecular weight polycarbonate and PMMA resins with the imbedded fiber retraction technique

Interfacial tensions between monodisperse polystyrene (PS) (M n = 2-200 kg/mol) and poly(methyl methacrylate) (PMMA) (M n = 24.4 kg/mol) were measured as a function of PS molecular weight at 190°C via imbedded-fiber retraction (IFR). The dynamic experiment enabled studied of much higher molecular weight resins than probed to date with equilibrium experiments, and this ability was applied to confirm the presence of the high-molecular-weight limiting interfacial-tension plateau for this system with direct measurements. The interfacial tension γ 12 followed a γ 12 ∞ M - C 1 M n -z dependence at 190°C. For M n exceeding about 48 kg/mol, γ 12 was independent of M n with a mean value (γ 12 ∞ M ) of 1.2±0.1 dyn/cm.

Arie Cohen

Papers

A Padé approximant to the inverse Langevin function

Estimation of Interfacial Tension Using Shape Evolution of Short Fibers

Analysis of a retraction mechanism for imbedded polymeric fibers

Evaluation of the interfacial tension between high molecular weight polycarbonate and PMMA resins with the imbedded fiber retraction technique

Molecular Weight Dependence of Polystyrene/Poly(methyl methacrylate) Interfacial Tension Probed by Imbedded-Fiber Retraction