P.D. Wu

On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers

Abstract T hree-dimensional molecular network theories are studied which use a non-Gaussian statistical mechanics model for the large strain extension of molecules. Invoking an affine deformation assumption, the evolution of the network — consisting of a large number of molecular chains per unit volume, which are initially randomly oriented in space — is shown to be governed by a balance equation in orientation space. Eulerian and Lagrangian type formulations of these balance equations are given, and the closed-form analytical solution for the so-called Chain Orientation Distribution Function is derived. This full network model is then used to describe the large strain inelastic behaviour of rubber-like materials. Detailed comparisons with experimental results and with two approximate models, namely the classical three-chain model and a very recently proposed eight-chain model, are provided for different types of deformation and rubbers. Finally, the network model is applied to describe the orientational hardening in amorphous glassy polymers, and confronted with experimental data for polycarbonate. The inherent physical limitations of the network theory for both applications are discussed.

Void growth in glassy polymers

This paper deals with a study of voids in amorphous glassy polymers that exhibit elastic-viscoplasticity with rate dependent yield, intrinsic softening and progressive strain hardening at large strains. The study is motivated by the plastic deformation in voided polymer-rubber blends caused by cavitation of the rubber particles, and thus attempts to contribute to the understanding of the toughening mechanisms in blends. Axisymmetric cell analyses are presented to study the plastic deformation around initially spherical voids and their resulting growth in terms of size and shape up to large overall strains. This void growth is demonstrated to inherit particular properties from the typical features of plasticity in glassy polymers, viz. small strain softening and large strain hardening. The role of strain localization into shear bands and their subsequent propagation in controlling void growth is highlighted. Furthermore, an approximate constitutive model is presented for the description of the macroscopic overall behaviour of porous glassy polymers. This model includes a modification of existing porous plasticity models to account for elasticity effects on the initiation of overall plasticity, which are important in polymers because of their relatively high yield strain. Its predictions are compared with the results from the numerical cell analyses.

Evaluation of anisotropic yield functions for aluminum sheets

Abstract The anisotropic behaviour of some rolled aluminum alloys is investigated using the phenomenological approach via some of the recently proposed 3-D yield functions. The directional variations of the yield stress and the so-called R-values in the plane of the sheet are predicted using the 3-D yield functions and the results are compared to previously reported experimental measurements. The hardening properties of the alloys examined are extracted from the experimental uniaxial curves along the rolling direction (RD). The forming limit curves are produced using the Marciniak and Kuczynski (Marciniak, Z., Kuczynski, K., 1967 Limit strains in the process of stret-forming sheet metals. Int. J. Mech. Sci. 9, 609–620) approach and the predictions of different yield functions are compared to the experimental curves. In general, although the refinements incorporated to the yield function proposals in chronological order produced better agreement with the experimental observations (at the expense of some complexity in the formulation), the overall performance of each criteria examined could nevertheless vary with the type of the application, as can be anticipated from the phenomenological nature of the approach.

On crystal plasticity FLD analysis

This paper is concerned with the computation of forming limit diagrams (FLDs) using a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski approach. Sheet necking is initiated from an initial imperfection in terms of a narrow band. The deformations inside and outside the band are assumed to be homogeneous and conditions of compatibility and equilibrium are enforced across the band interfaces. Thus, the polycrystal model needs only to be applied to two polycrystalline aggregates, one inside and one outside the band. Each grain is modelled as an FCC crystal with 12 distinct slip systems. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor‐type polycrystal model developed by Asaro and Needleman (1985). The effects of initial imperfection intensity and orientation, initial distribution of grain orientations, crystal elasticity, strain rate sensitivity, single slip hardening and latent hardening on the FLD are discussed in detail. The predicted FLD is compared with experimental data for an aluminium alloy sheet.

Pre-tension generates strongly reversible adhesion of a spatula pad on substrate

Motivated by recent studies on reversible adhesion mechanisms of geckos and insects, we investigate the effect of pre-tension on the orientation-dependent adhesion strength of an elastic tape adhering on a substrate. Our analysis shows that the pre-tension can significantly increase the peel-off force at small peeling angles while decreasing it at large peeling angles, leading to a strongly reversible adhesion. More interestingly, we find that there exists a critical value of pre-tension beyond which the peel-off force plunges to zero at a force-independent critical peeling angle. We further show that the level of pre-tension required for such force-independent detachment at a critical angle can be induced by simply dragging a spatula pad along a substrate at sufficiently low angles. These results provide a feasible explanation of relevant experimental observations on gecko adhesion and suggest possible strategies to design strongly reversible adhesives via pre-tension.

SIMULATION OF THE BEHAVIOUR OF FCC POLYCRYSTALS DURING REVERSED TORSION

Abstract Taylor-type polycrystal plasticity models with various single slip hardening laws are evaluated by studying the large strain behaviour of FCC polycrystals during reversed torsion. The hardening laws considered include the model of Asaro and Needleman (“Texture Development and Strain Hardening in Rate Dependent Polycrystals,” Acta Metall. (1985), 34 , 1553) as well as a power-law and an exponential version of that, and a more recent model by Bassani and Wu (“Latent Hardening in Single Crystals II. Analytical Characterization and Predictions,” Proc. R. Soc. Lond. (1991), A435 , 21). The material parameters for the various hardening laws are fitted to experimental compression data for copper and then used to predict reversed large strain torsion of tubes. Differences under “free-end” (axially unconstrained) or “fixed-end” (axially constrained) conditions between predictions and experimental observations are discussed in detail. In addition to the torque response, the Swift effects upon twist reversal are studied.

On Improved 3-D Non-Gaussian Network Models for Rubber Elasticity

Comparing these 3- and 8-chain representations with the actual 3-D initially random distribution of molecular chains, we intuitively felt that the 3-chain model would be likely to overestimate the actual stiffness of the network, while the 8-chain representation would probably give a lower bound. Therefore, we study here a more accurate non-Gaussian rubber model in which full account is taken of the orientation distribution of the individual chains in the network. The present treatment closely follows that in [3], but extends the latter to general 3-D deformations including arbitrary rotations of the principal stretch axes. The accuracy of the 3- and 8-chain models is assessed

Hierarchical modelling of attachment and detachment mechanisms of gecko toe adhesion

The mechanics of reversible adhesion of the gecko is investigated in terms of the attachment and detachment mechanisms of the hierarchical microstructures on its toe. At the bottom of the hierarchy, we show that a spatula pad of tiny thickness can be well absorbed onto a substrate with a large surface area and a highly constrained decohesion process zone, both of which are beneficial for robust attachment. With different peeling angles, the peeling strength of a spatula pad for attachment can be 10 times larger than that for detachment. At the intermediate level of hierarchy, we show that a seta can achieve a stress level similar to that in the spatula pad by uniformly distributing adhesion forces; as a consequence, the 10 times difference in the peel-off force of a single spatula pad for attachment and detachment is magnified up to a 100 times difference in adhesion energy at the level of seta. At the top of the hierarchy, the attachment process of a gecko toe is modelled as a pad under displacement-controlled pulling, leading to an adhesive force much larger than the geckos body weight, while the associated detachment process is modelled as a pad under peeling, resulting in a negligible peel-off force. The present work reveals, in a more systematic way than previous studies in the literature, that the hierarchical microstructures on the geckos toe can indeed provide the gecko with robust adhesion for attachment and reversible adhesion for easy detachment at the same time.

Analysis of shear band propagation in amorphous glassy polymers

Similar to the well-known neck propagation phenomenon, shearing of polymer materials often reveals the initiation and subsequent propagation of a shear band. Finite element analysis is used to numerically simulate large plane strain, simple shear tests, focussing attention on the initiation and propagation of the shear band. The mesh sensitivity and effects of initial imperfection, strain softening, orientation hardening, strain-rate as well as the edge effects are discussed in detail. It appears that the intrinsic softening is the driving force to promote initiation of the shear band and its propagation in the shear direction, while the orientation hardening is the driving force for widening of the shear band. The predicted numerical results are compared with experimental data for polycarbonate found in the literature.

A mesoscopic approach for predicting sheet metal formability

A mesoscopic approach for constructing a forming limit diagram (FLD) is developed. The approach is based on the concept of a unit cell. The unit cell is macroscopically infinitely small and thus represents a material point in the sheet, and is microscopically finitely large and thus contains a sufficiently large number of grains. The responses of the unit cell under biaxial tension are calculated using the finite element method. Each element of a mesh/unit cell represents an orientation and the constitutive response at an integration point is described by the single crystal plasticity theory. It is demonstrated that the limit strains are the natural outcomes of the mesoscopic approach, and the artificial initial imperfection necessitated by the macroscopic M–K approach is not relevant in the mesoscopic approach. The effects of strain-rate sensitivity, single slip hardening and latent hardening, texture evolution, crystal elasticity and spatial orientation distribution on necking are discussed. Numerical results based on the mesoscopic approach are compared with experimental data.