Rjm Robert Smit

Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling

An accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented. This method is based on the classical homogenization theory, assuming local spatial periodicity of the microstructure. Consequently, the microstructure is identified by a representative volume element (RVE) with conformity of opposite boundaries at any stage of the deformation process. The local macroscopic stress is obtained by applying the local macroscopic deformation (represented by the deformation tensor) on a unique RVE through imposing appropriate boundary conditions and averaging the resulting RVE stress field. If the assumption of local periodicity of the morphology is valid, this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation. The homogenization method was implemented in a multi-level finite element program with meshes on macroscopic level (mesh of entire structure) and microscopic level (meshes of RVEs). The performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet.

A Constitutive Equation for the Elasto-Viscoplastic Deformation of Glassy Polymers

Constitutive equations for finite elastic-plastic deformation of polymers and metals are usually formulated by assuming an isotropic relation between the Jaumann rate of the Cauchy-stress tensor and the strain-ratetensor. However, the Jaumann-stress rate is known to display spuriousnon-physical behaviour in the elastic region. Replacing the Jaumann-stress rate by a Truesdell-stress rate results in an adequate description in the elastic region, but gives rise to a volume decrease during plastic flow intensile deformation. In this paper a ’’compressible-Leonov model‘‘ is introduced, in which the elastic volume response is rigorously separated from the elasto-viscoplastic isochoric deformation. This has the advantage that the model can be extended in a straightforward way to include aspectrum of relaxation times. It is shown that in the limit of small elastic strains, the compressible Leonov model reduces to the Jaumann-stress rate model, but diverges from the Truesdell-stress rate model. Finally, a comparison is made of the above mentioned models in ahomogeneous uniaxial tensile test and a homogeneous plane-stress sheartest, using polycarbonate (PC) as a model system. All models considered in this paper are ’’single mode‘‘ models (i.e. one relaxation time), and, therefore, cannot describe the full (non)linear viscoelastic region, northe strain-hardening or strain-softening response.

Prediction of the large-strain mechanical response of heterogeneous polymer systems : local and global deformation behaviour of a representative volume element of voided polycarbonate

Abstract The mechanical behaviour of voided polycarbonate has been predicted by using a detailed finite element model of the microstructure and an accurate elasto–viscoplastic model for the glassy polymeric matrix material. On the microstructural level a spatially periodic plane strain matrix with irregularly distributed voids has been considered. The voids represent low-modulus non-adhering rubbery particles under negative pressure. The constitutive model for the homogeneous parts of the material reflects the typical yield and post-yield behaviour of glassy polymers: strain rate and history dependent yield, intrinsic strain softening and subsequent strain hardening. The finite element simulations show that the irregular void distribution causes a radical change in deformation behaviour. In particular the macroscopic strain softening disappears. This transformation in macroscopic behaviour originates from the arbitrary order in which local shear bands between the randomly distributed voids are formed and subsequently harden. In the averaged overall mechanical response the individual unstable yield and post-yield behaviour of the local shear bands is evened out, resulting in an overall stable macroscopic deformation behaviour. This mechanism is believed to be primarily responsible for the toughness enhancement of heterogeneous polymer systems through the addition of easily cavitating rubbery particles.

Predictive modelling of the properties and toughness of polymeric materials, Part I: Why is polystyrene brittle and polycarbonate tough?

The brittleness of polystyrene (PS) and the toughness but notch sensitivity of polycarbonate (PC) have been studied by the detailed finite element analyses of the stress and strain fields in a notched tensile bar with a minor defect. The defect represented a flaw or imperfection, generated during the test specimen production. The large-strain mechanical responses of both materials were approximated by an accurate elasto-viscoplastic constitutive model with appropriate material parameters. It was assumed that failure occurs instantaneously once the dilative stress exceeds a certain critical craze-initiation stress. The analyses show that the unstable post-yield mechanical response of both materials results in localisation of stresses and strains near the defect at a very low macroscopic strain (0.16%). As a result, a strong dilative stress concentration is formed just below the surface of the defect. For the polystyrene specimen, the critical stress is reached at the defect. For the polycarbonate, however, the effect of the stress concentrating defect was counteracted by a higher craze-initiation stress and stronger strain hardening. The PC craze-initiation resistance, however, did not suffice to overcome the dilative stress concentration raised by the notch tip.

Predictive modelling of the properties and toughness of polymeric materials Part II Effect of microstructural properties on the macroscopic response of rubber-modified polymers

The influence of microstructural properties on the macroscopic mechanical behaviour has been studied by finite element predictions of the response of different microstructures of polystyrene (PS) or polycarbonate (PC) containing voids or rubbery particles, subjected to unidirectional extension. The voids represent a low-modulus non-adhering dispersed phase. The rubbery inclusions, which are assumed to be pre-cavitated and perfectly adhering, idealise core-shell particles with a hard rubber shell and a soft non-adhering or pre-cavitated core. The predictions show that the inclusion properties strongly affect the averaged post-yield response of the heterogeneous systems. Especially the post-yield strain softening can be eliminated by the introduction of voids in PC or rubbery particles in PS. Since macroscopic strain softening is believed to be the main cause of catastrophical stress or strain localisations, the softening elimination is believed to be primarily responsible for toughness enhancement of the polystyrene or polycarbonate systems. The results and experiences are extrapolated in order to explain the influence of the absolute length scale of a sub-micron sized morphology on the macroscopic behaviour, especially toughness. Two potential sources of particle-size effects are presented that may result in a stabilised, and thus tougher, macroscopic mechanical response, i.e. the yield stress reduction near a surface or interface because of a locally enhanced mobility of the polymer segments, and the temporary excessive hardening because of a sufficiently small size of the yield zones which results in a reduced effective entanglement distance. The paper concludes with a discussion on the extension of this knowledge to all other, for the moment amorphous, polymers.

Predictive modelling of the properties and toughness of polymeric materials Part III Simultaneous prediction of micro- and macrostructural deformation of rubber-modified polymers

The deformation behaviour of heterogeneous tensile bars is investigated by using the recently developed multi-level finite element method (MLFEM) that allows for a numerical coupling between the microscopic and macroscopic stress-strain behaviour, combined with an accurate elasto-viscoplastic constitutive model (single-mode compressible Leonov model) and a detailed finite element model of the microstructure. The method is used to predict the influence of the microstructure on localisation phenomena in plane strain notched and hour-glass-shaped polycarbonate and polystyrene tensile specimen with different volume fractions of non-adhering or adhering rubbery particles. In Part I and II of this series it was already suggested that elimination of the unstable post-yield strain softening behaviour of a polymeric material by appropriate microstructural modifications may be essential for toughening. The results of the multi-level analyses presented in this paper confirm this statement. It is shown that a stable post-yield response, resulting from microstructural adaptations, is indeed a prerequisite for the distribution of plastic strains over the whole macro- and microstructure: massive shearing is promoted by the introduction of voids in the polycarbonate or load bearing pre-cavitated rubbery particles in the polystyrene. Furthermore, it is shown that the voids indeed reduce the macroscopic dilative stresses to safe values. The results suggest that localisations of strain and stress will always occur on a macro and/or micro level. Catastrophic failure, however, can be postponed by stabilisation of the post-yield behaviour of the material and reduction of the macroscopic dilative stresses through appropriate microstructural adjustments.