Schapery's Model Research Articles

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

AbstractThis study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.

Solution strategies for FEM analysis with nonlinear viscoelastic polymers

The Schapery model is a very common model for nonlinear viscoelastic behaviour of polymers. Therefore this model has been implemented with a user subroutine into the FEM package MARC. Since explicit integration is relatively fast but is often accompanied by numerical instabilities and implicit integration is relatively slow but numerically stable, combined strategies have been programmed. Results show that the viscoelastic response of complex products can be calculated without numerical instabilities and with short calculation times. Full viscoelastic calculations and calculations using creep isochrones on dead weight loading of an HDPE bottle crate only differ from the onset of creep buckling.

Thermal expansion anisotropy in a Ti–6Al–4V/SiC composite

We studied the thermal expansion behavior of a Ti–6Al–4V/35 vol.% continuous SiC fiber composite using in situ high temperature neutron diffraction (ND). The lattice expansion of constituent phases within the composite was monitored from axial (parallel to the unidirectionally aligned fibers) and transverse (perpendicular to the fibers) directions during heating from room temperature (RT) to 1170 K. The phase-specific thermal expansion of the Ti–6Al–4V matrix and SiC fibers in the composite is discussed in the context of thermal load partitioning between the matrix and fibers. In the axial direction, the matrix and the fiber share the thermal load and co-expand up to about 800–900 K, above which the thermal load transfer becomes ineffective. In the transverse direction, the matrix and fibers expand independently over the whole temperature range. Using the Schapery model (J. Comp. Mater. 2 (1968) 380) and the rule-of-mixtures (ROM), the macroscopic thermal expansion of the composite is predicted and compared with the experimental results.

Further development in thermodynamic approach for thermoviscoelastic materials

In this paper we derive linear and nonlinear theories ofthermoviscoelasticity for thermorheologically simple materials. Thisderivation is based on two theories: the Thermodynamic of IrreversibleTheory and the Generalized Viscoelastic Standard Theory. These twotheories, along with other assumptions, lead to three-dimensionalthermoviscoelastic models. The derivation we give here is an extensionof Schapery's models. We will soon state the relations expressing thebalance of energy, i.e., the specific heat capacity and the heatconduction equation.

Coefficients of thermal expansion of some laminated ceramic composites

Abstract The coefficients of thermal expansion (CTE) were determined for several non-fibrous alumina-based laminated ceramic composites. The results were compared with the CTE values predicted by the modified equations of Schapery and Chamis. Both models are identical in the longitudinal direction and showed differences from experimental CTE of 2.3 and 4.1% for alumina/barium zirconate (Al 2 O 3 /BaZrO 3 ) and alumina/tin dioxide (Al 2 O 3 /SnO 2 ) composites, respectively. For Al 2 O 3 /BaZrO 3 in the transverse direction, Schapery's model showed a 1.9% difference while Chamis’ model showed a 7.5% difference. For the Al 2 O 3 /SnO 2 in the transverse direction, Schapery's model had an 8.2% difference while Chamis’ model showed only a 2.3% difference. Results for alumina/calcium titanate (Al 2 O 3 /CaTiO 3 ) laminates showed larger differences, 27.6% in longitudinal CTE and differences of 9.6 and 19.8% transverse CTE for the Schapery and Chamis models, respectively. These differences were attributed to the formation of cracks that occurred in this composite system during processing because of the large CTE mismatch between Al 2 O 3 and CaTiO 3 . Results for Al 2 O 3 /BaZrO 3 and Al 2 O 3 /SnO 2 composites showed that for continuous laminae both the Schapery and Chamis models were adequate predictors of CTE for the systems.

Non-linear creep modelling of single-fibre model composites

This paper describes a method for determining the stress/strain fields in a single-fibre model composite subjected to creep loading conditions. The method is based upon Eshelby's thought experiment and the integration of interfacial tractions. The non-linear viscoelastic behaviour is modelled through an incremental stress/strain formulation based on the Schapery model. Comparisons have been made between the predictions from the model, a non-linear finite-element model, and experimental fibre strains measured by micro-Raman spectroscopy and good qualitative agreement is observed. The nature of the interfacial tractions controls the form of the interfacial shear stress which govern the shape of the longitudinal fibre stress profile. As a consequence the rates at which the strain is building up in the fibre are different when comparing calculations and experiments. However, the characteristics of the experiments are reflected in the theoretical predictions. Generally, it is seen that the initial residual stress in the fibre is restricting the level of the fibre strain during subsequent creep loading and can be beneficial in delaying tensile fibre fractures.

Thermal expansion behaviour of unidirectional carbon-fibre-reinforced copper-matrix composites

Abstract Continuous carbon-fibre-reinforced copper-matrix composites prepared by diffusion bonding technology have been used for investigation of the coefficient of thermal expansion. For reasons of economy and ease of availability, continuous Torayca T300 fibres have been used for sample preparation. They were coated continuously with copper (galvanically and then chemically) and unidirectional composites were prepared by diffusion bonding in vacuum at 873 K for 30 min. The linear coefficient of thermal expansion (CTE) of samples with different volume fractions of carbon fibres was measured in directions parallel and perpendicular to the fibre direction. The samples were heat-treated for one temperature cycle in the range 293–573 K or cycled three times in a temperature range from 253 to 573 K. Measured CTE values are compared with those predicted by the well-known Schapery model and the model derived by Kural and Min. Better agreement was achieved with the predictions of the longitudinal CTE of the composite. Prediction of the transverse CTE was more difficult because of a lack of knowledge of the transverse CTE of carbon fibres. Models including the transverse CTE of carbon fibres (Kural–Min) gave better results for prediction of the transverse CTE of the unidirectional composite.

Measurement and modelling of crack propagation velocity in a viscoelastic matrix composite

The steady-state propagation of a crack in a continuous carbon-fibre/epoxy-matrix composite was produced in a tapered double cantilever beam specimen subjected to a constant-velocity opening. Tests were performed at room temperature. Constant crack speed, driving force and stress intensity factor during crack extension were measured for each crosshead velocity. Viscoelastic creep functions of the material were also determined. Schapery's model of viscoelastic fracture was shown to describe the slow crack propagation phenomenon adequately, provided the dependence of fracture energy on crack velocity was taken into account.

Thermal expansion studies of prestrained Al 2O 3/Al metal matrix composite

The present study was undertaken to investigate the effect of prestraining on the coefficient of thermal expansion (CTE) of 10 and 20 volume percent (v/o) Al 2O 3 particle-reinforced composite over various temperature ranges using thermal mechanical analysis (TMA). The CTE of the composite was shown to be dependent on the volume fraction of the broken particle, plastic yielding and flow, and other transformation processes occurring at elevated temperatures. The experimentally measured values of the CTE were found to be in good agreement with Schapery's model at various values of plastic prestrain (0–10%) and with a modified Kerner's equation, at high prestraining ( ε p > 4%) for a wider temperature range.

Thermal expansion of a PRD-166 fibre-reinforced glass-matrix composite

Coefficients of thermal expansion (ctes) of PRD-166/glass and SnO2-coated PRD-166/glass composites were measured as a function of fibre volume fraction, and these experimental values of cte compared with those predicted using various models. Schapery's model was found to predict the cte values of the SnO2-coated fibre composites more accurately. None of the models were successful in predicting the cte of the uncoated fibre composites because of the chemical reactivity between the fibre and matrix. The effect of fibre misorientation and fibre breakage on the cte of the composites was also investigated. Fibre breakage induced during composite fabrication was found to affect the cte more significantly than the processing-induced fibre misorientation in these composite systems. This was because the constraint on expansion exerted by short (broken) fibres is less than that due to long continuous fibres.

On the application of viscoelastic fracture theories to the slow propagation of a crack in a carbon/epoxy composite

It is shown that an orthotropic extension of Christensen's model is potentially able to describe the slow growth of a crack in carbon/epoxy composites. The unexpected agreement of Christensen's and Schapery's models with each other allows them to play complementary roles, as they do not involve the same material parameters

A Possible Way of Applying Schapery's Viscoelastic Fracture Model to Creep Crack Growth in Carbon-epoxy Composites

Schapery and Christensen's models of viscoelastic fracture have been adapted to describe the phenomenon of transverse cracking in composites. In a second stage, the application of both models to a known composite allowed to determine the size of the “failure zone”, which is one of key parameters in Schapery's model.

On the characterization by Schapery's model of non-linear Viscoelastic materials

By carefully separating instantaneous and quasi-static responses while analyzing the data supplied by creep-recovery tests, the process of characterizing linear and non-linear parameters entering into the Schapery viscoelastic constitutive model can be made more systematic and reliable. Applications to the non-linearly viscoelastic mechanical response of polycarbonate on one hand and to the non-linearly viscoelastic mechanical response of an adhesive on the other are presented.