Generalised constitutive equations for densification of metal matrix coated fibre composites

Abstract

The present paper completes a study of constitutive equations for the consolidation processing of continuous fibre reinforced metal matrix composite materials. It builds on an earlier paper in which physically based constitutive equations were derived for the case of symmetrical, isostatic loading. In the present paper, constitutive equations are developed for in plane, general stress states. The total deformation of the consolidating composite is expressed as the sum of a conventional deviatoric creep term, together with a dilatational term, which was derived using a variational method previously published. The equations contain only two material parameters, which are the conventional creep coefficient and exponent for the fibre coating material (in this case, Ti-6Al-4V). The resulting equations have been implemented into finite element software enabling the simulation of practical consolidation processes. The model has been verified by comparing predicted results with those obtained from independent micromechanical models. A number of experimental tests have been carried out, and the model is used to predict the rates of densification for a range of experimental pressure and temperature histories. Good comparisons have been achieved.     

Simple model for consolidation of metal matrix coated SiC fibre composites

Abstract

A simple approach to modelling the consolidation of matrix coated fibre composites is presented. It employs an existing porous material constitutive model for monolithic materials. It is argued that in the consolidation of metal coated SiC fibres, the deformation primarily occurs in an outer layer of the fibre coating, and the internal core remains undeformed, largely because of the generally hydrostatic compressive loading, and because of the incompressible nature of the material in creep. The consolidation process is therefore not vastly different to that occurs for monolithic metal fibres, and similar equations can therefore be used for the composite consolidation. The constitutive equations have been implemented into general purpose non-linear finite element software within a large deformation formulation by means of two different user subroutines, one providing a general implementation, and the other a cpu time efficient approach. The manufacture and testing of SiC continuous fibre, Ti-6Al-4V metal matrix composite specimens is described and the results of the tests compared with the model calculations, showing that good agreement can be achieved with a simple model. The dependence of volume fraction of fibres and temperature can be introduced empirically through the specification of just two material constants. The model is therefore useful in the development of consolidation processes.     

Modelling hot deformation of Inconel 718 using state variables

Abstract

A dislocation density based state variable model has been developed to describe the characteristic flow stress behaviour during hot deformation of polycrystalline superalloy Inconel 718. Model equations have been formulated to describe the role of the evolving microstructures on the macroscopic flow stress response to deformation. Following a peak in the flow stress associated with strain hardening, the model utilises mechanisms associated with dynamic recovery and recrystallisation to explain the gradual decrease in flow stress with continued deformation. Incorporation of these microstructure based state variables also enables prediction of microstructures associated with a range of hot deformation conditions. Model flow stress predictions have been validated against isothermal uniaxial compression tests conducted over a range of temperatures and strain rates relevant to industrial forging conditions.     

INTERLAMINAR STRESSES IN SHELLS OF REVOLUTION

SUMMARY

A comparative study of the interlaminar stresses in shells of revolution has been made between first order shear deformation theory (FSDT), higher order shear deformation theory with thickness stretch (HSDT7), higher order shear deformation theory with higher order inplane displacement terms (HSDT9) and three-dimensional finite element (3D) models. A semi-analytical approach is used for all the models. Interlaminar stresses are evaluated using equilibrium equations in the cases of FSDT, HSDT7 and HSDT9 models as interlaminar stresses obtained from constitutive equations are not correct. For the 3D model an eight-noded quadratic quadrilateral semi-analytic solid element is used whereas for equivalent single layer (ESL) theories a three-noded isoparametric curved element is used. Crossply parabolic and hyperbolic caps subjected to uniform external pressure and a simply supported cylindrical shell subjected to an internal sinusoidal pressure are considered in the present study.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

Application of rigid–plastic meso-damage constitutive theory to superplastic bulge forming

Abstract

A rigid–plastic meso-damage constitutive theory is applied to predict the influence of void evolution on the superplastic forming process. Together with this constitutive theory, a simple analytical model is used to simulate the bulge forming process of a spherical dome. The distribution and evolution of damage, deformation, geometrical parameters, and mechanical properties such as strength and ductility throughout the bulging dome are systematically explored. Since a series of numerical results obtained are in good agreement with the actual phenomena and previous experimental results, the applicability of the rigid–plastic meso-damage constitutive theory to superplastic forming processes is considered proven.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

Analytical solutions to a network of standard linear solids

Various viscoelastic models, such as the standard linear solid, Maxwell model, and Kelvin–Voigt model, are frequently used to describe the behavior of biological materials from single cells to tissues. These models are expressed mathematically as simple differential equations, called constitutive equations, which relate the applied force (stress) to the resulting deformation (strain) of the material. Networks of these models, representing materials with heterogeneous mechanical properties, are described by systems of constitutive equations. We prove that the eigenvalues associated with such systems are all nonpositive real numbers, find bounds for them, and indicate how they can be estimated quickly and accurately. We then give formulas for the analytical solutions of the system of equations.     

Anisotropic viscoplasticity constitutive model for directionally solidified superalloy

Abstract

The macroscopic deformation behaviour of a Ni-based directionally solidified (DS) superalloy was experimentally investigated, and an anisotropic constitutive model of the material was developed. Monotonic and creep tests were performed on uniaxial test specimens machined from DS plates so that the angle between the loading direction and the solidified grain direction varied between 0 and 90°. Tension-torsion creep tests were also conducted to examine the anisotropic behaviour under multiaxial stress conditions. The material exhibited marked anisotropy under elastic and viscous deformation conditions, whereas it showed isotropy under plastic deformation conditions of high strain rates. Then crystal plasticity analyses were carried out to identify slip systems under creep loading conditions, assuming the anisotropic creep behaviour of the DS material. A viscoplastic constitutive model for expressing both the anisotropic elasticity-viscosity and the isotropic plasticity was proposed. The elastic constants were determined using a self-consistent approach, and viscous parameters were modelled by crystal plasticity analyses. The calculation results obtained using the constitutive model were compared with the experimental data to evaluate the validity of the model. It was demonstrated that the constitutive model could satisfactorily describe the anisotropic behaviour under uniaxial and multiaxial stress conditions with a given set of material parameters.     

Modeling nonlinear rate dependent behaviors of composite laminates

Abstract

This study aims to propose a simple explicit model for predicting the nonlinear rate dependent behaviors of composite laminates. Using one parameter plastic potential to describe the flow rule, the viscoplasticity model is expressed as a single master effective stress‐effective plastic strain curve in the form of a power law with a rate dependent amplitude. Based on the viscoplasticity model together with the laminated plate theory, the incremental form of the constitutive formulation is derived to model the nonlinear rate dependent behaviors of composite laminates. Symmetric glass/ epoxy and graphite/epoxy composite laminates were tested at three different strain rates and the experimental results were then compared with the model predictions. It was indicated that the proposed constitutive model is effective in characterizing the nonlinear rate dependent behaviors of composite laminates at strain levels up to 1%.     

Normalized load‐deflection curves of lateral piles in homogeneous elasto‐plastic soils

Abstract

In this paper, the procedure of normalizing load‐deflection curves of lateral piles in a homogeneous elasto‐plastic soil is presented. An analytical method using subgrade reactions reveals that the normalized relation of load‐deflection exists for force, moment or combined load. The normalized curves for force and moment load fit very well with the predictions using the equations. For the combined load, the normalized curves are distributed between the force and moment load curves, so it is easy to use interpolation from a family of curves. These normalized curves provide a very convenient way for calculating the pile load or deflection in the design stage.     

Evaluation of continuous alumina fibre reinforced composites based upon pure aluminium

Abstract

Composites of super purity aluminium unidirectionally reinforced with Altex or Nextel 610 continuous alumina basedfibre have been made by liquid metal infiltration. The composites were well consolidated, with fibre volume fractions V_f of 0.4 and 0.6 for the Altex composites and 0.7 for the Nextel composite, the higher values being obtained where preforming involved the use of sized fibre tows. Matrix porosity was very low and there was no evidence of any deleterious reaction product having formed at the fibre/matrix interface. Monotonic longitudinal tensile tests of the composites gave Youngs modulus values between 125 and 250 GPa, in line with rule of mixtures (ROM) predictions and evidence of effective load transfer between fibres. The onset of yielding in longitudinal composites was commensurate with the yield stress of unreinforced super purity aluminium for V_f = 0.4 (～20 MPa), but increased to 225 MPafor V_f =0.7. The tensile strengths of the Altex composites were 760 and 930 MPa, values in accord with ROM predictions based upon equal load sharing of fibres up to the mean filament failure stress. Although the Nextel composite had a higher tensile strength of 1250 MPa, this was significantly lower than the ROM value of 1650 MPa and was better described by fibre ‘bundle’ theory. Predictions of the accumulation of fibre damage, by statistical analysis, indicated that filament breakage commenced at an applied stress of ～50 MPa for the Altex composite and ～ 500 MPa for the Nextel composite. Despite damage at the lower stress, however, the Altex composites were able to tolerate many more ‘double’ fibre breaks than the Nextel composite, the failure of which coincided with the onset of the first double break. Transverse tensile tests of the composites gave Young's modulus values between 80 and 170 GPa, in line with ROM predictions. The yield stress increased with increasing V_f, from 10 to 60 MPa, this behaviour being attributed to plane strain deformation caused by the virtually non-deformable fibres constraining matrix flow. The tensile strengths showed a similar trend, with 84 MPa for V_f =0.4 increasing to 168 MPa for V_f = 0.7.     

Constitutive modeling for hot deformation behavior of ZA27 alloy

The hot deformation behavior of ZA27 alloy was investigated in the temperature range of 473–523 K with the strain rates in the range of 0.01–5 s⁻¹ and the height reduction of 60 % on Gleeble-1500 thermo mechanical simulator. Based on the experimental results, constitutive equations incorporating the effects of temperature, strain rate, and strain have been developed to model the hot deformation behavior of ZA27 alloy. Material constants, α, n, ln A, and activation energy Q in the constitutive equations were calculated as a function of strain. The results showed that the stress–strain curves of ZA27 alloy predicted by the constitutive equations are in good agreement with experimental results, which validates the efficiency of the constitutive equations in describing the hot deformation behavior of the material.     

ANALYTICAL AND EXPERIMENTAL MECHANICS OF WOVEN FABRIC COMPOSITES

SUMMARY

Mechanics of woven fabric composites are developed jointly by experimental laboratory tests and analytical models. The analytical scheme is based upon a micro- to mini- to macro-mechanics evolution which is representative of the relative physical scale. At the micro-mechanics level the interaction of an individual fibre with surrounding resin is evaluated by the composite cylinder assemblage (CCA) technique. Additionally, woven fabric requires a layered mini-mechanics analysis to evaluate the interaction of the undulated tows. Herein, we developed original and accurate three-dimensional surface equations to represent the fill and warp tows uniquely. Our approach eliminates the requirement of predetermining any elements in the stiffness matrices of the composite, and is a true three-dimensional representation of both the warp and fill tow undulation and cross-section. Finally, macro-mechanics analysis is used to represent the largest scale of the composite; the entire structure of the composite as stacked through the thickness of a plate or shell. The analytical predictions provide exceptional agreement when compared with experimental results of plain weave composites.     

The effect of heat developed during high strain rate deformation on the constitutive modeling of amorphous polymers

An adiabatic constitutive model is proposed for large strain deformation of polycarbonate (PC) at high strain rates. When the strain rate is sufficiently high such that the heat generated does not have time to transfer to the surroundings, temperature of material rises. The high strain rate deformation behavior of polymers is significantly affected by temperature-dependent constants and thermal softening. Based on the isothermal model which first was introduced by Mulliken and Boyce et al. (Int. J. Solids Struct. 43:1331-1356, 2006), an adiabatic model is proposed to predict the yield and post-yield behavior of glassy polymers at high strain rates. When calculating the heat generated and the temperature changes during the step by step simulation of the deformation, temperature-dependent elastic constants are incorporated to the constitutive equations. Moreover, better prediction of softening phenomena is achieved by the new definition for softening parameters of the proposed model. The constitutive model has been implemented numerically into a commercial finite element code through a user material subroutine (VUMAT). The experimental results, obtained using a split Hopkinson pressure bar, are supported by dynamic mechanical thermal analysis (DMTA) and Decompose/Shift/Reconstruct (DSR) method. Comparison of adiabatic model predictions with experimental data demonstrates the ability of the model to capture the characteristic features of stress–strain curve of the material at very high strain rates.     

A numerical approach for predicting the failure locus of fiber reinforced composites under combined transverse compression and axial tension

A computational model based on the finite element method is presented for the estimation of strength of a fiber-reinforced lamina subjected to a combination of the transverse compression and axial tension. A complex damage mechanism including fiber breakage, fiber/matrix debonding and matrix plastic deformation is reproduced in the proposed model by using appropriate constitutive equations. The numerical simulation of mechanical response of the unidirectional lamina under biaxial loading is used to obtained the failure locus. Subsequently, the model is verified against an analytical solution and experimental data. It was found that the numerical calculations agree better with experimental results than analytical predictions.     

Effect of consolidation pressure on volumetric composition and stiffness of unidirectional flax fibre composites

Unidirectional flax/polyethylene terephthalate composites are manufactured by filament winding, followed by compression moulding with low and high consolidation pressure, and with variable flax fibre content. The experimental data of volumetric composition and tensile stiffness are analysed with analytical models, and the composite microstructure is assessed by microscopy. The higher consolidation pressure (4.10 vs. 1.67 MPa) leads to composites with a higher maximum attainable fibre volume fraction (0.597 vs. 0.530), which is shown to be well correlated with the compaction behaviour of flax yarn assemblies. A characteristic microstructural feature is observed near the transition stage, the so-called local structural porosity, which is caused by the locally fully compacted fibres. At the transition fibre weight fraction, which determines the best possible combination of high fibre volume fraction and low porosity, the high pressure composites show a higher maximum performance in terms of tensile stiffness (40 vs. 35 GPa). The good agreement with the model calculations (fibre compaction behaviour, and composite volumetric composition and mechanical properties), allows the making of a property diagram showing stiffness of unidirectional flax fibre composites as a function of fibre weight fraction for consolidation pressures in the range 0–10 MPa.     

A study of reinforced concrete bridge columns retrofitted by steel jackets

Abstract

In this paper, theoretical and experimental results of two as‐built circular reinforced concrete (RC) bridge columns and two columns retrofitted with steel jackets are presented. A constitutive model for concrete confined by a steel jacket is proposed. The proposed model is implemented into a sectional analysis to predict the lateral load‐deformation relationship of retrofitted columns. 2/5 scaled RC bridge columns are designed based on the standard details of the existing bridge columns mostly built in late 1980s and early 1990s, in Taiwan (Ministry of Transportation and Communication, 1987; 1995). The columns are expected to have a flexural failure mode during severe ground shaking. Displacement‐controlled cyclic loading tests were conducted to obtain the seismic performance of the columns. The experimental results showed that the bridge column retrofitted with steel jacketing could greatly improve seismic performance measured based by the strength and ductility. The analytical results showed that the proposed constitutive model, implementing sectional analysis, could well capture the lateral force‐displacement relationship of the bridge columns retrofitted with steel jackets.     

Effects of transformation twin on Hall–Petch relationship in MnCu alloy

A new constitutive model based on the deformation mechanisms was developed for both bcc and fcc nanocrystalline metals over a wide strain rate range. Nanocrystalline metals were treated as composites consisting of grain interior and grain boundary phases, and the deformation mechanisms and physically based constitutive relations of grain interior and grain boundary phases over a wide strain rate range were analyzed and determined for both fcc and bcc nanocrystalline metals. Based on our recently established phase mixture method, a new mechanical model was built to calculate the stress–strain relations of nanocrystalline metals over a wide strain rate range; and the grain size and porosity effect was considered in the developed model, the predictions keep in good agreements with various experimental data in small plastic strain range, where uniform deformation can be assumed. Further discussion was presented for calculation results and relative experimental observations.