Computational implementation of a novel constitutive model for multidirectional composites

This paper details the numerical implementation of a constitutive model for unidirectional (UD) polymer-matrix fibre-reinforced composites, which is able to accurately represent the full non-linear mechanical response. Features such as hydrostatic pressure sensitivity, the effect of multiaxial loading and the dependence of the yield stress on the applied pressure are often neglected in constitutive modelling, but are included in this model.The constitutive model includes a novel yield function, non-associative flow rule and a non-linear kinematic hardening rule. It is combined with suitable failure criteria and associated damage model. The complete model is implemented in an explicit finite element code.Experimental test data is used to show that the model is able to predict the non-linear response of both unidirectional and multidirectional composite laminates. The model is shown to accurately predict the constitutive response under complex multiaxial loading and unloading, including significant hydrostatic pressure. Predictions are also shown to compare favourably for the evolution of matrix cracking after initial matrix cracking is detected by the failure criteria.     

A constitutive model for unsaturated soils: thermomechanical and computational aspects

This paper first presents a complete formulation of a constitutive model that deals with the irreversible behaviour of unsaturated soils under various loading and drying/wetting conditions. A standard form of incremental stress-strain relations is derived. The constitutive model is then cast into the thermodynamical theories and verified using the thermomechanical principles. It is shown that hydraulic hysteresis does not contribute to the plastic dissipation, though it contributes to the plastic work. All plastic work associated with a plastic increment of the degree of saturation is stored and can be recovered in a reversed plastic increment of saturation. The incremental constitutive equations are also reformulated for implementation in finite element codes where displacements and pore pressures are primary unknowns. Qualitative predictions of the constitutive model show that incorporating two suction related yield surfaces and non-associated flow rules into the Barcelona Basic Model opens a full range of possibilities in modelling unsaturated soil behaviour.     

A thermomechanical crystal plasticity constitutive model for ultrasonic consolidation

We present a micromechanics-based thermomechanical constitutive model to simulate the ultrasonic consolidation process. Model parameters are calibrated using an inverse modeling approach. A comparison of the simulated response and experimental results for uniaxial tests validate and verify the appropriateness of the proposed model. Moreover, simulation results of polycrystalline aluminum using the identified crystal plasticity based material parameters are compared qualitatively with the electron back scattering diffraction (EBSD) results reported in the literature. The validated constitutive model is then used to simulate the ultrasonic consolidation process at sub-micron scale where an effort is exerted to quantify the underlying micromechanisms involved during the ultrasonic consolidation process.     

A cubic spline implementation of non-linear shear behaviour in three-dimensional progressive damage models for composite laminates

An experimental study was carried out to characterise the constitutive response of carbon fibre-reinforced epoxy laminates. While maintaining essentially linear behaviour in the fibre and transverse directions, this material displays significant non-linear shear stress–strain behaviour to rupture. It is shown that the well known Hahn-Tsai non-linear shear model does not provide an acceptable fit for the strain range examined and so a novel approach was derived where a cubic spline interpolation method was used to capture the non-linear shear behaviour. The well known ply discount model, based on Hashin’s failure criteria, was also used to predict fibre and transverse matrix damage in the laminates. The spline approach is coupled with maximum strain failure criteria to predict the response in the in-plane and out-of-plane shear directions. The material Jacobian matrix is fully defined, thus allowing a full implicit material model to be implemented. Hence, the model is suitable for both implicit and explicit finite element codes. It is shown that the model accurately predicts the response of the material for load cases in which shear stresses dominate. The performance of the model is demonstrated by considering a number of laminate configurations and failure of an open-hole tension specimen.     

Micromechanical modelling of viscous sintering and a constitutive equation with sintering stress

The sintering stress is defined for viscous sintering, where the deformation of particles takes place, and its magnitude is computed by the viscoplastic finite element method using a micromechanical model. The computed sintering stress is compared with existing models for other sintering mechanisms. Although modelling of the sintering process is different, a similar tendency of the change in sintering stress with densification is observed. The influence of the sintering mechanism on the sintering stress is discussed. A constitutive law is developed by introducing the sintering stress, approximated by a simple equation, into the constitutive equation for viscous porous materials and applied to the sintering of polycrystalline materials. Grain boundary diffusion and grain growth are taken into consideration through the viscosity in the constitutive equation. The sintering curve calculated by the constitutive equation shows good agreement with the experimental data.     

Mechanical behaviour and 3D stress analysis of multi-layered wooden beams made with welded-through wood dowels

This paper presents experimental and numerical investigations on multi-layered timber beams using welded-through wood dowels in place of traditional poly(vinyl acetate) (PVAc)-adhesives (or metallic nails). Four-layer beams were constructed with varying numbers of dowels, in each, and then loaded using four-points bending tests to evaluate the mechanical performance of these beams. The practical difficulties encountered in constructing deeper multi-layer beams are discussed and possible solutions which have been employed for the purpose of this work, and proved successful are presented. In order to investigate thoroughly the full potential of multi-layered beams with a very limited number of experimental studies, a 3D FE model has been presented, validated against experimental results and then used to study some influential parameters. The results showed that a reasonable bending stiffness of multi-layered beams is achievable with a good combination of material and geometric parameters.     

Analysis of Ultra Low Cycle Fatigue problems with the Barcelona plastic damage model and a new isotropic hardening law

This paper presents a plastic-damage formulation and a new isotropic hardening law, based on the Barcelona plastic damage model initially proposed by Lubliner et al. (1989) [1], which is capable of predicting steel failure due to Ultra Low Cycle Fatigue (ULCF). This failure mechanism is obtained when the material is subjected to cyclic loads and breaks after applying a very low number of cycles, usually less than hundreds. The failure is driven by the plastic response of the material, and it is often predicted based on the plastic strains applied to it. The model proposed in this work has been formulated with the objective of predicting accurately the plastic behavior of the material, as well as its failure due to ULCF. This is achieved taking into account the fracture energy dissipated during the whole loading process. This approach allows the simulation of ULCF when it takes place due to regular cyclic loads or non-regular cyclic loads, as it is the case of seismic loads. Several simulations are conducted in order to show the capabilities of the formulation to reproduce the mechanical response of steel when it is subjected to regular and non-regular cyclic loads. The formulation is validated comparing the numerical results with several experimental tests made on X52 steel specimens. The agreement between the numerical and experimental results asses the validity of the proposed model to predict the plastic behavior of steel and its failure due to Ultra Low Cycle Fatigue.     

Application of a split-Hopkinson tension bar in a mutual assessment of experimental tests and numerical predictions

This paper shows how experimental test results from a split-Hopkinson tension bar (SHTB) and numerical simulations of the test set-up can be used for mutual verification. Firstly, a SHTB where the tension stress wave is generated by pre-stretching a part of the incident bar is briefly presented. This SHTB is used to carry out tensile tests of four aluminium alloys at high rates of strain, while tests at low to medium strain rates were performed in a servo-hydraulic tensile test machine. Using the test results, the parameters of an anisotropic thermoelastic-thermoviscoplastic constitutive relation and a one-parameter fracture criterion are identified for the materials at hand. Subsequently, the material model is used in explicit finite element analyses of the SHTB tests, including the entire experimental set-up and the stress wave propagation during the test. The numerical predictions were found to represent the observed behaviour in the experimental tests fairly well.     

A new constitutive model of austenitic stainless steel for cryogenic applications

A series of uniaxial tensile test under cryogenic temperature was carried out for AISI 304 and 316 austenitic stainless steels (ASS) in this study. Typical non-linear hardening phenomena under the cryogenic environment, such as transformation induced strain hardening and threshold strain for the 2nd hardening, has been observed in a quantitative manner.The important factors affecting the non linear material behavior of austenitic stainless steel including phase transformation, discontinuous yielding and micro-damage are modeled using constitutive equations system based on strain decomposition at the small strain formulation. A strong nonlinearity of strain hardening is described using the coupling of modified Bodner’s plasticity model and phase transformation induced strain model. The strain (threshold strain for onset of 2nd hardening) dependent plasticity model was proposed in the hardening function of Bodner’s model. In order to explicitly express the phase transformation induced strain, TI model (Tomita and Iwamoto model [Y. Tomita, T. Iwamoto, Constitutive modeling of TRIP steel and its application to the improvement of mechanical properties, International Journal of Mechanical Sciences 37 (1995) 1295–1305.]), which is a function of accumulated plastic strain and volume fraction factor of martensite, is selected in this study.Also the unified damage model, which can be connected with elasto-plastic constitutive equation developed in this study, is suggested, and the utility of proposed model was validated by the comparison between experiments and numerical evaluations.     

The application of a three-surface kinematic hardening model to repeated loading of thinly surfaced pavements

This paper examines the application of kinematic hardening to modelling the behaviour of thinly surfaced pavements dominated by the clay subgrade. An existing three-surface kinematic hardening model has been found to predict too much shear strain and therefore too much settlement under repeated loading for certain stress conditions. Under some stress conditions, the model also predicts an accumulation of negative shear strain with increasing number of cycles of load, leading to a pavement rut depth which decreases with increasing numbers of cycles. Consequently, a new model has been developed by modifying the flow rule and hardening modulus. The new model requires 10 parameters, most of which can be determined directly from simple triaxial tests. The new model is validated against drained cyclic triaxial results in order to determine model parameters, and it is shown that the new model predicts better the accumulation of shear strain and the problem of accumulation of negative shear strain is eliminated. This new model is applied to the repeated loading of a thinly surfaced pavement and is seen to predict realistic resilient and permanent deformations.This revised version was published online in September 2005 with a corrected sequence of authors.     

Plastic behavior and constitutive relations of DH-36 steel over a wide spectrum of strain rates and temperatures under tension

To understand the plastic characteristics of DH-36 steel, uniaxial tensile tests have been performed on dog bone samples. The strain rate range is from 0.001 to 3000/s, and the initial specimen temperatures are 293–800 K. To obtain the isothermal flow stress at high strain rates, dynamic recovery technique in Hopkinson Tension Bar has been used, and the interrupt and reloading tests have been performed. The value of strain rate sensitivity has been calculated based on the isothermal stresses at different strain rates. Similar to results from compressive tests, the dynamic strain aging has been observed under tension. Microstructure analysis of the samples after interrupt tests has been carried out by scanning electron microscopy (SEM). The results show that: (1) the strain rate sensitivity value is ∼0.0115 in terms of the isothermal flow stress (uncoupled with temperature) at a given strain, corresponding to 0.0045 coupled with temperature; (2) the 3rd dynamic strain aging (DSA) occurs at some relatively constant strain rates within certain temperature region under tension; DSA shifts to higher temperature or even disappears with increasing strain rates. Finally, in depth analysis of the data based on dislocation mechanisms, it leads to a physically based model which has taken into account the 3rd DSA effects. Good agreement between the theoretical prediction and experimental results has been obtained.     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs

The influence of the cell micro-topology on the in-plane dynamic crushing of honeycombs is studied by means of explicit dynamic finite element simulation using ANSYS/LS-DYNA. Firstly, under the assumption that the edge length and thickness are the same, the dynamic properties of the honeycombs filled by cells with different shapes (equilateral triangular or quadratic cells) and micro-arrangements (regular or staggered arrangement) are numerically analyzed. The full-scale in-plane dynamic crushing of the specimen, as well as the micro-structure transformation during the deformation, is discussed. Based on these, the influence of the cell micro-arrangement on the energy absorption ability of the honeycombs is clarified. The results show that owing to the differences in the micro-topology, triangular or quadratic honeycombs display different local deformation properties during the crushing. The variation of the cell arrangement patterns changes the local dynamic evolution characteristic of stress waves. ‘>’ and ‘<’ mode local deformation bands form at the sides of the stagger-arranged honeycombs, which results in lateral compression shrinkage during the crushing. The plateau stresses also increase with the impact velocity by a square law. The empirical equations for honeycombs filled with different cells (equilateral triangular or quadratic cells) and micro-arrangements (regular or staggered arrangement) at high impact velocities are formulated in terms of impact velocity, and the cell geometrical (edge length and thickness) and topology (edge connectivity) parameters.     

Refining constitutive relation by integration of finite element simulations and Gleeble experiments

Thermo-mechanical coupled finite element calculations were carried out to simulate the Gleeble compression of the samples of a titanium alloy (Ti60), and the results are analyzed and compared with the actual compression tests conducted on a Gleeble 3800 thermo-mechanical simulator. The changes in temperature, stress and strain distribution in the samples and the source of error on the constitutive relations from Gleeble hot compression test were analyzed in detail. Both simulations and experiments showed that the temperature distribution in the specimen is not uniform during hot compression, resulting in significant deformation inhomogeneity and non-ignorable error in the flow stress strain relation, invalidating the uniform strain assumption commonly assumed when extracting the constitutive relation from Gleeble tests. Based on the finite element simulations with iterative corrections, we propose a scheme to refine the constitutive relations from Gleeble tests.     

FEM analysis of dynamic flexural behaviour of composite sandwich beams with foam core

The dynamic flexural behaviour of sandwich beams, with composite face-sheets and a foam core, was analysed by developing a 3D finite-element model. To model the core behaviour, a crushable foam model was used. The Hou criteria were used to predict the failure of the face-sheets. Dynamic bending tests were performed to validate the numerical model. The comparison between numerical and experimental results in terms of contact-force histories, peak-force values, absorbed energy, and maximum displacement of both face-sheets was satisfactory. It was revealed that the collapse of the foam core under the impact region favoured the failure of the upper face-sheet.     

Towards a virtual functionally graded foam: Defining the large strain constitutive response of an isotropic closed cell polymeric cellular solid

Functionally graded materials have been defined by Hirai [1] to be “a new generation of engineered materials wherein the microstructural details are spatially varied through a non-uniform distribution of the reinforcement phase(s)…”. Extending this paradigm to the field of cellular solids, a functionally graded foam material (FGFM) may be thought of as a foam for which microstructural features such as cell size, and strut and wall (for closed cell foams) thickness vary in a continuous manner across the volume of the foam. These features may be varied globally by variation of the foam’s density, ρ_f. Cui et al. [2] and Kiernan et al. [3] have shown some potential benefits of FGFMs under dynamic conditions using discretely layered finite element foam models. In their work the ABAQUS crushable foam model defines the constitutive response of each element layer of a regularly shaped specimen. The density, and corresponding Young’s modulus, E_f, and hardening law of each layer is unique, thus defining a quasi-graded response. Motivated by their results, this paper attempts to describe the large strain compressive response of a gas filled EPS foam such that it may, in a future work, be incorporated into the framework of a user written finite element in which constitutive properties may be graded according to simple polynomials. Experimental data is gathered from a number of expanded polystyrene foam specimens of different densities, and important foam characteristics are defined as functions of ρ_f. Results compare excellently to those of the ABAQUS foam model, and limitations of the modelling approach are discussed.     

An implicit algorithm using explicit correctors for the kinematic hardening model with multiple back stresses

This paper presents an efficient mathematical algorithm for a class of non‐linear kinematic hardening models with multiple back stresses, as an extension of the implicit integration algorithm for a single back stress hardening model. Explicit formulations for general three‐dimensional stress states as well as plane stress and plane strain are given. The new formulation is implemented in a general‐purpose finite element code, ABAQUS, and is verified by comparison with the existing formulation for the single back‐stress constitutive model. Comparison is also made with the experimental results obtained from a plate containing a circular hole subjected to cyclic loading, demonstrating the validity of new method. Copyright © 2001 John Wiley & Sons, Ltd.     

An implementation of the stiffness derivative method as a discrete analytical sensitivity analysis and its application to mixed mode in LEFM

In this work, an improvement in the stiffness derivative method based on a shape design sensitivity analysis is proposed, so that the error inherent in the finite difference procedure is avoided. For a global estimation of G from a given finite element solution, this approach is shown to be equivalent to the well-known J-integral when the latter is numerically implemented through its equivalent domain integral. However, it is verified that its direct application to 2D mixed mode problems of linear elastic fracture mechanics through the field decomposition technique yields estimates for G_I and G_II which are in general more accurate for the proposed method. The importance of the velocity field is also remarked and some suggestions for its choice are given.     

Nonlinear fracture dynamics of laminates with finite thickness adhesives

Finite thickness interfaces, such as structural adhesives, are often simplified from the modeling point of view by introducing ideal cohesive zone models that do not take into account the finite thickness properties in the evaluation of the interface stiffness and inertia. In the present work, the nonlinear dynamic response of those layered systems is numerically investigated according to the finite element method. The weak form of the dynamic equilibrium is written by including not only the contribution of cohesive interfaces related to the virtual work exerted by the cohesive tractions for the corresponding relative displacements, but also considering the work done by the dynamic forces of the finite thickness interfaces resulting from their inertia properties. A fully implicit solution scheme both in space and in time is exploited and the numerical results for the double cantilever beam test show that the role of finite thickness properties is not negligible as far as the crack growth kinetics and the dynamic strength increase factor are concerned.