Strain-based anisotropic damage modelling and unilateral effects

A new continuum damage mechanics model is developed to describe the behaviour of quasi-brittle materials under general path loading. The induced damage is represented by a second rank symmetric tensor. The constitutive equations are based on irreversible thermodynamics theory. The strain-based model covers in an unified way the unsymmetrical behaviour in tension and in compression and the unilateral response due to crack closure effect. Uniaxial stress tests (in tension as in compression) show realistic non-linear responses in the stress–strain space. The different behaviour in both domains is covered by a single set of equations. A significant volume dilatation is noticed in compression. The model can be generalised to time-dependent phenomena.     

Development of an anisotropic constitutive model for single-crystal superalloy for combined fatigue and creep loading

A unified anisotropic viscoplastic constitutive model for single-crystal superalloys is developed based on a modification of a phenomenological isotropic model. Orientation-dependent viscoplastic behaviour was observed in experiments. The model is used to simulate the orientation and cyclic mechanical response of single-crystal SRR99 under combined fatigue and creep conditions at 950°C. The results from the simulations are then used to estimate the fatigue-creep lives of the single-crystal nickel-base superalloy SRR99 using a new life prediction methodology. The predicted fatigue-creep lives are in good agreement with the experimental results.     

The prediction of creep rupture of pressurized tubes using a model for void growth and coalescence

A phenomenological formulation based on a theory of plasticity for voided solids, and power-law creep, to estimate the rupture times and strains for pressurized tubes is presented. Realistic values of the damage parameters involved are selected by satisfying the creep rupture uniaxial data for the same materials. Void growth and coalescence and hence the loss of load-carrying capacity of the tube is taken as a failure criterion. Computed values of rupture times for thin- and thick-walled tubes are shown to be mostly in close agreement with the experimental data found in the literature. Only fair agreement among predicted and experimental fracture strains is observed.     

Nonlinear cumulative damage model for multiaxial fatigue

On the basis of the continuum fatigue damage theory, a nonlinear uniaxial fatigue cumulative damage model is first proposed. In order to describe multiaxial fatigue damage characteristics, a nonlinear multiaxial fatigue cumulative damage model is developed based on the critical plane approach. The proposed model can consider the multiaxial fatigue limit, mean hydrostatic pressure and the unseparated characteristic for the damage variables and loading parameters. The recurrence formula of fatigue damage model was derived under multilevel loading, which is used to predict multiaxial fatigue life. The results showed that the proposed nonlinear multiaxial fatigue cumulative damage model is better than Miner’s rule.     

Sheet metal formability analysis for anisotropic materials under non-proportional loading

Sheet metal formability is conventionally assessed in a two-dimensional plot of principal strains or stresses in comparison to a forming limit curve. This method of assessment implicitly assumes that the forming limit is isotropic in the plane of the sheet. While the assumption of isotropy in the forming limit is perhaps a good engineering approximation, it is intrinsically inconsistent with the use of material models that are anisotropic. Since the trend today is to utilize models with full anisotropy in order to more accurately capture the physics of material behavior, the issue of anisotropy of forming limits must also be addressed. The challenge is that the forming limit is no longer defined by a curve but requires the definition of a surface in strain or stress space, and therefore it is no longer appropriate to view these limits with the convenience of two-dimensional diagrams. Furthermore, recent developments in the characterization of sheet forming limits under non-proportional loading suggest that is advantageous to view forming limit behavior in terms of stresses rather than strains, a view that is adopted in this paper. A solution to the challenge of assessing formability for an anisotropic material is proposed that rescales the stresses by a factor so that the scaled stresses have the same relationship to a single forming limit curve in a 2D plot in stress-space, as the actual stresses have to the true anisotropic forming limit in 3D space. The rescaling enables engineers to accurately view the formability of all the elements at the same time for a given finite element analysis of an application. This paper also discusses other challenges of using stresses in the assessment of formability, focusing on an analysis of the 2-Stage Forming Benchmark highlighted in the Numisheet ’99 Conference. Stresses are found in this application to unload to non-critical values after reaching critical levels earlier in a forming process, which suggests that a full integration of the stress-based forming limit criterion with FE simulation is required to detect critical states that may temporarily occur during the forming process.     

A theory of smeared continuum damage mechanics

This paper considers smeared continuum damage mechanics based on the equivalent elliptical crack representation of a local damage. This approach provides a means of utilizing the crack energies derived in fracture mechanics, and of identifying the local damage state from local stress and strain information. The strain energy equivalence principle is used to derive the effective continuum elastic properties of a damaged solid in terms of the undamaged elastic properties and a scalar damage variable. The scalar damage variable is used to develop a consistent damage evolution equation. The combination of representing local damage as an equivalent elliptical crack, the determination of effective elastic properties using a strain energy equivalence principle, and a consistent damage evolution equation yields a simple, yet powerful local approach for continuum damage analysis     

An approximate method for computing structural creep deformation due to non-proportional cyclic loading

A method is proposed for estimating structural creep deformation due to histories of non-proportional cyclic loading. The method applies to structures composed of materials whose creep strain is given by an equation of the form: . Knowledge of the form of the creep law for time varying stress is not required, as use is made of data obtained from a single cyclic creep test.     

High-temperature rupture of 5083-Al alloy under multiaxial stress states

High-temperature rupture behavior of 5083-A1 alloy was tested for failure at 548K under multiaxial stress conditions : uniaxial tension using smooth bar specimens, biaxial shearing using double shear bar specimens, and triaxial tension using notched bar specimens. Rupture times were compared for uniaxial, biaxial, and triaxial stress conditions with respect to the maximum principal stress, the von Mises effective stress, and the principal facet stress. The results indicate that the von Mises effective and principal facet stresses give good correlation for the material investigated, and these parameters can predict creep life data under the multiaxial stress states with the rupture data obtained from specimens under the uniaxial stress. The results suggest that the creep rupture of this alloy under the testing condition is controlled by cavitation coupled with highly localized deformation process, such as grain boundary sliding. It is also conceivable that strain softening controls the highly localized deformation modes which result in cavitation damage in controlling rupture time of this alloy.     

A novel evolutionary algorithm for determining unified creep damage constitutive equations

The determination of material constants within unified creep damage constitutive equations from experimental data can be formulated as a problem of finding the global minimum of a well defined objective function. However, such an objective function is usually complex, non-convex and non-differentiable. It is difficult to be optimised by classical gradient-based methods. In this paper, the difficulties in the optimisation are firstly identified. Two different objective functions are proposed, analysed and compared. Then three evolutionary programming algorithms are introduced to solve the global optimisation problem. The evolutionary algorithms are particularly good at dealing with problems which are complex, multi-modal and non-differentiable. The results of the study shows that the evolutionary algorithms are ideally suited to the problem. Computational results of using the algorithms to determine the material constants in a set of physically based creep damage constitutive equations from experimental data for an aluminium alloy are presented in the paper to show the effectiveness and efficiency of the three evolutionary algorithms.     

A variationally based nonlocal damage model to predict diffuse microcracking evolution

We explore a variationally based nonlocal damage model, based on a combination of a nonlocal variable and a local damage variable. The model is physically motivated by the concept of “nonlocal” effective stress. The energy functional which depends on the displacement and the damage fields is given for a one-dimensional bar problem. The higher-order boundary conditions at the boundary of the elasto-damaged zone are rigorously derived. We show that the gradient damage models can be obtained as particular cases of such a formulation (as an asymptotic case). Some new analytical solutions will be presented for a simplified formulation where the stress–strain damage law is only expressed in a local way. These Continuum Damage Mechanics models are well suited for the tension behaviour of quasi-brittle materials, such as rock or concrete materials. It is theoretically shown that the damage zone evolves with the load level. This dependence of the localization zone to the loading parameter is a basic feature, which is generally well accepted, from an experimental point of view. The computation of the nonlocal inelastic problem is based on a numerical solution obtained from a nonlinear boundary value problem. The numerical treatment of the nonlinear nonlocal damage problem is investigated, with some specific attention devoted to the damageable interface tracking. A bending cantilever beam is also studied from the new variationally based nonlocal damage model. Wood’s paradox is solved with such a nonlocal damage formulation. Finally, an anisotropic nonlocal tensorial damage model with unilateral effect is also introduced from variational arguments, and numerically characterized in simple loading situations.     

A lower bound to the creep rupture time of pressurised thick cylinders

A lower bound to the creep rupture time of internally pressurised thick cylinders is derived. Material behaviour is described by a phenomenological creep rupture theory that accounts for all phases of creep, and for the full coupling between the deformation and damage processes. To obtain the desirable lower bound, the effective stress and the equivalent rupture stress, which represent the effects of multiaxial stress states on the creep strain and damage rates, respectively, were approximated by the maximum shear stress in the constitutive equations. By comparing the lower bound estimations for a wide range of cylinder dimensions and different engineering materials with the rupture times determined from accurate finite element calculations, it is shown that the lower bound estimations provide quite conservative lifetime predictions.     

Creep damage evolution equations expressed in terms of dissipated power

Evolution equations of continuum damage mechanics are usually expressed in terms of stresses; some authors introduce other quantities, like strains or elastic energy. The present paper introduces evolution equations expressed in terms of specific dissipated power: for uniaxial stationary creep such equations may simply be derived by some substitutions. It turns out that basing on available experimental data such an approach results in reduction of the number of parameters. Then the extensions to nonstationary creep and to multiaxial states are proposed as hypotheses subject to experimental verification. Finally, an extension to biomechanics is proposed, namely to biological materials subject to recovery. For such materials the damage rate is not necessarily positive: it may also be negative as a result of recovery.     

A thermal creep model for the contact of nominally flat surfaces: Jeffreys' linear visco-elastic model

This work considers the mechanics of contact of thermo-visco-elastic materials. In particular the creep behavior of a nominally flat rough surface in contact with a rigid half space is studied. The rough surface is modeled using fractal geometry. A synthesized profile, a Cantor structure, is utilized to model the surface. Such a profile has two scaling parameters and different heights for each generation of asperities. The effect of temperature will be included through the concept of activation energy using the Arrhenius equation.The objective of this model is to study the normal creep approach of the surface (punch) as a function of the applied creep load, time, and temperature. The material of the punch is assumed to behave according to Jeffreys' model. Such a model is an arrangement of springs and dashpots in parallel and/or in series.The creep approach of linearly visco-elastic materials is explored using elastic-visco-elastic correspondence analysis. An asymptotic power law is obtained, which relates the force and the bulk temperature acting on the punch to its approach. This model is valid only when the approach between the punch and the half space is in the range of the roughness size. The proposed model admits an analytical solution for the case when the deformation is linear thermo-visco-elastic. The obtained model shows a good agreement when compared with experimental results from literature.     

Discussion of cyclic plasticity and viscoplasticity of single crystal nickel-based superalloy in large strain analysis: comparison of anisotropic macroscopic model and crystallographic model

The inelastic behavior of nickel-based superalloy is investigated in detail by application of a macroscopic anisotropic plasticity model developed here, and the results are compared to predictions based on crystal plasticity, which incorporates the kinematic hardening. Uniaxial deformation processes and simple shear deformations at large strain are considered. The plastic spin concept coupled with an anisotropic Chaboche model is provided in the framework of macroscopic viscoplasticity. The plastic spin formulation used here is based on the concept of the noncoaxiality between the stress and plastic rate of deformation. The present model succeeds in reproducing the inelastic behavior during large deformation. It is shown that the plastic spin associated with the anisotropic flow rule plays a key role in the macroscopic model. Simulation results find these two different scale models provide similar predictions under uniaxial deformation for [0 0 1] and [1 1 1] orientation, while their predictions for simple shear deformation at large strain exhibit quantitative difference, but their trends are the same. The interpretations for simulation results are pursued in detail.     

Constitutive modelling of the anisotropic creep behaviour of nickel-base single crystal superalloys

This paper focuses on the modelling of primary, secondary and tertiary creep of nickel-base single crystal superalloys at high temperatures. In particular, we propose an extension of the Cailletaud single crystal plasticity model [Méric L, Poubanne P, Cailletaud G. Single crystal modeling for structural calculations: part I—model presentation. Transactions of the ASME 1991;133:162-170] to include tertiary creep. This is achieved by introducing an additional evolution equation for a scalar damage variable per slip system. In addition, a methodology for the calibration of the material parameters of the model to fit the results from experiments has been implemented. The parameter identification rests upon a two-membered evolution strategy. The comparison with uniaxial and multiaxial test data shows a good agreement between model and experiment. The structural simulations have been performed by means of a special element technology which enables efficient and accurate finite element computations.     

Failure prediction for multi-material creep test specimens using a steady-state creep rupture stress

The steady-state stress distributions in single- and multi-material notched and waisted specimens were investigated for practical creep test specimens using material properties obtained for materials from a service-aged CrMoV pipe weldment. The tri-axial stress conditions existing in notched and waisted specimens machined from welded pipes were identified. By using a steady-state effective failure stress, it has been shown that an approximate method, based on steady-state stress, for the prediction of rupture life and failure position can produce reasonably accurate results. The applicability of the approximate method was confirmed by comparing the results obtained using it with those obtained from corresponding creep continuum damage modelling. These results indicate that use of steady-state stress analysis, as an approximate technique, may be useful for assessing creep failure behaviour, for determining the effect of specimen size and for generating material properties for welded components.     

A new cyclic anisotropic model for plane strain sheet metal forming

A finite difference model was developed for sheet metal subjected to plane strain cyclic bending under tension. The model was used to describe the effects on the mechanics of the deformation of the sheet due to mixed isotropic-kinematic cyclic hardening curves. The analytical results were compared with experimental results obtained under testing conditions closely representative of the analytical model. Two tests were used: a pure bending moment device and a bending/unbending under tension device consisting of three cylindrical pins. The model was used to determine a constitutive curve that best characterize the cyclic behavior of the material tested, as compared with the experimental results. The significance of these results were discussed in relation to the prediction of the restraining forces in the sheet as it is drawn through the blank holder drawbeads.     

The application of an approximate analysis of the diffusion process for a description of creep and creep rupture

An analysis of the environmental effect on creep and creep rupture of metals is presented. It is pointed that the investigation of the diffusion processes leads to significant computational difficulties. These difficulties arise on representing the solution of the diffusion equation with variable boundaries in the form convenient for analysis.The process of damage accumulation is modelled for a material, which is subjected to the combined action of mechanical loads and the aggressive environment. The Rabotnov kinetic theory is applied in which two parameters are taken into account: the damage of the material and the concentration of chemical elements reducing the resistance of the material to mechanical loads. An approximate solution of the diffusion equation is suggested for rod or shell subjected to axial tension. This solution is based on dividing the cross-section of rod or shell into the disturbed and undisturbed parts and determining the motion of the boundary between these parts. The system of constitutive equations which describe the interaction between the diffusion and rupture fronts during the creep process until failure is obtained. A good quantitative agreement between experimental and theoretical results has been received.     

Numerical modelling of cyclic plasticity and fatigue damage in cold-forging tools

Industrial cold-forging tools with complex geometry are very likely to be exposed to local plastic deformation near stress concentrating details. The accumulation of plastic deformation resulting from the cyclic loading conditions leads to fatigue damage and eventually to generation of a crack in the surface of the die. To study the effect of die prestressing on fatigue damage development, a plane strain finite element model of a cold-forging die is analysed. A complex material model, combining kinematic and isotropic hardening with continuum damage mechanics, is used to simulate the elastic–plastic material behaviour and damage development. Furthermore, a range of uncoupled damage measures are applied in the comparison of conventional and new prestressing concepts.