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In this paper, the high-temperature constitutive behaviour of an additively manufactured ductile nickel-based superalloy is investigated and modelled, with application to thermomechanical fatigue, low-cycle fatigue and creep conditions at temperatures up to ${800^ \circ }$ 800 ∘ C. Thermomechanical fatigue tests have been performed on smooth specimens in both in-phase and out-of-phase conditions at a temperature range of $100 - {800^ \circ }$ 100 − 800 ∘ C, and creep tests at ${625^ \circ }$ 625 ∘ C, ${700^ \circ }$ 700 ∘ C, ${750^ \circ }$ 750 ∘ C and ${800^ \circ }$ 800 ∘ C. Additionally, low-cycle fatigue tests at different strain ranges and load ratios have been performed at ${700^ \circ }$ 700 ∘ C, and tensile tests have been performed at ${600^ \circ }$ 600 ∘ C, ${700^ \circ }$ 700 ∘ C and ${800^ \circ }$ 800 ∘ C. A clear anisotropic mechanical response is obtained in the experiments, where the anisotropic effects are larger at high stress levels in creep loadings. To capture this behaviour, a rate-dependent strain based on a double-Norton model has been adopted in the model, by which the creep and mid-life response of the thermomechanical fatigue tests can be simulated with good accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a temperature-dependent viscoplasticity model is presented that describes thermal and cyclic softening of the hot work steel X38CrMoV5-3 under thermomechanical fatigue loading. The model describes the softening state of the material by evolution equations, the material properties of which can be determined on the basis of a defined experimental program. A kinetic model is employed to capture the effect of coarsening carbides and a new isotropic cyclic softening model is developed that takes history effects during thermomechanical loadings into account. The temperature-dependent material properties of the viscoplasticity model are determined on the basis of experimental data measured in isothermal and thermomechanical fatigue tests for the material X38CrMoV5-3 in the temperature range between 20 and 650 ∘ C. The comparison of the model and an existing model for isotropic softening shows an improved description of the softening behavior under thermomechanical fatigue loading. A good overall description of the experimental data is possible with the presented viscoplasticity model, so that it is suited for the assessment of operating loads of hot forging tools. [ABSTRACT FROM AUTHOR]

• Cyclic experiments of 316LN SS are conducted at different temperatures. • The effect of dynamic strain aging on cyclic deformation is analyzed. • A unified constitutive model is proposed to describe DSA at different temperatures. • A hardening parameter is introduced to improve the forecasting accuracy of DSA. Dynamic strain aging (DSA) widely exists in austenitic stainless steel at elevated temperature. The materials will be drastically strengthened by this phenomenon. A series of symmetric strain-controlled and asymmetric stress-controlled cyclic experiments were conducted at different temperatures from 293 K to 823 K for 316LN stainless steel. The DSA phenomenon was observed in all the tests under elevated temperatures and it became more significant with the increase of temperature until reached the maximum at 823 K. There were a series of manifestations of DSA such as the anomalous cyclic hardening with temperature increasing in the strain-controlled tests and the quick-shakedown of ratcheting strain in the stress-controlled tests. An appropriate constitutive model to describe the cyclic behavior at different temperatures is necessary. The proposed kinematic hardening model is based on Ohno-Wang model and modified by introducing a hardening index into dynamic recovery term. The index accounts for DSA hardening degree of material under high temperature, therefore the performance mentioned above can be simulated well by this proposed model. By relating the parameters with temperature and plastic strain amplitude, a unified visco-plastic constitutive model was proposed. Simulation results of the proposed model agree much better with experimental values. [ABSTRACT FROM AUTHOR]

In this work, we use polycrystal modeling to study the interactions between slip and twinning during load reversals of commercially pure titanium. The constitutive response incorporates anisotropic elasticity, crystal plasticity, a dislocation density based hardening law for prismatic slip, basal slip, and pyramidal type I 〈 c + a 〉 slip, and micromechanical model for twin reorientation on two types: { 101 2 ¯ } extension twinning and { 11 2 ¯ 2 } contraction twinning. The key feature of the model is the inclusion of slip-system level backstress development due to dislocation density accumulation. To demonstrate, the model is used to simulate the stress-strain response and texture evolution in a series of compression-tension and tension-compression tests carried out to different strain levels and applied in two different load directions to a strongly textured CP-Ti plate. Material parameters associated with the slip strengths for the three slip modes are reported. The model identifies the few systems within the pyramidal 〈 c + a 〉 slip mode as developing the most backstress among the three slip modes. It also indicates that the backstresses that develop in the forward loading path promote pyramidal slip in the reversal loading path. We also find that reverse loading changes negligibly the relative slip mode contributions from monotonic loading but it strongly affects the twinning-detwinning behavior. This work highlights the ability of polycrystal modeling to account for the co-dependent nature of multiple crystallographic slip and twinning modes, the hysteresis in plastic response during the forward-reversal cycle, and the two sources of hardening engendered by history-dependent dislocation density accumulation. • A polycrystal model is presented to study interactions between slip and twinning during strain reversal path change tests of commercially pure titanium (CP-Ti). • The model incorporates anisotropic elasticity; the prismatic, basal and pyramidal type I slip modes extension and contraction twinning mode. • A single set of material parameters are presented that reproduce the eight distinct strain reversal sequences. • The model identifies the pyramidal slip mode as developing the most backstress among the three slip modes. • The model indicates that the backstress results in a Bauschinger effect and can promote contraction twinning. [ABSTRACT FROM AUTHOR]

The onset of macroscopic strain localization limits the ductility of many ductile materials. For porous ductile materials, two distinct mechanisms of macroscopic localization have been identified: void growth induced softening and void coalescence. In this work we focus on analyzing the influence of material's strain rate sensitivity (SRS) on the two mechanisms of macroscopic localization or ductile failure as a function of the imposed stress triaxiality. To this end, three dimensional finite element calculations of unit cells have been carried out to model void growth and coalescence in an infinite block containing a periodic distribution of initially spherical voids in a band. The matrix material of the unit cell is considered to follow a strain rate dependent elastic perfectly plastic flow response. The unit cell calculations are carried out for a range of SRS parameter, imposed stress triaxiality and initial orientations of the voided band. Our results show that both the critical porosity and strain at the onset of localization and coalescence are strongly influenced by the SRS parameter and the imposed stress triaxiality values. Furthermore, the relative effect of the SRS parameter is found to increases with the increasing value of the imposed stress triaxiality. • Increasing SRS delays the onset of localization and void coalescence. • SRS affects more the onset of localization than the void coalescence. • Effect of SRS increases with increasing stress triaxiality. [ABSTRACT FROM AUTHOR]

In order to investigate the effect of non-proportional loading history on cyclic deformation of 316L stainless steel, a series of multiaxial cyclic experiments with variable loading path are conducted at room temperature. It is found that the materials exhibit cyclic hardening at initial stage, followed by a long period of cyclic softening. Significant cyclic softening is observed when non-proportional loading path changes into proportional loading path and the rate of cyclic softening is related to non-proportional loading history. A visco-plasticity constitutive model based on Ohno-Wang kinematic hardening rule and Marquis isotropic hardening rule is used to characterize non-proportional cyclic behavior. Simulation results agree with experiments very well in terms of cyclic hardening effects. However, this model is found to be less effective in simulating cyclic softening effect, especially considering the influence of non-proportional loading history. A modified isotropic hardening model is thus proposed to simulate the cyclic softening behavior of 316L stainless steel by introducing a memory non-proportionality and qualifying a partial recoverable softening term of isotropic hardening rule. Simulation results of the proposed model agree much better with experimental values. • Multiaxial cyclic tests of 316L stainless steel are conducted for variable loading. • The effect of non-proportional loading on subsequent cyclic softening is analyzed. • Non-proportional cyclic behavior is described by a visco-plastic constitutive model. • Cyclic softening with non-proportional loading is characterized by a modified model. [ABSTRACT FROM AUTHOR]

Abstract Stress hotspots are regions of stress concentrations that form under deformation in polycrystalline materials. We use a machine learning approach to study the effect of preferred slip systems and microstructural features that reflect local crystallography, geometry, and connectivity on stress hotspot formation in hexagonal close packed materials under uniaxial tensile stress. We consider two cases: a hypothetical HCP material without any preferred slip systems with a critically resolved shear stress (CRSS) ratio of 1:1:1, and a second with CRSS ratio 0.1:1:3 for basal: prismatic: pyramidal slip systems. Random forest based machine learning models predict hotspot formation with an AUC (area under curve) score of 0.82 for the Equal CRSS and 0.81 for the Unequal CRSS cases. The results show how data driven techniques can be utilized to predict hotspots as well as pinpoint the microstructural features causing stress hotspot formation in polycrystalline microstructures. Graphical abstract Image 1 Highlights • Data driven methods can provide insights into stress hotspot formation. • Machine learning models should be trained separately for each material. • Stress hotspots are more pronounced under limited number of slip systems. • Crystal plasticity determines the grain textures prone to hotspot formation. • Stress hotspots tend to form in smaller grains. [ABSTRACT FROM AUTHOR]

Abstract It is important to investigate the constitutive behavior of high density polyethylene (HDPE) under cyclic loading since it is widely used for industrial applications. However, there are only a few preliminary investigations on the constitutive behavior of HDPE under cyclic loading. This work aims at investigating the deformation mechanism, developing the constitutive model especially for the cyclic-loading behavior, and proposing an efficient approach for model calibration. Firstly, the deformation mechanism was successfully explored by conducting a special designed relaxation-unloading test by which the time-independent elastic-plastic behavior was decoupled from the time-dependent overall deformation behavior. Then, a nonlinear elastic-viscoplastic constitutive model was developed based on the parallel rheological framework (PRF) composed of an elastic-plastic network in parallel with multiple nonlinear viscoelastic networks. Finally, an approach was proposed to calibrate the constitutive model by means of relaxation-unloading tests and material parameter optimizations. The agreement between the prediction and the test data successfully demonstrated that the model could be used to precisely simulate the constitutive behavior of HDPE under static, quasi-static and dynamic cyclic loading. The developed constitutive model is beneficial to the long-term durability evaluation of HDPE. The methodology for exploring the deformation mechanism and calibrating the constitutive model may also be applicable to other polymers. Highlights • The elastic-plastic behavior is identified from overall elastic-viscoplastic behavior by the relaxation-unloading test. • The parallel rheological framework is developed to predict cyclic-loading behavior of high density polyethylene. • An approach is proposed based on tests and parameters optimization to calibrate the constitutive model accurately. [ABSTRACT FROM AUTHOR]

A dislocation-based creep model combined with a continuum damage formulation was developed and implemented in the finite element method to simulate high temperature deformation behavior in modified 9Cr-1Mo steel welds. The evolution of dislocation structures was considered as the main driving mechanism for creep. The effect of void growth, precipitate coarsening, and solid solution depletion were considered to be the operating damage processes. A semi-implicit numerical integration scheme was developed and implemented in the commercial finite element code ABAQUS-Standard as a user material subroutine. Furthermore, several creep tests of modified 9Cr-1Mo steel welded specimens were conducted at temperatures between 550–700 °C and stresses between 80–200 MPa. The accuracy of the model was verified by comparing the finite element results with experiments. The comparison between the experimental and computational results showed excellent agreement. The model can be used to simulate and predict the creep-damage behavior of Cr-Mo steel components used as structural applications in power plants.

Abstract We investigate the formation of stress hotspots in polycrystalline materials under uniaxial tensile deformation by integrating full field crystal plasticity based deformation models and machine learning techniques to gain data driven insights about microstructural properties. Synthetic 3D microstructures are created representing single phase equiaxed microstructures for generic copper alloys. Uniaxial tensile deformation is simulated using a 3-D full-field, image-based Fast Fourier Transform (FFT) technique with rate-sensitive crystal plasticity, to get local micro-mechanical fields (stress and strain rates). Stress hotspots are defined as the grains having stress values above the 90th percentile of the stress distribution. Hotspot neighborhoods are then characterized using metrics that reflect local crystallography, geometry, and connectivity. This data is used to create input feature vectors to train a random forest learning algorithm, which predicts the grains that will become stress hotspots. We are able to achieve an area under the receiving operating characteristic curve (ROC-AUC) of 0.74 for face centered cubic materials modeled on generic copper alloys. The results show the power and the limitations of the machine learning approach applied to the polycrystalline grain networks. Graphical abstract Image 1 Highlights • Machine learning can predict grains that become stress hotspots under deformation. • Data driven insights can delineate the factors responsible for hotspot formation. • Both the texture and grain geometry contribute to stress hotspot formation. • Hotspots form in small grains with low misorientation and compatible slip systems. • Simulation artifacts are also captured by data driven insights. [ABSTRACT FROM AUTHOR]

Abstract In this paper a new multiscale approach is presented for the analysis of structures made of composite material characterized by elastoplastic or viscoplastic nonlinear response. Scale separation is assumed so that the homogenization theory can be applied: at the structural scale (macroscale) the material appears as homogeneous, while at the microscale it is characterized by a heterogeneous microstructure that affects the global behavior of the structure. The resolution of the micromechanical problem is performed developing a reduced order homogenization technique based on a mixed variational formulation, named Mixed Transformation Field Analysis (MxTFA). The microscopic reduced internal variables are the stress and plastic multiplier parameters of RVE subsets, whose evolution is computed enforcing the weak form of the governing equations for every subset. Some numerical tests are performed to show the accuracy and efficiency of the proposed multiscale technique. [ABSTRACT FROM AUTHOR]

A crystal viscoplasticity (CVP) model for the creep-fatigue interactions of nickel-base superalloy CMSX-8 is proposed. At the microstructure scale of relevance, the superalloys are a composite material comprised of a γ phase and a γ′ strengthening phase with unique deformation mechanisms that are highly dependent on temperature. Considering the differences in the deformation of the individual material phases is paramount to predicting the deformation behavior of superalloys at a wide range of temperatures. In this work, we account for the relevant deformation mechanisms that take place in both material phases by utilizing two additive strain rates to model the deformation on each material phase. The model is capable of representing the creep-fatigue interactions in single-crystal superalloys for realistic 3-dimensional components in an Abaqus User Material Subroutine (UMAT). Using a set of material parameters calibrated to superalloy CMSX-8, the model predicts creep-fatigue, fatigue and thermomechanical fatigue behavior of this single-crystal superalloy. Finally, a sensitivity study of the material parameters is done to explore the effect on the deformation due to changes in the material parameters relevant to the microstructure. [ABSTRACT FROM AUTHOR]

We consider a model of a dynamic frictional contact between the body and the foundation. In this model the contact is bilateral. The behaviour of the material is described by the elastic-viscoplastic constitutive law with thermal effect. The variational formulation of this model leads to a system of two evolution hemivariational inequalities. The aim of this paper is to prove that this system of inequalities has a unique solution. The proof is based on the Banach fixed point theorem and some results for hemivariational inequalities. [ABSTRACT FROM AUTHOR]

Existing continuum level phase field plasticity theories seek to solve plastic strain by minimizing the shear strain energy. However, rigorously speaking, for thermodynamic consistency it is required to minimize the total strain energy unless there is proof that hydrostatic strain energy is independent of plastic strain which is unfortunately absent. In this work, we extend the phase-field microelasticity theory of Khachaturyan et al. by minimizing the total elastic energy with constraint of incompressibility of plastic strain. We show that the flow rules derived from the Ginzburg-Landau type kinetic equation can be in line with Odqvist's law for viscoplasticity and Prandtl-Reuss theory. Free surfaces (external surfaces or internal cracks/voids) are treated in the model. Deformation caused by a misfitting spherical precipitate in an elasto-plastic matrix is studied by large-scale three-dimensional simulations in four different regimes in terms of the matrix: (a) elasto-perfectly-plastic, (b) elastoplastic with linear hardening, (c) elastoplastic with power-law hardening, and (d) elasto-perfectly-plastic with a free surface. The results are compared with analytical/numerical solutions of Lee et al. for (a-c) and analytical solution derived in this work for (d). In addition, the J integral of a fixed crack is calculated in the phase-field model and discussed in the context of fracture mechanics. [ABSTRACT FROM AUTHOR]

A new homogenization technique, based on the Transformation Field Analysis, able to determine the overall behavior of viscoplastic heterogeneous materials, is proposed. This method is derived developing a variational formulation of the compatibility and evolution equations governing two classical elastic–viscoplastic models, i.e. Perzyna and Perić models. A representation form for the stress field and the plastic multiplier is defined on the representative volume element (RVE) of the composite material, resulting in a significant reduction of the history variables ruling the evolution problem. A numerical procedure is developed integrating the evolution equation by a backward-Euler algorithm and solving the single time step developing a specific form of the predictor corrector technique. The accuracy of the proposed homogenization method is assessed through some numerical examples for elastic–viscoplastic composites. The applications aim to verify the effectiveness of the presented technique in modeling the elastic–viscoplastic composite behavior adopting a reduced number of history variables. The homogenization results are compared with the ones obtained by nonlinear finite element micromechanical analyses. [ABSTRACT FROM AUTHOR]

The coupled thermo-mechanical strain gradient plasticity theory that accounts for micro-structure based size effects is outlined within this work. This incorporates spatial gradients of selected micro-structural fields based on length-scales that describe the evolving dissipative mechanisms. In the mechanical part, the model problem of von Mises plasticity with gradient-extended hardening/softening response is considered as discussed in Miehe et al. (2013, 2014a). In the thermal part, we follow the investigations of Simó and Miehe (1992) that demonstrate the effect of temperature on the mechanical fields resulting in a thermal expansion. To this end, two classes of solution schemes for the coupled problem are considered: (i) Global product formula algorithm arising from operator split which leads to a two step solution procedure, and (ii) an implicit coupled algorithm which employs simultaneous solution of the coupled system of equations. In the product formula algorithm, the mechanical and thermal problems are solved separately, resulting in a symmetric problem. However, in the implicit coupled algorithm, a simultaneous solution of the coupled system of equations for gradient thermo-plasticity is employed. A noteworthy drawback of this solution scheme arises from the high computational efforts in comparison with the product formula algorithm. From the computational viewpoint, the standard Galerkin finite element method fails in the context of isochoric plastic flow due to the over-constrained pressure field. To circumvent these difficulties, we extend the well-known Q1P0-type and MINI-type mixed finite elements design of gradient plasticity to account for thermal effects. The performance of the formulation is demonstrated by means of some representative examples. [ABSTRACT FROM AUTHOR]

With the advantages of high-formability, low-cost and unique physical properties, polymers have been widely used in microforming of polymeric components for a large scale of applications in many fields including micro-optics, microfluidic and sensors, etc. In micro-scale, the deformation behaviors of polymers are observed to be size-dependent. Conventional constitutive models of polymers, however, cannot predict and represent those size-dependent behaviors well. To address this issue, a constitutive model with consideration of size effect for amorphous polymers in micro-scale was developed in this research. Firstly, on the basis of the couple stress theory, the impact of rotational gradients was taken into consideration and a strain gradient “elastic-viscoplastic” constitutive model was proposed to quantitatively describe the size-dependent behaviors of amorphous polymers in micro-scale. After that, four point micro-bending experiments were implemented on poly (methyl methacrylate) (PMMA) films with thickness varying from the millimeter scale to micrometer scale. The size effect of PMMA in micro-scale was further illustrated and the proposed strain gradient “elastic-viscoplastic” model was finally validated and verified for the capability of modeling of the size effect of amorphous polymers in micro-scaled deformation. This research thus advances the understanding of the size effect and the strain gradient based mechanical behaviors of amorphous polymers and facilitates its applications in industries. [ABSTRACT FROM AUTHOR]

Overall principles of build-up of the physical equations are offered. Essential modeling hypotheses are made. The statement of a defining relation for considered model of a material is given. The example of a numerical modeling of process of simple shearing strain is resulted.

The large strain behavior of single-phase OFHC copper under different deformation processes are studied based on various polycrystal plasticity models including the classic Taylor-type model and various popular self-consistent models. The models are evaluated by comparing the predictions for the evolution of crystallographic texture and the stress–strain response in uniaxial compression and tension, plane strain compression, and simple shear against corresponding experiments. It is found that while the Taylor-type model is in reasonable first-order agreement with the experiments for the evolution of texture and the overall stress–strain response, the self-consistent models with grain interaction stiffness halfway between those of the limiting secant (stiff) and tangent (compliant) approximations give better results. [ABSTRACT FROM AUTHOR]

Recent experiments and atomic scale computations indicate that the standard continuum models of diffusion in stressed solids do not accurately describe transport, deformation and stress in Li–Si alloys. We suggest that this is because classical models do not account for the irreversible changes in atomic structure of Si that are known to occur during a charge–discharge cycle. A more general model of diffusion in an amorphous solid is described, which permits unoccupied Si lattice sites to be created or destroyed. This may occur as a thermally activated process; or as a result of irreversible plastic deformation under stress. The model predicts a range of phenomena observed in experiment that cannot be captured using classical models, including irreversible changes in volume resulting from a charge–discharge cycle, asymmetry between the tensile and compressive yield stress, and a slow evolution in mechanical and electrochemical response over many charge–discharge cycles. [ABSTRACT FROM AUTHOR]

A dislocation-based creep model combined with a continuum damage formulation was developed and implemented in the finite element method to simulate high temperature deformation behavior in modified 9Cr-1Mo steel welds. The evolution of dislocation structures was considered as the main driving mechanism for creep. The effect of void growth, precipitate coarsening, and solid solution depletion were considered to be the operating damage processes. A semi-implicit numerical integration scheme was developed and implemented in the commercial finite element code ABAQUS-Standard as a user material subroutine. Furthermore, several creep tests of modified 9Cr-1Mo steel welded specimens were conducted at temperatures between 550-700 °C and stresses between 80-200 MPa. The accuracy of the model was verified by comparing the finite element results with experiments. The comparison between the experimental and computational results showed excellent agreement. The model can be used to simulate and predict the creep-damage behavior of Cr-Mo steel components used as structural applications in power plants. [ABSTRACT FROM AUTHOR]

In this paper, the issue of the identification of constitutive parameters of the Anand visco-plastic model is addressed using the Virtual Fields Method (VFM) in an infinitesimal deformation framework. By using VFM, one can take advantage of heterogeneous strain fields obtained by full-field experimental techniques, such as Digital Image Correlation (DIC). Since a wide range of strains and strain rates are sampled in a typical heterogeneous strain field, the number of experiments required to reliably estimate constitutive parameters, especially of rate-dependent materials, is significantly smaller than that needed if conventional experiments (such as uniaxial tension or pure shear configurations) leading to nominally homogeneous strain states were used. However, for such an approach to be successful, the test configuration and loading program should be such that all the constitutive parameters play a significant role (are ‘activated’) in the resulting strain fields. An analysis of the Anand constitutive model shows that 4 of the 8 parameters can only be found to within a multiplicative constant from full-field kinematic data. Therefore, one of these 4 constants is arbitrarily chosen and the activation of the remaining 7 material parameters is investigated by performing a series of one-element models. Detailed sensitivities of the VFM cost function to these material parameters are derived for a variety of normal stress to shear stress ratios and loading rates. Two main conclusions are drawn based on this one-element study: (i) the VFM cost function sensitivities to the material parameters do not vary significantly with loading ratios or rates, and (ii) 2 of the 7 material parameters are not activated for any of the loading ratios or rates considered. Based on the results of the finite-element study, a modified single lap-shear test configuration is designed to yield heterogeneous strains in the joint. Deformation data from a finite-element analysis of this experiment are used as inputs to a VFM routine to compute the Anand material parameters. Our results highlight that non-uniqueness of the identified parameters is a significant issue. The effect of the choice of the cost function and the loading profile on the inverse technique is also thoroughly investigated. [ABSTRACT FROM AUTHOR]

For a compression-shear mixed mode interface crack, it is difficult to solve the stress and strain fields considering the material viscosity, the crack-tip singularity, the frictional effect, and the mixed loading level. In this paper, a mechanical model of the dynamic propagation interface crack for the compression-shear mixed mode is proposed using an elastic-viscoplastic constitutive model. The governing equations of propagation crack interface at the crack-tip are given. The numerical analysis is performed for the interface crack of the compression-shear mixed mode by introducing a displacement function and some boundary conditions. The distributed regularities of stress field of the interface crack-tip are discussed with several special parameters. The final results show that the viscosity effect and the frictional contact effect on the crack surface and the mixed-load parameter are important factors in studying the mixed mode interface cracktip fields. These fields are controlled by the viscosity coefficient, the Mach number, and the singularity exponent. [ABSTRACT FROM AUTHOR]

In this study, the mechanical response of an additively manufactured nickel-based combustor alloy, subjected to thermomechanical fatigue (TMF) loadings has been investigated and modelled. TMF tests were performed in both in-phase and out-of-phase conditions with different strain ranges and temperature ranges of 100 °C–450 °C and 100 °C–600 °C, respectively. The smooth specimens were manufactured in two different orientations to study the influence of anisotropy, and the specimens were machined to final dimensions with conventional techniques. A constitutive model with focus on describing the mid-life behaviour was developed where the total inelastic strain was divided into one plastic (rate-independent) and one creep (rate-dependent) part, to be able to describe both the rate-dependent effects from TMF conditions as well as rate-independent responses. A cycle jumping procedure was used, which enables to simulate the mid-life response of the material for TMF as well as low-cycle fatigue conditions within three simulated loading cycles. • The thermomechanical behaviour of an AM combustor alloy was studied. • A constitutive model for rate-dependent and rate-independent responses was set up. • Hardening was modelled with the first Ohno–Wang model and an additional rate term. • Temperatures up to 600 °C with different loads were considered. • The model predicts the experimental mid-life behaviour with good agreement. [ABSTRACT FROM AUTHOR]

Abstract: Monotonic tension, isothermal/anisothermal fully reversed strain cycling and zero-to-tension cyclic tests were conducted within the temperature domain from room temperature to 823K to investigate the mechanical behavior of Z2CND18.12N austenitic stainless steel under various uniaxial loading conditions. Interesting results were observed from these tests, including obvious rate-dependence at room temperature but lack of rate-dependence at elevated temperatures with the occurrence of serrated flow stress in tensile tests, more cyclic hardening at higher temperature in strain cycling tests, and tendency to reach shakedown condition at elevated temperatures in zero-to-tension cyclic tests. Dynamic strain aging (DSA) effect was presumably believed to contribute to these characteristics of the material. A thermo-viscoplastic constitutive model was proposed to describe the mechanical behavior of the material under uniaxial loading conditions at small strains. Kinematic hardening rule with two components of back stress and isotropic hardening rule incorporating DSA effect are the novel features of the proposed model. The simulated and predicted results show reasonable agreement with the experimental data. [Copyright &y& Elsevier]

Abstract: In this paper, a constitutive model for the unified description of rate dependent and rate independent material behavior is proposed. It is applicable to isotropic metals subjected to arbitrary thermomechanical loading conditions at small strains. The focus of the model formulation is its validity for the full range of thermal and mechanical loading conditions to be covered in an industrial context by e.g. modern aero-engine designs. Consequently, the proposed model describes the material behavior on a macroscopic level covering the full temperature range from room temperature to the upper application limit under monotonic as well as cyclic loading. Special emphasis is also put on the correct representation of the observed ratcheting behavior. The most prominent features of the model are the combined treatment of both, rate dependent, as well as rate independent inelasticity through a limit surface concept, the description of primary, secondary and tertiary creep behavior and the application of an appropriate backstress evolution equation. A fully implicit integration algorithm for the proposed model is developed and implemented in connection with uniaxial integration point drivers as well as three alternative Finite Element packages. The parameters of the proposed model can be identified based on a limited number of complex cyclic tests and monotonic creep tests. After reporting on the fitting results for such tests for the Nickel-base superalloy IN718 the predictive capabilities of the proposed model are assessed for a number of isothermal and nonisothermal tests. Finally, the performance of the algorithmic implementation into the Finite Element packages is briefly addressed. [Copyright &y& Elsevier]

Abstract: Ductile damage is a consequence of large strains more or less localized inside bands. Taking into account damage in constitutive behaviour of metallic materials is necessary to model various engineering problems involved in forming processes (stamping, punching, shearing…). Damage can be described at macroscopic level with continuum mechanics theories but introducing microstructural features can lead to more accurate predictions. In the present study, two polycrystalline plasticity models including damage effects in the framework of scale transition methods are investigated. These models are based on different approaches with direct application to duplex stainless steel. The first one is a variant of the Lipinski–Berveiller model in which ductile damage effects have been introduced. The second one is a generalized Cailletaud model taking into account ductile damage. Because of the microstructural complexity of the chosen materials, some particular developments of the micro-mechanical approaches are considered. Moreover, continuous damage mechanics is used at grains scale including its coupling with plastic or elastic–plastic flow. The modelling is justified from previous experimental results obtained by neutrons diffraction on duplex stainless steels. The developed models allow then deducing from the grains behaviour the macroscopic behaviour of the aggregate with damage effects. [Copyright &y& Elsevier]

An original two-step 'three phase' elastic-viscoplastic scale transition model is developed based on the combined self-consistent and Mori-Tanaka schemes. A coated inclusion is embedded within a matrix, wherein the inclusion represents grain interiors and the coating of the inclusion mimics the effects of grain boundaries and triple junctions. The predominant behavior within the grain interiors is captured through dislocation glide, whereas grain boundary (GB) dislocation emission and absorption, as well as thermally assisted GB sliding, describe the deformation processes within the coating describing the GB affected zone. Furthermore, an imperfect interface is assumed between the inclusion and the coating to account for viscoplastic grain boundary sliding along a stick-slip mechanism. Results and discussion focus on the competitive roles of GB sliding, GB dislocation emission/absorption, dislocation sweep in grain cores and collective dislocation plasticity, and the origins of the pronounced strain rate sensitivity of fcc NC materials. [ABSTRACT FROM AUTHOR]

Abstract: A specific flow rules and the corresponding constitutive elasto-viscoplastic model combined with new experimental strategy are introduced in order to represent a spheroidal graphite cast-iron behaviour on a wide range of strain, strain rate and temperature. A “full model” is first proposed to correctly reproduce the alloy behaviour even for very small strain levels. A “light model” with a bit poorer experimental agreement but a simpler formulation is also proposed. These macroscopic models, whose equations are based on physical phenomena observed at the dislocation scale, are able to cope with the various load conditions tested – progressive straining and cyclic hardening tests – and to correctly describe anisothermal evolution. The accuracy of these two models and the experimental databases to which they are linked is estimated on different types of experimental tests and compared with the accuracy of more standard Chaboche-type constitutive models. Each test leads to the superiority of the “full model”, particularly for slow strain rates regimes. After developing a material user subroutine, FEM simulations are performed on Abaqus for a car engine exhaust manifold and confirm the good results obtained from the experimental basis. We obtain more accurate results than those given by more traditional laws. A very good correlation is observed between the simulations and the engine bench tests. [Copyright &y& Elsevier]

The aim of this paper is to study the process of frictional contact with adhesion between a body and an obstacle. The material's behavior is assumed to be elastic-viscoplastic, the process is quasistatic, the contact is modeled by the Signorini condition and the friction is described by a non local Coulomb law coupled with adhesion. The adhesion process is modelled by a bonding field on the contact surface. We derive a variational formulation of the problem, then, under a smallness assumption on the coefficient of friction, we prove an existence and uniqueness result of a weak solution for the model. The proof is based on arguments of timedependent variational inequalities, differential equations and Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

Abstract: Although homogenization methods based on the Eshelby inclusion problem well capture the effects of heterogeneous local behaviours, volume fractions and morphologies of constituents on the macroscopic behaviour, inclusion size is still not considered. However, grain size effects are known to mainly participate on experimental results. In this contribution, we propose a new micromechanical approach based on the representation of the material as a two-phase composite: the inclusion phase which corresponds to the grain core region for which statistically stored dislocations (i.e., with net Burgers vector equal to zero) mainly participate in the plastic flow of the material, and, the matrix phase which is a region close to grain boundaries where plastic strain gradients and associated geometrically necessary dislocations (i.e., dislocations with net Burgers vector different from zero) are present. The macroscopic material behaviour is retrieved by applying a relevant self-consistent modelling for elastic–viscoplastic materials based on the “translated fields” technique and using secant viscoplastic compliances for each phase. By accounting for plastic strain gradient effects, the present modelling is able to well capture the observed grain size effect on the overall strain hardening. The model is applied to polycrystalline ferritic steels with different grain sizes and different chemical compositions. Numerical results in terms of macroscopic behaviours, local mechanical fields, evolution of dislocation densities as well as Bauschinger effect are discussed and compared with experimental ones. [Copyright &y& Elsevier]

We study a quasistatic frictionless contact problem with normal compliance, adhesion and damage for elastic-viscoplastic material. The adhesion of the contact surfaces is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. The mechanical damage of the material, caused by excessive stess or strains, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation of the problem and prove the existence and uniqueness of a weak solution. The proofs are based on time-dependent variational equalities, classical results on elliptic and parabolic variational inequalities, differential equations and fixed point arguments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the contact between an elastic–viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to tension or compression, is also considered. The variational formulation leads to a coupled system of evolutionary equations for which an existence and uniqueness result is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained and, as a consequence, the convergence of the algorithm is deduced. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper the influence of constitutive relation has been studied in numerical simulations of the perforation of 12-mm thick Weldox 460 E steel plates impacted by blunt-nosed projectiles in the sub-ordinance velocity regime. A modified version of the well-known and much used constitutive relation proposed by Johnson-Cook and both the bcc- and hcp-version of the Zerilli-Armstrong constitutive relation were combined with the Johnson-Cook fracture criterion. These models were implemented as user-defined material models in the non-linear finite element code LS-DYNA. Identification procedures have been proposed, and the different models were calibrated and validated for the target material using available experimental data obtained from tensile tests where the effects of strain rate, temperature and stress triaxiality were taken into account. Perforation tests carried out in a compressed gas gun on 12-mm-thick circular Weldox 460 E steel plates were then used as base in a validation study of plate perforation using LS-DYNA and the proposed constitutive relations. The numerical study indicated that the physical mechanisms during perforation can be qualitatively well predicted by all constitutive relations, but quantitatively more severe differences appear. The reasons for this are discussed in some detail. It was concluded that for practical applications, the Johnson-Cook constitutive relation and fracture criterion seems to be a good choice for this particular problem and excellent agreement with the experimental results of projectile impact on steel plates were obtained under the conditions investigated. [Copyright &y& Elsevier]

Abstract: Numerical simulations of high strain rate and high temperature deformation of pure metals and alloys require realistic plastic constitutive models. Empirical models include the widely used Johnson–Cook model and the semi-empirical Steinberg–Cochran–Guinan–Lund model. Physically based models such as the Zerilli–Armstrong model, the Mechanical Threshold Stress model, and the Preston–Tonks–Wallace model are also coming into wide use. In this paper, we determine the Mechanical Threshold Stress model parameters for various tempers of AISI 4340 steel using experimental data from the open literature. We also compare stress–strain curves and Taylor impact test profiles predicted by the Mechanical Threshold Stress model with those from the Johnson–Cook model for 4340 steel. Relevant temperature- and pressure-dependent shear modulus models, melting temperature models, a specific heat model, and an equation of state for 4340 steel are discussed and their parameters are presented. [Copyright &y& Elsevier]

Abstract: A novel cyclic deformation test program was undertaken to characterize macroscopic time dependent deformation of a titanium alloy for use in viscoplastic model development. All tests were conducted at a high homologous temperature, 650°C, where there are large time dependent and loading rate dependent effects. Uninterrupted constant amplitude tests having zero mean stress or a tensile mean stress were conducted using three different control modes: strain amplitude and strain rate, stress amplitude and stress rate, and a hybrid stress amplitude and strain rate. Strain ratcheting occurred for all cyclic tests having a tensile mean stress and no plastic shakedown was observed. The shape of the strain ratcheting curve as a function of time is analogous to a creep curve having primary, steady state and tertiary regions, but the magnitude of the ratchet strains are higher than creep strains would be for a constant stress equal to the mean stress. Strain cycles interrupted with up to eight 2-h stress relaxation periods around the hysteresis loop, including hold times in each quadrant of the stress–strain diagram, were also conducted. Stress relaxation was path-dependent and in some cases the stress relaxed to zero. The cyclic behavior of these interrupted tests was similar even though each cycle was very complex. These results support constitutive model development by providing exploratory, characterization and validation data. [Copyright &y& Elsevier]

Abstract: In this paper, an experimental investigation and a constitutive modeling of the mechanical response of an interstitial-free (IF) steel over a wide range of strain rates (from 0.001/s to 750/s) are presented. Tensile tests at relatively high strain rates, exceeding 100/s, are performed at an initial room temperature, using the so-called one bar technique developed on the basis of the Hopkinson bar method. At a high strain rate, a distinct upper yield limit is observed, and the subsequent flow stress increases remarkably. Furthermore, the ductility is reduced significantly in comparison to the case of low strain rate tension. In order to express such a complicated material response of IF steel, we develop a new constitutive model that takes into account effects of a change in the mobile dislocation density and thermal softening. The model can be easily applicable to large-scale engineering computations, because it is macroscopically formulated. We try to reproduce the tensile response including a diffuse neck formation at high strain rates, using the proposed constitutive model and finite element method. The results indicate that a change in the mobile dislocation density, together with thermal softening, has substantial effects on apparent work hardening behavior at high strain rates, although the change in the mobile dislocation density is transcribed at macroscopic scale in the model. Finally, we discuss characteristics of true stress–true strain curves at various strain rates, and their correlation with the plastic instability behavior. [Copyright &y& Elsevier]

A rate-dependent inelastic constitutive equation is formulated by extending the elastoplastic constitutive equation so as to retain the latter''s mathematical structure and thus reduce to the latter equation at an infinitesimal rate of deformation. That structure differs substantially from that of the over-stress model, the best-known rate-dependent inelastic constitutive model. The proposed constitutive model is a type of superposition model, which is premised on the additive decomposition of the inelastic strain rate into the plastic and creep strain rates. The plastic strain rate is formulated so as to become suppressed as the rate of deformation increases but is induced even at the infinite rate of deformation. This is the distinguishing features of this model from the existing superposition models. The present model can describe realistically the rate-dependent inelastic deformation for a wide range of strain rates. On the other hand, the over-stress model cannot predict appropriately the difference of mechanical response due to the rate of deformation, especially being inapplicable to the description of deformation at high rate of deformation as known from the unrealistic prediction of the infinite strength at an infinite rate of deformation. The proposed model is applied to various metals, and its adequacy is verified through comparisons with various test data under a wide variety of strain rates and temperatures. [Copyright &y& Elsevier]

The results from a systematic study of the response of a Ti–6Al–4V alloy under quasi-static and dynamic loading, at different strain rates and temperatures, are presented. The correlations and predictions using modified Khan–Huang–Liang (KHL) viscoplastic constitutive model are compared with those from Johnson–Cook (JC) model and experimental observations for this strain rate and temperature-dependent material. Overall, KHL model correlations and predictions are shown to be much closer to the observed responses, than the corresponding JC model predictions and correlations. Similar trend has been demonstrated for other titanium alloys using published experimental data [Mech. Mater. 33(8) (2001) 425; J. Mech. Phys. Solids 47(5) (1999) 1157]. [Copyright &y& Elsevier]

The motion of a crack in an elastic-(plastic )-viscoplastic medium is studied in terms of an energetic analysis. Combined with the stress and velocity fields obtained in Part 1, Kishimoto's energy integral, Ĵ, is used as a crack driving force to determine its motion. The major results obtained are: (1) dependence of crack speed on a modified near-field parameter, KI tip, (or equivalently, a modified dynamic energy release, GI tip), which is different from the usual stress intensity factor KI of an elastic crack-tip field but is related to it; (2) influence of inelastic effect, such as the viscoplastic exponent n, on the motion of the crack; and (3) stability condition of crack motion. In particular, for the last point, it has been found that, for a given loading and material coefficients, there exist two possible motions of the crack: one is stable crack growth and the other is unstable fracture. The lower and upper bounds of crack motion are also discussed. It is finally shown that the maximum crack velocity is lower than the Rayleigh wave speed, and is dependent on the viscoplastic exponent of the material. [ABSTRACT FROM AUTHOR]

This work is focused on the theoretical development and numerical implementation of a viscoplastic law. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed. Based on this dissipation inequality and the principle of maximum dissipation a finite strain viscoplastic model described also in the rotated material coordinate system is formulated. The evolution equations are expressed in terms of the material time derivatives of the rotated elastic logarithmic strain, the accumulated plastic strain and the strain-like tensor conjugate to the rotated back stress. The mathematical structure of this theory is concise and similar to that of the infinitesimal viscoplastic theory. These characteristics make the numerical implementation of this theory easy. The stress integration algorithm and the algorithmic tangent moduli for the infinitesimal theory can be applied to the numerical implementation of the present finite strain theory with a little reformulation. The complicated algorithmic formulations for most of other finite plastic laws can be therefore circumvented. In order to check the effectivity of the present finite strain theory a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present viscoplastic material law at describing the finite strain elastoplastic and viscoplastic problems. [Copyright &y& Elsevier]

A uniaxial viscoplastic deformation is motivated as a discrete sequence of stable and unstable equilibrium states and approximated by a smooth family of stable states of equilibrium depending on the history of the mechanical process. Three-dimensional crystal viscoplasticity starts from the assumption that inelastic shearings take place on slip systems, which are known from the particular geometric structure of the crystal. A constitutive model for the behaviour of a single crystal is developed, based on a free energy, which decomposes into an elastic and an inelastic part. The elastic part, the isothermal strain energy, depends on the elastic Green strain and allows for the initial anisotropy, known from the special type of the crystal lattice. Additionally, the strain energy function contains an orthogonal tensor-valued internal variable representing the orientation of the anisotropy axes. This orientation develops according to an evolution equation, which satisfies the postulate of full invariance in the sense that it is an observer-invariant relation. The inelastic part of the free energy is a quadratic function of the integrated shear rates and corresponding internal variables being equivalent to backstresses in order to consider kinematic hardening phenomena on the slip system level. The evolution equations for the shears, backstresses and crystallographic orientations are thermomechanically consistent in the sense that they are compatible with the entropy inequality. While the general theory applies to all types of lattices, specific test calculations refer to cubic symmetry (fcc) and small elastic strains. The simulations of simple tension and compression processes of a single crystal illustrates the development of the crystallographic axes according to the proposed evolution equation. In order to simulate the behaviour of a polycrystal the initial orientations of the anisotropy axes are assumed to be space-dependent but piecewise constant, where each region of a constant orientation corresponds to a grain. The results of the calculation show that the initially isotropic distribution of the orientation changes in a physically reasonable manner and that the intensity of this process-induced texture depends on the specific choice of the material constants. [Copyright &y& Elsevier]

Classes of flow rules for finite viscoplasticity are defined by assuming that certain measures for plastic strain rate and plastic spin depend on the state variables but not on the plastic deformation. It is shown that three of these classes are mutually exclusive for finite elastic strains. For small elastic shear strains, two of the three classes are approximately equivalent. A number of exact and approximate kinematic relations between the various measures for plastic strain rate and plastic spin are derived. Some inconsistent flow rules encountered in the literature are also discussed. Throughout the paper, arbitrarily anisotropic materials are considered, and some of the simplifications resulting from the assumption of isotropy are noted. [Copyright &y& Elsevier]

Adopting an elastic-viscoplastic, the asymptotic problem of mode I propagating crack-tip field is investigated. Various asymptotic solutions resulting from the analysis of crack growing programs are presented. The analysis results show that the quasistatically growing crack solutions are the special case of the dynamic propagating solutions. Therefore these two asymptotic solutions can be unified. [ABSTRACT FROM AUTHOR]

The mechanical behaviour of a superplastic alloy is characterised by both the sinh-law, ϵ˙p∝sinh(βσ), and power-law, ϵ˙p∝(σ/K)1/m, constitutive equations together with the hardening variables containing significant physical meanings. The material constants in the power-law equation are determined using two different methods: the genetic algorithm (GA)-based optimization technique and the more recently developed physically-based multiple step method. The two techniques give different values for the material constants and thus result in different magnitudes for the strain rate sensitivity parameter m. Together three SETs of determined unified constitutive equations are obtained and all enable the experimental stress–strain curves and grain growth data of the material to be well captured. In this paper, studies are carried out to investigate the effects of the three determined material models on necking predictions for the superplastic gas-blow forming of a structural component. In addition the predicted gas pressure cycles, required to ensure that the maximum strain rate over the workpiece approximately equals the pre-defined target deformation rate, are compared and analysed for the three cases. [Copyright &y& Elsevier]