Time-independent Plasticity Research Articles

Rate dependent but time independent plasticity formulation with application to sand

A novel rate dependent but time independent plasticity constitutive framework is proposed by rendering the flow rule and the internal variables evolution equations, appropriate functions of the strain rate, while maintaining a rate independent yield surface. This new framework is applied to a full-fledged rate independent and Critical State Theory compatible constitutive model for sands, transforming it into a rate dependent to address the cases where rate dependence is the main interest in comparison with creep and relaxation of secondary importance and manifestation. A qualitative comparison is made with available rate dependent data for sands.

Fatigue crack initiation life prediction using different plasticity models in cold expanded AL-alloy plates utilized in double shear lap joints

Cold expansion of fastener holes is a recognized technique for enhancing the fatigue life of holed plates utilized in detachable joints. This method involves introducing compressive residual stress around the holes, which has proven to be effective in improving the structural integrity and longevity of such components. In this research, the role of using different plasticity models in finite-element simulations of cold expansion (CE) in determining residual stress distribution and predicting the fatigue life of lap joints was investigated. Finite-element simulations were conducted to analyze the distribution of residual stress in cold-expanded Al-alloy 2024-T3 plates utilized in double-shear lap joints. Three time-independent plasticity models, specifically multilinear kinematic hardening, multilinear isotropic hardening (M.L.I.H) based on monotonic tensile tests, and nonlinear combined hardening models derived from saturated hysteresis stress and strain loops, were employed in the simulations. These models allowed for an accurate determination of the residual stress distribution in the plates. In finite-element simulations, two CE sizes of 1.5% and 4.7% were employed to create residual stresses. In the simulations, after creating residual stress by CE, remote sinusoidal loads are applied to the joints corresponding to the previously conducted fatigue tests in order to obtain stress and mechanical strain distributions. In order to predict the fatigue life, four different multiaxial fatigue criteria (Smith–Watson–Topper, Glinka, Kandil–Brown–Miller, and Fatemi–Socie) were employed, using the stress and strain distributions obtained from the finite element simulations with the different plasticity models. The simulations yielded varying stress and strain distribution results for the multilinear kinematic and M.L.I.H models, while the results of the nonlinear combined hardening model fell between the other two models. Notably, the fatigue life prediction based on the nonlinear combined hardening plasticity model closely matched the experimentally observed fatigue lives, with an absolute percentage deviation of 24.8%.

Rate of change of J‐integral in creep‐fatigue condition

AbstractThe operational conditions in next generation power plants and other high temperature applications such as turbine blades in jet engines demand the component to perform under extreme conditions where metallic materials show time dependent deformation under cyclic loading conditions. Under creep‐fatigue loading condition, the crack tip is exposed to both time dependent and independent plastic deformation. Conventional crack characterizing parameters such as (C ) has shown good correlation with dominant damage‐based crack velocity, (d /d ) . However, the true definition or prediction of crack driving forces under such scenario are vague due to limited theoretical validity of the conventional crack tip characterizing parameters, such as or C . In this work, the concept of configurational forces are applied for the first time to understand the creep‐fatigue crack growth behavior. The crack growth is simulated using node‐release technique and the configurational forces are calculated using post processing the finite element results for calculation of d /d .

The effects of internal stresses on the creep deformation investigated using in-situ synchrotron diffraction and crystal plasticity modelling

In this study we investigated the evolution of meso-scale internal stresses and those effects on local deformation behaviour during incremental plastic and creep deformation in type 316H stainless steel at 550 °C, using in-situ X-ray synchrotron diffraction and crystal plasticity modelling. Owing to the fast data accusation rate of synchrotron diffraction technique, for the first time, the transient behaviour of different grain families was captured during initial fast stress relaxation period of the displacement-controlled creep dwells. Significantly it is found that the evolution of internal stresses during time independent plastic deformation is distinct from that during time dependent creep deformation. During plastic deformation, lattice strains in the {311} grain family exhibit linear behaviour whereas during creep deformation it exhibits non-linear behaviour, instead the {111} grain family exhibit linear behaviour. A novel unified constitutive law was devised within crystal plasticity framework based on the theoretical physics; the model successfully predicts the macroscopic deformation behaviour as well as the distinction between the evolution of meso-scale internal stresses during plastic and creep deformation, therefore, correctly accounting for the effect of internal stresses generated during plastic deformation on the subsequent creep deformation. The validated model has elucidated the grain-neighbouring effects on individual grain deformations.

A constitutive model for municipal solid waste considering mechanical creep and biodegradation-induced compression

A constitutive model is developed to describe the stress–strain–time behavior for decomposing municipal solid waste (MSW) within a critical state soil mechanics framework. The model is an extension of the Modified Cam-Clay plasticity model. In this model, three sources contribute to the hardening of MSW due to volumetric strain: time-independent plastic volumetric strain, time-dependent volumetric mechanical creep strain, and time-dependent volumetric strain due to the biodegradation (decomposition) of MSW. The MSW model was evaluated through numerical analyses of large-scale one-dimensional compression tests in the laboratory and the field and the reported vertical and horizontal deformations of a MSW landfill. The associated model parameters were obtained by compositional analysis of the waste, from values reported in the literature, and by fitting numerical results to observed behavior. For the laboratory compression test, the best-fit numerical simulation over-predicted the early settlement but converged on the experimental values after 200 days. Initially, the calculated vertical strains in the field-scale test deviated from the measured strains by up to 10% over the 398-day period of the test. However, numerical results after adjusting model parameters to provide the best fit with the measured strains resulted in a maximum deviation of less than 3% over the test duration. The calculated vertical displacements of the MSW landfill were consistent with field measurements. However, the calculated horizontal displacements were significantly lower than the measured values. Sensitivity studies showed that the time-dependent settlement predicted by the model is highly sensitive to the biodegradation rate of MSW. The good agreement between numerical values and observed vertical deformations for the SWEAP section on the MSW landfill suggests that the model has the potential to assess the performance of subsystems in landfills (e.g., the performance of a side slope liner system subject to landfill settlement). However, the discrepancy between predicted and observed horizontal displacements suggests that the numerical model can be improved by incorporating deviatoric creep deformations in the constitutive model.

Time-Independent Plasticity Formulated by Inelastic Differential of Free Energy Function

Abstract The authors presented a time-independent plasticity approach, where a typical plastic-loading process is viewed as an infinitesimal state change of two neighboring equilibrium states, and the yield and consistency conditions are formulated based on the conjugate forces of the internal variables. In this paper, a stability condition is proposed, and the yield, consistency, and stability conditions are reformatted by the inelastic differential form of the Gibbs free energy. The Gibbs equation in thermodynamics with internal variables is a representation to the differential form of the Gibbs free energy by a single Gibbs free energy function. In this paper, we propose the so-called extended Gibbs equation, where the differential form may be represented by multiple potential functions. Various associated and nonassociated plasticity with a single or multiple yield functions can be derived from various representations based on the reformulated approach, where yield and plastic potential functions are in the form of inelastic differentials of the potential functions. The generalized Drucker inequality can only be derived from the one-potential representation as a stability condition. For a multiple-potential representation, the stability condition can be ensured if the multiple potentials are concave functions and possess the same stationary point.

Multiphysics modeling of continuous casting of stainless steel

Abstract During the solidification of stainless steel, the mechanical behavior of the solidifying shell follows nonlinear elastic-viscoplastic constitutive laws depending on metallurgical phase fraction calculations (liquid, ferrite and austenite). A multiphysics model that couples thermal and mechanical behavior in a Lagrangian reference frame, including both classical time-independent plasticity and creep, with turbulent fluid flow in the liquid phase in an Eulerian frame, is applied to determine realistic temperature and stress distributions in the solidifying shell of stainless steel in a commercial continuous caster. Compositional effects are incorporated through the use of phase diagrams to define the phase fraction variations with temperature during the process. The behavior at these high temperatures can be adequately captured using specific constitutive equations for each phase and careful decisions about switching between them. Results for a 409L ferritic stainless steel show that, due to its phase fraction history, solidification stresses differ significantly from those in plain carbon steels. Specifically, they include a secondary sub-surface compression peak due to phase change expansion between γ-austenite and δ-ferrite through the thickness of the shell.

A micro-macro confined compressive fatigue creep failure model in brittle solids

Fatigue properties under cyclic static compressive loading and unloading (i.e., compressive fatigue/cyclic creep properties) have an important meaning for evaluation of lifetime in brittle solids containing numerous microcracks. The total deformation during cyclic static compression consists of time-independent elastic, time-dependent viscoelastic, time-dependent viscoplastic and time-independent plastic deformations. In this study, the viscoplastic and plastic deformations are collectively called as plastic deformation below. The plastic deformation dominates the lifetime of brittle solids. The rebound of elastic and viscoelastic deformations at static unloading is widely studied. However, the rebound of plastic deformation at static unloading is rarely studied under constant lateral confining pressure, and the total time-dependent deformation during cyclic static compressive failure also is rarely studied in theory. Furthermore, the rebound of plastic deformation has a great significance for cyclic static compressive failure, and microcrack variable influences seriously plastic mechanical properties of brittle solids. However, the theoretical relationship between plastic deformation rebound and microcrack variable in brittle solids during a cyclic static compressive failure is rarely established. In this study, an analytical solution is proposed by coupling the confined cyclic static loading and unloading path, the Hooke-Kelvin viscoelastic model and the formulated micro-macro model, which explains the total time-dependent visco-elastic-plastic deformation caused by microcracks variable during cyclic static compressive failure. Effects of residual static axial stress after unloading and confining pressure on cyclic creep failure are studied. Irreversible elastic and viscoelastic deformations caused by residual static stress after unloading are illustrated. An interesting rebound phenomenon of plastic deformation induced by microcrack recovery is found. A critical value of axial stress causing a rebound of plastic deformation is also found. Effects of parameters Δσ1a, Δσ1b, and Δt in cyclic static loading and unloading path on the mechanical properties of cyclic static compressive failure are discussed. The proposed theoretical model provides a certain help for evaluation and prediction of fatigue lifetime in brittle solid engineering.

Modelling of creep and plasticity deformation considering creep damage and kinematic hardening

This study addresses creep behavior under constant load with the evolution of damage and plastic deformation by load change. A creep strain rate equation with one damage variable is adopted. Two kinematic hardening models related to time-independent plasticity and viscoplasticity, respectively, are incorporated. Combining the creep equation and kinematic hardening theory, two models are proposed: a creep-plasticity model and a nonunified viscoplasticity model. Experiment data for Al 5083 and AISI type 316H are used to verify the utility of the proposed models. Due to the creep part with damage, both models precisely predict the creep strain curve and the rupture time and strain. However, the creep-plasticity model fails to describe the first-stage creep and strain rate sensitivity in monotonic tensile testing that can be captured with the nonunified viscoplasticity model. Finally, the parameter determination is discussed.

Effect of geometrically necessary dislocations on inelastic strain rate for torsion stress relaxation of polycrystalline copper in micro scale

To characterize the torsion stress relaxation in micro scale, the relation between the inelastic strain rate and the geometrically necessary dislocations (GNDs) is proposed. Based on the Orowan equation, which provides the description for strain rate evolution with the influence of density and velocity of mobile dislocation, the effect of GNDs is considered when materials exhibit non-uniform deformation at small scale. Additionally, the inelastic strain rate is presented as a power form of stress. Consequently, from the analysis about the test results for polycrystalline copper, the parameters for the evolution of inelastic strain rate are determined. Moreover, a time-independent plasticity model is introduced to describe the plastic hardening during the repeated stress relaxation. Finally, the activated volume from simulation and experiments are discussed in detail to interpret the underlying mechanism.

A combined model to simulate the powder densification and shape changes during hot isostatic pressing

Under high temperature and pressure during Hot Isostatic Pressing (HIP), various deformation and sinter mechanisms such as plastic yielding, power law creep, and diffusion occur. They contribute either simultaneously or consecutively to the densification process depending on temperature and pressure levels. Some published works have shown that neither the pure plastic model nor the pure viscoplastic model sufficiently models the densification process during HIP, therefore a combined model which includes both a time independent plasticity and a rate dependent plasticity (creep) should be used. An innovative aspect of this work in comparison with available models in literature is the accommodation for model inaccuracies in the initial creep stage. The densification model incorporates the initial creep mechanism which occurs for a short time with a higher rate as compared to the steady stage creep rate. In this study, a combined model which consists of the influence of plasticity, primary creep, and secondary creep mechanisms is used. The authors believe that this approach can improve the quality of “near-net-shape” simulation.Innovative Aspects: The proposed model is more robust and predicts the final shape of HIP-ed components better which will support the production of Selective Net Shape or Net Shape HIP components. The reported results are of importance for companies producing big and complex shaped components through powder-HIP.

Mapping Viscoelastic and Plastic Properties of Polymers and Polymer-Nanotube Composites using Instrumented Indentation.

An instrumented indentation method is developed for generating maps of time-dependent viscoelastic and time-independent plastic properties of polymeric materials. The method is based on a pyramidal indentation model consisting of two quadratic viscoelastic Kelvin-like elements and a quadratic plastic element in series. Closed-form solutions for indentation displacement under constant load and constant loading-rate are developed and used to determine and validate material properties. Model parameters are determined by point measurements on common monolithic polymers. Mapping is demonstrated on an epoxy-ceramic interface and on two composite materials consisting of epoxy matrices containing multi-wall carbon nanotubes. A fast viscoelastic deformation process in the epoxy was unaffected by the inclusion of the nanotubes, whereas a slow viscoelastic process was significantly impeded, as was the plastic deformation. Mapping revealed considerable spatial heterogeneity in the slow viscoelastic and plastic responses in the composites, particularly in the material with a greater fraction of nanotubes.

Time-Independent Plasticity Based on Thermodynamic Equilibrium and Its Stability

Within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 433–455), Yang et al. (2014, “Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional,” ASME J. Eng. Mater. Technol., 136(2), p. 021001) established a model of time-independent plasticity of three states. In this model, equilibrium states are the states with vanishing thermodynamic forces conjugate to the internal variables, and correspond to critical points of the free energy or its complementary energy functions. Then, the conjugate forces play a role of yield functions and further lead to the consistency conditions. The model is further elaborated in this paper and extended to nonisothermal processes. It is shown that the incremental stress–strain relations are fully determined by the local curvature of the free energy or its complementary energy functions at the critical points, described by the Hessian matrices. It is further shown that the extended model can be well reformulated based on the intrinsic time in the sense of Valanis (1971, “A Theory of Viscoplasticity Without a Yield Surface, Part I. General Theory,” Arch. Mech., 23(4), pp. 517–533; 1975, “On the Foundations of the Endochronic Theory of Viscoplasticity,” Arch. Mech., 27(5–6), pp. 857–868), by taking the intrinsic time as the accumulated length of the variation of the internal variables during inelastic processes. It is revealed within this framework that the stability condition of equilibrium directly leads to Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) inequalities, by introducing the consistency condition into the work of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25(3), pp. 448–461). Generalized inequalities of Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) for nonisothermal processes are established straightforwardly based on the connection.

Modeling of Low Cycle Behavior of P92 Steel Based on Cyclic Plasticity Constitutive Equations

Strain controlled uniaxial low cycle fatigue (LCF) tests of P92 steel were conducted at strain amplitudes of 0.4%, 0.6% and 0.8% in fully reversed manner with strain rate of 1.0×10-3s-1 at high temperature of 650 °C. Cyclic softening behavior was studied and time-independent cyclic plasticity model was used to represent the cyclic mechanical behavior of this steel. Material parameters were determined step by step at higher strain amplitude of 0.8%, experimental data with lower strain amplitude were used to validate the extrapolation of the model. Comparison of the simulated and experimental results shows that the proposed model can give a reasonable prediction of stress-strain hysteresis loop for P92 steel at high temperature.

Finite element modelling of tensile deformation and failure of aluminium plate exposed to fire

This paper presents a coupled thermal–mechanical finite element model for analysing the tensile softening, deformation and failure of aluminium plate exposed to fire. The model consists of two parts: thermal analysis followed by mechanical analysis of aluminium under combined one-sided heating and axial tensile loading. The thermal analysis computes the temperature rise in the aluminium when exposed to one-sided transient radiant heating (e.g. fire simulation) and the mechanical analysis calculates the strength loss, plastic deformation (including necking) and failure of the aluminium under tensile loading. The mechanical model analyses the combined effects of elastic softening, time-independent plastic softening, time-dependent (creep) plastic softening, and thermal expansion on the tensile failure of aluminium plate. The model is validated by comparing theoretical predictions of the tensile deformation and stress rupture times against values measured from fire structural tests performed on aluminium plates. The model predicts the temperature, plastic deformation and failure with good accuracy.

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

Elastoplastic–viscoplastic modelling of metal powder compaction: application to hot isostatic pressing

A constitutive model is proposed for the densification of metal powder during hot isostatic pressing. The model considers an inelastic deformation resulting from time dependent (viscoplastic) and time independent (plastic) mechanisms during loading. With employing the Abouaf’s formalism for viscoplastic part, the paper is focused mainly on the plasticity contribution including the hardening effects of both relative density and the equivalent plastic strain. The proposed plastic–viscoplastic model is summarised to an equation expressing the densification rate under hydrostatic loading. The gas atomised 316LN stainless steel powder was used in this study. Model parameters necessary for simulation of HIP process during a given pressure ramp at constant temperature are identified using previously published data. The identified model is then verified using experimental data obtained from HIP trials performed at another fixed temperature but with varied pressure ramp rates. A good agreement was found between the model prediction and the experimental results.

The Reliability of Solder Joints in Fine Pitch 3-D Stack Package by Taguchi Method

To achieve high density and high performance, through-Silicon Vias (TSVs) have recently aroused much interest because it is a key enabling technology for three-dimensional (3-D) integrated circuit stacking and silicon interposer technology. In this study, a 3-D 1/8th symmetrical nonlinear finite element model of a stack die TSV package was developed using ANSYS finite element simulation. The model was used to optimize the package for robust design and to determine design rules to enhance 3-D stack package in view of bump reliability. An L8(2×7) Taguchi matrix was developed to investigate the effects of interposer thickness, TSV diameter, insulation (SiO2) thickness, chip thickness, substrate thickness, bump height, and bump diameter on bumps reliability. A temperature cycling test in the range of 0 °C to 100 °C was conducted by three cycles. The mechanical property of SAC leadless solder included time independent plastic and time dependent creep behaviors. The parameter of inelastic strain range of the third cycle was used to evaluate the bump life prediction. Two levels were chosen for each parameter to cover the ranges of interest. The results show that the smaller insulation (SiO2) and substrate thickness and the larger dimension for the other factors provide the best combination. These could be used as guides for further similar 3-D stack packages design.

Effect of Interfacial Microstructures on the Bonding Strength of Sn–3.0Ag–0.5Cu Pb-Free Solder Bump

The effect of interfacial microstructures on the bonding strength of Sn–3.0Ag–0.5Cu Pb-free solder bumps with respect to the loading speed, annealing time, and surface finish was investigated. The shear strength increased and the ductility decreased with increasing shear speed, primarily because of the time-independent plastic hardening and time-dependent strain-rate sensitivity of the solder alloy. The shear strength and toughness decreased for all surface finishes under the high-speed shear test of 500 mm/s as a result of increasing intermetallic compound (IMC) growth and pad interface weakness associated with increased annealing time. The immersion Sn and organic solderability preservative (OSP) finishes showed lower shear strength compared to the electroless nickel immersion gold (ENIG) finish. With increasing annealing time, the ENIG finish exhibited the pad open fracture mode, whereas the immersion Sn and OSP finishes exhibited the brittle fracture mode. In addition, the shear strength of the solder joints was correlated with each fracture mode.

Accurate modeling of the cyclic response of structural components constructed of steel with yield plateau

A time-independent constitutive model for structural steels with a yield plateau is presented. The model is based on the basic principles of time-independent plasticity. The model couples the nonlinear kinematic hardening concept with a memory surface in the plastic strain space, and a pseudo memory surface in the deviatoric stress space. A brief description of the constitutive model, which is capable of capturing the response of the material for monotonic, proportional and non-proportional cyclic loading paths are given. The performance of the proposed constitutive model is verified at the structural component level, by means of finite element simulations of large-scale structural columns or bridge pier elements under constant axial load and uni-directional quasi-static cyclic loading.