Rate-dependent Formulation 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.

Nonlinear integrated dynamic analysis of drill strings under stick-slip vibration

Stick-slip torsional vibration raising from bit-formation contact is a catastrophic dynamic phenomenon occurring in deep-water oil and gas drilling systems. The torsional vibrations can result in premature fatigue failure of drilling equipment and reduce drilling efficiency. Thus, studying the dynamics of the system is a vital key to identify the roots of the vibration and establish appropriate mitigation and remediation methods. In this paper, an efficient approach for finite element (FE) modeling of stick-slip vibrations of the full drill strings was proposed. The model was developed based on a rate-dependent formulation of bit-rock interaction, in which the cutting process was integrated through the frictional contact. The nonlinear effects of the large rotations and the geometrically nonlinear axial-torsional coupling were taken into account. The effect of energy dissipation due to the presence of drill mud along the drill pipes and drill collars was incorporated through Rayleigh viscous damping. Modal analysis was conducted to determine the natural frequencies and the mode shapes of the drill string. Furthermore, a five degree-of-freedom lumped parameter model was developed. The performance of the developed models was verified through comparisons with example stick-slip results from a field test. The study showed that the self-excited torsional vibrations may also occur at fundamental frequencies lower than the first natural torsional frequency of the drill string, depending on operational parameters. The conducted research work resulted in a robust and practical integrated FE model to simulate the entire drill string system dynamics under torsional vibrations.

Strain rate dependent formulation of the latent heat evolution of superelastic shape memory alloy wires incorporated in multistory frame structures

Due to their unique hysteretic energy dissipation capacity, shape memory alloy (SMA) wires are particularly interesting for the development of new-type of intelligent vibration control systems for structures. However, in structural control, most of the vibrations occur in high strain rate regimes, which interfere the release of self-generated heat and thus influence the hysteretic dissipation. This paper proposes a strain rate dependent formulation of the latent heat evolution and aims to improve the accuracy of existing macroscopic modeling approaches developed for SMA wires particularly for the dynamic load cases. The proposed formulation is determined phenomenologically and implemented in a continuum thermomechanical framework based constitutive SMA wire model without impairing the simplicity and robustness of the solution process. The proposed formulation is validated by cyclic tensile tests conducted on SMA wires. Results show that the calculations using the formulation can predict the wire response more accurately than the strain rate independent formulation. For the simulation of multistory frame structures incorporating multiple SMA wires, the governing equations are driven. Shaking table tests are conducted on a 3-story frame structure under harmonic and seismic excitation. The responses of the structure are successfully replicated using the strain rate dependent latent heat formulation.

Non-smooth evolutive laws in multisurface elastoplasticity with experimental evidence by infrared thermography

Plastic and viscoplastic evolutive laws and the constitutive behaviour of materials represent a topic of interest for the mechanical modeling of many multisurface materials and structures. In the present paper non-smooth evolutive laws and constitutive relations for multisurface elastoplasticity are illustrated. This formulation is functional for modeling non-smooth elastoplastic problems as well as problems characterized by non-smooth yield criteria and non-differentiable functions. In this treatment the plastic dissipation is expressed by including the indicator function of the convex elastic domain in a form usually not reported in the literature. However, it is shown that this is a crucial aspect which enables for a regular extension of the rate-independent plastic formulation into a rate-dependent formulation. Accordingly, in the present work extended forms of non-smooth evolutive laws and constitutive relations are illustrated within the adopted mathematical framework which proves to be well-suited for multisurface structural problems. The connections between the proposed formulations and the classical forms are discussed by highlighting the advantages for the definition of extended forms of evolutive laws and constitutive relations in non-smooth multisurface elastoplasticity. Furthermore, the present treatment is complemented by experimental investigations. In order to give experimental evidence to the present constitutive modeling of multisurface elastoplasticity and to exemplify the illustrated formulation some elastoplastic evolutive processes are described by the application of infrared thermography. The reported experimental examples bear witness for the usefulness of infrared thermography to visualize elastoplastic and damage deformations and, accordingly, the capability to validate alternative approaches and extended formulations of evolutive laws and constitutive behaviour in non-smooth multisurface elastoplasticity.

Experimental and numerical investigations of dynamic strain ageing behaviour in solid solution treated Inconel 718 superalloy

PurposeThe purpose of this paper is to systematically investigate the dynamic strain aging (DSA) effect in solid solution treated IN718 at different temperatures through experiments and simulations to gain an understanding of the inelastic deformation mechanisms.Design/methodology/approachIn the present work, uniaxial tensile tests have been carried out in conjunction with finite element (FE) simulations to investigate the behaviour of the solid solution treated Inconel 718 superalloy at different temperatures and strain rates. Dynamic strain aging (DSA) effects, which manifested during the tests in the form of a negative strain rate sensitivity and stress serrations, are investigated. The most significant DSA effect occurs at 500°C and at a strain rate of 10–4 s-1. In a newly proposed rate-dependent constitutive formulation, the DSA model, proposed by McCormick, Kubin and Estrin, was introduced into slip-assisted solute hardening, and an activation energy-dependent exponential flow rule was adopted.FindingsThe observed negative strain rate sensitivity and stress serrations are well predicted by a 3 D FE. The FE results indicate that the equivalent plastic strain rate distribution in the specimen gauge length is as highly inhomogeneous as in the other materials exhibiting DSA effects such as aluminium and titanium alloy. During inelastic deformation, propagating high strain rate bands can be closely correlated to the stress serrations.Originality/valueFor the DSA effect in solid solution treated IN718, the existing researching mainly focuses on the mechanical properties experiment and microstructure observation. In this study, a constitutive formulation, combined with the DSA model, has been proposed, and the mechanical behaviors, including the DSA effect, have been well predicted by a finite element model.

A comparative study of integration algorithms for finite single crystal (visco-)plasticity

Computational aspects of finite crystal plasticity have been discussed in the literature for some time, yet open challenges regarding the robustness and uniqueness of predictions for different algorithms remain. In this contribution, a careful comparison between different formulations of single crystal plasticity is therefore presented, covering both rate-dependent and rate-independent behavior in the finite deformation setting. To this end, algorithms are developed for an augmented-Lagrangian formulation and a formulation based on nonlinear complementary functions in the rate-independent case. For the rate-dependent case, algorithmic treatments are considered for two commonly employed viscoplastic formulations. The robustness of the different algorithms is assessed by means of simulations at the material point level, involving proportional fully deformation-controlled tests and non-proportional loading cycles in stress space. The augmented-Lagrangian formulation and a particular viscoplastic formulation are found to be superior in terms of robustness compared to the other algorithm considered in their respective category. It is then further observed that these algorithms of choice, which perform well for the ideally-plastic case and for Taylor-type hardening, induce a significant reduction in acceptable deformation increment size for an anisotropic, phenomenological hardening law, due to the rather complex determination of the active slip systems.Additionally, it is found from a sensitivity analysis that increasing the degree of anisotropy in the hardening law amplifies the sensitivity of the model’s response to small initial misorientations, both in the rate-dependent and the rate-independent case. A sensitivity analysis is also applied in the ideally-plastic case to demonstrate that for common choices of viscous parameters, the response of a particular rate-dependent formulation of the crystal plasticity model not necessarily possesses the same characteristic features as the response obtained by the corresponding rate-independent augmented-Lagrangian formulation. Therefore, it is recommended to apply the augmented-Lagrangian formulation to determine the rate-independent model response, rather than employing a rate-dependent formulation with limiting values of the viscous parameters due to the superior robustness of the former.The application of the augmented-Lagrangian formulation in the solution of an inhomogeneous fixed-end torsion problem by means of finite element analysis demonstrates the beneficial numerical properties of this formulation and the ability of the material model to reproduce characteristic features observed in experiments on single crystal copper samples.

Operational safety assessment of offshore pipeline with multiple MIC defects

Microbiologically influenced corrosion (MIC) creates multiple defects. The interaction of MIC defects and their time dependence need to be considered for robust safety assessment of the asset.This paper presents a methodology for the dynamic safety assessment of the assets under the influence of MIC. The methodology is built by integrating the Bayesian Network (BN)–Markov Mixture (MM) technique with Monte Carlo simulation. The integration of BN and MM provides an empirical model to probabilistically predict the effective defect growth rate based on the multiple defects’ interaction. A rate-dependent stochastic formulation is also developed for the remaining strength and safe operating pressure prediction using the Monte Carlo simulation. The proposed methodology dynamically predicts and captures the evolving effect of corrosion defects’ interaction and effective defect growth rate on the remaining strength and survival likelihood of an in-service corroding asset. The methodology is tested on an offshore pipeline, and the dynamic effects of corrosion influencing parameters and defects’ interaction on the pipeline survivability were predicted. Critical safety influencing factors of the pipeline under complex microbial biofilm architecture were identified. The proposed methodology provides a parametric-based condition monitoring tool for effective management of MIC and ensuring safety in offshore systems.

A Numerical Study on Ductile Failure of Porous Ductile Solids With Rate-Dependent Matrix Behavior

Abstract This work examines the effects of loading rate on the plastic flow and ductile failure of porous solids exhibiting rate-dependent behavior relevant to many structural metals. Two different modeling approaches for ductile failure are employed and numerical analyses are performed over a wide range of strain rates. Finite element unit cell simulations are carried out to evaluate the macroscopic mechanical response and ductile failure by void coalescence for various macroscopic strain rates. The unit cell results are then used to assess the accuracy of a rate-dependent porous plasticity model, which is subsequently used in strain localization analyses based on the imperfection band approach. Strain localization analyses are conducted for (i) proportional loading paths and (ii) non-proportional loading paths obtained from finite element simulations of axisymmetric and flat tensile specimens. The effects of strain rate are most apparent on the stress–strain response, whereas the effects of strain rate on ductile failure is found to be small for the adopted rate-dependent constitutive model. However, the rate-dependent constitutive formulation tends to regularize the plastic strain field when the strain rate increases. In the unit cell simulations, this slightly increases the strain at coalescence with increasing strain rate compared to a rate-independent constitutive formulation. When the strain rate is sufficiently high, the strain at coalescence becomes constant. The strain localization analyses show a negligible effect of strain rate under proportional loading, while the effect of strain rate is more pronounced when non-proportional loading paths are assigned.

A finite strain FE-Implementation of the Fleck-Willis gradient theory: Rate-independent versus visco-plastic formulation

This paper presents a numerical basis for finite strain analysis using the higher order flow theory by Fleck and Willis [2009, A mathematical basis for strain-gradient plasticity theory - part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057]. The rate-dependent (visco-plastic) formulation shares similarities with its rate-independent (generalized J2-flow) counterpart, and a combined presentation is laid out. The visco-plastic formulation [2018, A homogenized model for size-effects in porous metals, J. Mech. Phys. Solids, DOI: 10.1016/j.jmps.2018.09.004] is revisited to set the scene for the development of the rate-independent model, but the two formulations differ substantially in the numerical implementation as the rate-independent framework possesses a challenge in determining the yield of individual plastic zones. The framework developed readily accounts for repeated loading/unloading and handles the interaction of multiple plastic zones. The method presented exploits the image analysis techniques that was first combined with the Fleck-Willis theory in Nielsen and Niordson [2014, A numerical basis for strain-gradient plasticity theory: rate-independent and rate-dependent formulations, J. Mech. Phys. Solids 63, 113–127], but extends it to finite strains by adopting an updated Lagrangian framework. Besides somewhat standard extensions to deal with finite deformations, the framework largely resembles that of small strains. In fact, the adopted image analysis can be conducted in the undeformed configuration without any approximations as it relates only to the connectivity of plastic zones on the finite element mesh. The new framework is focused on the deformation and localization in both uniform and notched round bars by formulating the numerical model in an axi-symmetric setup. Comparisons between the visco-plastic and the rate-independent formulations are presented.

Deformation behaviour of small scale cp‐titanium specimen with large grains

AbstractTo describe the material behaviour of the microstructure of commercially pure (cp‐) titanium with the Finite Element (FE) Method, a crystal plastic material model is used. The anisotropic plastic deformation is considered by specifying the slip systems of the hexagonal‐closed‐packed (hcp) crystal structure. To circumvent the difficulties associated with possible non‐uniqueness of the set of active slip systems, a rate‐dependent formulation is used. Because of the volume preserving plastic deformation, locking effects can arise, which are dealt with by a modified F‐bar deformation gradient. To investigate the material behaviour of cp‐titanium small scale tensile test simulations are performed. Electronic backscatter diffraction (EBSD) investigations of the sample provides the specific grain orientations of the microstructure to perform realistic simulations.

Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: A rate-dependent anisotropic crack phase-field model

A deeper understanding to predict fracture in soft biological tissues is of crucial importance to better guide and improve medical monitoring, planning of surgical interventions and risk assessment of diseases such as aortic dissection, aneurysms, atherosclerosis and tears in tendons and ligaments. In our previous contribution (Gültekin et al., 2016) we have addressed the rupture of aortic tissue by applying a holistic geometrical approach to fracture, namely the crack phase-field approach emanating from variational fracture mechanics and gradient damage theories. In the present study, the crack phase-field model is extended to capture anisotropic fracture using an anisotropic volume-specific crack surface function. In addition, the model is equipped with a rate-dependent formulation of the phase-field evolution. The continuum framework captures anisotropy, is thermodynamically consistent and based on finite strains. The resulting Euler–Lagrange equations are solved by an operator-splitting algorithm on the temporal side which is ensued by a Galerkin-type weak formulation on the spatial side. On the constitutive level, an invariant-based anisotropic material model accommodates the nonlinear elastic response of both the ground matrix and the collagenous components. Subsequently, the basis of extant anisotropic failure criteria are presented with an emphasis on energy-based, Tsai–Wu, Hill, and principal stress criteria. The predictions of the various failure criteria on the crack initiation, and the related crack propagation are studied using representative numerical examples, i.e. a homogeneous problem subjected to uniaxial and planar biaxial deformations is established to demonstrate the corresponding failure surfaces whereas uniaxial extension and peel tests of an anisotropic (hypothetical) tissue deal with the crack propagation with reference to the mentioned failure criteria. Results favor the energy-based criterion as a better candidate to reflect a stable and physically meaningful crack growth, particularly in complex three-dimensional geometries with a highly anisotropic texture at finite strains.

Numerical strategies for damage assessment of reinforced concrete block walls subjected to blast risk

Computational tools and numerical strategies for the determination of the response of masonry walls under blast overpressure often rely on oversimplifying and conservative assumptions, which, although justifiable for design purposes, are not as warranted when fragility analysis and risk assessment are the objectives. In addition, due to the composite nature of reinforced masonry construction, the evaluation of multivariate fragility functions may likely require significant computational effort, owing to the multiplicity of variables describing the constituent materials and their associated uncertainty. The problem is further compounded by the effects of the high strain rates typically induced by blast loading, which make the understanding of masonry behaviour even more challenging; although efforts are being made in order to account for strain rate effects at the macro-scale, no well established models are available for reinforced masonry walls. Therefore, in order to perform accurate yet expedient fragility analyses that can effectively capture rate dependent phenomena, a dynamic model based on single-degree-of-freedom (SDOF) approach is developed. The SDOF model accounts for the nonlinear stress–strain behaviour obtained from standard prism tests and integrates strain rate dependent formulations provided in the open literature. The model predictions are corroborated using data—including pressure and displacement histories—from field testing of six scaled concrete block walls subjected to the detonation of live explosives. Within the scope of the current study, the proposed model is found to be a reasonable trade-off between computational efficiency and numerical accuracy and is an improvement upon a basic SDOF approach, which is typically based on the fixed dynamic increase factors recommended by modern design standards and technical manuals for blast protection. The results presented in this study are expected to contribute to the ongoing development of a comprehensive framework for the probabilistic risk assessment of structures subjected to explosive loading.

Predictive modeling of interfacial damage in substructured steels: application to martensitic microstructures

Metallic composite phases, like martensite present in conventional steels and new generation high strength steels exhibit microscale, locally lamellar microstructures characterized by alternating layers of phases or crystallographic variants. The layers can be sub-micron down to a few nanometers thick, and they are often characterized by high contrasts in plastic properties. As a consequence, fracture in these lamellar microstructures generally occurs along the layer interfaces or within one of the layers, typically parallel to the interface. This paper presents a computational framework that addresses the lamellar nature of these microstructures, by homogenizing the plastic deformation at the mesoscale by using the microscale response of the laminates. Failure is accounted for by introducing a family of damaging planes that are parallel to the layer interface. Mode I, mode II and mixed-mode opening are incorporated. The planes along which failure occurs are captured using a smeared damage approach. Coupling of damage with isotropic or anisotropic plasticity models, like crystal plasticity, is straightforward. The damaging planes and directions do not need to correspond to crystalline slip planes, and normal opening is also included. Focus is given on rate-dependent formulations of plasticity and damage, i.e. converged results can be obtained without further regularization techniques. The validation of the model using experimental observations in martensite-austenite lamellar microstructures in steels reveals that the model correctly predicts the main features of the onset of failure, e.g. the necking point, the failure initiation region and the failure mode. Finally, based on the qualitative results obtained, some material design guidelines are provided for martensitic and multi-phase steels.

Numerical simulations of crack propagation in screws with phase-field modeling

In this work, we consider a phase-field framework for crack propagation problems in elasticity and elasto-plasticity. We propose a rate-dependent formulation for solving the elasto-plastic problem. An irreversibility constraint for crack evolution avoids non-physical healing of the crack. The resulting coupled two-field problem is solved in a decoupled fashion within an augmented Lagrangian approach, where the latter technique treats the crack irreversibility constraint. The setting is quasi-static and an incremental formulation is considered for temporal discretization. Spatial discretized is based on a Galerkin finite element method. Both subproblems of the two-field problem are nonlinear and are solved with a robust Newton method in which the Jacobian is built in terms of analytically derived derivatives. Our algorithmic developments are demonstrated with several numerical tests that are motivated by experiments that study failure of screws under loading. Therefore, these tests are useful in practice and of high relevance in mechanical engineering. The geometry and material parameters correspond to realistic measurements. Our goal is a comparison of the final crack pattern in simulation and experiment.

Anisotropic viscoelastic–viscoplastic continuum model for high-density cellulose-based materials

A continuum material model is developed for simulating the mechanical response of high-density cellulose-based materials subjected to stationary and transient loading. The model is formulated in an infinitesimal strain framework, where the total strain is decomposed into elastic and plastic parts. The model adopts a standard linear viscoelastic solid model expressed in terms of Boltzmann hereditary integral form, which is coupled to a rate-dependent viscoplastic formulation to describe the irreversible plastic part of the overall strain. An anisotropic hardening law with a kinematic effect is particularly adopted in order to capture the complex stress–strain hysteresis typically observed in polymeric materials. In addition, the present model accounts for the effects of material densification associated with through-thickness compression, which are captured using an exponential law typically applied in the continuum description of elasticity in porous media.Material parameters used in the present model are calibrated to the experimental data for high-density (press)boards. The experimental characterization procedures as well as the calibration of the parameters are highlighted. The results of the model simulations are systematically analyzed and validated against the corresponding experimental data. The comparisons show that the predictions of the present model are in very good agreement with the experimental observations for both stationary and transient load cases.

A numerical basis for strain-gradient plasticity theory: Rate-independent and rate-dependent formulations

A numerical model formulation of the higher order flow theory (rate-independent) by Fleck and Willis [2009. A mathematical basis for strain-gradient plasticity theory – part II: tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 1045-1057.], that allows for elastic–plastic loading/unloading and the interaction of multiple plastic zones, is proposed. The predicted model response is compared to the corresponding rate-dependent version of visco-plastic origin, and coinciding results are obtained in the limit of small strain-rate sensitivity. First, (i) the evolution of a single plastic zone is analyzed to illustrate the agreement with earlier published results, whereafter examples of (ii) multiple plastic zone interaction, and (iii) elastic–plastic loading/unloading are presented. Here, the simple shear problem of an infinite slab constrained between rigid plates is considered, and the effect of strain gradients, strain hardening and rate sensitivity is brought out. For clarity of results, a 1D model is constructed following a procedure suitable for generalization to 2D and 3D.

Size effects in the conical indentation of an elasto-plastic solid

The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix–Gao model, which have been extensively used to predict size effects in indentation hardness.

A new strain hardening model for rate-dependent crystal plasticity

This paper presents a crystal plasticity based finite element analysis employing the new microstructure-based strain hardening model recently presented by Saimoto and Van Houtte (2011) [7] to simulate formability and texture evolution in the commercial aluminum alloy 5754. Simulations are performed to compare the predictive capability of the new hardening model against the common work hardening models using a rate-dependent plasticity formulation. The parameters in the numerical models are calibrated using the X-ray data published by Iadicola et al. (2008) [9] for the aluminum sheet alloy 5754. The predictions of the model for balanced biaxial tension and in-plane plane-strain tests are compared against experimental observations presented in Iadicola et al. (2008) [9]. It is concluded that the new model provides the best predictions of the large strain behavior of Aluminum sheet alloy 5754 subjected to various strain paths.

Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.

Numerical analysis of stick-slip instability by a rate-dependent elastoplastic formulation for friction

In various fields of engineering, it is important to clarify friction-induced vibration, such as stick-slip motion, for a wide range of scales from microscopic elements to continental plates. In the present study, we apply a rate- and state-dependent friction model [30] (Hashiguchi and Ozaki, 2008), which can rationally describe the reciprocal transition between the static friction and the kinetic friction by a unified formulation, to the simulation of stick-slip instability for a one-degree-of-freedom spring–mass system under various conditions. It is verified that the various basic experimental findings on stick-slip motion can be pertinently described by the present approach. Moreover, the effect of the dynamic characteristics of the system, such as the mass, stiffness and driving velocity, is discussed, and parameters prescribing the rate of reciprocal transition of static–kinetic frictions and the preliminary microscopic sliding on the instability of the stick-slip motion are also discussed.