Viscoplastic Model Research Articles

Anand Model Constants of Sn-Ag-Cu Solders: What Do They Actually Mean?

Abstract Finite element simulations are broadly utilized to calculate the thermally induced mechanical deformations of interconnecting solder materials in electronics. The Anand unified viscoplasticity model is a prevalent choice for simulating the mechanical responses of solder joints, comprising nine parameters whose individual effects are not fully addressed in the literature. For this reason, this study examines the influence of each Anand parameter on the mechanical response of leadless Tin-Silver-Copper solder alloys with varying silver content through comprehensive nonlinear finite element simulations. Anand model coefficients for different SAC alloys, including SAC105, SAC205, SAC305, and SAC405, were sourced from existing literature and used to establish a systematic study matrix for each parameter. These coefficients were then integrated into thermomechanical simulations to induce inelastic deformations in the solder interconnections. The response of the solder interconnects is analyzed using equivalent inelastic strains and inelastic strain energy density. Results indicated that specific Anand parameters could skew the solder joint stress-strain response towards brittle behavior, while other parameters could lead to a more ductile response. Through statistical factorial analysis, the significance of each parameter is found to vary considerably, ranging from negligible to highly significant. The findings of this study are crucial for understanding the behavior of different SAC solder configurations and their expected thermal fatigue performance based on Anand creep constants. Furthermore, this paper provides foundational insights into the interpretation of Anand coefficients and their influence on the mechanical response of solder materials, a topic not yet explored in the existing literature.

A deformation-diffusion interactive model to study crack-tip behaviour and predict crack growth rate under fatigue and hydrogen-embrittlement conditions

Hydrogen is accepted as a cleaner and more sustainable energy source, since it carries abundant energy and does not produce any effluent gas from burning when compared to fossil fuels. However, long-term degradation in mechanical properties, attributed to hydrogen embrittlement, is reported for metallic materials exposed to gaseous hydrogen, leading to accelerated crack growth behaviour under fatigue. In this paper, a visco-plastic deformation-diffusion interactive model is developed to study fatigue crack growth behaviour in a nickel-based superalloy under hydrogen-embrittlement condition. In the model, cyclic deformation is described by visco-plasticity constitutive model, while hydrogen diffusion is simulated by a modified form of Fick’s first law of diffusion which can address hydrogen diffusion driven by the hydrostatic stress gradient. In particular, trapped hydrogen, expressed as dependent on both diffusible hydrogen concentration and inelastic strain, is considered in the modelling of hydrogen diffusion. The model also describes the effects of hydrogen concentration on mechanical deformation by considering its influence on the evolution of isotropic hardening as well as on the strain state. Interaction between cyclic deformation and hydrogen diffusion is therefore studied for a crack tip in a compact tension specimen subjected to fatigue and hydrogen attack, showing a strong dependency on both loading frequency and stress intensity factor range. Subsequently, a two-parameter criterion is proposed for fatigue crack growth prediction, which accounts for the contributions of both cyclic deformation and hydrogen diffusion. The predictions show a good agreement with experimental data in terms of crack growth rate under fatigue and hydrogen-embrittlement conditions.

Прогноз толщины ледяного покрова в прибрежных областях Карского и Охотского морей

Based on the CICE viscoplastic sea ice model, the ice cover of the Gulf of Ob and the northwestern Okhotsk Sea was simulated for the winter seasons of 2021/2022, 2022/2023, and 2023/2024. The description of the seasonal variability of sea ice characteristics in these areas and examples of their numerical modeling are given. Using atmospheric forcing based on the WRF-ARW nonhydrostatic atmospheric model, the sea ice thickness in the northwestern Okhotsk Sea was predicted for April 2024. A comparison of the forecasts with the ice maps of the Hydrometcentre of Russia and sea ice thickness measurements at the Ayan and Bolshoy Shantar stations gave satisfactory results.

Features and Constitutive Model of Hydrate-Bearing Sandy Sediment’s Triaxial Creep Failure

In the longstanding development of hydrate-bearing sediment (HBS) reservoirs, slow and permanent deformation of the formation will occur under the influence of stress, which endangers the safety of hydrate development projects. This paper takes hydrate-bearing sandy sediment (HBSS) as the research object and conducts triaxial compression creep tests at different saturation degrees (20%, 30%, and 40%). The results show that the hydrate-containing sandy sediments have strong creep characteristics, and accelerated creep phenomenon will occur under the long-term action of high stress. The longstanding destructive power of the specimen progressively raises with the increase in hydrate saturation, but the difference in the triaxial strength of the specimen progressively increases. This indicates that the damage to the hydrate structure during long-term loading is the main factor causing the strength decrease. Further, a new nonlinear creep constitutive model was developed by using the nonlinear Burgers model in series with the fractional-order viscoplastic body model, which can well describe the creep properties of HBSS at different saturation levels.

A numerical study of contemporary crustal deformation partitioning across the Southwestern Tian Shan orogen

The Tian Shan region is a typical example of crustal deformation within an intracontinental environment, where the lithospheric rheological properties are marked by significant spatial heterogeneity. However, the role of lithospheric rheology in crustal strain partitioning within the interior of the Tian Shan orogenic belt remains nebulous. Here, we utilized a two-dimensional viscoplastic model with contact elements constrained by GPS velocities and fault slip rates to investigate the influence of the block's strength on crustal deformation in the southwestern Tian Shan region. Our founding suggests that a weaker lower crust beneath the northern Tian Shan region offers a potential mechanism for the contemporary deformation patterns. Contemporary crustal deformation at Tian Shan is mainly concentrated along the piedmont thrust-and-fold belts and diffuses in the orogen's interior. This pattern of crustal strain partitioning is closely linked to the interplay between fault slips and lithospheric deformation. This finding emphasizes the significant role that the lithospheric rheology and pre-existing faults played in the deformation partitioning in the interior of the Tian Shan orogeny.

Orientation-dependent deformation and failure of micropillar shape memory ceramics: A 3D phase-field study

Some microscopic samples of zirconia-based shape memory ceramics (SMCs) have shown full martensitic phase transformation (MPT) over multiple loading cycles without cracking. However, the occurrence of MPT is strongly influenced by grain orientation. Depending on the specific grain orientation relative to the loading direction, alternative mechanisms such as plastic slip and fracture may emerge. This study introduces a phase-field (PF) based framework that integrates a PF-MPT model, a PF fracture model, and a crystal viscoplasticity model to investigate the effects of grain orientation on MPT, plastic slip, and fracture mechanisms in SMC micropillars. Single crystal micropillars are created to distinguish the orientations that facilitate each mechanism. A wide range of grain orientations are found to predominantly exhibit MPT. Micropillars with grain orientations close to the (100) and (001) directions primarily experience fracture, with minimal plastic slip. Additionally, samples oriented along the (110) direction show a significant amount of plastic slip. A pole figure is constructed to elucidate the interplay between MPT, cracking, and plastic slip under compressive loading conditions. This research provides valuable insights into the intricate behavior of SMCs under different loading scenarios, crucial for optimizing their performance in practical applications.

A thermal performance study on magnetic dipole based viscoplastic nanomaterial deploying distinct rheological aspects

Magnetic nanoliquids stand inimitable when compared with conventional liquids as their distinctive magnetic attributes can be regulated utilizing magnetic fields. For this reason, heat transference can be stimulated or regulated subjected to externally imposed magnetic fields. Magnetic nanoliquids are useful in comparison to orthodox or non-magnetic nanoliquids. These encompasses, energy conversion, electronics, hydraulics, thermal engineering and bioengineering. This study elaborates the magnetic dipole impact on convectively heated rheological nanomaterial confined by stretchy surface. Mathematical modeling is based on thermally radiative viscoplastic (Casson) model. Porous medium features are scrutinized through Darcian Forchheimer (DF) relation. Two-component Buongiorno nanomaterial model which captures Brownian diffusive together with thermophoretic diffusion is under consideration. Energy and solutal transportation expressions capture thermal source and chemical reaction effects. The dimensionalized nanomaterial flow model for stretching flow is obtained by deploying similarity variables. Numerical solutions are computed through Bvp4c scheme. The physical outcomes are elucidated graphically and arithmetically on dimensionless quantities (i.e., Nusselt number, temperature, skin-friction, velocity, Sherwood number and concentration streams). A benchmark is reported to authenticate the acquired numeric solutions. It is further visualized that nanomaterial temperature escalates subject to higher estimations of thermal Biot number, Curie temperature factor, radiation factor, thermophoresis parameter, heat source, ferrohydrodynamic interaction factor and Brownian diffusive parameter while it diminishes with Prandtl number and dissipation factor.

Anisotropy of plastic flow in Zr-2.5Nb pressure tube material analysed using a viscoplastic self-consistent approach

Plastic anisotropy is observed during plastic flow in samples from the axial and transverse directions of Zr-2.5Nb pressure tubes used in CANDU11CANada Deuterium Uranium - Pressurized Heavy Water power reactors. nuclear reactors tested in uniaxial tension. Plastic anisotropy was also measured in shear by testing in torsion ‘mini’ pressure tubes from the same material. Room temperature results from these experiments were analysed using a visco-plastic self-consistent model that takes into account the crystallographic texture of the material and allows for the single crystal work hardening behaviour to be described by means of a deformation law specific to each slip system.The visco-plastic self-consistent model was used to derive: (i) the evolution with strain of the critical resolved shear stress values consistent with prismatic, basal and pyramidal dislocation glide, (ii) the evolution of slip system activity as a function of strain, and (iii) the values of Hill's plastic anisotropy coefficients that are consistent with the observed anisotropy of yielding and their dependence on accumulated strain. The model also allows for the prediction of the components of the flow stress tensor that cannot be measured experimentally and their dependence on the work hardening behaviour of pressure tube material. Moreover, the yield surface after different amounts of plastic strain was calculated using the visco-plastic self-consistent model, compared to the one that results from using the Hill's anisotropy coefficients. Our work exposes the limitations of the Hill ellipsoid for describing plastic yield when microstructural evolution is present.

Unraveling anisotropic mechanical behaviors of lithium-ion battery separators: Microstructure insights

The mechanical properties of separators significantly affect the electrochemical stability and potential short circuit risks in lithium-ion batteries. An important aspect of their mechanical behavior is their anisotropy, which is predominantly influenced by the microstructures formed during manufacturing process. This study aims to bridge the gap between the anisotropic mechanical features of the separators and their microstructures caused by the manufacturing methods. Initially, we delve into the characterization of separators, featuring their heterogeneous components and orientated arrangement of fibers. Then, we conduct uniaxial tensile tests to measure the stress-strain relationships of separators along the machine direction (MD), diagonal direction (DD), and transverse direction (TD), revealing pronounced anisotropy in both strength and rate sensitivity. Subsequently, image processing techniques is adopted to obtain a representative configuration of separators, which is further divided into fibers and lamellae. According to the manufacturing process of separators, a viscoplastic model is used to describe the mechanical behavior of lamellae while a strengthened viscoplastic model is utilized to mimic the mechanical response of fibers. The finite element analyses underscore the dominant role of orientated fibers in determining anisotropic mechanical properties. Furthermore, we explore the effects of manufacturing and geometry parameters on the separator's anisotropic mechanical behavior. This research provides valuable insights for optimizing manufacturing parameters and enhancing safety measures for lithium-ion batteries.

A hot deformation constitutive model applicable for complete austenite and dynamic ferrite transformation interval

Dynamic ferrite transformation is a vital microstructure evolution mechanism for carbon alloy steels and an essential technology for manufacturing high-performance steel materials. In practice, carbon steel hot working processes are usually carried out at various temperatures, involving the evolutionary behavior of the single-phase austenite and the ferrite dynamic transformation process. Few models can simultaneously predict the microstructural state and deformation resistance in the single-phase austenitic interval and the dynamic ferrite transformation interval. The deformation mechanisms involve work-hardening, dynamic recovery, and dynamic recrystallization of austenite and ferrite phases, as well as the dynamic transformation of austenite to ferrite and coordinated deformation between the two phases. Therefore, this paper extends the application range of the model based on the viscoplastic unified constitutive model by considering the above deformation mechanisms. The constitutive model proposed in this paper applies to both the single-phase austenite interval and the dynamic ferrite transformation interval. Moreover, it can accurately predict carbon alloy steel's microstructure state and deformation resistance in different intervals. This paper can help to construct a hot deformation constitutive model in a wide range of deformation temperature intervals of steel materials.

Finite element modeling for cohesive/adhesive failure of adhesive structures with a thermosetting resin

In this study, we established a novel method based on the finite element method (FEM) for predicting the strength of adhesive structures. Viscoplasticity, void growth, and cohesive zone model were introduced into the FEM to create a nonlinear damage growth (NDG) model. This model was used to comprehensively analyze the process zones within the adhesive layer. Furthermore, the embedded process zone approach was used to develop an interface constitutive law that averages the mechanical response of the adhesive layer. This modified Ma–Kishimoto (MMK) model can accurately represent the adhesive layer as an interface element and is computationally efficient. Furthermore, the study obtained the necessary interface properties for the MMK model from the NDG model, creating a numerical material test that can approximate the effect of the process zone. To validate the proposed method, single-lap shear tests were performed, and the accuracy of the predicted strength and deformation field was evaluated. The damage evolution in the NDG model and the MMK model were compared, and the scope of application of the MMK model was discussed. The results of this study can be used as a reference for the failure mechanism of thermosetting adhesives and establishment of design indices for adhesive structural strength.

Nonlinear mechanics of horseshoe microstructure-based lattice design

Enhancing buffering capacity, flexibility, and energy absorption to withstand large deformations in structure remains a challenge. Bio-inspired horseshoe lattice structures, with their curved trusses, exhibit distinct mechanical characteristics compared to conventional metamaterials. However, their mechanical properties under in-plane compression have been rarely explored. This study characterised and modelled three types of novel 3D-printed horseshoe lattice structures, totalling 12 configurations, with unit cell geometry varying based on cell-wall angles ranging from 120°to 210°. The implementation of the FE simulation based on the three-network viscoplastic (TNV) model showed good agreement with the experiments. The results demonstrated that the cell-wall angle in the geometry and the cross-lap joint topology were significantly associated with the failure mechanism of the unit cell and the overall non-linear mechanical behaviour. Increasing the cell-wall angles can prevent beams from failing due to bending and buckling fractures, facilitate the initiation of internal contacts and stretching during in-plane compression. This reveals a configurable mechanism where the flexibility and stability of the lattice structure can trigger strain hardening, resulting in an increase in load-bearing capacity. The sensitivity to strain hardening varies depending on the order of cross-laps within the topology. A colour-pattern tracking method was employed to monitor the progressive stabilisation of lattice structures, and offering a novel approach for the future design of flexible, configurable, and programmable horseshoe-based lattice structures.

Thermo-mechanical fatigue analysis of ferritic stainless steel STS444LM for exhaust manifold application

This work presents thermo-mechanical fatigue (TMF) life analysis of ferritic stainless steel STS444LM recently developed to enhance the durability of the automotive exhaust manifolds. To determine cyclic material properties, isothermal tensile and low cycle fatigue (LCF) tests at temperature ranging from 200 °C to 900 °C under two different strain rates of 0.1 %/s and 0.01 %/s are performed. Based on these tests, the unified Chaboche visco-plasticity model is determined. Then, the TMF test using the V-shaped specimen subjected to the temperature cycle from 250 to 850 – 950 °C with three different dwell times at the maximum temperature is performed. An energy-based model based on the Morrow model is determined using TMF test data by incorporating effects of the dwelling time and maximum temperature.

Study of the one-dimensional consolidation and creep of clays with different thicknesses using different hypotheses and three elastic visco-plastic models

The one-dimensional consolidation analysis of clays considering creep compression is a classical issue in soil mechanics and geotechnical design. The major debate lies in how to predict the consolidation settlement for a thick layer in the field using parameters obtained from a thin specimen from the laboratory. Different hypotheses have been advocated, based on which various methods and constitutive models have been developed. However, there are still some questions unaddressed and concepts inconsistently used, which may mislead engineers in the selection of methods/models and may result in settlements underestimated on a risk design side. In this paper, a state-of-the-art review and a thorough comparison study are performed on the existing methods and models for the consolidation analysis of clays exhibiting creep, from theoretical derivations to numerical simulations in comparison with soil test data. An in-depth discussion is carried out on several key issues related to the thickness effects on the time-dependent compression behaviour of clays. The arguments of Hypothesis A and Hypothesis B are revisited based on the current development of constitutive theories. Three existing elastic visco-plastic (EVP) models that consider the creep compression implicitly during the whole consolidation process can perform well in predicting the settlement of clay layers with different thicknesses, and are in line with Hypothesis B. It is concluded that using existing EVP models based on porous-media continuum mechanics is a rigorous scientific method (also called “rigorous” Hypothesis B method), which is superior to the old Hypothesis A method which has logic errors and may result in unsafe underestimation of settlements.

Non-isothermal Fractional Order Two-surface Viscoplastic Model for Stiff Clays

Stiff clay exists widely in the world, but its significant time- and temperature-dependent mechanical features have not been fully modelled. In the context of fractional consistency viscoplasticity and bounding/subloading surface theory, this study proposes a novel non-isothermal fractional order two-surface viscoplastic model for stiff clays. First, with proposing a generalized plastic strain rate, the isotach viscosity is modified and extended to both over-consolidated and non-isothermal conditions that takes into consideration the effects of temperature and OCR on thermal accelerated creep. Then, two strain rate and temperature dependent yield surfaces are proposed with isotropic and progressive hardening rules to consider thermal collapse, strain rate effects, and smooth transition from elastic to viscoplastic behaviors. Next, the stress-fractional operator of the loading surface, according to the principle of the fractional consistency viscoplasticity, is introduced to describe the non-associativity of stiff clays. Finally, the predictive ability of the model is validated through simulating triaxial tests on Boom clay with various stress paths considering the temperature- and time-dependent features of stiff clays.

Anisotropic formability and deformation mechanism of near-α titanium alloy sheet under continuous nonlinear strain paths at high temperature

In the present study, the anisotropic formability and underlying deformation mechanism of near-α TA32 titanium alloy sheet under continuous nonlinear strain paths (CNSPs) at high temperature were investigated in depth. To achieve this goal, a new experimental method combining hot gas bulging with step-combined dies was proposed, and the hot CNSPs of metal sheets can be flexibly and conveniently realized by changing the number and shape of step-combined dies. Based on this method, the anisotropic deformation behavior and forming limits of TA32 sheet under fifteen CNSPs were tested at 800 ℃ with a strain rate of 0.001 s−1. Then, two advanced constitutive models with different scales were embedded into the classical Marciniak-Kuczyński (M-K) theory to predict the forming limits of TA32 sheet under different strain paths: the macro-scale viscoplastic model coupled with Hill48 yield criterion and non-associated flow rule (NAFR) as well as the meso-scale three-dimensional crystal plasticity finite element (CPFE) model coupled with cellular automata (CA). The results demonstrate that the CPFE-CA-MK coupled model exhibits higher accuracy in predicting the forming limits of TA32 sheet under linear and continuous nonlinear strain paths. Especially for the tension-tension strain paths, the CPFE-CA-MK coupled model improves accuracy by at least 3.1 % compared to macro-scale models. Due to the material anisotropy, the initial inclination angle of the groove in the CPFE-CA-MK model is closely related to the strain path and significantly affects the prediction accuracy. Based on the CPFE simulation, the effects of anisotropy and strain path change on the dislocation slip mode of different texture components was analyzed in depth, which provides a theoretical guidance for the optimization of hot forming process of TA32 titanium alloy complex components.

Thixoviscoplastic flow simulations based on Houska thixotropic and Bingham viscoplastic models

Thixoviscoplastic flow simulations based on Houska thixotropic and Bingham viscoplastic models

Unified viscoplastic constitutive model for creep–fatigue behavior of austenitic stainless steel 304 under axial–torsional loading

Unified viscoplastic constitutive model for creep–fatigue behavior of austenitic stainless steel 304 under axial–torsional loading

A polycrystal plasticity-cellular automaton integrated modeling method for continuous dynamic recrystallization and its application to AA2196 alloy

Continuous dynamic recrystallization usually dominates the microstructural evolution in hot working of aluminum alloys, in which the high-angle grain boundaries of new grains mainly originate from the gradual increase in subgrain misorientation angles. In this work, an integrated computational method is proposed to simulate continuous dynamic recrystallization process of aluminum alloys by coupling three-dimensional cellular automaton and visco-plastic self-consistent models. The stress response, dislocation accumulation and recovery, and evolution of crystal orientations are computed in the context of polycrystal plasticity; the formation and rotation of subgrains, followed by stored energy and curvature-driven boundary migration, are captured and visualized by cellular automaton. The non-octahedral slip mode {110}<110> is additionally introduced to capture the 〈001〉 texture during hot compression. A universal cell topology deformation method is adopted to achieve an effective track of grain morphology evolution during plastic deformation. The proposed simulation framework is validated through simulating the isothermal uniaxial compression process of AA2196 alloy under different temperatures and strain rates. The orientation dependence of CDRX during compression is numerically reproduced by correlating the subgrain formation and rotation process with the activation state of slip systems. The simulated macroscopic flow stress, 3D microstructure and inherent microstructural characteristics such as subgrain size, subgrain boundaries and textures are in good agreement with the experimental results. The proposed method provides an effective and efficient tool for multi-scale simulation of hot forming process of aluminum alloys.

Investigation of hot deformation behavior and three-roll skew rolling process for hollow stepped shaft of Al–Zn–Mg–Cu alloy

The increasing demand for high-strength lightweight hollow shafts in transportation highlights the need for advanced fabrication techniques. Al–Zn–Mg–Cu alloys, noted for their superior properties, are selected for three-roll skew rolling (TRSR). In TRSR, the material undergoes combined axial tensile and radial compressive stresses. This study evaluates the feasibility of TRSR for producing high-strength lightweight hollow stepped shafts from Al–Zn–Mg–Cu alloy. An integrated approach, including constitutive modeling, hot processing map development, and TRSR numerical simulations/experiments, is employed to optimize the TRSR forming process. The constitutive model was established based on 300°C–450 °C & 0.01–10 s−1 hot compression and 350°C–430 °C & 0.1–5 s−1 high-temperature tensile test data. The established Johnson-Cook optimization by genetic algorithms (GA-JC) model and unified viscoplastic constitutive model, accurately capture the alloy's hot deformation behavior, exhibiting minimal average absolute relative errors (AARE) of 5.431% and 5.808%, respectively. Microstructure evolution analyses shed light on the predominant softening mechanisms, emphasizing dynamic recovery (DRV) at elevated strain rates and diminishing texture intensity with escalating deformation temperatures. The composite hot processing map delineates optimal process parameters (400°C–450 °C & 0.1s−1-1s−1), facilitating informed decision-making in manufacturing practices. The validation of numerical simulations through TRSR forming experiments with initial temperature of 450 °C for the billet and axial moving speed of 10 mm/s for the chuck in affirms the feasibility of producing hollow stepped shafts from high-strength Al–Zn–Mg–Cu alloy. Close agreement was found between simulated and experimental wall thickness variations. This study enhances understanding and optimization of TRSR forming for high-strength lightweight alloys, advancing industrial manufacturing methodologies.