Thermomechanical Constitutive Model Research Articles

Dynamic thermomechanical response and constitutive modeling of eutectic high-entropy alloy

Since the eutectic high-entropy alloys (EHEAs) have been expected to be candidates for superalloys at the beginning of their invention, getting insights into their high-temperature mechanical performance under impact loadings is of particular significance for exploring their application potential in extreme service environments with high temperature and high strain rate. Meanwhile, additive manufacturing methods have been found to be very suitable for preparing EHEAs in the past two or three years. Therefore, this work, as the first attempt, carries out a comparative study on the microstructural and dynamic thermomechanical performances of the Ni32Co30Cr10Fe10Al18 (Ni32Al18) EHEA manufactured by the laser metal deposition (LMD) and arc melting. The uniaxial compressive responses are tested over a strain rate range of 0.001/s∼7000/s and a temperature range of 77 K∼1123 K. The difference in microstructures and plastic flow behaviors between the LMD and the arc melted samples are revealed. The results show that the LMD samples feature a combination of the primary FCC phases and the typical lamellar eutectic structures, while the arc melted samples possess only ultrafine lamellar eutectic structures. The LMD samples exhibit lower strength and higher ductility than the arc melted ones. The high strength in the arc melted samples is attributed to the high athermal resistance of the dense lamellar structures (composed of ultrafine FCC and B2 phases) to mobile dislocations, while the primary FCC phases lead to high ductility and strain hardening ability in the LMD samples. The anisotropy in flow stress of the LMD sample is found at each strain rate and attributed to the different phase boundary densities generated by the LMD route in different directions. Then, a viscoplastic constitutive model considering the microstructural features is developed, which can reflect the size effect of grains and phases on flow stress, as well as the influence of the phase content on the rate-temperature coupling effect. This model is demonstrated to successfully predict the dynamic plastic flow behavior of both the LMD and arc melted Ni32Al18 EHEA over a wide range of temperature. Furthermore, the Ni32Al18 EHEA is found to have superior high-temperature dynamic specific yield strength compared to several existing typical superalloys.

Coupled thermo-mechanical constitutive damage model for sandstone

Underground rock dynamic disasters are becoming more severe due to the increasing depth of human operations underground. Underground temperature and pressure conditions contribute significantly to these disasters. Therefore, it is important to understand the coupled thermo-mechanical (TM) behaviour of rocks for the long-term safety and maintenance of underground tunnelling and mining. Moreover, investigation of the damage, strength and failure characteristics of rocks under triaxial stress conditions is important to avoid underground rock disasters. In this study, based on Weibull distribution and Lemaitre’s strain equivalent principle, a statistical coupled TM constitutive model for sandstone was established under high temperature and pressure conditions. The triaxial test results of sandstone under different temperature and pressure conditions were used to validate the model. The proposed model was in good agreement with the experimental results up to 600 °C. The total TM damage was decreased with increasing temperature, while it was increased with increasing confining pressure. The model’s parameters can be calculated using conventional laboratory test results.

Thermomechanical Constitutive Models of Shape Memory Polymers and Their Composites

Abstract Shape memory polymers (SMPs) and SMP composites (SMPCs) have been widely employed in several fields and exhibit excellent self-actuation, deformation, and self-adaption. Establishing reasonable constitutive models is vital for understanding the shape memory mechanism and expanding its applications. Moreover, the mechanical response of SMPs under different conditions can be predicted, facilitating their precise control. The internal mechanism for the shape memory behavior in most SMPs is thermal actuation. This study reviews the theories of thermally actuated SMPs, rheological and phase transition concept models, and models combining the rheology and phase transition concepts. Furthermore, the constitutive models of particulate-reinforced SMPCs, carbon-fiber-reinforced SMPCs, and the buckling behavior of SMPCs are summarized. This study is expected to help solve the remaining issues rapidly and contribute to the establishment of rational constitutive models for SMPs and SMPCs.

A Constitutive Model for Polycrystalline U–Nb Alloy Considering Multiple Phase Transition

AbstractThis study proposes a thermomechanical constitutive model for the shape memory effect of U–12.8 at.% Nb (U–Nb) alloy. The phenomenological model uses two variables for transformation. A transitional evolution function based on dissipative force is constructed, which removes the consistency condition. This improvement leads to a natural smoothness of all transitions, without any specifically defined hardening function. In addition, hysteresis is a natural result of the same reference force being converted forward and backward, and defining two different mixing energies is not necessary. The effect of strain rate is captured in the same frame by a function of the transition coefficient. This paper designs a robust and efficient numerical algorithm for the ordinary differential equation system. The model is validated by several special load cases. The simulation results are then compared with previous experimental data in the paper.

Deep learning for predicting the thermomechanical behavior of shape memory polymers

Thermomechanical constitutive modeling is essential for shape memory polymers (SMPs) to be used in engineering structures and devices. However, the classical method of deriving constitutive models is difficult, time consuming, and relies heavily on trial and error. In this work, we aim to decrease the time and resources needed to develop new thermomechanical models for SMPs. The method proposed in this work uses deep learning (DL) to predict the thermomechanical behavior of SMPs under thermomechanical cycles. Particularly, a semicrystalline two-way shape memory polymer (2W-SMP) is selected as an example. Predicting such behavior will give insight on the SMP properties and help validate its characteristics. In this paper, we have compared several DL models to find which one can predict the experimental thermomechanical behavior with the highest accuracy within a reasonable time frame. The results reveal that the fully connected neural network (FCNN) and the convolutional neural network (CNN) were the most accurate DL models. Overall, using one of the selected DL models, we can predict the results of new iterations of the experiment without spending as much time and resources.

Mechanics of shape-locking-governed R2G adhesion with shape memory polymers

Shape memory polymers (SMPs), with unique properties such as tunable elastic modulus, temporary shape locking and shape recovery upon external stimulation, are emerging as a new class of smart materials with switchable adhesion capabilities. A prominent feature of the adhesion between SMP and a spherical indenter is the so-called R2G adhesion, defined as making contact in the rubbery state to a certain indentation depth followed by detachment in the glassy state. While it has been demonstrated that the R2G adhesion with SMPs can achieve orders of magnitude higher adhesive strength compared to conventional elastic adhesive systems with similar geometrical and adhesion parameters, the fundamental mechanics of R2G adhesion and why it leads to such tremendous adhesion enhancement remain largely mysterious at this point. Here, combined experimental testing, theoretical analysis, and finite element analysis (FEA) based on a thermomechanical constitutive model of SMP are performed to investigate the mechanics of R2G adhesion with a rigid spherical indenter. It is shown that the orders of magnitude enhancement of R2G adhesion over conventional elastic adhesion systems is mostly governed by the shape locking effect during the transition from the rubbery to glassy states. The shape locking effect freezes the deformed configuration of the SMP substrate, resulting in nearly conformal contact between the spherical indenter and the glassy-state SMP substrate, thus greatly increases the effective radius of curvature of the contact surface. Our experimental measurements and FEA simulations demonstrate that the net effect of shape locking leads to a pull-off force of a sphere nearly the same as that of a flat punch on an elastic half-space with the same contact radius. An explicit expression of the pull-off force for R2G adhesion is proposed based on flat-punch adhesion. Our results from combined experimentation, modeling and simulations reveal the fundamental mechanics of R2G adhesion. Such understanding provides guidance for the design of SMP smart adhesives.

Numerical investigation on Sn-rich solder joints failure mode: Effects of the thermomechanical anisotropic performance

In this paper, a thermomechanical anisotropic constitutive model was employed to investigate the failure mode of multiple Sn-rich solder joints in system in package under thermal cycle loading. As a comparison, traditional isotropic constitutive model was also employed to investigate the failure mode. Results showed that the anisotropic constitutive model predicted a new failure mode that failure would occur earlier at inner solder joint rather than the outmost solder joint, which was different from traditional perspective. This new failure mode indicated that traditional fatigue life prediction model just based on the outermost solder joint was not enough in view of the thermomechanical anisotropic performance of Sn-rich solder.

Thermomechanical coupling effect on the phase transition wave propagation in an SMA TiNi bar subjected to shock loading

Phase transition can strongly affect the stress wave propagation features. In this paper, the influence of temperature on the phase transformation wave (PTW) propagation was investigated theoretically and numerically with a thermo-mechanical coupling constitutive model and kinetic relation of the phase transition. The results showed that the temperature interface is also the phase transition threshold stress discontinuity. For shock wave propagating in the rod with fixed temperature interfaces, the transformation from shock wave to phase transition wave, then to an elastic wave was predicted due to the temperature interface effect, and an unloading elastic wave during loading was discovered induced by the boundary effect, which has barely been studied before. Moreover, for the adiabatic shock loading, phase transition wave front is not only a material interface, but also a temperature discontinuity. A loading shock wave with phase transition was predicted due to the second phase strengthening and an unloading shock wave with reverse transformation was generated arising from unloading path changing effect induced by adiabatic temperature rise across the transformation shock front. The theoretical analysis and simulation discloses the evolution of phase transition waves affected by temperature interfaces and reflect the intrinsic characteristic of phase transition materials with strong nonlinear behavior due to the dynamic thermo-mechanical coupling.

Insight in thermomechanical constitutive modeling of shape memory polymers

Shape memory polymers (SMPs) are a new member of the smart materials family. SMPs have found wide applications or potential applications in almost all manmade structures and devices. In order to better design smart structures and devices using SMPs, thermomechanical constitutive modeling is essential. In this insight paper, we will focus on presenting several multi-length-scale and multi-physics modeling frameworks, including the thermodynamics consistent model, elasto-viscoplastic model, statistical mechanics model, and phase evaluation law model. The SMPs modeled will include amorphous one-way shape memory polymers, semicrystalline one-way shape memory polymers, semicrystalline two-way shape memory polymers, and functional and mechanical damage effects on SMPs. Finally, we will give some in-depth perspectives on future development in this area of study.

A thermodynamically consistent two surface/bubble thermo-mechanical model considering thermal and mechanical cyclic behaviour of fine-grained soils

The formulation of a two surface/bubble thermo-mechanical constitutive model consistent with the principles of thermodynamics is presented. This allows plastic deformations inside the outer yield surface, resulting in a smooth stress–strain prediction and progressive cyclic deformations. This is achieved by the translation of the inner yield surface (also known as the bubble surface) with the stress state of the soil, inside the outer yield surface, by using a kinematic rule. The constitutive equations, including the hardening rules, are derived by specifying a Gibbs-type energy potential and a rate of dissipation potential function, ensuring thermodynamic consistency. The kinematic rule is divided into isothermal and non-isothermal parts. With the isothermal component, the model is capable of capturing the hysteresis behaviour of soils during cyclic mechanical loading. With the non-isothermal part, the model is able to predict the shakedown behaviour of soils observed when they are subjected to heating–cooling cycles. The performance of the model is compared with various experimental data for isothermal and non-isothermal conditions, and is shown to be in good agreement.

Fatigue behavior of laminated composites with embedded SMA wires

Embedding pre-strained shape memory alloy (SMA) wires within laminated composites can enhance their fatigue performance due to pseudoelastic and shape memory effects. In the present research, an experimental program was conducted to investigate the fatigue behavior of carbon/epoxy laminated composites with/without embedded SMA wires. Moreover, a novel fatigue model was proposed to predict the behavior of laminated composites with embedded SMA wires. The model is an integration of three parts. The first part is the progressive fatigue damage model to simulate the fatigue behavior of composites. The second part is the self-heating model for the prediction of the self-temperature rise in laminated composites subjected to cyclic loading. The third part is the thermomechanical constitutive model of the mechanical behavior of the SMA wire. Effects of the stacking sequence, the applied stress level, and the pre-strain level of SMA on fatigue behavior of laminated composites were studied. Good agreement between theoretical and experimental results revealed the ability of the present model.

A thermomechanical constitutive model for investigating the fatigue behavior of Sn‐rich solder under thermal cycle loading

AbstractBased on crystal plastic theory, a novel thermomechanical constitutive model was proposed for Sn‐rich solder. Material parameters of the proposed constitutive model were determined by fitting the stress–strain curves under uniaxial tensile testings. Combined with the proposed model and the material parameters, crystal plastic finite element method was used to investigate the effects of the microstructure features of grain orientation, grain boundary, and intermetallic compound on the fatigue behaviors of Sn‐rich solder joint under the condition of thermal cycle loading. Results showed that the mean absolute percentage error between the simulation and experimental tensile curves was 3.50%, and the three microstructure features affected the fatigue behaviors of Sn‐rich solder from different aspects, which were analyzed in detail. Besides, it was concluded that the proposed constitutive model could be effectively used to investigate the fatigue behaviors of Sn‐rich solder joint from micro perspective.

Broaching force prediction of nickel-based superalloy based on material parameter identification

The broaching process of the nickel-based superalloy determines the aero-engine turbine disc mortises performance; thus, the broaching force should be analyzed principally. The aim of the study is to reveal the material constitutive law for the nickel-based GH4169 alloy broaching force prediction. However, the GH4169 material constitutive model may just be verified by the hot compression tests at the limited high-temperature conditions. The residual stress-based finite element updating (FEU) inverse method was used to optimize the GH4169 material parameters. And the finite element (FE) model for the whole multi-stage processing of the turbine disc billet was simulated with high efficiency and accuracy, where the wide forming temperature range would be the critical factor for the consequent broaching analysis. The cutting conditions were tested to measure the residual stresses in the multi-stage formed GH4169 alloy disc billet by the layer removal method. The objective function (OF) designed with the experimental-numerical residual stresses were optimized successfully. The GH4169 alloy broaching force predicted with the identified thermo-mechanical material constitutive model in this study was satisfied with the validation test results, and the influence of cutting parameters on the broaching force distribution design could be effectively controlled, which could guarantee the high dimensional accuracy and the excellent reliability of the aero-engine turbine disc.

A variational formulation of thermomechanical constitutive update for hyperbolic conservation laws

In this paper, a variational framework is proposed for the constitutive update of thermomechanical constitutive models in the special case where their input results from quantities directly updated by hyperbolic conservation laws. Both a continuum and a consistent first order accurate discrete settings are derived. The originality of this work lies in that the constitutive update is driven by the rates of some strain measure and the internal energy density in the continuum setting, leading to a rate-type description of the local constitutive problem, and by the updated values at some discrete time of these strain measure and internal energy density in the discrete setting. These quantities are updated by the solution of a system of discrete conservation laws including the first principle of thermodynamics, ensuring that the right shock speeds will be computed. This point is of crucial importance when simulating impact on structures for instance. The proposed variational approach is illustrated for thermo-hyperelastic-viscoplastic solid media, especially using the parameterization of the flow rule direction based on pseudo-stresses proposed by Mosler & co-workers. The proposed discrete variational solver is then coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the set of conservation laws. Comparisons are performed on a set of test cases with numerical solutions obtained with finite elements coupled to an explicit time-stepping and to a temperature-driven variational constitutive update. They allow to show the good behavior of the proposed approach.

A thermo-mechanical constitutive model for fine-grained soils based on thermodynamics

The formulation of a new thermo-mechanical constitutive model consistent with the principles of thermodynamics is presented. The model is capable of predicting the rate-independent thermo-mechanical behavior of fine-grained soils. The constitutive equations are derived by defining only a Gibbs-type free energy and a dissipation potential, in accordance with the hyperplasticity method. The addition of thermo-elasticity to the energy potential, and the embedding of the identified thermo-mechanical mechanisms into a newly proposed dissipation potential, enables the model to describe the thermo-mechanical behavior. The proposed dissipation potential eliminates the application of shift stress, which results in a simpler formulation in the context of hyperplasticity. The step-by-step procedure of deriving the equations, as well as a detailed analysis of the parameters of the model, is presented. The performance of the model is shown to be in good agreement with experimental data. A qualitative description of the possible micro-scale mechanisms for fine-grained soils, when subjected to temperature variation, is presented, as a step towards including the mechanisms in the formulation.

Tutorial: Thermomechanical constitutive modeling of shape memory polymers

Shape memory polymers (SMPs) are one of the intriguing functional materials and have been widely and intensively studied. In order to apply these new polymers to load bearing engineering structures and devices, developing physics-based thermomechanical constitutive models is mandatory. The aim of this Tutorial is to demonstrate how to establish a thermomechanical constitutive model for SMPs. It begins with classifications of SMPs, followed by a discussion on the underlying physics for different SMPs. After that, three classical SMP thermomechanical modeling frameworks are introduced, which include the visco-elasto-plastic based rheological framework, the storage strain-based phase transition framework, and the representative unit cell based multi-branch framework. Next, three commonly adopted new model establishment methods are presented within these frameworks with detailed examples. Finally, future perspectives on this research direction are discussed. We hope that this Tutorial will help readers understand the roadmap from physics to mathematical modeling of SMPs.

A Hybrid Local/Nonlocal Continuum Mechanics Modeling of Damage and Fracture in Concrete Structure at High Temperatures

This paper proposes a hybrid peridynamic and classical continuum mechanical model for the high-temperature damage and fracture analysis of concrete structures. In this model, we introduce the thermal expansion into peridynamics and then couple it with the thermoelasticity based on the Morphing method. In addition, a thermomechanical constitutive model of peridynamic bond is presented inspired by the classic Mazars model for the quasi-brittle damage evolution of concrete structures under high-temperature conditions. The validity and effectiveness of the proposed model are verified through two-dimensional numerical examples, in which the influence of temperature on the damage behavior of concrete structures is investigated. Furthermore, the thermal effects on the fracture path of concrete structures are analyzed by numerical results.

Damage Evolution Constitutive Behavior of Rock in Thermo-Mechanical Coupling Processes

For thermal and loaded rock in engineering structures for some projects, triple-shear Drucker–Prager yield criteria, compaction coefficient K, damage variable correction factor δ, and thermal damage variable DT are introduced in a new thermomechanical (TM) constitutive model for the entire process. The compaction stage of rock in uniaxial compression test and the strain softening of rock caused by thermal attack are considered in this article. The damage evolution of rocks is described by a damage variable and a constitutive equation, which are in agreement with the actual thermal experimental breakage. The uniaxial compressive strength of granite subjected to a TM coupling effect can be predicted properly by this new unified constitutive model. The new TM unified constitutive model considering the compaction stage and post-failure stage is in good agreement with the test curves throughout the entire process. The coupling effect of heat and load in the total damage of rock has obvious nonlinear properties, but the coupling effect significantly weakens the specimens. By using the new TM unified constitutive model, the whole process of changes in rock damage with strain after high temperature can be calculated. Meanwhile, the model well represents the stress–strain curve at the post-failure stage. It is expected that this model can provide references for studying the mechanical response of the rock damage propagation characteristics in the future.

Atomistic investigation on the conversion of plastic work to heat in high-rate shear deformation

The conversion of work into heat is one of the most significant characteristics of plastic deformation, especially in a high strain rate regime. The quantitative characterization of the fraction of plastic work converted into heat plays a crucial role in understanding the deformation process and mechanism and establishing an accurate and reasonable thermomechanical constitutive model. However, the fraction of plastic work converted into heat is experimentally found to span over a wide range of values between 0.1 and 1. In this paper, the conversion of plastic work into heat in high-rate shear deformation is systematically investigated through the relations between temperatures, stresses, elastic and plastic strains using molecular dynamics simulation. The results show that the temperature rise in the shear deformation is caused by the heat dissipation of the plastic work rather than the thermo-elastic coupling effect due to invariance of the volume. Dislocation nucleation and glide are the main mechanisms of shear deformation and are also the origins of heat generation. The continuous slip of dislocations also results in the formation of deformation twins. Interestingly, the work-to-heat conversion coefficient is revealed to be independent of crystal orientation, in contrast to the plastic deformation itself. From the present study, we conclude that almost all the plastic work in shear deformation is converted into heat dissipation for all studied samples.

A new phenomenological constitutive model for shape memory alloys

A new thermomechanical constitutive modeling approach for shape memory alloys (SMAs) that undergo a martensite to austenite phase transformation and the associated pseudoelastic and shape-memory responses is presented. The novelty of this new formulation is that a single transformation surface is implemented in order to capture the forward and reverse phase transformations, as well as the reorientation and detwinning of the martensite phase. The framework is akin to the usual flow theory plasticity with kinematic hardening, however in addition to the transformation strain there is also a transformation entropy that is directly related to the martensite volume fraction appearing in prior theories. A transformation surface in effective stress and effective temperature space is introduced and an associated flow rule governs the evolution of the transformation strain and entropy. In order to capture the multitude of SMA behaviors, a transformation potential function is introduced in transformation strain and entropy space for the derivation of the back stresses and back temperatures that define the kinematic hardening behavior. It is this potential function that governs all of the important behaviors within the model. After the description of the general theory, specific forms for the transformation surface and the transformation potential are devised and results for the behaviors captured by the model are provided for a range of thermomechanical loadings. Finally, the model is implemented in the finite element method and applied to the pseudoelastic and shape memory effects of a beam in pure bending.