Thermomechanical Constitutive Model Research Articles

A unified thermo-mechanical bounding surface model for saturated clay and sand

To accurately predict soil thermal effects is of great importance for simulations of complex boundary value problems such as the energy foundations and nuclear waste disposal. Existing thermo-mechanical constitutive models only account for clay, and cannot simulate the sand’s behaviour under thermo-mechanical conditions. In this study, a unified thermo-mechanical bounding surface (UTMBS) model is proposed for saturated clay and sand. Based on the thermal effects on the isotropic compression line, the model proposes a new unified thermal softening relationship and a plastic modulus for clay and sand, with the thermal cyclic behaviour replicated by a memory surface. The unified model considers the thermal effects on the critical state line and the shape of the bounding surface, accounting for both drained and undrained shearing of clay and sand at different temperatures. In addition, the non-linear elasticity relationship represents the hysteresis loops of the stress–strain relationship in the mechanical cycles. The performance of the proposed model is evaluated against existing experimental results for clay and sand in terms of their thermal cyclic behaviour, drained/undrained triaxial compression, and mechanical cyclic behaviour at different temperatures. It is evident that the UTMBS model is able to simulate various thermo-mechanical behaviours of clay and sand.

A thermo-mechanical constitutive model for triple-shape and two-way shape memory polymers

Semicrystalline polymers often have diverse molecular configurations in response to the variation of temperature, making it easier to realize multiple or two-way shape memory effects (SMEs). In order to provide guidance for the design of relevant smart structures, it is necessary to develop corresponding constitutive models to better characterize the thermal-mechanical behavior of such shape memory polymers (SMPs). For the first time, a thermo-mechanical finite deformation constitutive model is presented to reveal the deformation mechanism of the triple-shape two-way SME of semicrystalline SMPs in our work, based on the theory of thermodynamics with internal state variables. To verify the validity of the model, the model results are compared with the test data for a typical shape memory cycle. It is found that the model can be employed to describe the non-equilibrium response of semicrystalline SMPs with two types of crystallites in the vicinity of crystallization and melting. Since the model results fit well with the test data, the effectiveness of the model in predicting the triple two-way shape memory behavior is demonstrated.

A thermoplastic clay constitutive model with temperature dependent evolution of stress anisotropy

This paper presents a thermomechanical constitutive model that captures temperature dependent evolutions of preconsolidation stress and stress anisotropy in normally consolidated and lightly overconsolidated saturated clays. Following a non-associative flow rule, the model was formulated to account for the rate of evolution of stress anisotropy as a function of temperature. A temperature-dependent rotational hardening parameter was introduced and calibrated employing a simple optimization algorithm for four different clays. The developed model was further implemented in a finite element (FE) analysis software for use in boundary value problems. Success of such numerical implementation and predictive performance of the constitutive model was further verified through FE simulations of drained and undrained triaxial tests on saturated clays at reference and elevated temperature. FEA results obtained from these simulations agreed very well with test data reported in the literature.

Experimental and numerical analysis of bending performance for a glass-epoxy composite with integrated sheet-shaped shape memory alloy actuator elements

Shape memory alloys (SMA) belong to the group of smart materials and show particularly a promising potential for an active functionalization of lightweight composite structures with respect to enabling smart structures. In this work, various manufacturing concepts for the integration of thin sheet-shaped SMA actuator elements in a thin glass-epoxy composite beam are experimentally and numerically studied with regard to its bending deformation performance realized by thermal induced shape memory effect of the SMA actuator element. For numerical modeling, the shape memory effect of the SMA on structural level, a three-dimensional thermomechanical constitutive material model is applied. In the experiments, the same deformation pattern of the composite beam was observed for all integration concepts investigated, with the degree of deformation showing strong differences both when the shape memory effect is activated and deactivated. The complex material behavior of the SMA actuator under bending deformation is modeled via the finite element method with sufficient accuracy including existing tension and compression loads.

Multiphysics Study of Thermal Profiles and Residual Stress in Welding.

One of the effects of welding is residual stress. Welding involves complex tests concerning differences in values of the mechanical parameters of its regions as an effect of residual stress. Such multiphysics characteristics of welding pose a challenge in predicting residual stress. In the present study, a thermo-mechanical constitutive model considering phase transformation and transformation plasticity is implemented in the numerical model in ABAQUS user subroutines. In order to consider phase evolution in welding, the metallurgical parameters for Leblond's phase equation were obtained from the calibration of DH36 steel with a CCT diagram. In addition, the effects of welding speed on thermal profiles and residual stress generation were investigated. Analysis has suggested that the width of the heat-affected zone (HAZ) decreases with an increase in welding speed, and the phase fraction is significantly affected by this kind of parameter. Such phase transformation has led to the generation of a compressive stress in the fusion zone (FZ) and HAZ. The volume difference between coexisting phases produces a compressive stress in cooling, and its magnitude was increased with martensite increasing.

Complete constitutive model of CFRP including continuous damage in low strain rate compression and temperature generation in high strain rate impact

AbstractThis paper reports a method to construct the complete constitutive model of carbon fiber‐reinforced plastic (CFRP) by distinguishing the strain rate state and modifying the material stiffness. In quasistatic (low strain rate) compression, the initial nonlinear effect caused by the material defects was described by introducing an nonlinear increasing factor. In dynamic (high strain rate) impact, the nonlinear strengthening effect was described by introducing the dynamic amplification factors of stress and strain based on the reference state. Considering the instantaneous temperature rise obtained from the impact work–temperature generation equation and the influence of temperature on modulus and strength, the coupled thermomechanical constitutive models combining the strain rate effect and the temperature effect were established. The user‐defined material subroutines (VUMAT) were developed, and then these constitutive models were verified by finite element analysis (FEA). Finally, the developed material subroutine was applied to predict the impact penetration of CFRP laminates, and the results show that the opening holes, damage and energy dissipation are in good agreement with the reference experiment. This work comprehensively analyzed the construction method of CFRP constitutive models, which would provide a guidance for the coupled thermomechanical behavior under dynamic impact.Highlights An anisotropic continuum mechanical constitutive behavior considering strain rate effect and damage evolution behavior was established. A macroscopic anisotropic constitutive model including the coupled impact temperature generation‐mechanical behavior was established. Combined with the incremental finite element method, the three‐dimensional user‐defined material subroutines (VUMAT) were developed and verified. The thermomechanical impact penetration behaviors of CFRP laminates was predicted, and the stress field, temperature field and failure characteristics were visualized.

Rate induced thermomechanical interactions in NiTi tensile tests on strips

The paper uses tensile experiments on NiTi strips at different displacement rates to establish and simulate the thermomechanical interactions caused by the latent heat of the reversible transformation between the austenitic and martensitic phases. The evolution of deformation in the specimen is synchronously monitored with digital image correlation, and the temperature field through infrared imaging, essential for structural modeling. Transformation leads to localized deformation that propagates through the specimen, while the latent heat released/absorbed at the propagating fronts locally heats/cools the specimen. The sensitivity of the transformation stress to temperature results in a complex interaction between the heat transfer conditions and the nucleation and evolution of transformation in the specimen. At low rates of loading, the alternate phase propagates nearly isothermally with a small number of fronts producing relatively flat stress plateaus. Higher rates lead to significant heating/cooling that results in progressive nucleation of multiple fronts and apparent "hardening" responses. The experiments are simulated in a three-dimensional static displacement transient temperature finite element analysis, using a new fully coupled thermomechanical constitutive model. Transformation strain and entropy are its internal variables whose evolution is governed by the motion in the stress-temperature space of a single transformation surface governing both transformations. The prevailing localization is captured by the introduction of softening over the unstable branches of the recorded isothermal material response. The results demonstrate how the important role of the thermal interaction between the specimen and the environment can be addressed. This, together with appropriate calibration of the constitutive and structural model, enable the analysis to reproduce the effect of rate on the recorded response, the evolution of localization patterns, and the associated thermal fields. The results can guide the development of constitutive and structural models of phase transforming materials with strong thermomechanical interactions.

Themomechanical Response of an Additively Manufactured Hybrid Alloy by Means of Powder Bed Fusion

The laser powder bed fusion (L-PBF) technique was utilized to manufacture a hybrid M789-N709 alloy by depositing M789 steel on wrought N709 steel. The tensile strength of the M789-N709 interface generated during the process has been established to be higher than that of the base materials. In the previous work of the current authors, extensive characterization of the M789-N709 interface (before and after heat treatment) was performed by means of electron backscatter diffraction, electron probe microanalysis, transmission electron microscopy with energy dispersive spectroscopy, and atom probe tomography analyses, to understand the mechanisms associated with its superior strength. In the present work, since the application of the hybrid alloy is targeted towards an elevated temperature environment, the individual high-temperature mechanical properties of M789 and N709 steels were acquired at various temperatures and strain rates using a Gleeble 563 thermomechanical system. Then, based on the flow curves, phenomenological-, and physical-based constitutive material models were established. These constitutive models can be utilized to accurately assess the high-temperature response of the hybrid alloy system using finite element analysis programs. This work demonstrates the application of thermomechanical processing and constitutive modeling in the field of metal additive manufacturing.

Thermomechanical behaviour of soils and soil-structure interfaces in thermo-active geo-structures

In recent years, the demand for sustainable, low carbon-emission energy solutions has risen, leading to an increased interest in shallow geothermal energy systems. These systems, such as ground source heat pump (GSHP) systems, harness the Earth's stable temperature at shallow depths as a heat sink and source of energy. Energy piles are a type of GSHP system that performs both structural and thermal energy-exchange functions [e.g. 1]. Despite the technological advantages of these systems, understanding the thermomechanical behavior of soils and soil-structure interfaces remains a challenge, limiting safe, efficient, and cost-effective designs. This research aimed to address these challenges by developing thermomechanical constitutive models for soils, implementing these models in boundary-value solvers, and analyzing the experimental results from direct shear tests performed on soils to understand their thermomechanical behavior at the soil-structure interface. By comprehensively examining the complex interactions between energy piles and surrounding soils, the research results offer a foundation for improved design standards and implementation.
 The methodology followed the following three main directions:
 Development of thermomechanical constitutive models for soils: A thermomechanical constitutive model based on the thermodynamic framework of Hyperelasticity-Hyperplasticity was developed to capture the major thermomechanical behaviors of fine-grained soils when subjected to temperature variations [2]. By utilizing a flexible yield surface within the framework, a more precise representation of soil behavior can be simulated. This framework was extended to develop a dual-surface thermomechanical model with an additional yield surface and a temperature-dependent kinematic rule to capture the shakedown behavior of soils subjected to heating-cooling cycles more accurately [3], as illustrated in Figure 1 (left) where two triaxial tests are simulated. The enhanced model addresses the need for improved representation of diverse soil behaviors and their implications for the performance of energy pile systems.
 Numerical aspects to improve simulation capability: A new flexible yield function was introduced that addresses the common issue of undesired elastic domains and erratic or divergent gradients in the numerical implementation of constitutive models with flexible yield surfaces and plastic potentials for use in implicit stress integration schemes [4]. By providing a robust and efficient framework for return mapping algorithms, this new yield function supports more precise simulations of soil behavior. An advanced numerical algorithm was devised to implement the thermomechanical constitutive model within a boundary-value solver such as the finite-element method [5]. The algorithm incorporates Gibbs energy potential, Lode angle dependency, and additional features to ensure accuracy, robustness, effectiveness, and convergence during simulations.
 Experimental analysis of the soil-structure interface including the impact of initial shear stress and thermal cycles: Laboratory-scale direct shear tests were conducted on fine- and coarse-grained soils to investigate their thermomechanical behavior at the interface with a concrete structure [6, 7]. A focus here was on the impact of thermal cycles; in particular, under realistic working stress conditions. Figure 1 (right) presents example results, where thermal cycles are applied in a temperature controlled shear box, with a clay-concrete interface which had a constant shear stress. Thermal creep is observed, which also has a hardening effect on future shear loading. By exploring the variations in soil behavior under differing temperature and loading conditions, the research deepens the understanding of the complexity inherent in soil-pile interactions for energy pile applications.
 This comprehensive investigation of the thermomechanical behavior of soils and soil-structure interfaces in the context of thermo-active geo-structures represents an essential step forward in optimizing energy pile systems for diverse applications. The developed constitutive models, numerical algorithms, and experimental analyses contribute valuable insights for improving energy pile design, ultimately leading to more efficient, safe, and cost-effective solutions.
 Future research should continue refining our understanding of the complex processes governing soil behavior in energy pile systems, with emphasis on consolidating the developed models and their implementation for a range of soil types and conditions. Additionally, further experimental studies should focus on elucidating the interactions between soils, piles, and the broader environment to identify potential opportunities for maximizing the performance of energy pile systems through innovative design and implementation.

A thermo-mechanical coupling model for concrete including damage evolution

In the current study, a coupled thermo-mechanical constitutive model of concrete was newly developed to characterize the mechanical responses of concrete at different temperatures within thermodynamics framework. The coupling thermo-mechanical contributions among thermal effect (up to 800°C), damage evolution and thermo-plastic hardening were creatively taken into account in energy potential function. The thermal damage evolution law and load induced plastic mechanical damage evolution law was developed respectively and were coupled in the dissipation function. The thermo-mechanical hardening rule was considered through adopting back stress and hardening parameters. A particular advantage of the proposed model is that fiber influence coefficients can be employed in potential functions to upgrade for modeling fiber reinforced concrete. The impact of types and volume fractions of fibers on the reinforcement mechanism was calculated in the fiber influence coefficients. Besides, an orthotropic triaxial compression model was developed for concrete and fiber reinforced concrete to predict multiaxial stress behavior at high temperature. The proposed model was coded and implemented into the Fortran program and the nonlinear behaviors of concrete and fiber reinforced concrete were predicted. Good effectiveness and accuracy have been validated through comparing the modelling results and six groups experimental results in literatures, which illustrates that the proposed model is robust and reliable. Finally, a discussed study was conducted to investigate the effect of hardening parameters and thermal effect on the yield surface.

A thermomechanical constitutive model for super-stretchable multi-walled carbon nanotube-enforced polyimide aerogel

A thermomechanical constitutive model for super-stretchable multi-walled carbon nanotube-enforced polyimide aerogel

Multiaxial shape memory effect of thermo-induced shape memory polyurethane under proportional tension-torsion loading

Shape memory behavior of thermo-induced shape memory polyurethane (TSMPU) under proportionally multiaxial loading is investigated experimentally. The shape memory effect is discussed under different axial and torsional loading angles. By comparing the equivalent stress-equivalent strain curves under different loading angles, it is found that the difference in thermo-mechanical behavior is mainly reflected in the shape memory effect. The equivalent characteristic quantities describing the shape memory effect are extracted from the experimental results. By analyzing their evolutions in the strain and stress spaces, it is found that the shape memory effect shows obvious anisotropy under different loading angles, i.e. the equivalent recovery rate of TSMPU increases with the increase of the loading angles. These results are helpful to establish a thermo-mechanical constitutive model for describing the shape memory effect under multiaxial loading.

Phase-field fracture modeling for unidirectional fiber-reinforced polymer composites

In the present work, a phase-field fracture model is developed to create an experimentally validated, physically motivated, and computationally tractable framework to predict the orientation-dependent complex fracture response of the unidirectional fiber-reinforced polymer matrix composites (PMC). The damage mechanism in PMC is many-fold, such as matrix micro-cracking, fiber breakage or pullout, fiber debonding, and delamination. The progressive damage and failure in the composite can be attributed to the competition of all these different damage mechanisms individually or in concert, depending on fiber volume fraction and orientation of the plies. Moreover, other important factors at the constituent level, such as viscoelasticity or plasticity of the matrix, can significantly influence the constitutive behavior of the composite. In this work, a homogenized, coupled thermo-mechanical constitutive model is developed considering the viscoelasticity of the polymer matrix for the fiber-reinforced PMCs. Subsequently, a phase-field fracture model is adopted to capture the essential damage mechanisms as matrix cracking and fiber pullout/breakage for unidirectional composite lamina. The phase-field variable considers an anisotropic tensorial term in its gradient to account for the orientation-dependent fracture of fiber-reinforced PMC. The model is numerically implemented by writing an ABAQUS user-element subroutine (UEL). The model can predict the direction-dependent damage propagation and the final load-deformation response at fracture in commercially acquired unidirectional glass-fiber-reinforced epoxy composites at different fiber orientations in reasonable agreement with the experiments. Based on the decoupling of the driving energies the model can also identify the dominant damage mechanism among the two such as fiber breakage or matrix cracking, for a given fiber orientation. Simulations are also conducted to validate the model predictions for carbon-fiber reinforced epoxy lamina with available experimental data in the literature.

Thermo-mechanical modeling of semicrystalline triple shape memory polymers

Compared with amorphous shape memory polymers (SMPs), semicrystalline SMPs have more diverse shape memory effects (SMEs) which promotes their application in smart structures. To reveal the driving mechanism of the triple SMEs of semicrystalline SMPs, our study focuses on developing the constitutive model under the condition of finite deformation. In the paper, a thermo-mechanical constitutive model under consideration of the second law of thermodynamics is developed based on the theory of thermodynamics with internal state variables. The model can be used to describe the nonequilibrium response of the amorphous and semicrystalline components in the vicinity of the glass transition, melting, and crystallization. To verify the validity of model, numerical simulation is carried out for a thermo-mechanical shape memory cycle which can be divided into a two-step programing process and a two-step recovery process. The comparison between the model results and the test data shows good agreement.

Characterization of Thermo-mechanical parameters of Nickel-Based single crystal superalloys under high-temperature by modified I-DIC method

Understanding the evolution law of thermo-mechanical parameters of nickel-based single crystal superalloys under high-temperature is significant for modeling, application, and structural optimization. In this study, a modified integrated digital image correlation (I-DIC) method is developed to characterize its thermo-mechanical properties. Particle Swarm Optimization (PSO) and Gradient-Based optimization algorithm are introduced to the modified I-DIC method. The orthotropic thermo-mechanical constitutive model for nickel-based single crystal superalloys is evaluated and the relative five constitutive parameters at temperatures ranging from 600 ℃ to 1000 ℃ are inversed simultaneously by the modified I-DIC method. Comparison between experiments and simulations confirms that the suggested model is reliable and the modified parameter inversion method is accurate and effective. The results show that the parameters vary with temperature. The coefficient of thermal expansion (CTE) and Poison’s ratio increased with increasing temperature, while Young’s modulus and shear modulus decreased with increasing temperature.

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.