Micromechanical modeling and evaluation of CNTs reinforced magnesium matrix composites based on stress softening modified constitutive model by Mingjie Shen, Baojian Han, Tao Ying, Jingya Wang, Liping Zhou, Xiangzhen Xue, Yang Gao, Zongyang Yang

“…By introducing the recrystallization softening factor and strain parameter into the constitutive equation, the accurate prediction of the global rheological response of the composites was realized, and the accuracy of the new constitutive model was proved. …”
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Transient Interaction of a Spherical Shell with an Underwater Explosion Shock Wave and Subsequent Pulsating Bubble by H. Huang, K.C. Kiddy

“…The nonlinear interaction problem is analyzed by simultaneously solving the mass, momentum, and energy conservation equations together .with appropriate material constitutive equations governing the fluid dynamics of the explosion gaseous product and the water and the structural dynamics of the compliant shell. …”
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Phase Characterization of Cucumber Growth: A Chemical Gel Model by Bo Li, Xuejing Liu, Junshi Zhang

“…Based on thermodynamics, constitutive equations, incorporating biological quantities, are established. …”
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Free Vibration of Composite Cylindrical Shells Based on Third-Order Shear Deformation Theory by Muneerah Saad AL Nuwairan, Saira Javed

“…The focus of this study is to analyse the free vibration of cylindrical shells under third-order shear deformation theory (TSDT). The constitutive equations of the cylindrical shells are obtained using third-order shear deformation theory (TSDT). …”
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Similarity Solution for High Weissenberg Number Flow of Upper-Convected Maxwell Fluid on a Linearly Stretching Sheet by Meysam Mohamadali, Nariman Ashrafi

“…Upon proper scaling and by means of an exact similarity transformation, the nonlinear momentum and constitutive equations of each layer transform into the respective system of highly nonlinear and coupled ordinary differential equations. …”
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Effect of Couple-Stress on Peristaltic Transport of Sutterby Fluid in an Asymmetric Channel by Asmaa Mohammed, Liqaa Zeki Hummady

“…In the presence of rotation and couple–stress, mathematical modeling is developed using constitutive equations based on the model of Sutterby fluid. …”
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The Laplace Virtual Fields Method for the direct extraction of viscoelastic properties of materials by Marcot, Quentin, Fourest, Thomas, Langrand, Bertrand, Pierron, Fabrice

“…The Laplace transform enables the use of theory of elasticity in the Laplace domain. The constitutive equations are expressed in the plane stress framework with the 2D plane stress stiffness coefficients. …”
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Free Vibration of Embedded Porous Plate Using Third-Order Shear Deformation and Poroelasticity Theories by Ali Ghorbanpour Arani, Zahra Khoddami Maraghi, Mehdi Khani, Iman Alinaghian

“…To obtain the Biot formulation of the constitutive equations for a porous material, linear poroelasticity theory is used. …”
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Collapse Pressure Analysis of Transversely Isotropic Thick-Walled Cylinder Using Lebesgue Strain Measure and Transition Theory by A. K. Aggarwal, Richa Sharma, Sanjeev Sharma

“…The concept of transition theory based on Lebesgue strain measure has been used to simplify the constitutive equations. Results have been analyzed theoretically and discussed numerically. …”
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Implementation of a practical sand constitutive model coupled with the high cycle accumulation framework in PLAXIS by Pishun Tantivangphaisal, David M.G. Taborda, Stavroula Kontoe

“…A reference 15 MW offshore wind turbine monopile foundation subject to lateral cyclic wave loading is used in an engineering design scenario at three different load levels to verify the current numerical implementation.Details include: • Modifications made to the HCA framework to couple it with a practical sand constitutive model, • Implementation of an efficient workflow to switch between low and high cycle constitutive equations in PLAXIS, and • Verification of the implementation at single element and boundary value problem scales.…”
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Numerical Modeling and Simulation of Wind Blown Sand Morphology under Complex Wind-Flow Field by Xamxinur Abdikerem, Lei Wang, Afang Jin, Mamtimin Geni

“…The flow field and the sand flow field constitutive equations are analyzed at first, then the different desert highway numerical models are established by considering the crossroad and by changing the road surface height and air stream flow field, then three kinds of different models with different complex air flow fields are made for simulating the sand ripple formation process by weak coupling of air and sand flow field, and finally the numerical simulations of these models are conducted and the affect process of sand morphology under complex air flow fields are discussed. …”
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Dufour Effect on Transient MHD Double Convection Flow of Fractionalized Second-Grade Fluid with Caputo–Fabrizio Derivative by Imran Siddique, Sehrish Ayaz, Fahd Jarad

“…This article presents the problem, in which we study the unsteady double convection flow of a magnetohydrodynamics (MHD) differential-type fluid flow in the presence of heat source, Newtonian heating, and Dufour effect over an infinite vertical plate with fractional mass diffusion and thermal transports. The constitutive equations for the mass flux and thermal flux are modeled for noninteger-order derivative Caputo–Fabrizio (CF) with nonsingular kernel, respectively. …”
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Unsteady MHD Flow for Fractional Casson Channel Fluid in a Porous Medium: An Application of the Caputo-Fabrizio Time-Fractional Derivative by Pongsakorn Sunthrayuth, A. A. Alderremy, Fazal Ghani, Ayékotan M. J. Tchalla, Shaban Aly, Yasser Elmasry

“…A mix of Laplace and Fourier transformations is used to get the exact solution for the momentum equation. The constitutive equations for Caputo-Fabrizio’s time-fractional derivative are also incorporated for recovering the exact solutions of the flow problem under consideration. …”
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Study on Nonlinear Damage Creep Model for Rocks under Cyclic Loading and Unloading by Liangliang Zhang, Xiaojian Wang

“…The differential damage constitutive equations of the proposed creep model under one-dimensional and three-dimensional stress states are derived based on the creep mechanics and elasticity theory. …”
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Implementation of Virtual Crack Closure Technique for Damaged Composite Plates Using Higher-Order Layerwise Model by Kwang S. Woo, Jae S. Ahn

“…In assumed displacement fields, strain-displacement relations and three-dimensional constitutive equations of layers are obtained by combination of two- and one-dimensional shape functions. …”
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Strength Model for Concrete in Near-Reinforcement Region by Vladimir I. Kolchunov, Natalia V. Fedorova, Tatiana A. Iliushchenko

“…New functionals of reinforced concrete are constructed, which are consistent with the physical interpretations of the strength of cross-sections of bar elements in near-reinforcement zones. Constitutive equations for the concrete matrix, which models zones between transverse cracks, are written. …”
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Nonlinear Failure Analysis of Critical Area of Transmission Towers Based on the Continuum Damage Theory by Nailong Zhang, Jie Chen, Chao Gao, Xiao Tan, Hongze Li

“…A material subroutine for elastic-plastic-damage constitutive equations is developed using commercial software, and thorough verification ensures the accuracy of both the subroutine and the algorithm. …”
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Construction and application of a physically-based constitutive model for superplastic deformation of near-α TNW700 titanium alloy by Lixia Ma, Dingyong He, Bangjian Yang, Xiuquan Han, Jichun Zhang, Weidong Li

“…Then, a series of constitutive equations based on microstructure evolution (covering dislocation density, phase ratio, grain size and plastic damage) are established, accounting for the disparate deformation behavior of primary α and β phases. …”
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A Damage Constitutive Model for the Effects of CO2-Brine-Rock Interactions on the Brittleness of a Low-Clay Shale by Qiao Lyu, Xinping Long, P. G. Ranjith, Jingqiang Tan, Yong Kang, Wenbin Luo

“…Therefore, a series of strength tests were conducted to obtain the stress-strain characteristics of shale soaked in different phases of CO2 including subcritical or supercritical CO2 with formation of water for different time intervals (10 days, 20 days, and 30 days). Two damage constitutive equations based on the power function distribution and Weibull distribution were established to predict the threshold stress for both intact and soaked shale samples. …”
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Modelling Hysteresis in Shape Memory Alloys Using LSTM Recurrent Neural Network by Mohammad Reza Zakerzadeh, Seyedkeivan Naseri, Payam Naseri

“…The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. …”
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