Three-dimensional Constitutive Equations Research Articles

Creep damage constitutive model of rock based on the mechanisms of crack-initiated damage and extended damage

Since the classical element model cannot describe the nonlinear characteristics of rock during the entire compressive creep process, nonlinear elements and creep damage are generally introduced in the model to resolve this issue. However, several previous studies have reckoned that creep damage in rock only occurs in the accelerated creep stage and is only described by the Weibull distribution. Nevertheless, the creep damage mechanism of rocks is still not clearly understood. In this study, a reasonable representation of the damage variables of solid materials is presented. Specifically, based on the Gurson damage model, the damage state functions reflecting the constant creep stage and accelerated creep stage of rock are established. Further, the one-dimensional and three-dimensional creep damage constitutive equations of rock are derived by modifying the Nishihara model. Finally, the creep-acoustic emission tests of phyllite under different confining pressures are conducted to examine the creep damage characteristics of phyllite. And the proposed constitutive model is verified by analyzing the results of creep tests performed on saturated phyllite. Overall, this study reveals the relationship between the creep characteristics of rocks and the corresponding damage evolution pattern, which bridges the gap between the traditional theory and the quantitative analysis of rock creep and its damage pattern.

A Three-Dimensional Velocity Field Related to a Generalized Third-Grade Fluid Model

In this work, we propose a new three-dimensional constitutive equation related to a third-grade fluid. This proposal is based on experimental work for which the viscosity term and the terms related to viscoelasticity may depend on the shear rate, in accordance with a power-law type model. The numerical implementation of this fluid model is rather demanding in terms of computational calculation and, in this sense, we use the Cosserat theory related to fluid dynamics, which makes the transition from the three-dimensional fluid model to a one-dimensional fluid model for a specific geometry under study which, in this case, is a straight tube with constant circular cross-section. Based on this approximation theory, the one-dimensional fluid model is solved by assuming an ordinary differential equation involving: an unsteady mean pressure gradient; an unsteady volume flow rate; the Womersley number; and the viscosity and viscoelasticity parameters. Consequently, for specific data, and using the Runge–Kutta method, we can obtain the solution for the unsteady volume flow rate and we can present simulations to the three-dimensional velocity field.

Nonlinear Nishihara model of soft rock based on damage mechanics and its parameter identification

The traditional Nishihara model is difficult to describe the nonlinear primary and accelerated creep characteristics of soft rock. Based on this, an improved nonlinear Nishihara model is established by modifying the unsteady dashpot and a nonlinear damaged viscoplastic body in series. The one-dimensional and three-dimensional creep constitutive equations of soft rock are derived. Levenberg-Marquardt algorithm is used to identify model parameters of creep test data of Cretaceous soft rock and red sandstone. The results show that the correlation coefficient between the fitted results of the improved nonlinear Nishihara model and the creep test data is above 0.97. The sensitive parameters λ,η′1 and a of the model are all within a reasonable range. It shows that the model can accurately reflect the characteristics of the unsteady primary creep stage and the nonlinear damage accelerated creep stage of soft rock, which is better than the traditional Nishihara model. The research results can provide theoretical guidance for similar soft rock creep model research.

No yield stress required: Stress-activated flow in simple yield-stress fluids

An elastoviscoplastic constitutive equation is proposed to describe both the elastic and rate-dependent plastic deformation behavior of Carbopol® dispersions, commonly used to study yield-stress fluids. The model, a variant of the nonlinear Maxwell model with stress-dependent relaxation time, eliminates the need for a separate Herschel–Bulkley yield stress. The stress dependence of the viscosity was determined experimentally by evaluating the steady-state flow stress at a constant applied shear rate and by measuring the steady-state creep rate at constant applied shear stress. Experimentally, the viscosity’s stress-dependence was confirmed to follow the Ree–Eyring model. Furthermore, it is shown that the Carbopol® dispersions used here obey time-stress superposition, indicating that all relaxation times experience the same stress dependence. This was demonstrated by building a compliance mastercurve using horizontal shifting on a logarithmic time axis of creep curves measured at different stress levels and by constructing mastercurves of the storage- and loss-modulus curves determined independently by orthogonal superposition measurements at different applied constant shear stresses. Overall, the key feature of the proposed constitutive equation is its incorporation of a nonlinear stress-activated change in relaxation time, which enables a smooth transition from elastic to viscous behavior during start-up flow experiments. This approach bypasses the need for a distinct Herschel–Bulkley yield stress as a separate material characteristic. Additionally, the model successfully replicates the observed steady-state flow stress in transient-flow scenarios and the steady-state flow rate in creep experiments, underlining its effectiveness in capturing the material’s dynamic response. Finally, the one-dimensional description is readily extended to a full three-dimensional finite-strain elastoviscoplastic constitutive equation.

A three-dimensional constitutive equation for shape memory polymer composites considering energy dissipation under single loading–unloading condition

Shape memory polymer composites (SMPCs), composed of shape memory polymers (SMPs) and fabrics, can recover their deformed shapes to initial configurations by changing temperature. The shape recovery energy of the SMPs and the elastic energy of the fabrics are regarded as key points to predict the restored configuration of the SMPCs. In this research, a three-dimensional constitutive equation for SMPCs is newly developed, considering the energy dissipation of fabrics under a single loading–unloading condition. The constitutive equation is derived from Mooney–Rivlin, viscoelastic, and stored strain energy models for SMPs, as well as polynomial functions for fabrics. The Mooney–Rivlin and viscoelastic models describe the time- and temperature-dependent hyperelastic properties, and the stored strain energy model can explain the shape recovery. The polynomial functions are newly proposed to take into account the dissipated energy of fabrics, such as residual deformation and hysteresis effect. Because the elastic energy of fabrics considerably influences the shape recovery of SMPCs, the proposed constitutive equation can accurately predict the recovered configuration. Analyses of SMPCs are conducted using the finite element method and validated by comparing them with the results obtained from experiments. The constitutive equation is used to analyze the shape recovery behaviors of an SMPC skin. The skin can be largely stretched, and the deformed shape can be restored to its initial shape by applying electricity. The SMPC skin is applied to a morphing flap, and its shape-changing behaviors are demonstrated experimentally.

Subloading-elastoplastic constitutive equation of glass

Elastoplastic constitutive equation of glass is proposed in this article, which is formulated based on the subloading surface model which possesses the various distinguished advantages for the description of the plastic deformation, i.e., the smooth elastic-plastic transition, the continuous variation of the tangent stiffness modulus tensor, the automatic controlling function to attract the stress to the yield surface during the plastic deformation process, etc. It would be the firstly proposed three-dimensional elastoplastic constitutive equation of glass, which is furnished with the basic properties, e.g. the ellipsoidal yield surface with the dependence of the third deviatoric stress invariant, undergoing the flattering to the deviatoric direction with the plastic compression and the plastic volumetric hardening with the associated flow rule. The validity of the description of the deformation behavior of glass will be verified by comparisons with some test data for silica glass specimens.

Bending and free vibration analyses of CNTRC shell structures considering agglomeration effects with through-the-thickness stretch

The present work investigates an enhanced finite element model and numerical analysis to illustrate the agglomeration effect of Carbon Nanotubes (CNTs) on static and free vibration responses of reinforced composite plate and cylindrical panel shell. The developed model is based on the kinematics of the improved first-order-shear deformation theory (FSDT), assuming a parabolic variation of the transverse shear strains and fulfilling the zero circumstance of the transverse shear stresses at the extreme structure’ surfaces omitting the use of shear correction factors. The material properties of the nanocomposite are determined using a developed micromechanical model based on the Mori–Tanaka approach. The developed finite element formulation is based on three-dimensional constitutive equations without reduction involving the addition of one parameter of thickness change combined with the Enhanced Assumed Strain (EAS) method. Four enhancement models are established and discussed. In a first attempt, the improvement of the third bending strain using one or four parameters are presented. In a second attempt, the membrane strain is enhanced with four parameters. Due to lack of investigations on structural behavior of composite structures reinforced with agglomerated CNTs, new static and free vibration results for shell structures with three level of CNT agglomeration are examined. Parametric studies are here in developed to present the influences of the characteristic parameters on the structural behavior of the CNTRC shell in a new graphic way; such as the volume fraction and agglomeration of nanofibers, geometrical parameters, and boundary conditions. Based on numerical results, it is found that the agglomeration and distributions of the reinforcing fibers are really crucial and play a significant role on the static and vibrational performances of the structure.

A layer-wise plasticity-based approach for the analysis of reinforced concrete shell structures using a mixed finite element

The analysis of reinforced concrete shell structures accounting for material nonlinearity is addressed. The structural response is numerically evaluated using a mixed shell finite element and a plasticity-based material behaviour. The finite element is a quadrilateral with four nodes and is based on self-equilibrated assumed stresses. The kinematic fields are interpolated only along the element boundary by polynomials up to the third order. The generalised shell stresses are evaluated through layer-wise integration of the Cauchy pointwise stresses. This allows appropriated three-dimensional elastoplastic constitutive equation to be employed and to include multiple reinforcing layers. The integration of the constitutive laws is performed using a dual decomposition method which preserves the assumed stress interpolation. Results, including civil engineering applications, show that the proposed approach is reliable and accurate in evaluating the static nonlinear equilibrium path. Additionally, the mixed finite element shows a quadratic rate of convergence in the collapse load and low error even for coarse meshes.

Three-dimensional thermodynamic behaviors for a FML structure subjected to 3-D moving dual-ellipse distribution laser heat source

In present work, the three-dimensional transient heat transfer equation for a fiber metal laminated (FML) plate is established, and temperature field under dual-ellipse distribution laser processing is obtained through the Newton-Cotes method. Then the three-dimensional constitutive equations are derived, and the transient thermodynamic governing equations are obtained based on the Hamilton variational principle subsequently. Finally, the finite difference method and Newmark method are utilized to solve the thermodynamic equations in both space and time domains, respectively. The research aims at studying three-dimensional thermodynamic behaviors for a FML structure subjected to dual-ellipse distribution laser heat source and giving the effects of Fourier series term, finite difference point, aspect ratio, fiber arrangement angle and laser moving speed on the thermodynamic behaviors for the FML structure.

An Improved Nonlinear Creep Model for Rock Applied to Tunnel Displacement Prediction

Traditional creep models cannot describe the nonlinear creep characteristics of rock; thus, considerable efforts have been made to establish better rock creep models. However, many of the previously presented improved models are too complicated, leading to difficulties in mathematical derivation and engineering application. To address this problem, this study proposes a simple model that can comprehensively describe rock creep characteristics and that can be effectively applied in practical engineering. Based on the variable-order fractional derivatives and continuum damage mechanics, an improved Burgers creep model is established. The use of this proposed model can be divided into two stages based on whether the accumulated creep strain of the rock reaches the threshold of starting accelerated creep. Then, both the one-dimensional and three-dimensional constitutive equations of the improved model are provided. In addition, the compression creep test results of six groups of rock samples are used to verify the effectiveness and reliability of the improved Burgers model proposed in this study. Furthermore, an analytical solution for tunnel displacement is derived on the basis of the improved model. This analytical solution is applied to the time-dependent displacement prediction of an auxiliary tunnel in Jinping hydropower station, and a good agreement between the theoretical results and monitoring data is achieved. The outcome of this study can provide useful guidance for related engineering designs.

An isogeometric finite element formulation for geometrically exact Timoshenko beams with extensible directors

An isogeometric finite element formulation for geometrically and materially nonlinear Timoshenko beams is presented, which incorporates in-plane deformation of the cross-section described by two extensible director vectors. Since those directors belong to the space R3, a configuration can be additively updated. The developed formulation allows direct application of nonlinear three-dimensional constitutive equations without zero stress conditions. Especially, the significance of considering correct surface loads rather than applying an equivalent load directly on the central axis is investigated. Incompatible linear in-plane strain components for the cross-section have been added to alleviate Poisson locking by using an enhanced assumed strain (EAS) method. In various numerical examples exhibiting large deformations, the accuracy and efficiency of the presented beam formulation is assessed in comparison to brick elements. We particularly use hyperelastic materials of the St.Venant-Kirchhoff and compressible Neo-Hookean types.

Three-Dimensional constitutive equations for hyper viscoelastic particulate reinforced composite materials based on damage parameter

The aim of the article is mathematical modeling of behavior of hyper viscoelastic particulate reinforced composite materials. Three dimensional hyper viscoelastic constitutive equations, which include an inherent damage parameter and its functions, have been developed. The model was used to demonstrate physical phenomenon such as the so called dewetting effect. New strain rate, damage rate and stress state sensitive dewetting criterion is proposed. At the same time dewetting criterion is the first order ordinary differential equation with respect to damage parameter and serves as its evolution law. The softening of the material due to dewetting during the loading is considered by means of the strain energy density function augmented by the damage parameter and its functions. The proposed material model simulates volumetric strains caused by the hydrostatic pressure, as well as the dilatation caused by the deviatoric part of the stress, namely deviatoric-bulk coupling is considered. The material model proposed in the framework of this article takes into account the behavior of particulate reinforced hyper viscoelastic composite materials such as strong nonlinearity, dewetting, complex coupling of the deviatoric bulk behavior. Mullins effect and temperature dependence have not been covered by this research. The proposed material model has been implemented into Finite Element package MSC. Marc by means of FORTRAN user subroutines “elevar” and “uelastomer” and it has been shown good correlation with test results.

Buckling Analysis of Laminated Composite Beams by Using an Improved First Order Formulation

In this work, a finite element model based on an improved first-order formulation (IFSDT) is developed to analyze buckling phenomenon in laminated composite beams. The formulation has five independent variables and takes into account thickness stretching. Three-dimensional constitutive equations are employed to define the material properties. The Trefftz criterion is used for the stability analysis. The finite element model is derived from the principle of virtual work with high-order Lagrange polynomials to interpolate the field variables and to prevent shear locking. Numerical results are compared and validated with those available in literature. Furthermore, a parametric study is presented.

A Sequential Approach for Simulation of Thermoforming and Squeeze Flow of Glass Mat Thermoplastics

In this study, a sequential thermoforming and squeeze flow simulation approach for Glass Mat Thermoplastic (GMT) material is proposed and applied to a hat section geometry using input properties based upon Tepex flowcore, a long glass fiber reinforced polyamide (PA/GF) mat manufactured by Lanxess. First, a fully-coupled thermomechanical simulation is conducted based on a purely Lagrangian description, to efficiently capture thermoforming. Subsequently, relevant state variables are mapped and initialized for a Coupled-Eulerian-Lagrangian (CEL) approach. The CEL approach is adopted to accurately capture squeeze flow, which is not possible by a purely Lagrangian description. While numerical techniques differ, both approaches use the same three-dimensional and thermomechanical constitutive equations including an equation of state, a nonlinear viscosity model, and crystallization kinetics, implemented through a material user-subroutine (VUMAT) for the commercially available simulation software package ABAQUS/Explicit.

A three-dimensional viscoelastic analysis of thermoplastic resin matrix composite laminates during hot stamping

In this paper, a three-dimensional thermo-viscoelastic constitutive equation is used to simulate and analyze the hot stamping process of thermoplastic resin matrix composites. Dynamic mechanical analyzer (DMA) was used to measure the relaxation behavior of resin matrix at different temperatures. Then, the relaxation times and the weight factors of the resin matrix were obtained by regression analysis of the data of the resin matrix relaxation behavior using genetic intelligent algorithm. According to the classical Maxwell model with N Maxwell elements and representative volume element (RVE), the integral constitutive model was established, and the corresponding integral finite element program was incorporated into commercial software ABAQUS with an UMAT subroutine.Thermal expansion was also taken into account in the present model. The constitutive model and its finite element program in this paper were verified by experiments with a self-designed hot stamping die for thermoplastic composites.In addition, the simulation results were compared with the actual failure areas to further verify the applicability of the constitutive model adopted in this paper. The results indicate that the ball edge of the hemispherical part is easy to be damaged.

Large deformation analysis of fully incompressible hyperelastic curved beams

The aim of this contribution is to develop a nonlinear formulation for modelling the large deformation of elastic curved beams made of fully incompressible hyperelastic materials. The basic idea is to apply the incompressibility constraint besides the plane stress assumption in the directions perpendicular to the centreline of the beam. Accordingly, a system of three nonlinear algebraic equations for the hydrostatic pressure as well as normal strain components in the beam cross section are obtained. By solving the system of equations, either analytically or numerically, it is possible to propose constitutive equations for the force and moment resultants present in the formulation. The main advantage of this strategy is that it allows for using three-dimensional incompressible constitutive equations. Due to highly nonlinear nature of the differential equations, a Total Lagrangian finite element formulation is developed. Performance and accuracy of the formulation are investigated through several numerical examples.

Study on the Triaxial Unloading Creep Mechanical Properties and Damage Constitutive Model of Red Sandstone Containing a Single Ice-Filled Flaw

The freezing method is an effective approach for constructing coal mine shafts through water-rich soft rock strata. The frozen wall produced by this technique stops the movement of water and offers temporary support, ensuring the safety and stability of the shaft working face. However, most frozen rock masses generally comprise ice-filled flaws and frozen intact rock. A frozen fissured rock mass undergoing the long-term effects of excavation unloading under in situ stress gradually accumulates damage or creep deformation over time; this damage is mainly responsible for the significant deformation, cracking and even instability of the frozen rock wall. To study the creep characteristics of ice-filled fissured rock masses under unloading conditions, a series of conventional triaxial compression tests and triaxial unloading creep tests were performed on ice-filled flawed red sandstone specimens from the Shilawusu mine in Northwest China using a self-developed DRTS-500 subzero rock triaxial testing system. In conjunction with the experimental data, the instantaneous strength and deformation characteristics of the rock specimens were analyzed, and their creep deformation characteristics and damage evolution characteristics were discussed. In this paper, based on the nonlinear creep characteristics of frozen rock, fractional calculus theory and damage theory, a new nonlinear creep damage model of frozen fissured red sandstone was defined in series with the improved Burgers model and elastoplastic damage model. The proposed creep model can reasonably describe the creep deformation of frozen fissured red sandstone. Classical elastoplastic mechanics and creep theory were used to derive the three-dimensional creep damage constitutive equation, and the results of creep experiments and simulation results of the creep damage constitutive model were very consistent in this study. The new creep model not only reflects the whole creep deformation process of frozen fissured red sandstone but also exhibits better performance than the classical Burger model and improved Nishihara model in describing the primary creep stage and accelerating creep stage. Therefore, the proposed nonlinear creep damage model is suitable for studying the mechanical creep properties of frozen fissured rock under unloading conditions. The results can provide an important reference for the long-term stability assessment of frozen rock walls of coal mine shafts.

From rheological to original three-dimensional mechanical modelling of semi-crystalline polymers: Application to a wide strain rate range and large deformation of Ultra-High Molecular Weight PolyEthylene

Ultra-High Molecular Weight semi-crystalline polymers, such as Ultra-High Molecular Weight PolyEthylene (UHMWPE) exhibit strong wear and impact resistance, making them good candidates for structural applications in many industrial fields. At high strain rate and large strain, mechanisms of deformation are quite different from those involved in classical semi-crystalline polymers, mainly because chain disentanglements are almost impossible for very long macromolecules even at temperature far above the melting point. Thus, there is a need to develop specific models for these materials and, from the works of Deplancke and her co-workers (Deplancke et al., 2019; Deplancke et al., 2015) who developed a scalar description based on polymer physics, three-dimensional constitutive equations are developed in this work. The developed model proposes an innovative way to take into account the repartition of strain for a semi-crystalline polymer and more generally for a two-phase material. Moreover, by modelling the evolution of microstructure during the plastic deformation of the material, the model is able to reproduce quite fairly the mechanical behavior of UHMWPE for both loading and unloading.

A three-dimensional semi-analytical solution of hygrothermal dynamic behavior for a FML structure subjected to moving laser heat source

In present work, the three-dimensional constitutive equations considering hygrothermal conditions are derived firstly, and governing equations are established subsequently. Then the finite difference method and Newmark method are used to solve the hygrothermal dynamic equations in both space and time domains, respectively. Finally, the effects of moist coefficient, fiber type and fiber arrangement angle on the hygrothermal dynamic behaviors for the FML structures are analyzed. The amplitude of the normal stress for the FML structures increase with the increasing of the moist coefficient, which can be attributed to fact that the fiber materials (S2-glass fiber epoxy resin and glass-polymer) absorb more water and the volume expand greater when increasing the moist coefficient firstly, then the normal strain and normal stress of the FML structures increase subsequently. And for two different FML structures (one only considering S2-glass fiber epoxy resin for fiber layer and the other only considering glass-polymer) with the same workpiece sizes, the former FML structure has the better capability to resist deformation in the in-plane displacement as the S2-glass fiber epoxy resin is of greater density and mass than that of glass-polymer. The deflection profile of the 0°/45° fiber orientation is most affected when the temperature is high, both the fiber orientation and laminated interaction between the layers are two important factors determining the order of magnitude for normal stress.

Modeling and identification of the constitutive behavior of embedded non-persistent joints using triaxial creep experiments

Joints are the boundary that controls slope failure of rock slope. Most of the joints are non-persistent and embedded in the rock slope. The creep mechanism of non-persistent joints is of great significance to reveal the creep deformation behavior of the slope over time. However, the embedded non-persistent joints are difficult to sample and to replicate uniformly in large numbers. In the study, a modeling method for embedded non-persistent joints is proposed, and the shear creep test on embedded non-persistent joints was conducted. Based on uniaxial compression testing and shear creep testing, the phase characteristics of creep testing are analyzed, a nonlinear viscous acceleration element and damage variable suitable for structural surfaces are introduced, and a shear-creep-damage constitutive model for non-persistent joints is established. Three-dimensional shear creep constitutive equation is derived using the Laplace transform. Furthermore, Levenberg-Marquardt (L-M) algorithm and global optimization method are used to identify the creep curves and solve the model parameters. After identification and comparison of experimental results and model fitting results, the new constitutive model can clearly reflect the creep characteristics of the non-persistent joints, which provide reference for deformation characteristics of non-persistent joint rock slopes.