Nonlinear Elastic Constitutive Model Research Articles

Study on fragmentation characteristics of carbon nanotube-enhanced concrete and a comparative evaluation of the ZWT and ottosen constitutive model

In order to study the difference between the crushing characteristics of carbon nanotubes (CNTs) concrete and ordinary concrete, this paper conducts a test between ordinary concrete and CNTs concrete with a volume content of 0.3 % at different impact speeds through the Hopkinson test, using fragmentation and particle size distribution as evaluation indexes. Crushed fragments are analyzed from the perspective of concrete pores. Finally, the nonlinear viscoelastic constitutive model and the nonlinear elastic constitutive model of CNTs concrete are then developed and compared. Based on the results, the average pores of the CNTs concrete decreased from 42.0 nm to 38.0 nm, but the pore area increased from 5.743 m2/g to 6.305 m2/g, indicating that CNTs can fill the internal pores. Meanwhile, the average particle size of broken CNTs concrete is significantly larger than that of ordinary concrete. However, the relationship between the crushing particle size and impact velocity is approximately y=A-Bln(x+C) for both concretes. According to the study of concrete constitutive models, compared to the Loand damage model, the combination of Loand model and Mazars model is more accurate in describing concrete damage characteristics, and the average correlation coefficient increases from 0.84 to 0.92. Meanwhile, compared to the Zhu-Wang-Tang (ZWT) model, Ottosen model can better capture the dynamic mechanical properties of CNT concrete, with a correlation coefficient over 0.98.

Displacement and pressure of surrounding rock during shield tunnelling and supporting in low water content loess

Using a self-developed micro shield machine, a laboratory-scale experimental scheme for shield tunnelling and supporting was designed, and the soil displacement law and earth pressure on the segments during tunnel construction were studied. Based on a nonlinear elastic constitutive model describing the characteristics of loess, the UMAT subroutine was redeveloped and verified by conventional triaxial tests. It was then applied to the numerical simulation of shield tunnelling and supporting. The following are the findings: 1) The disturbance modes of surrounding rock directly above the shield tunnel under different boundaries are similar. The main disturbances occur approximately 4 m above the shield tunnel, extending to a distance of one segment outer diameter on either side of the tunnel's central axis. 2) The pressures on the top and bottom of the segments can be divided into rapid increase, slow increase, and stable fluctuation. These stages correspond to the processes of the segments getting out of the shield shell and the soil collapsing. 3) The largest surrounding rock pressure is at the vault, followed by that at the arch bottom, and the smallest is at both sides of the arch waist, which is consistent with the law obtained based on laboratory-scale model tests. The surrounding rock pressure values of the vault in the numerical simulation are consistent with the laboratory-scale model test results. These verify the reliability of the laboratory-scale model test and the UMAT subroutine based on the nonlinear elastic constitutive model of loess in simulating the shield tunnel construction in loess strata.

Nonlinear consolidation finite element analysis of a layered soft soil foundation under multistage loading based on the continuous drainage boundary

To comprehensively consider the impacts of stratification, residual pore water pressure, soil nonlinearity, and boundary permeability on consolidation settlement of soft soil foundations for accurate prediction, a continuous drainage boundary condition is proposed in this study that reflects the residual pore pressure under multistage loading, and a nonlinear elastic constitutive model based on the double logarithmic model is adopted to account for the nonlinear consolidation behaviour of soils. A UMAT subroutine is developed based on the proposed boundary condition and nonlinear elastic constitutive model. Subsequently, the developed subroutine is compared with the built-in linear elastic soil constitutive model in ABAQUS and engineering examples. The application of continuous drainage boundaries in stratified foundations is analysed, as well as the influence of factors such as the loading rate and soil nonlinearity on consolidation settlement. The results indicate that, compared to the built-in model, the subroutine developed in this study can be employed to more accurately calculate the nonlinear consolidation of multilayered foundations under multistage loading. By adjusting the loading rate parameter αk, consolidation under different loading conditions can be predicted. Additionally, the proposed boundary condition simplifies the calculations for soft soil foundations with sand layers, providing a novel computational approach for the design of construction loading schemes and long-term settlement predictions in soft soil foundations.

Laboratory Test and Constitutive Model for Quantifying the Anisotropic Swelling Behavior of Expansive Soils

Expansive soils exhibit directionally dependent swelling that traditional isotropic models fail to capture. This study investigates the anisotropic swelling characteristics of expansive soil with a medium swelling potential through the use of modified oedometric testing. Vertical swelling strains can reach up to 1.71 times that of the horizontal movements, confirming intrinsic anisotropy. A nonlinear elastic constitutive model incorporates vertical and horizontal elastic moduli with respect to matric suction to characterize anisotropy. Three elastic parameters were determined through the experiments, and predictive equations were developed to estimate the unsaturated moduli. The constitutive model and predictive techniques provide practical tools to better assess expansive soil pressures considering anisotropy, offering guidelines for utilization and design. The outcomes advance understanding of these soils’ directionally dependent behavior and stress–strain–suction response.

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number ${W}$ , polymer stress diffusivity $\varepsilon$ , solvent-to-total viscosity ratio $\beta$ and Reynolds number ${Re}$ , considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with ${Re}$ . For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$ , it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$ . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

Development and evaluation of a practical nonlinear elastic constitutive model for rockfill dam deformation simulation based on monitoring results

Development and evaluation of a practical nonlinear elastic constitutive model for rockfill dam deformation simulation based on monitoring results

Nonlinear Elastic Constitutive Model for Pvdf

This paper concerns the development of a constitutive model of Polyvinylidene fluoride (PVDF), a piezoelectric polymer. Suitable assumptions are made based on experimental data available in literature. The model is proposed for Form II of the PVDF material. Mooney-Rivlin and Neo-Hookean models are assumed to predict the finite elastic deformation characteristics of the material and experimental data are fitted to test the range of validity of the model. Predictions based on the Mooney-Rivlin and the Neo-Hookean models show that the Mooney-Rivlin model correlates very well with the experimental data within reasonably finite strain ranges (deviation upto 3-5Vo) whereas the Neo-Hookean model deviates largely from the experimental (about 50Vo).

Nonlinear elastic constitutive model of clayey sand reservoirs during depressurization exploitation of natural gas hydrate

Accurate evaluation of the mechanical response of hydrate-bearing reservoirs is a prerequisite for safe and efficient exploitation of natural gas hydrate resources. Therefore, it is necessary to develop a suitable constitutive model and to determine reasonable model parameters for hydrate-bearing sediments (HBSs) during exploitation. In this study, based on the strain-hardening behavior of clayey sand reservoirs published in the previous study, Duncan–Chang model was used as the basic model to establish constitutive models for HBSs during exploitation. The model parameters, initial elastic modulus and ultimate deviatoric stress, were determined by modifying existing formulas and establishing empirical formulas. Four nonlinear elastic constitutive models were proposed by combining the model parameters determined by different methods. Multi-stage and single-stage triaxial test data were used to verify the applicability of the model and determine the optimal model suitable for the sediments under study.

Microstructural modeling of moisture-thermal and mechanical behavior within concrete during drying at elevated temperatures

This article presents a numerical study for the analysis of the moisture-thermal and mechanical behavior of concrete subjected to elevated temperatures. Firstly, the moisture-thermal coupled model, is proposed to better describe the strong coupling between heat and moisture transfer. Secondly, based on the mechanics of partially saturated porous media, a nonlinear elastic damage constitutive model is formulated. The simulation results of the coupled moisture-thermal model at each time step were adopted to simulate the distributions of porous behavior, thermo-chemical damage, and moisture damage within concrete under the corresponding moisture-thermal conditions. Consequently, the evolution of the mechanical properties of concrete at elevated temperatures during the entire drying process could be simulated. Finally, an application concerning drying of concrete samples at elevated temperatures was presented. The results obtained show that the proposed moisture-thermal-mechanical model has potential capacity in numerically modeling the complex coupling process of concrete structures under elevated temperatures.

Experimental investigation and modeling of the mechanical behaviour of Yanji swelling mudstone subjected to wetting-drying-freezing-thawing cycles

Rocks and soils in seasonally frozen regions are subjected to cyclic wetting–drying-freezing-thawing (WDFT) process. The influence mechanism of WDFT process on their mechanical behaviour is not fully understood and the existing constitutive models are not capable of capturing the strain-softening behaviour under WDFT cycles. In the present study, a series of consolidated drained triaxial compression and mercury intrusion porosimetry tests were performed on Yanji mudstone subjected to WDFT cycles. Results indicated that the WDFT cycles could induce a large amount of cracks and large pores between aggregates and compress the aggregates with a decrease of the intra-aggregate pores. After cyclic WDFT process, the specimens exhibited more significant strain-softening and dilative behaviour. The elastic modulus, peak and residual shear strengths of the mudstone decreased significantly with increasing WDFT cycles, especially in the case of low confining pressures. These decreases were more pronounced in the first four cycles and became negligible after 4 cycles. As the cyclic number increased, the cohesion of specimens decreased remarkably owing to the presence of cracks and the destruction of bonds between aggregates. By contrast, the effective friction angle, Poisson’s ratio and dilatancy angle increased with the WDFT cycles due to the more compact aggregates induced by the high suction during drying process. A nonlinear elastic constitutive model was developed for describing the strain-softening behaviour of Yanji mudstone subjected to WDFT cycles. The deviator stress-axial strain relationship predicted by the proposed model agreed well with the measured one, indicating the relevance of the predictive model.

Mechanical Constitutive and Seepage Theoretical Model of Water Storage Media Based on Fractional Derivative

Using an abandoned underground goaf in coal mines as water reservoirs has been successfully applied for protecting mine water resources in western China. The water storage media are composed of broken rock masses and the voids between the rock masses. It is critical for reservoir capacity calculation that the deformation characteristics and seepage evolution of the water storage media under triaxial stress are theoretically described. In this study, broken rock masses and the space among the rock masses are simplified as two springs in a series. The mechanical behavior of the broken rock mass is described by Hooke’s law of linear elasticity, and the deformation characteristics of the space among the rock masses are represented by a nonlinear elastic constitutive model. The nonlinear stress-strain constitutive model of the water storage media is established by combining Hooke’s law and the fractional derivative stress-strain model. Similarly, a non-Darcy seepage model of the water storage media is obtained. The nonlinear stress-strain model is verified by mechanical experiments, physical simulation tests, and field measured data, and parameter sensitivity analysis is performed. The non-Darcy seepage equation is fitted and analyzed by using the seepage experimental data of the broken rock mass under triaxial compression conditions. The fractional non-Darcy model rather than the Forchheimer equation can more accurately describe the nonlinear seepage process in water storage media.

A model for multiphase moisture and heat transport below and above the saturation point of deformable and swelling wood fibers-II: Hygro-mechanical response

A non-linear elastic constitutive model for a swelling/shrinking orthotropic wood matrix is proposed. The model is thermodynamically consistent, derived based on the principles of continuum mechanics and the hybrid mixture theory. Moisture induced strains are introduced considering finite deformations by assuming a multiplicative split of the deformation gradient tensor into a swelling and an elastic part. Novel definitions of the Cauchy stress tensor and the moisture-dependent elastic material tangent matrix are obtained. The model is coupled to a multi-phase transient mass and heat transport model developed in Part I of this work. In this part of the work 2-D and 3-D test examples are used to describe the ability of the model to simulate moisture-induced distortions when drying wood within the hygroscopic and also from the over-hygroscopic moisture ranges. Despite deriving the model considering wood, the obtained constitutive relations can be suitably adopted to other orthotropic porous materials displaying properties similar to that of wood.

Turbulent planar wakes of viscoelastic fluids analysed by direct numerical simulations

Direct numerical simulations employing the finitely extensible nonlinear elastic constitutive model closed with Peterlin's approximation (FENE-P) are used to investigate the far-field region of turbulent planar wakes of viscoelastic fluids and to develop the theory describing these flows. The theoretical results display excellent agreement with the simulations and provide new scaling laws for the evolution of the shear layer thickness $\delta (x) \sim x^{1/2}$ , mean velocity deficit ${\rm \Delta} U(x) \sim x^{-1/2}$ and, for very high Deborah numbers, of the maximum polymer shear stresses $\sigma ^{[p]}_c(x) \sim x^{-2}$ and averaged polymer chain extension $\textrm {tr}(\bar {C}(x) - \boldsymbol{\mathsf{I}}) \sim x^{-2}$ , where $x$ is the streamwise distance from the solid body generating the wake. The theory is able to show the existence of self-similarity for the profiles of mean velocity, mean polymer shear stress, averaged polymer chain extension and the conditions for similarity of the turbulent shear stress, and is very well supported by the numerical simulations. Similarly to the case of viscoelastic turbulent planar jets (Guimarães et al., J. Fluid Mech., vol. 899, 2020, p. A11), when the inlet Weissenberg and Deborah numbers are sufficiently large, turbulent viscoelastic wakes exhibit a considerable reduction of the spreading rate and of the normalised Reynolds stresses. However, for very large downstream locations the turbulent viscoelastic wake recovers the classical evolution laws observed for Newtonian turbulent planar wakes.

Non-linear elastic behavior and constitutive model of coal during compression and its application

Non-linear elastic behavior and constitutive model of coal during compression and its application

Nonlinear elastic analysis of 2D materials of arbitrary symmetries with application to black phosphorus

Murnaghan’s polynomial based nonlinear elastic constitutive model has been previously applied to 2D materials of hexagonal symmetry. We present a general approach for determining the nonlinear elastic constants of 2D materials of arbitrary symmetries and any constitutive polynomial order. The methodology, which is based on ray sampling of the strain energy density in strain space, is independent of the energy calculation method. The ray based methodology is verified by evaluating the elastic constants of graphene which is a hexagonally symmetric 2D material previously considered in the literature. The methodology is then applied to determine the elastic constants of black phosphorus, an orthorhombic 2D material whose comprehensive nonlinear elastic behavior has not been previously considered in the literature. The energy calculations are carried out using plane-wave density functional theory. Detailed convergence analyses are performed to assess the accuracy of the nonlinear elastic constants. The linearized mechanical properties of black phosphorus are obtained from the elastic constants for comparison with the results published in the literature.

Analytical solutions for geosynthetic-encased stone column-supported embankments with emphasis on nonlinear behaviours of columns

This paper presents an analytical approach to predict the behaviours of geosynthetic-encased stone column (GESC)-supported embankments. The soil arching in the embankment and the nonlinear behaviours of stone columns are considered. Based on nonlinear elastic and elastoplastic constitutive models of stone columns, the nonlinear behaviours of GESCs, including settlement and radial deformation, are analysed. The deformations of GESCs, the surrounding soil, and the overlying embankment fill are compatible by applying stress continuity and volume deformation continuity at the bottom of the embankment fill. This method is verified via comparison with literature data and numerical analysis. The influences of parameters of the GESC, including encasement stiffness and column friction, on the performance of the embankment are investigated. Without considering the nonlinear behaviours of the column, the column-soil stress ratio is overestimated. It is more appropriate that the nonlinear characters of the column be considered in the analysis of GESC-supported embankments.

Effect of viscoelasticity in polymer nanofiber electrospinning: Simulation using FENE-CR model

The uniaxial elongational flow of a polymer solution in the electrospinning process was investigated numerically and experimentally. The numerical model was constructed taking into account the viscoelasticity of the material. Therefore, governing equations were supported by a finite extensible non-linear elastic constitutive model and for the first time a FENE-CR model was used. This version of FENE model allows a good description of the rheology of extensional flows. For validation purposes, experimental studies were conducted using polymeric solutions of polyamide 6 (PA6). The main emphasis was made in determining the final fiber diameter and how initial polymer concentration acts on it. Further, the effect of the different concentration regimes on thinning dynamics was also examined. The comparative results of the numerical and experimental outcomes are sufficient to prove the predictive ability of the simulations. It can also be observed that the numerical approach adopted in this study captures not only the viscoelastic behavior of the material but also solidification of nanofibers due to solvent evaporation. As predictive quantitative models for electrospinning are lacking, this study gives good insights into the simulation of this process.

Topology optimization considering the Drucker–Prager criterion with a surrogate nonlinear elastic constitutive model

We address material nonlinear topology optimization problems considering the Drucker–Prager strength criterion by means of a surrogate nonlinear elastic model. The nonlinear material model is based on a generalized J2 deformation theory of plasticity. From an algorithmic viewpoint, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. The objective function of the optimization problem consists of maximizing the strain energy of the system in equilibrium subjected to a volume constraint. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved through direct minimization of the structural strain energy using Newton’s method with an inexact line search strategy. Four numerical examples demonstrate features of the proposed nonlinear topology optimization framework considering the Drucker–Prager strength criterion.

Meso-scale compaction simulation of multi-layer 2D textile reinforcements: A Kirchhoff-based large-strain non-linear elastic constitutive tow model

This paper presents a non-linear, large strain constitutive approach to modelling the mechanics of fibrous tows. The isotropic and linear St. Venant hyperelastic model was modified to extend its applicability to a transversely isotropic continuum. From this model, a set of large-strain material parameters which resemble the linear-elastic moduli was derived. The non-linear constitutive response of fibrous tows was accounted for through a modelling scheme that incorporates the instantaneous microstructural arrangement of the tows. Tow samples were isolated from the studied textile roll and subjected to a compression study utilising a custom-made tow compression rig, from which a set of material functions that fully defined the constitutive model were generated. The developed model was then implemented as a material subroutine and utilised in a mesoscopic compaction simulation of a multi-layer 2D woven reinforcement. It was found that the constitutive model performed very well in predicting the shape changes and stresses in deforming tows.

Nonlinear solutions for laterally loaded piles

A nonlinear analysis framework for laterally loaded piles is presented that is as accurate as equivalent three-dimensional nonlinear finite element analysis, but computationally one order of magnitude faster. The nonlinear behavior of sands and clays are account for by using hyperbolic modulus–reduction relationships. These nonlinear–elastic constitutive models are used to calculate the reduced modulus at different points in the soil based on the soil strains induced by lateral pile displacement. The reduced modulus at different points in the soil domain are spatially integrated to calculate the reduced soil resistance parameters associated with the differential equation governing the lateral pile displacement. The differential equations governing the lateral displacements of pile and soil under equilibrium are obtained by applying the principle of virtual work to a continuum-based pile–soil system. These coupled differential equations are solved using the one-dimensional finite difference method following an iterative algorithm. The accuracy of the analysis is verified against equivalent three-dimensional nonlinear finite element analysis, and the validity of the analysis in predicting the field response is checked by comparisons with multiple pile load test results. Parametric studies are performed to gain insights into the lateral pile response.