Nonlinear Stress-strain Relationship Research Articles

Nonlinear dynamic study of FG-GFRC-NPR structures

Abstract The functionally graded glass fibre-reinforced polymer (FG-GFRP) composite is one of the contemporary sophisticated materials that may be employed for a variety of technical purposes. The shock absorption capacity of FG-GFRP laminates can be improved by incorporating auxetic property (i.e., negative PR - NPR) in the structures, which can be attained by a particular stacking sequence of glass fibres (GF) and a different percentage of GF distribution along the thickness direction within the FG-GFRP structure. The nonlinear dynamic behaviour of such structures must be investigated. The clamped-clamped FG-GFRP-NPR structure is stimulated under a uniform transverse load distribution in the current work. Based on the higher-order shear deformation theory (SDT), the displacement field and nonlinear stress-strain relationship are derived. The nonlinear differential equation with cubic non-linear components were derived using Hamilton’s principle and transformed using Galerkin’s technique and obtained governing equation of motion. MATLAB has been used to conduct all numerical simulations. A time series, a phase picture and a PCM diagram (PCM) were generated to characterize the vibration behavior of such auxetic structures.

An efficient machine learning-based model for predicting the stress-strain relationships of thermoplastic polymers with limited testing data

Thermoplastic polymers used in aeronautical structures such as poly-ether-ether-ketone (PEEK) usually exhibit nonlinear stress-strain relationships, which can be usually predicted by the physical and phenomenological models. For different thermoplastic polymers, however, existing models may encounter difficulties in reasonably predicting the stress-strain relationships. By reasonably using experimental data, the machine learning-based model can accurately predict the stress-strain relationships. In this paper, an efficient machine learning-based model is built to predict the stress-strain relationships of thermoplastic polymers by using the Kriging model, where limited testing data are merely used. Experiments of 69 specimens for PEEK are firstly performed under uniaxial tensile, where the temperatures range from 23 °C to 140 °C and the strain rates range from 10−3 s−1 to 10−1 s−1. Genetic algorithm is employed to train the proposed model based on the stress-strain relationships obtained from experiments. Moreover, the results predicted by the proposed Kriging-based model are compared with those obtained from the DSGZ (Duan-Saigal-Greif-Zimmerman) model. The results indicate that the Kriging-based model possesses better accuracy, lower complexity, and better robustness than the selected model, because the prior estimation is introduced by the assumption of Gaussian stochastic processes. In addition, the proposed model is also applied to Poly-Propylene (PP), PEEK at high temperatures, and polycarbonate (PC), it can be found that the Kriging-based model can also predict the stress-strain relationships of other amorphous and semi-crystalline thermoplastic polymers.

A non-linear stress–strain relationship for soil and rock

While many employ a hyperbolic stress–strain relationship for soils, it is known that such a relationship is accurate over the small-strain range as encountered in earthquake and soil dynamics problems. For large strains, a relationship with different input parameters is needed, as is required for finite-element analyses of large-deformation behaviour. These separate characterisations do not carry over into the other application. Here, a proposed power relationship is presented that was developed to characterise the triaxial test stress–strain behaviour of cohesionless material from lubricated or ‘frictionless’ cap and base tests (144 tests) covering a range in the natural variation in particle size, particle shape and surface roughness, over low to high confining pressure. This relationship covers the strain range from 10−6 to soil failure. To date, it has been successfully used in laterally loaded pile response characterisation (the strain wedge model) and shallow-foundation load–settlement–bearing capacity response. Most recently, it has been extended to assess the behaviour of rock-like material (caliche). The relationship and its comparison with the hyperbolic relationship for large strain and the shear modulus reduction curve for seismic behaviour are presented.

Effect of the incorporation of plastic waste on the mechanical properties of composite materials

Through this scientific research, we have tried to study the effect of the incorporation of plastic waste on the mechanical properties of concrete, in order to obtain good concrete with high resistance at a lower cost. To carry out this work, we adopted the following steps: Knowledge of the properties of concrete, which contains plastic waste of High Density Polyethylene (HDPE). Thus, the natural aggregate was replaced by concrete formulated with plastic waste in partial substitution varying between 0%, 10% and 20%. In all the concrete mixtures, the components, water, cement, gravel 3/8 and 8/15 and sand 0/3, remained constant while the HDPE waste varied according to the substitution rate. Through this process, the mechanical properties of concrete in fresh and hardened states were determined. The analysis of the results of the study allowed us to know the incorporation of plastic waste and its effect on the behaviour of the manufactured concrete. The results showed that the density of concrete made from plastic waste is lighter than the reference concrete without waste (C0), which contains only natural aggregates, in a fresh state. In the case of hardening, the results showed a decrease in the compressive strength of composite concrete produced from plastic waste compared to (C0) the reference concrete. In the study's second part, a numerical model for precision and effectiveness is constructed using the finite element (FE) method. Furthermore, manufacturing experiments are scheduled to use computer simulations that account for labour, materials, tests, and time. The nonlinear stress-strain relationship for time-dependent concrete deformations and tension cracks presents a challenge for concrete modelling. ANSYS software is used to use three-dimensional nonlinear finite elements in order to determine this kind of complex mechanical behavior.

Contact formulations for analysis of micropolar media with finite continuum beam elements

Microscale beam-like structures are standard components of micromechanical systems in many devices. However, the small dimensionality affects their deformation characteristics and leads to misinterpretations of the results. Presenting such size dependency is possible with advanced continuum descriptions via introducing additional variables compared to the classical one. Hence, the problems where contact occurs require the readjustment of boundary conditions considering these extra parameters. That burdens the already challenging task of contact problem calculations and restricts most demonstration examples to two-dimensional problems and geometrical linearity. The resolution of the imposed restrictions within finite element modeling further emphasizes the usage of advanced media in design and facilitates its application to various problems, which is the aim of this paper. The delivery of that task is as follows. Firstly, we present the kinematics and material descriptions for the micropolar media. The authors propose to use a newly developed continuum-based micropolar beam formulation to avoid an overwhelming computational burden and, at the same time, deal with nonlinear stress–strain relations. Secondly, the work develops a contact approach within the micropolar theory from two-dimensional to three-dimensional elasticity, although the contact is considered frictionless. Finally, it compares two existing contact formulations, including the developed one, for the contact beam problems, using the examples of two collinear sliding beams’ bending.

Investigating the Constitutive Model of Frozen Supersulfate Saline Soil: Insights from Fractional Calculus

The study of the mechanical properties of frozen saline soil is one of the key issues in addressing the design of infrastructure in cold regions. This research focuses on the supersulfated saline soil of the Ningxia Yellow River Irrigation Area in China, conducting triaxial tests under negative temperatures (−5, −10, −15 °C, and − 20 °C) with varying water contents (12%, 16%, 20%). Based on fractional calculus theory and incorporating an exponential decay factor, this study proposes a novel fractional-order constitutive model for a unified description of the softening and hardening behaviors in frozen saline soil. The model treats frozen saline soil as a composite blend of ideal solids and ideal fluids in varying proportions, taking into account the material's inherent time-dependency and non-linear stress-strain relationships. Finally, the validity of the model is verified by the calculated values of the model and the triaxial tests. The results indicate that, based on preliminary judgment, due to the presence of salt solutes, a large amount of liquid water remains in the supersulfated saline soil at temperatures ranging from 0 to −10 °C, forming an unstable state called “warm frozen saline soil”. The mechanical properties of frozen saline soil depend on the relative content of unfrozen water, ice crystals, and salt crystals and the formation of ice and salt crystals significantly enhances the strength of frozen saline soil. The computational results of the improved fractional constitutive model align well with experimental results, effectively describing the stress-strain relationship of frozen supersulfate saline soil. In the model, the parameter κ functions analogously to an elastic modulus and exhibits a linear relationship with temperature, and the parameter α characterizes the strain hardening of saline soil, while β describes its softening behavior. The proposed fractional constitutive model, with only three parameters having clear physical significance, is convenient for practical engineering applications.

Comparative analysis of nonlinear dynamic behaviors of hyperelastic curved structure modelled by different constitutive laws

This work focuses on conducting numerical studies and comparisons on the nonlinear dynamic behaviors of hyperelastic curved structures subjected to external dynamic loadings. A numerical model of the hyperelastic structure is developed using the nonlinear finite element method, which considers both the geometric nonlinearity resulting from large deformations and the material nonlinearity due to hyperelasticity. Four commonly used hyperelastic constitutive laws, namely the Mooney-Rivlin model, the neo-Hookean model, the Yeoh model, and the Ogden model, are employed to describe the nonlinear stress-strain relationship of the hyperelastic material. The convergence performance of the numerical model with respect to grid element and time step is thoroughly examined through several numerical examples. Subsequently, based on the developed numerical model, the nonlinear dynamic behaviors of the hyperelastic curved structures excited by uniform pressure loadings with single and dual frequencies are studied and compared using various constitutive laws. The material constants used in these constitutive laws are estimated based on the well-known Treloar's test data on vulcanized rubber material. Additionally, comprehensive parametric analyses are conducted on the nonlinear dynamic behaviors of the hyperelastic structure simulated by different constitutive laws. The deformation evolution process, resonance curves, time-frequency analysis, bifurcation diagrams, and phase portraits are employed to demonstrate the similarities and differences in the nonlinear dynamic behaviors (i.e., super- and sub-harmonic resonance, internal resonance, chaotic motion and bifurcations) of the hyperelastic structure modeled by different constitutive laws.

Inverse design of periodic microstructures with targeted nonlinear mechanical behaviour

This paper introduces an inverse design framework for the precise tailoring of desired nonlinear mechanical responses in periodic microstructures, with particular focus on prescribed nonlinear stress–strain relationships. The topology optimization hinges on minimizing the error between the target and realized properties of the microstructures. A deformation-driven homogenization framework is setup. The periodic constraints needed for the microscale equilibrium equation are imposed through strongly enforced periodic boundary conditions and the removal of the translational nullspace, avoiding the need for Lagrange multipliers, greatly simplifying the implementation. Automatic differentiation is leveraged to efficiently calculate the necessary sensitivities for the gradient-based optimization. To further aid the design of discrete designs a intermediate density penalty constraint is proposed. Numerical examples underscore the efficacy of our methodology, showcasing microstructures that demonstrate targeted softening and stiffening as well as distinctive directional behaviour.

P-M INTERACTION DIAGRAMS FOR REINFORCED CONCRETE WALLS RETROFITTED WITH EXTERNALLY BONDED CARBON-FIBER REINFORCED POLYMER

This research develops analytical axial load-bending moment (P-M) interaction diagrams for reinforced concrete (RC) walls retrofitted with carbon fiber reinforced polymer (CFRP). A 12-inch thick, 1-foot strip of RC wall section is considered with varying reinforcement ratios, varying axial compressive loading, and varying number of CFRP layers. The CFRP material is treated as externally bonded onto the tension face of the RC wall to investigate its impact on the flexural capacity of the section. Each P-M interaction diagram was generated considering a discretized compression zone and by satisfying principles of static equilibrium and strain compatibility. An elastic-plastic steel stress-strain relationship is used for Grade 60 reinforcement; a uniaxial nonlinear compressive stress-strain relationship is used for concrete; and a linear stress-strain relationship is used for CFRP in tension. The failure modes considered are steel yielding, concrete crushing, and CFRP debonding. The computed P-M interaction diagrams are normalized for their general use in the retrofit of existing RC walls using CFRP. Results show that as the axial compressive force on the RC wall increases, the less effective CFRP is. CFRP is more effective in sections with beam-like behavior, where the reinforcement ratio tends to be the smallest.

Mechanistic prediction of folding angles in 4D printed shape memory polymers under varied loading conditions

Four-dimensional printing technology empowers 3D-printed structures to change shapes upon external stimulation. However, most studies did not consider recovery under loaded conditions. This paper introduces a mechanistic prediction model for forecasting recovery angles in 4D printing utilizing shape memory polymer under various loads. The model integrates Neo–Hookean model to describe the non-linear stress–strain relationship with experimentally determined force density data to characterize polymer restoration properties under various loads. Validation was demonstrated by the recovery experiment of a 3D-printed polylactic acid-thermoplastic polyurethane composite structure loaded by means of a cord and pulley mechanism. The predictive outcomes exhibited reasonable agreement with experimental results, demonstrating a trend of more accurate forecasts as the applied load increased. The model can accommodate various active materials provided that the pertaining force density data is accessible. The predictive model supports the design, optimization and material selection for 4D-printed structures to meet specific performance requirements.

Open-system force-elongation relationship of collagen in chemo-mechanical equilibrium with water

A significant deformation mechanism of collagen at low loads is molecular uncoiling and rearrangement. Although the effect of hydration and cross-linking has been investigated at larger loads when collagen undergoes molecular sliding, their effects on collagen molecular reorganization remain unclear. Here we develop two thermodynamic models that use the notion of open-system elasticity to elucidate the effect of swelling due to water uptake during deformation of collagen networks under low and high cross-linking conditions. With low crosslinking, entropic contributions dominate resulting in rejection of solvent from the polymer network leading to reduced collagen stiffness with increased loads. Contrarily, high cross-linking inhibits initial coiling and structural kinking and the mechanical behavior is dominated by elastic energy. In this configuration, the solvent content depends on the sign of the applied load resulting in a non-linear open-system stress-strain relationship. The models provide insight on the parameters that impact the stress-strain relationships of hydrated collagen and can inform the way collagenous matrices are treated both in medical and laboratory settings.

A new constitutive relation to describe the response of bones

Trabecular bone, a solid that has a heterogeneous porous structure, demonstrates nonlinear stress–strain relationship, even within the small strain region, when subject to stresses. It also exhibits different responses when subject to tension and compression. This study presents the development of an implicit constitutive relation between the stress and the linearized strain specifically tailored for trabecular bone-like materials. The structure of the constitutive relation requires the solution of the balance of linear momentum and the constitutive relations simultaneously, and in view of this, a two-field mixed finite element model capable of solving general boundary value problems governed by a system of coupled equations is proposed. We investigate the effects of nonlinearity and heterogeneity in a dogbone-shaped sample. Our study is able to capture the significant nonlinear characteristics of the response of the trabecular bone undergoing small strains in experiments, in both tension and compression, very well.

Machine-learning-based asymptotic homogenisation and localisation of spatially varying multiscale configurations made of materials with nonlinear stress-strain relationships

Machine-learning-based asymptotic homogenisation and localisation of spatially varying multiscale configurations made of materials with nonlinear stress-strain relationships

Simplified modelling of full-strain-range non-linearity of cyclically loaded undrained clays

Constitutive modelling of cyclically loaded undrained clay is of significant importance for various branches of geotechnical engineering exemplified by offshore wind turbine (OWT) foundation design and prevention and mitigation of earthquake hazards. Fine-grained soils can display non-linear stress-strain relations from small (10-5) to relatively large (10-1) strain levels. This full-strain-range non-linearity can remarkably affect the cyclic response. Effective stress-based constitutive models have achieved great success in modelling clayey soils, whereas they can be overly complex for practicing engineers, in particular, when considering non-monotonic loading and full-strain-range non-linearity. This paper explores the possibility of modelling, in simplified manner, the full-strain-range nonlinearity of cyclically loaded undrained clay, which can directly utilize outcomes of in situ site exploration and routine laboratory tests. For this purpose, we idealize soils under undrained conditions as single-phase materials governed by total stress. Considering that most current total stress-based models are proposed for metals and may be limited to capture the non-linear stress-strain relations of soil, a novel generalized non-linear (GNL) hardening law is proposed that can describe versatile stress-strain relations of undrained clays. Bounding surface and a mapping rule considering update of projection centre are introduced to reflect the influences of maximum past stress history and recent stress history, respectively, on the stress-strain nonlinearity. The simplified constitutive model is first validated at the element level by simulating monotonic and cyclic loading laboratory tests on undrained clays. Later, the proposed soil model is applied to the finite element analyses of OWT pile foundation subjected to cyclic lateral loading.

Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

Damage-Induced Softening of the Sclera: A Pseudo-Elastic Modeling Approach

Abstract The biomechanical properties of the sclera such as the stiffness, anisotropic behavior, and nonlinear stress–strain relationship have been extensively investigated for the pathogenesis study of ocular diseases. Even so, scarce mechanical investigations have been conducted on the damage in the sclera when subjected to large and repetitive deformations. Hence, the aim of this study is to quantify microstructural damage of the posterior and anterior sclera, through mechanical testing and model fitting. We performed uniaxial mechanical tests on scleral strips dissected from African green monkeys. Samples were subjected to strain-driven cycles of 5%, 10%, 15%, and 20% to evaluate the damage behavior commonly known as the Mullins effect. Experimental results showed qualitative changes in the stress–stretch curves when higher loading cycles were applied. A pseudo-elastic model accurately captured the curve trends across all tested samples, as indicated by a coefficient of determination above 0.96 and a subsequent finite element analysis (FEA) validation. Damage evolution and resultant permanent set demonstrated that considerable microstructural failure was attainable even at small strain levels and that the inherent plasticity had a similar contribution to stress-softening as the Mullins effect. Computed material and damage properties are expected to provide a broader understanding of the underlying mechanisms of ocular diseases and the development of more effective approaches for their treatment.

Multi-scale model to study gas transport processes in clay materials

In the field of radioactive waste confinement, the question of gas transfers in clay formations is a crucial issue [1]. A certain amount of gas, such as Hydrogen may be generated during the exploitation phase in the nearfield by the deterioration of the metal components of the system. Since the host medium is characterised by a very low permeability, the mechanisms of gas transport by advection and diffusion within the groundwater remain insufficient to evacuate the gas generated and a free gas phase is formed. If the gas pressure keeps increasing and reaches the minimum principal stress acting on the rock, micro-fractures, known as preferential gas pathways could develop through the rock mass [2], and affect the clay barrier integrity.
 There is a growing body of experimental evidences [3, 4] that separation planes such as bedding planes or pre-existing fractures and heterogeneities in clay-rich materials represent preferred weaknesses for the process of opening discrete gas-filled pathways. Capturing the related transport mechanisms therefore requires to go from macroscopic to microscopic scale. In this context, the adopted numerical approach would ideally take into account the effect of each constituent of the material microstructure on the macroscopic gas flow. Yet, direct modelling of the entire microstructure using small-scale models is usually not possible due to the huge computational costs it would require at the scale of a repository. Conversely, indirect modelling of the behaviour of all the micro-constituents by collective closed-form macroscopic constitutive equations using large-scale models [5] has also limits in terms of assumptions formulation and parameters identification. Hence, the use of a multi-scale approach that models the micro-scale effects explicitely on their specific length scale and couples their homogenized effects to the macro-scale is proposed in the present work. Based on a periodicity assumption of the microstructure, the physical and geometrical properties of the microstructure are embedded on a Representative Element Volume (REV) which contains a detailed model of the microstructure constituents, i.e. the pore network, the bedding planes and the bridging planes, while eluding a complex description of the whole microstructure over the domain. From this local description of the material, the macroscopic response to loading can be derived using numerical homogenisation techniques [6].
 The multi-scale model proposed here has been implemented in the finite element code Lagamine, and validated against a well established and documented macro-scale THM coupled model [7]. The size and structure of the REV comes from experimental data acquired from scanning images in clay material, like for instance the Boom Clay formation which is envisaged as a potential host rock for a deep geological disposal in Belgium. Such scans reveal the opening of large aperture with a repeated distance in-between, which make it possible to extrapolate a physical idealisation of the microstructure (Figure 1a), built with one of these fractures corresponding to the bedding planes (red) and the pore network substituted by an assembly of tubes (green). Narrow-aperture fissures, known as the bridging planes (blue) were also identified to contribute to the flow normal to the predominant direction of fissure [4], by connecting the natural discontinuities which were initially closed. These micro-scale constituents are defined in such a way as to satisfy the conditions of pore size distribution, macroporosity and intrinsic permeability. For the mechanical behaviour, a simple linear elastic constitutive law was introduced for the tube network while a non-linear stress-strain relationship was used for the interfaces defining the bedding and bridging planes. Concerning the hydraulic behaviour, gas diffusive fluxes were considered and modelled by a Fick’s law while a channel flow based on the Navier-Stokes equations was used for the multiphase flow in the different constituents.
 The model has been subsequently applied to simulate a gas injection test parallel and perpendicular to the bedding of initially saturated samples of Boom Clay [4]. This analysis provides a rather good agreement with the experimental results in terms of injection and outflow pressure response, outflow volume and average axial strain along the sample height. In addition, it allows to simulated the creation of a preferential flow pathway along the sample axis (Figure 1b, top), which serves as basis to numerically reproduce the development of random pathway through the sample in plane strain state (Figure 1b, bottom), and aims to improve the mechanistic understanding of the gas transport processes at play in clayey barriers [8].

Measurement and analysis of winding stresses in dry-wound pancake coils considering nonlinear compressive behaviors

Considering coil stress management, conductor thermal stability, and susceptibility to delamination, the dry-winding approach has been gradually applied and tested in high-temperature-superconducting coils wound with REBa2Cu3O (REBCO) coated conductors (CCs). In dry-wound coils, the winding tension can be a useful tool for managing the mechanical stresses and maintaining the structural stability. It is also related to the contact resistivity characteristics of no-insulation and metal-as-insulation coils. In this paper, we present a study on the winding stresses in dry-wound pancake coils considering nonlinear compressive behaviors in the radial direction. Coils wound with stainless-steel (SS) tapes and REBCO CCs were prepared, respectively, with various sizes (90–140 turns) and winding tensions (26–61 MPa). The radial pressure was obtained by applying the pressure measurement films, which cover a pressure range up to 50 MPa. A relatively moderate radial pressure buildup was observed in coils wound with SS tapes, indicating that the effective radial modulus of the coil as an entity is considerably lower compared to the tensile modulus in the azimuthal direction. This is consistent with the nonlinear stress–strain relationship of stacked thin sheets under transverse compression, which can be further explained by the contact behaviors between two rough surfaces. The average radial pressure in CC coils shared similar characteristics as that in SS coils, whereas the radial stress along the conductor edges were significantly higher compared to the middle. Cross-sectional microscopy of the CCs suggests that this can be primarily attributed to the thickness variations across the conductor width. These results suggest that the microscopic rough-surface contact and thickness variation of the wound tape play a critical role in the coil stiffness as well as the winding stress distributions.

Effects of 3D deformation and nonlinear stress–strain relationship on the Brazilian test for a transversely isotropic rock

To improve the accuracy of indirect tensile strength for a transversely isotropic rock in the Brazilian test, this study considered the three-dimensional (3D) deformation and the nonlinear stress–strain relationship. A parametric study of a numerical Brazilian test was performed for a general range of elastic constants, revealing that the 3D modeling evaluated the indirect tensile strength up to 40% higher than the plane stress modeling. For the actual Asan gneiss, the 3D model evaluated the indirect tensile strength up to 10% higher and slightly enhanced the accuracy of deformation estimation compared with the plane stress model. The nonlinearity in stress–strain curve of Asan gneiss under uniaxial compression was then considered, such that the evaluated indirect tensile strength was affected by up to 10% and its anisotropy agreed well with the physical intuition. The estimation of deformation was significantly enhanced. The further validation on the nonlinear model is expected as future research.

Research on the bearing behavior of single pile in self-weight collapsible loess areas.

The negative skin frictional caused by loess collapse will decrease the bearing capacity of single pile, which is essential to the design of pile foundations in loess areas. In this study, a method for estimating the subsidence of soil layer at any depth is firstly proposed based on the total self-weight collapse value. Secondly, a new load transfer constitutive model for pile-soil interface is developed, which considers the nonlinear stress-strain relationship and the ultimate shear strength of soil. Then, a load transfer calculation model for pile foundation is established, which can calculate the pile axial force, the pile skin frictional, neutral point position and the settlement of a single pile. The calculation results are compared with the test data that obtained from a pile foundation on-site immersion test and the effectiveness of the calculation method is verified well. This calculation method may be useful for designing pile foundations in collapsible loess regions.