Nonlinear Stress-strain Relationship Research Articles

3D polycatenated architected materials

Architected materials derive their properties from the geometric arrangement of their internal structural elements. Their designs rely on continuous networks of members to control the global mechanical behavior of the bulk. In this study, we introduce a class of materials that consist of discrete concatenated rings or cage particles interlocked in three-dimensional networks, forming polycatenated architected materials (PAMs). We propose a general design framework that translates arbitrary crystalline networks into particle concatenations and geometries. In response to small external loads, PAMs behave like non-Newtonian fluids, showing both shear-thinning and shear-thickening responses, which can be controlled by their catenation topologies. At larger strains, PAMs behave like lattices and foams, with a nonlinear stress-strain relation. At microscale, we demonstrate that PAMs can change their shapes in response to applied electrostatic charges. The distinctive properties of PAMs pave the path for developing stimuli-responsive materials, energy-absorbing systems, and morphing architectures.

Study on the scale effect and mechanical properties of gravel materials in riverbed cover layers

To address the challenges of performing in-situ tests on riverbed overburden gravel, this study employs three scaling methods—equal mass substitution, similar gradation, and the mixed method—to investigate the original gradation of the gravel. Large-scale triaxial consolidated drained shear tests were conducted to evaluate the effects of the maximum particle size reduction ratio (M) and confining pressure on the stress–strain behavior, fractal dimension, particle breakage, and the parameters of the Duncan-Chang model (an elastic model describing nonlinear stress-strain relationships). The study explores how scaling, based on fractal dimension and particle breakage rate, impacts the strength and deformation characteristics of gravel materials. The results show that increasing confining pressure typically leads to higher material strength. As confining pressure increases, particle breakage becomes more pronounced across the various scaling methods. Gravel processed using the similar gradation method exhibited the highest strength, the smallest change in fractal dimension, and the lowest breakage rate, followed by the mixed method. Furthermore, as the M value increases, peak stress also rises, with the modulus coefficient (k) and bulk modulus coefficient (Kb) in the Duncan-Chang model increasing by 4.5 times and 3.3 times, respectively, for M = 15 compared to M = 0. A clear relationship was observed between the change in fractal dimension before and after testing and the breakage index (Bg). As the M value increases, both the difference in fractal dimension and the breakage rate decrease, suggesting that larger M values lead to reduced particle breakage and improved mechanical stability.

Bending test and numerical analysis of inflatable tubes

In this paper, bending test of an inflatable, polyvinylchloride (PVC)-coated fabric tube is carried out to study the deformation, bearing capacity, and wrinkling characteristics. In order to illustrate the effect of hoop stress, two nonlinear stress-strain relationships are measured through the uniaxial tension and the biaxial tension tests. Meanwhile, a computing method based the fiber beam model is developed for the inflatable tube. This method can consider nonlinearity of membrane materials and wrinkling effect easily. The numerical results using the stress-strain relationship considering hoop stress effect are shown to more accurately predict the load-displacement response in bending test. As the test results shown, the cross-section remains basically circular and the strain distribution in no wrinkling region approximately satisfies the plane section assumption before and after wrinkling. Furthermore, the wrinkling load of the inflatable tube in bending test is compared with the numerical results. Lastly, wrinkling expansion process and wrinkling angle at ultimate load are detailed.

A Quasi-Two-Dimensional Approach in Laminar Flow of Lubricating Oil and Grease

<div>The overarching objective of the present study is to apply a quasi-two-dimensional approach to analyze the laminar flow of lubricating oil. Lubricating oils are non-Newtonian by nature. For these types of oils, the Sisko fluid model is the most suitable model of the nonlinear stress–strain relationship for these types of oils. It is hoped that by omitting the dependence of flow quantities in one direction, more qualitative information can be obtained on the characteristics of the purely three-dimensional boundary layer flow of lubricating oils. Some of the most familiar flow geometries discussed are steady flow over a flat plate, a corner of a wedge, and a stagnation region; steady flow in a convergent and divergent channel; and impulsively started flow over an infinite flat plate and semi-infinite flat plate. The governing equations of all flow geometries are transformed into nonlinear ordinary differential equations (ODE) using the free parameter transformation. The results are discussed briefly in the graphical presentation.</div>

An experimental and numerical analysis study of roller-compacted concrete based on recycled asphalt pavement

The contribution of this study is to examine the potential of using reclaimed asphalt pavement (RAP) as a replacement for natural aggregates (NA) in the production of roller-compacted concrete (RCC). In several countries, recycling old asphalt pavements has gained interest due to its economic and environmental advantages. However, in Algeria, large quantities of RAP are generated annually, yet they are rarely quantified or reused. This research investigates the use of RAP as coarse aggregates, testing five different replacement levels (0%, 25%, 50%, 75%, and 100%). The properties of fresh concrete, such as density, were measured, along with the impact of curing temperatures (20°C, 40°C, and 60°C) on the hardened properties, including compressive strength and modulus of elasticity. The results reveal that it is possible to produce RCC with up to 50% RAP, providing significant environmental benefits by reducing waste and preserving natural resources. These studies confirm that using up to 50% RAP in RCC is both feasible and environmentally beneficial, while maintaining good concrete performance. As part of the study, an accurate numerical model simulating the mechanical behavior of RCC, including time-dependent deformations and the non-linear stress-strain relationship, was created using finite element analysis in ANSYS-FEM. The purpose of this model is to highlight how RCC with RAP can be used in pavement applications.

Modeling nonlinear stress-strain model for sulfate dry-wet cycle erosion of concrete: considerations for the initial compaction stage

Sulfate dry-wet cycle erosion significantly affects the mechanical properties of concrete. Investigating the uniaxial compressive stress-strain relationship under these conditions is essential for developing accurate constitutive models. This study analyzes the uniaxial stress-strain curves of concrete subjected to dry-wet cycles in 5% and 15% sulfate solutions. The results show that the initial compaction phase in the stress-strain relationship is particularly pronounced under increasing sulfate concentrations and cycle counts. The concrete experiences an extended compaction phase, which accounts for up to 35.71% of the total strain process. This finding challenge traditional constitutive models, which struggle to accurately describe this phase. To address this issue, the study develops a nonlinear stress-strain model for concrete, incorporating the initial damage caused by sulfate dry-wet cycle erosion, based on Weibull statistical damage mechanics principles. The research indicates that the effects of sulfate concentration and cycle count are predominantly reflected in the pronounced nonlinearity of the skeleton strain function’s opening size (a) and shape characteristics (b), modeled using a fourth-degree polynomial. The model demonstrates an excellent fit to experimental data with an R2 value of 0.99989, showing that the proposed nonlinear stress-strain relationship effectively captures the uniaxial mechanical behavior of concrete under sulfate dry-wet cycle erosion and provides a robust framework for developing constitutive models in such environments.

A Comparative Study of Geometric Phase Change- and Sideband Peak Count-Based Techniques for Monitoring Damage Growth and Material Nonlinearity.

This work presents numerical modeling-based investigations for detecting and monitoring damage growth and material nonlinearity in plate structures using topological acoustic (TA) and sideband peak count (SPC)-based sensing techniques. The nonlinear ultrasonic SPC-based technique (SPC-index or SPC-I) has shown its effectiveness in monitoring damage growth affecting various engineering materials. However, the new acoustic parameter, "geometric phase change (GPC)" and GPC-index (or GPC-I), derived from the TA sensing technique adopted for monitoring damage growth or material nonlinearity has not been reported yet. The damage growth modeling is carried out by the peri-ultrasound technique to simulate nonlinear interactions between elastic waves and damages (cracks). For damage growth with a purely linear response and for the nonlinearity arising from only the nonlinear stress-strain relationship of the material, the numerical analysis is conducted by the finite element method (FEM) in the Abaqus/CAE 2021 software. In both numerical modeling scenarios, the SPC- and GPC-based techniques are adopted to capture and compare those responses. The computed results show that, from a purely linear scattering response in FEM modeling, the GPC-I can effectively detect the existence of damage but cannot monitor damage growth since the linear scattering differences are small when crack thickness increases. The SPC-I does not show any change when a nonlinear response is not generated. However, the nonlinear response from the damage growth can be efficiently modeled by the nonlocal peri-ultrasound technique. Both the GPC-I and SPC-I techniques can clearly show the damage evolution process if the frequencies are properly chosen. This investigation also shows that the GPC-I indicator has the capability to distinguish nonlinear materials from linear materials while the SPC-I is found to be more effective in distinguishing between different types of nonlinear materials. This work can reveal the mechanism of GPC-I for capturing linear and nonlinear responses, and thus can provide guidance in structural health monitoring (SHM).

Nonlinear dynamic study of FG-GFRC-NPR structures

The functionally graded glass fibre-reinforced polymer composite(FG-GFRC) 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 Poisson's ratio 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 Poincare 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