Nonlinear Stress-strain Curves Research Articles

Characterizing nonlinear constitutive behaviors of fiber metal laminates

Abstract This study characterized the nonlinear tensile behavior of fiber metal laminates (FMLs). FMLs comprise layers of thin metallic sheets and fiber-reinforced composite layers, and a constitutive FML model includes the constitutive relationships of the FML’s constituent materials; however, nonlinear behavior is typically only considered for the metal components of an FML. In this study, a nonlinear constitutive relationship for the unidirectional fiber composites was modeled using a one-parameter plastic model. The nonlinear constitutive law for the metal was formulated using the J2 flow rule. These relationships were summed for each layer in accordance with laminated plate theory to obtain a constitutive FML model, which was then used for numerical predictions of nonlinear stress–strain curves. The model was validated by comparing its predictions with experimental results from the literature. Moreover, the effect of the inclusion of nonlinear fiber composite behavior on the model predictions was investigated. Results revealed that the difference between the model predictions and the experimental results was less than 4%. These predictions with nonlinear fiber composite behavior were substantially more accurate than those of the model without this behavior for FMLs with angle-ply fiber composites.

The structural strain method for fatigue evaluation of welded components: Analytical treatment of reversed plasticity

The traction structural stress method has been adopted by various industry sectors, in addition to ASME Boiler and Pressure Code (B&PV), for fatigue life evaluation and assessment of welded components, especially for high-cycle fatigue applications. Its effectiveness has been validated and demonstrated by showing good data correlation in the form of the master S–N curve scatter band. For low cycle fatigue with the involvement of elastic-plastic deformation, the structural strain method has been recently developed by simply extending the current elastic-deformation-based structural stress method to more general elastic-plastic deformation. Closed-form structural strain solutions for elastic-perfectly plastic materials and numerical solutions for nonlinear strain hardening materials have been obtained under simple load-controlled loading and unloading cycling conditions. In this paper, the equations of nonlinear cyclic structural stress-strain curves under fully-reversed fatigue loading are analytically derived for the first time with a 3-bar model, which consists of three elastic-perfectly plastic bars. The general trends and patterns of the loading paths of the constructed cyclic structural stress-strain curves with the 3-bar model are validated with finite element analysis (FEA). Based on the insights gained from the 3-bar model and the FEA results, a simple procedure for constructing cyclic structural stress-strain curves from their corresponding monotonic loading curves are developed for both elastic-perfectly plastic model and the modified Ramberg-Osgood nonlinear strain hardening model. The effectiveness of this procedure is demonstrated by correlating low-cycle fatigue data of welded structures under pulsating and fully-reversed loading conditions.

Experimental and constitutive model investigation on the tensile mechanical properties of polyurea at wide strain-rate range

Polyurea is extensively utilized as a protective coating material for structures under impact and blast loadings, owing to its significant large tensile deformation capability and strain hardening effect. In this study, the tensile performance of polyurea was experimentally and theoretically investigated. Firstly, rational geometric shapes and dimensions were designed, and a series of quasi-static and dynamic uniaxial tensile tests were conducted to obtain the nonlinear stress-strain curves at different strain rates. The results indicated a notable strain-rate effect in polyurea, with an increase in elastic modulus and tensile stress as the strain rate increased. Moreover, as the strain rate escalates, the strain hardening effect becomes more pronounced, thereby enhancing the protective capabilities of polyurea. Subsequently, an existing hyperelastic model, namely the Mooney-Rivlin (MR) model, was modified to incorporate the strain rate effect, resulting in a non-linear visco-hyperelastic constitutive model. Furthermore, the proposed constitutive model and parameters were validated by simulating existing tensile and blast tests using the finite element (FE) program LS-DYNA. This work could provide a reference for the design of structural protection.

Implementing a simple 2D constitutive model for rocks into finite element method

Our research group has developed a 2D constitutive model for rocks that incorporates a strain-dependent elastic modulus. This model efficiently represents nonlinear stress–strain curves using just three equations and four parameters. In this study, we have implemented the simple 2D model into finite element codes to tackle engineering problems. We have calculated and compared stresses and displacements around a pressurized thick-walled hollow cylinder, a lined non-circular tunnel, and a rock slope using the elastic, simple 2D, and elasto-plastic models. The results confirmed that the simple 2D model closely matches the stress distribution obtained from the elastic solution at low-stress levels. Additionally, the simple 2D model produces a plastic zone with a larger inward displacement than the elasto-plastic model at higher stress levels. Only the simple 2D model captured sidewall convergence, roof sag, and floor heave in non-circular tunnels, and the toppling failure for 90° slope and toe translational slide for 60° slope in rock slopes with smooth critical slip surface. We did not encounter any convergence difficulties while solving the simple 2D model.

The stress distribution around a thick-walled cylinder by a proposed constitutive model for rocks.

To simulate the nonlinear stress-strain curve of rocks under static loads and contribute to the design and construction of rock engineering structures, a constitutive model has been proposed based on the elastic modulus E decreasing with the increase in longitudinal cracks. This constitutive equation offers numerous advantages, with the most noteworthy being that the simulation of stress-strain curves for rocks necessitates only three equations (Eqs 1-3) and four parameters (A, k0, C and εs). Following this, we employ the constitutive equation to analyze the stress distribution around a thick-walled cylinder and explore the impact of its four parameters on the stress distribution surrounding the thick-walled cylinder. Parameter A primarily affects the range of the plastic zone and the magnitude of the maximum tangential stress; parameter C mainly influences the magnitude of the maximum tangential stress; parameter εs mainly affects the range of the plastic zone and the magnitude of the maximum tangential stress; parameter k0 primarily influences the magnitude of the maximum tangential stress. We got the similar results with Bray model, but distribution of stress around the tunnel are different present that the shape of stress-strain curves are different.

Hyper-elastic behavior of soft-tissue like microgels in two-phase converging microchannel flow

Deformation of biological cells, tissues, and similar soft materials is often considered linearly elastic; however, the assumption is only valid in a very limited stress range and often leads to significant errors in mechanical evaluation. We demonstrated the hyper-elastic behavior of ultra-soft poly(N-isopropyl acrylamide) (PNIPAm) microgels (USPNMs) in a converging channel flow, as a representation for biological tissues. The hyper-elasticity of USPNMs in response to a broad range of deformation was characterized at the centerline of the converging flow. We introduced a carrier fluid consisting of baby hydrogels (avg. diameter, 10 μm) and oil that carried the hydrophilic USPNM sample (avg. diameter, 100 μm) on the centerline of oil background fluid. By “baby hydrogel,” we mean small PNIPAm particles obtained during USPNM synthesis, using which, enabled settling-free flow, prevented wall contact, and enhanced carrier fluid viscosity for increased stresses at lower flowrates. Furthermore, drastic reduction of interfacial tension was observed in the converging area due to contact of baby gels with USPNM particles in the carrier fluid. The shear and elongational stresses were balanced with the elastic stress and interfacial Laplace pressure. As a result, we obtained a stress–strain curve from the microscopic images during flow. The non-linear stress–strain curve was characterized by conventional hyper-elastic models. The elastic modulus of the synthesized USPNM was 24 Pa, which is as low as animal brain tissue. This method holds great potential for implementing in similar hyper-elastic systems, enabling accurate mechanical evaluations in the field of soft materials, biology, and medicine.

Ytterbia – praseodymia co-stabilized TZP with high toughness and transformation limited strength

Tetragonal zirconia polycrystals (TZP) with high strength and toughness offer the potential to compensate for manufacturing or application related defects. In this study aiming at ultimate toughness a co-stabilized TZP was manufactured by coating of monoclinic zirconia by a wet chemical process with 1.5 mol% ytterbia and 1.5 mol% praseodymia. The powder was hot-pressed at 1300–1400 °C in 25 °C increments for 1 h by 60 MPa axial pressure. The materials exhibit an ultrafine microstructure with grain sizes ranging from 130 to 200 nm. Mechanical properties were measured in a 3-point bending setup, the material shows a combination of moderate strength (600–950 MPa) and high toughness (11–12 MPa√m). Non-linear stress strain curves and occurrence of transformation bands prior to fracture indicate transformation induced failure behaviour. The transformation stress determined from the location of transformation bands in fractured samples increases with rising sintering temperature and is about 100–250 MPa lower than the bending strength.

In situ damage propagation and fracture in notched cross-ply SiC/SiC composites: Experiment and numerical modeling

In this paper, the damage progression and fracture behavior in notched cross-ply SiC/SiC composites were investigated using the in situ digital image correlation (DIC) and acoustic emission (AE) techniques. Under tensile loading, the mechanical properties of composite’s elastic modulus, proportional limit stress (PLS), ultimate tensile strength (UTS), and fracture strain were obtained for non-notched, single-hole, double-holes, and four-holes samples. Based on the analysis of composite’s tangent modulus, the nonlinear tensile stress-strain curves were divided into three main domains, relating with internal multiple damage mechanisms observed using the in situ DIC and AE. Effects of the number and location of the circular holes on the full-filed composite’s damage and strain evolution, AE energy and ring-down count were investigated. After tensile fracture, the composite’s macro fracture morphology and micro damage mechanisms were observed under the scanning electronic microscopy (SEM). Numerical modeling of tensile damage and fracture process of different notched samples were conducted using the extend finite element method (XFEM). Relationships between the notch types, tensile mechanical properties, macro strain evolution, and micro damage mechanisms were established.

Experimental studies and modeling of strain rate- and temperature-dependent springback behavior of hot-deformed aluminum alloys

The unloading process of hot-deformed aluminum alloys exhibits obvious strain rate- and deformation temperature-dependent nonlinearity and the springback is generated by internal stress relaxation, which affects the forming accuracy of aluminum alloy components. The strain hardening and dynamic softening during deformation as well as static softening during unloading affect the springback of hot-deformed aluminum alloys. In this study, the homogeneous aluminum alloys samples were used to perform the single-pass isothermal hot compression and unloading tests with different deformation temperatures, loading strain rates and strains. The strain-, strain rate- and deformation temperature-dependent unloading and springback behavior of hot-deformed aluminum alloys were studied. The effects of strain hardening, dynamic and static softening on the springback behavior of alloys were revealed. The relationships between unloading stress, springback strain and instantaneous tangent modulus in unloading were analyzed. A polynomial-type equation was proposed according to the nonlinear unloading stress-strain curves. The microscopic mechanism of nonlinear unloading of hot-deformed aluminum alloys was also clarified based on the physical mechanism of internal stress relaxation and anelastic strain accumulation. A micro-mechanism-based model was developed to describe the springback and internal stress relaxation in unloading after hot deformation of aluminum alloys. The comparison of calculated unloading curves with experimental results indicates that the proposed polynomial-type equation and developed micro-mechanism-based model can describe accurately the nonlinear springback and stress decay in unloading of hot-deformed aluminum alloys, which provide a prerequisite for analyzing the final geometry and residual stress of precision hot-deformed alloy components.

Kink band orientation of 3D printed continuous carbon fiber composites under compressive loading

For continuous carbon fiber-reinforced plastic (cCFRP), in the case of loading in the longitudinal direction, occurrence of kink band failure causes compression strength to be lower than the tensile strength. Furthermore, 3D-printed cCFRP may exhibit intrinsic compressive behavior depending on the printing orientation. In this study, two types of rectangular specimens were fabricated using the Markforged Mark Two > 3D printer. The first was the in-plane specimen, whose width direction was parallel to the print bed and thickness direction was along the layup direction. The second was the out-of-plane specimen, whose width direction was the layup direction and thickness direction was parallel to the print bed. Compression tests were performed, and the experimental results were compared. As a result, the compressive strengths of both types were similar, and the kink bands always grew in the lay-up direction. For the out-of-plane specimens, nonlinear stress–strain curves were observed. This may be due to the delamination crack growth caused by the low strength in the lay-up direction and the existence of initial defects. This study experimentally investigated the effect of two different printing orientations of 3D printed cCFRP to its strength and failure mode, and this work shows the importance of considering structural deformation due to compression loading in 3D printed cCFRP.

Mechanical Properties of Highly Deformable Elastomeric Gyroids for Multifunctional Capacitors

Triply periodic minimal surface lattices have mechanical properties that derive from the unit cell geometry and the base material. Through computation software like nTopology and Abaqus, these geometries are used to tune nonlinear stress–strain curves not readily achievable with solid materials alone and to change the compliance by two orders of magnitude compared to the constituent material. In this study, four elastomeric TPMS gyroids undergo large deformation compression and tension testing to investigate the impact of the structure's geometry on the mechanical properties. Among all the samples, the modulus at strain ε varies by over one order of magnitude (7.7–293.4 kPa from FEA under compression). These lattices are promising candidates for designing multifunctional systems that can perform multiple tasks simultaneously by leveraging the geometry's large surface area to volume ratio. For example, the architectural functionality of the lattice to bear loads and store mechanical energy along with the larger surface area for energy storage is combined. A compliant double‐gyroid capacitor that can simultaneously achieve three functions is demonstrated: load bearing, energy storage, and sensing.

Integration and calibration of UBCSAND model for drained monotonic and cyclic triaxial compression of aggregates

We report on results from triaxial drained monotonic and cyclic compression tests for four different aggregates (gabbro, limestone, demolished concrete, steel slag), and discuss the applicability of UBCSAND model for simulating the relevant non-linear stress-strain curves. First, UBCSAND is integrated analytically for the elastic and plastic volumetric and deviatoric strains, leading to a pair of novel closed-form solutions expressed in terms of hypergeometric functions. These solutions are inherently free of numerical errors and can be used for the evaluation of results from numerical integration. Second, three calibration techniques for the model parameters are presented and compared: (1) a graphical “chart” solution based on results from forward numerical simulations; (2) a numerical approach based on minimization of residuals; and (3) an informed trial-and-error approach based on the analytical solution. Errors in numerical integration of the model are explored by comparing the analytical solution against results from an Euler integration. Variations in the calibrated parameters are discussed for drained monotonic and cyclic triaxial loading considering different types of aggregates.

Modeling the nonlinear responses of soft network materials assisted by masked-fusion artificial neural network

Soft network materials (SNMs) have attracted intensive attentions due to the merits of lightweight, high stretchability, and highly tunable biomechanical properties and responses. In this context, the SNMs have been widely used in the fields of flexible electronics, health monitoring, and tissue engineering etc. in the last decade. To assist the modeling of nonlinear J-shaped stress-strain curve of SNMs with complex curved filaments, a phenomenological framework assisted by machine learning (ML) approach was developed in the last year. However, the overfitting problem affected the computational efficiency and numerical stability. Herein, a novel masked-fusion artificial neural network (ANN) has proposed to replace the conventional feed-forward ANN in ML approach in this research. Compared to the conventional strategy, the masked-fusion ANN has fewer hidden layer which leads to smaller test loss and less computational burden. The risk of overfitting can also be avoided. Besides, the predicted J-shaped stress-strain curves of randomly generated SNMs from masked-fusion ANN are closer to those obtained from finite element (FE) analyses compared to the conventional ANN.

Designing Viscoelastic Gelatin-PEG Macroporous Hybrid Hydrogel with Anisotropic Morphology and Mechanical Properties for Tissue Engineering Application

The mechanical properties of scaffolds play a vital role in regulating key cellular processes in tissue development and regeneration in the field of tissue engineering. Recently, scaffolding material design strategies leverage viscoelasticity to guide stem cells toward specific tissue regeneration. Herein, we designed and developed a viscoelastic Gel-PEG hybrid hydrogel with anisotropic morphology and mechanical properties using a gelatin and functionalized PEG (as a crosslinker) under a benign condition for tissue engineering application. The chemical crosslinking/grafting reaction was mainly involved between epoxide groups of PEG and available functional groups of gelatin. FTIR spectra revealed the hybrid nature of Gel-PEG hydrogel. The hybrid hydrogel showed good swelling behavior (water content > 600%), high porosity and pore interconnectivity suitable for tissue engineering application. Simple unidirectional freezing followed by a freeze-drying technique allowed the creation of structurally stable 3D anisotropic macroporous architecture that showed tissue-like elasticity and was capable of withstanding high deformation (50% strain) without being damaged. The tensile and compressive modulus of Gel-PEG hybrid hydrogel were found to be 0.863 MPa and 0.330 MPa, respectively, which are within the range of normal human articular cartilage. In-depth mechanical characterizations showed that the Gel-PEG hybrid hydrogel possessed natural-tissue-like mechanics such as non-linear and J-shaped stress-strain curves, stress softening effect, high fatigue resistance and stress relaxation response. A month-long hydrolytic degradation test revealed that the hydrogel gradually degraded in a homogeneous manner over time but maintained its structural stability and anisotropic mechanics. Overall, all these interesting features provide a potential opportunity for Gel-PEG hybrid hydrogel as a scaffold in a wide range of tissue engineering applications.

Representation of brittle rock plasto-elastic behaviour based on tangible cracks in discontinuous deformation analysis

Micro-cracks, which are prevalent in crustal rock masses, can significantly affect the physical and mechanical properties of rocks. The closure of micro-cracks explains the initial non-linear stress–strain curves observed during compression in brittle rocks. In this study, the plasto-elastic deformation was implemented through crack closure in the discontinuous deformation analysis (DDA). Tangible micro-cracks were formed by updating the vertex positions as opposed to setting the virtual gap and width. These micro-cracks were arranged randomly, but still adhered to specific crack parameters (crack intensity, aspect ratio, and dip-angle distribution). The uniaxial compression simulation results verified that these micro-cracks could open and close under external stress and execute the crack-closure stage. The results indicated that the increase in crack intensity primarily decreased the ratio of the initial to elastic moduli, and the crack aspect ratio was approximately equal to the crack-closure strain. An inhomogeneous dip-angle distribution led to anisotropic plasto-elastic deformation owing to the different numbers of cracks perpendicular to the axial stress. According to previous theories and reasonable assumptions, a function between the crack parameters and crack-closure stress was first established. Finally, the experimental results for two specific rocks were reproduced using this method, which can be applied to research on rock properties or rock mass engineering influenced by crack closures.

A constitutive model of liquid crystal elastomers with loading-history dependence

Liquid crystal elastomers (LCEs) are a class of active polymers of increasing interest, because of their excellent actuation performances. To quantify the actuation performances and understand the thermomechanical behavior, many theoretical models were developed, mainly based on the free energy framework with uniaxial order tensor, which, however, cannot well capture the biaxial liquid crystal alignment under biaxial stretching or shearing. The LCE with the shape memory cycle effect exhibits an evident loading-history dependence of the thermomechanical process, which cannot be described without introducing the evolution condition of the alignment. In this work, a constitutive model of LCE is established by introducing a triaxial order tensor to describe the complex alignment under uniaxial/biaxial loading conditions. Inspired by the classical theory of plasticity, an equivalent order parameter is proposed to characterize the order degree of the alignment, and its evolution condition is introduced to specify the condition of the alignment evolution. Experiments of LCE under a variety of thermomechanical loading conditions were carried out to validate the developed model, showing the capability of the model in capturing complex thermomechanical responses of the polydomain LCE with shape memory cycle effect under uniaxial/biaxial stretching. Furthermore, the constitutive model can be extended to predict nonlinear stress-strain curves and actuation performances of LCE lattices, which hold promising potential for applications in soft actuators and flexible robots.

Effect of assembly pressure on the performance of proton exchange membrane fuel cell

The assembly force is a crucial factor in the process of proton exchange membrane fuel cell (PEMFC) stacking, and has significant effects on the fluid flow, mass transfer, and water and thermal management, which affect the fuel cell performance. In this study, from the most deformable component, the gas diffusion layer (GDL), combining with a finite-element analysis, and computational fluid dynamic method, the impact of the assembly force on a single-channel PEMFC is analyzed. A nonlinear stress–strain curve obtained from a microanalysis is creatively introduced into the two-dimensional compression model. The gas diffusion coefficient in the three-dimensional model is also obtained from the microscopic simulation. The simulated effective oxygen diffusion coefficient of the compressed GDL is approximately 0.86 times the Bruggemann estimated value. When the contact resistance is ignored, the output voltage at 2.5 MPa is decreased by approximately 15.4% at 1.7 A·cm−2 compared with that at 0.5 MPa. After the contact resistance is considered, the effects of the assembly pressure on the cell performance (V–I curve) are qualitatively different. The pressure drop of the 2.5 MPa case is 20% higher than that of the 1.4 MPa case at 1.7 A cm−2. O2 is hard to flow into the region under the rib where the porosity and permeability are lower. The results indicate that both liquid water and membrane water contents increase when the assembly force increases. The effect of the assembly force on the temperature is also analyzed.

Cervical Spine Finite Element Models for Healthy Subjects: Development and Validation

Abstract Finite element modeling is a popular method for predicting kinematics and kinetics in spine biomechanics. With the advancement of powerful computational equipment, more detailed finite element models have been developed for the various spine segments. In this study, five detailed finite element models of the cervical spine are developed and validated. The geometric boundaries of the vertebrae are determined from computed tomography (CT) scans of five female subjects. The models include the C2–C7 vertebrae, intervertebral discs, nuclei, endplates, and five major ligaments (anterior longitudinal ligament (ALL), posterior longitudinal ligament (PLL), ligamentum flavum (LF), interspinous ligament (ISL), and capsular ligament (CL)). The ligaments follow nonlinear stress–strain curves whereas all other parts adopt linear material properties. All the material properties are taken from existing literature. The mesh convergence test is performed under flexion/extension. For flexion/extension motion, a pure moment is applied at the top surface of the odontoid process of the C2 vertebra while nodes at the bottom surface of the C7 vertebra are fixed in all directions. The models are extensively validated in flexion/extension, lateral bending, and axial rotation against experimental and finite element studies in the literature. Intervertebral rotation and range of motion are studied under different loading conditions found in the literature. This research also investigates intersubject variability for the cervical spine among five finite element models from five different subjects. Predicted angular displacements and ranges of motion of the current models are consistent with the literature. The validated models are expected to be applicable to simulate neck-related trauma like whiplash and high-g acceleration, among other scenarios.

Thermoelectric transport in holographic quantum matter under shear strain

We study thermoelectric transport under shear strain in two spatial dimensional quantum matter using the holographic duality. General analytic formulae for the DC thermoelectric conductivities subjected to finite shear strain are obtained in terms of black hole horizon data. Off-diagonal terms in the conductivity matrix also appear at zero magnetic field, resembling an emergent electronic nematicity, which cannot nevertheless be identified with the presence of an anomalous Hall effect. For an explicit model study, we numerically construct a family of strained black holes and obtain the corresponding nonlinear stress–strain curves. We then compute all electric, thermoelectric, and thermal conductivities and discuss the effects of strain. While the shear elastic deformation does not affect the temperature dependence of thermoelectric and thermal conductivities quantitatively, it can strongly change the behavior of the electric conductivity. For both shear hardening and softening cases, we find a clear metal-insulator transition driven by the shear deformation. Moreover, the violation of the previously conjectured thermal conductivity bound is observed for large shear deformation.

A Unified Nonlinear Elastic Model for Rock Material

Under conditions of low or medium stress, rocks that are in the compression state exhibit perceivable nonlinear elastic characteristics. After a comprehensive review of the existing nonlinear elastic models of rocks and joints, we proposed a new unified nonlinear elastic model. This new model contains two parameters with clear definitions, namely, the initial elastic modulus Eni and the modulus change rate m. This model covers a variety of existing models, including the simple exponential model, BB model and two-part Hooke’s model, etc. Based on the RMT experimental system, a large number of uniaxial compression tests for dolomite, granite, limestone and sandstone were performed, and their nonlinear deformation stress‒strain curves were obtained and fit with the unified nonlinear elastic model. The results show that the rocks have obvious nonlinear elastic characteristics in their initial compression stage, and that the unified nonlinear elastic model is able to describe these phenomena rather well. In addition, an empirical formula for predicting the uniaxial compressive strength of the rock was constructed, corresponding to the unified nonlinear elastic model.