Nonlinear Stress Research Articles

Reference Power Cable Models for Floating Offshore Wind Applications

The present study aims to address the knowledge gaps in dynamic power cable designs suitable for large floating wind turbines and to develop three baseline power cable designs. The study includes a detailed database of structural and mechanical properties for three reference cable models rated at 33 kV, 66 kV, and 132 kV to be readily used in global dynamic response simulations. Structural properties are obtained from finite element method (FEM) models of respective cable cross-sections built in UFLEX v2.8.9—a non-linear stress analysis program. Extensive mesh sensitivity studies are performed to ensure the accuracy of the predicted structural properties. The cable’s structural design is investigated using global response simulations of an OC3 5MW reference wind turbine coupled with the dynamic power cable in a lazy wave configuration. The feasibility of the present reference cable in floating offshore wind applications is assessed through a simplified analysis of cable fatigue life and structural integrity analysis of the cable in extreme environmental conditions. The analysis results suggest that the dynamic power cable does not significantly affect the response characteristics of the floating wind turbine in the analyzed lazy wave configuration. Furthermore, a simplified fatigue analysis demonstrates that the proposed cable design can sustain representative environmental loading scenarios and shows favorable dynamic performance in a lazy wave configuration.

Numerically assisted calibration procedure of nonlinear in-plane shear properties of unidirectional composite laminae based on off-axis tensile experiments

In this paper, a novel methodology is presented to evaluate the true nonlinear shear response of continuous fiber-reinforced plastic (CFRP) unidirectional laminae. It requires simple off-axis tensile experiments to be conducted and a finite element (FE) representation of them with a material constitutive law capable of handling nonlinearity in shear. Upon successful evaluation of the normal stiffness properties, the shear stress-strain response is derived via numerical calibration of the underlying FE models. The presented approach is also demonstrated by processing the raw data of an extensive material characterization test campaign conducted on a thermoplastic matrix CFRP. The outcome is compared to the conventional methods for shear response derivation using relevant standards such as ASTM D3518 and ASTM D5379. It has been successfully demonstrated that the pseudo-hardening phenomenon obtained from standard shear experiments as a result of fibers aligning with load direction can be eliminated with off-axis specimen tension experiments, thus, the true shear stress versus deformation response can be extracted up to failure. The main purpose of current work is to demonstrate the inconsistency in the available standard methods related to mechanical testing-based derivation of nonlinear in-plane shear behavior of UD plies. In addition, a novel technique is presented to achieve a more accurate prediction of nonlinear shear stress and strain along the entire representative loading range that contributes to more accurate simulations of composite parts up to failure and thus, better strength predictions.

Nonlinear stress growth and failure characteristics of gangue-cemented backfill

The stability of a filling body is the main control index of the filling effect, which is significantly affected by the strength and damage characteristics, and its stress increases nonlinearly with damage evolution. In this study, the uniaxial compression test and digital image correlation (DIC) method were used to investigate the stress and damage characteristics of a gangue-cemented backfill, and the stress–strain characteristics and crack propagation laws of the backfill under different curing ages were analyzed. A modified damage model of the gangue-cemented backfill based on the Weibull distribution characteristics of the microelement damage of the backfill and pore closure characteristics in the compression stage was constructed. Through numerical simulations, the nonlinear growth characteristics of stress in the specimen were revealed from the perspective of energy dissipation. The results showed that the nonlinear increase in stress originated mainly from pore closure and crack extension. The crack propagation in the backfill was divided into intermediate crack budding and symmetric crack propagation at the corners. The key signs of the peak stress and fracture of the specimen are the connection of the surface cracks and fracture surfaces on both sides. The proposed modified damage model was proven to be in relatively close agreement with the experimental results. The stress growth in the backfill exhibited a nonlinear-linear-nonlinear growth pattern before the peak stress. This study provides a theoretical basis for studying the strength characteristics and damage evolution of backfill, which is important for optimizing the material proportions and analyzing the stability of the quarry and other aspects.

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.

A Generalized Slab Model

Abstract We construct a generalized slab model to calculate the ocean’s linear response to an arbitrary, depth-variable forcing stress profile. To introduce a first-order improvement to the linear stress profile of the traditional slab model, a nonlinear stress profile, which allows momentum to penetrate into the transition layer (TL), is used [denoted mixed layer/transition layer (MLTL) stress profile]. The MLTL stress profile induces a twofold reduction in power input to inertial motions relative to the traditional slab approximation. The primary reduction arises as the TL allows momentum to be deposited over a greater depth range, reducing surface currents. The secondary reduction results from the production of turbulent kinetic energy (TKE) beneath the mixed layer (ML) related to interactions between shear stress and velocity shear. Direct comparison between observations in the Iceland Basin, the traditional slab model, the generalized slab model with the MLTL stress profile, and the Price–Weller–Pinkel (PWP) model suggest that the generalized slab model offers improved performance over a traditional slab model. In the Iceland Basin, modeled TKE production in the TL is consistent with observations of turbulent dissipation. Extension to global results via analysis of Argo profiling float data suggests that on the global, annual mean, ∼30% of the total power input to near-inertial motions is allocated to TKE production. We apply this result to the latest global, annual-mean estimates for near-inertial power input (0.27 TW) to estimate that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production.

Static and free vibration analysis of delaminated composite plates using secant function based shear deformation theory

In this work, the static and free vibration analysis of delaminated composite plates are studied. The secant function based shear deformation theory (SFSDT), is used for the analysis of multiple delaminated models. This theory exhibits a non-linear shear stress distribution and fulfills the traction-free boundary conditions at the top and bottom surfaces of the plates, hence there is no need for a shear correction factor. The finite element method, a numerical solution-based approach is used here considering an eight-noded isoparametric quadratic element. The investigation explores the effects of delamination size, position and stacking sequence on the flexural and dynamic characteristics of the composite plates. Additionally, other geometric factors such as delamination thickness and shape are investigated to understand their impact. The study aims to identify the optimal configurations that minimize the detrimental effects of delaminations. The results provide valuable insights into the static and free vibration behaviour of delaminated composite plates, contributing to design guidelines and optimization strategies for structurally sound and efficient composite structures. Overall, this research offers a comprehensive investigation of the static and free vibration analysis of delaminated composite plates, advancing knowledge in the field and facilitating improved design practices.

The displacement mechanism of the cracked rock – a seismic design and prediction study using XFEM and ANNs

Materials with sufficient strength and stiffness can transfer nonlinear design loads without damage. The present study compares crack propagation speed and shape in rock-like material and sandstone when subjected to seismic acceleration. The nonlinear extended finite element method (NXFEM) has been used in numerical simulation. It assumes the model has a pre-existing crack at 0° from the horizontal. The mechanical properties of the model, crack propagation shape, and crack speed were selected as the main parameters. The nonlinear stress and strain along the crack have been compared in two simulated models. NXFEM and Artificial Neural Networks (ANNs) were used to predict the displacement. The simulation results illustrate that the materials’ crack propagation mechanism and mechanical properties control the stress, strain, and displacement at the selected points in the model. In addition, crack propagation in materials is related to elastic-plastic stresses and strains along the crack path. The speed and shape of the crack are associated with the mechanical properties of the materials. The prediction of crack paths helps to understand failure patterns. Comparison of the seismic response of the rock-like material with sandstone helps to assess the stress, strain, and displacement levels during cracking. This study’s findings agree with the literature report and field observations.

Load Assessment Method for Multi-Layer Oceanographic Winch with Synthetic Fibre Ropes Based on Non-Rotation Symmetric Cylindrical Model

Offshore winches are crucial in marine engineering, particularly in marine scientific research and deep-sea exploration. The use of fibre ropes presents significant opportunities for the weight reduction of winches as a consequence of the low length–strength ratio and characteristics of corrosion resistance. Nonetheless, a challenge arises in underestimating the stress load levels in load assessments of multi-layer winch systems using synthetic fibre ropes. Traditional computational methods reliant on symmetrically rotational models fall short in accurately predicting and assessing practical applications. This paper introduces a finite element analysis model based on a non-rotationally symmetric approach with four surfaces subjected to various radial pressure on account of the deformation of the fibre ropes. In the design model, sixteen stress detection paths have been incorporated to identify and confirm non-linear stresses. The outcomes of the finite element simulations have been compared with experimental results with two synthetic fibre ropes, each with distinct deformation characteristics utilised. The findings demonstrate that the application of the model aligns well with experimental results, showcasing its relevance and practical value in real-world scenarios. Precise theoretical calculations and experimental validation are pivotal to ensuring that equipment reliability and safety are maintained alongside the pursuit of light-weighting.

Analysis of heat and mass transfer of a non-linear convective heat generating fluid flow in a porous medium with variable viscosity

The present study addresses the significance of analyzing the heat and mass transfer characteristics of a non-linear convective fluid flow featuring variable viscosity within a porous medium. Our focus lies on understanding the impact of varying viscosity on the hydromagnetic coupling stress, particularly in the presence of internal heat generation. The fluid is confined within a porous medium, and convection cooling takes place at both wall of the fluid. A key assumption in our investigation is the nonlinearity nature of the buoyancy force, specifically when there is a substantial difference in temperature and concentration within the fluid and its surroundings, the buoyancy force becomes quadratic. The exploration of couple stress fluids, alongside considerations of temperature-dependent internal heat generation and mass transfer, contributes to advancing our understanding of thermal system performance. By accounting for nonlinearity in density, temperature, and concentration, the research aims to provide more accurate results than previous linear models, crucial for optimizing energy consumption in practical applications such as nuclear reactors and heat exchangers. Dealing with this type of problem often pose significant challenges, as they often lack analytical solutions, necessitating the use of more advanced numerical methods or approximation techniques. We employ Spectral Quasilinearisation Method (SQLM) to solve the coupled non-linear boundary value problems (PDE) arising from the model, momentum equation, energy equation, concentration equation, and entropy generation. The obtained results are then validated using the Runge–Kutta method and a good agreement is achieved. The findings for the dimensionless velocity, temperature, mass concentration and entropy generation are presented in graphics and in table manner, and effects of controlling parameters are extensively explained. The results highlight that an increase in variable viscosity leads to an enhancement in the velocity profile, while simultaneously reducing the temperature in the system. Furthermore, increments in the non-linear convection parameter and coupling stress parameter result in an appreciation of the velocity profile, temperature profile, and entropy generation rate.

Probabilistic relative entropy in elasto‐plasticity using the iterative generalized stochastic stress‐based Finite Element Method

AbstractThe generalized iterative stochastic perturbation approach to the stress‐based Finite Element Method has been proposed in this work. This approach is completed using the complementary energy principle, Taylor expansion of the general order applicable to all random functions and parameters as well as nodal polynomial response bases determined with the use of the Least Squares Method. The main aim of this elaboration is the usage of such a probabilistic approach to determine Bhattacharyya relative entropy for some nonlinear engineering stress analysis with uncertainty. Mathematical apparatus with its numerical implementation has been used to study elastoplastic torsion of some prismatic bar with Gaussian material uncertainty and the corresponding reliability measures. This problem has been solved using the Constant Stress Triangular (CST) plane finite elements, the modified Newton–Raphson algorithm, whereas the first four probabilistic characteristics resulting from the iterative generalized stochastic perturbation method have been contrasted with these obtained with the crude Monte‐Carlo sampling and the semi‐analytical probabilistic approach.

A mathematical model for two solutes transport in a poroelastic material and its applications

Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of five nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multiparameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.

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.

Quantification of long-term nonlinear stress relaxation of bovine trabecular bone

The reliability of computational models in orthopedic biomechanics depends often on the accuracy of the bone material properties. It is widely recognized that the mechanical response of trabecular bone is time-dependent, yet it is often ignored for the sake of simplicity. Previous investigations into the viscoelastic properties of trabecular bone have not explored the relationship between nonlinear stress relaxation and bone mineral density. The inclusion of this behavior could enhance the accuracy of simulations of orthopedic interventions, such as of primary fixation of implants. Although methods to quantify the viscoelastic behavior are known, the time period during which the viscoelastic properties should be investigated to obtain reliable predictions is currently unclear. Therefore, this study aimed to: 1) Investigate the duration of stress relaxation in bovine trabecular bone; 2) construct a material model that describes the nonlinear viscoelastic behavior of uniaxial stress relaxation experiments on trabecular bone; and 3) implement bone density into this model. Uniaxial compressive stress relaxation experiments were performed with cylindrical bovine femoral trabecular bone samples (n = 16) with constant strain held for 24 h. Additionally, multiple stress relaxation experiments with four ascending strain levels with a holding time of 30 min, based on the results of the 24-h experiment, were executed on 18 bovine bone cores. The bone specimens used in this study had a mean diameter of 12.80 mm and a mean height of 28.70 mm. A Schapery and a Superposition model were used to capture the nonlinear stress relaxation behavior in terms of applied strain level and bone mineral density. While most stress relaxation happened in the first 10 min (up to 53 %) after initial compression, the stress relaxation continued even after 24 h. Up to 69 % of stress relaxation was observed at 24 h. Extrapolating the results of 30 min of experimental data to 24 h provided a good fit for accuracy with much improved experimental efficiency. The Schapery and Superposition model were both capable of fitting the repeated stress relaxation in a sample-by-sample approach. However, since bone mineral density did not influence the time-dependent behavior, only the Superposition model could be used for a group-based model fit. Although the sample-by-sample approach was more accurate for an individual specimen, the group based approach is considered a useful model for general application.

The shape function method of nonlinear thermal stress of granite fracture tips in a high-temperature environment

Exposed rock masses in tunnel portals are susceptible to thermal deterioration in southern China, where temperatures are relatively high. The thermal stress field of rock masses is affected by fracture shape and distribution as fractures near the surface are channels for solar radiation energy to be converted into rock thermal energy. In this study, a function expression is developed for triangular heat sources of fractured rock masses in a tunnel portal in a high-temperature environment. By the function expression, the temperature field and thermal stress field are calculated, and the influence of fracture shape parameters and multi-fracture interaction is analyzed. The results are as follows: (1) the temperature field and thermal stress field of exposed rocks are redistributed by fractures. The internal temperature of the fractured rocks is higher than that of non-fractured rocks, and thermal stress near the fracture tip increases. (2) For triangular fractures of the same length, thermal stress increases as the apex angle increases. (3) When the spacing between parallel fractures or coplanar fractures is close, the superposition effect of thermal stress becomes significant. (4) In a high-temperature environment, temperature field and thermal stress field of a fractured rock are both nonlinear as temperature and thermal stress around fractures increase significantly. The results provide effective reference for stability evaluation of fractured rock masses in tunnel portals and offer theoretical foundation for thermal diseases analysis and protection measures of tunnel engineering in high-temperature environments of southern China.

Transient and Steady‐State Friction in Non‐Isobaric Conditions

AbstractThe frictional properties of faults control the initiation and propagation of earthquakes and the associated hazards. Although the ambient temperature and instantaneous slip velocity controls on friction in isobaric conditions are increasingly well understood, the role of normal stress on steady‐state and transient frictional behaviors remains elusive. The friction coefficient of rocks exhibits a strong dependence on normal stress at typical crustal depths. Furthermore, rapid changes in normal stress cause a direct effect on friction followed by an evolutionary response. Here, we derive a constitutive friction law that consistently explains the yield strength of rocks from atmospheric pressure to gigapascals while capturing the transient behavior following perturbations in normal stress. The model explains the frictional strength of a variety of sedimentary, metamorphic, and igneous rocks and the slip‐dependent response upon normal stress steps of Westerly granite bare contact and synthetic gouges made of quartz and a mixture of quartz and smectite. The nonlinear normal stress dependence of the frictional resistance may originate from the distribution of asperities that control the real area of contact. The direct and transient effects may be important for induced seismicity by hydraulic fracturing or for naturally occurring normal stress perturbations within fault zones in the brittle crust.

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.

A new method to characterize the nonlinear magneto-viscoelasticity behavior of magneto-active elastomers under large amplitude oscillatory axial (LAOA) loading

The nonlinear viscoelasticity of magneto-active elastomers (MAEs) under large amplitude oscillatory shear (LAOS) loading has been extensively characterized. A reliable and effective methodology, however, is lacking for such characterizations under large amplitude oscillatory axial (LAOA) loading. This is partly due to complexities associated with experimental compression mode characterizations of MAEs and in-part due to their asymmetric stress–strain behavior leading to different elastic moduli during extension and compression. This study proposes a set of new nonlinear measures to characterize nonlinear and asymmetric behavior of MAEs subject to LAOA loading. These include differential large/zero strain moduli and large/zero strain-rate viscosity, which could also facilitate physical interpretations of the inter- and intra-cycle nonlinearities observed in asymmetric and hysteretic stress–strain responses. The compression mode stress–strain behavior of MAEs was experimentally characterized under different magnitudes of axial strain (0.025 to 0.20), strain rate (frequency up to 30 Hz) and magnetic flux density (0 to 750mT). The measured stress–strain responses were decomposed into elastic, viscous and viscoelastic stress components using Chebyshev polynomials and Fourier series. The stress decomposition based on Chebyshev polynomials permitted determination of equivalent nonlinear elastic and viscous stress components, upon which the proposed measures were obtained. An equivalent set of Fourier coefficients was also obtained for estimating equivalent elastic/viscous stress, thereby facilitating faster calculation of the proposed material measures. The proposed methodology is considered to serve as an effective tool for deriving constitutive models for describing nonlinear and asymmetric characteristics of MAEs.

On the dynamic performance of additively manufactured visco-elastic meta-materials

Additive manufacturing (AM) has revolutionized the production of structures with tailored material properties, including elastomer polyurethanes (EPU) which exhibit exceptional mechanical performance. EPU possesses unique characteristics, such as high elongation at break, efficient energy dissipation, and superior specific strength, making it well-suited for applications requiring resilience to dynamic loadings. By combining the advantages of AM and EPU, enhanced and customized meta-materials can be created, surpassing the mechanical performance of traditional bulk materials. However, because of the non-linear stress–strain response of both the constitutive material and the structure, designing such meta-materials for high strain-rates can be challenging. In this work, therefore, quasi-static and dynamic experiments were conducted to evaluate a meta-material architecture. The investigation revealed a strong positive rate dependency. The mechanical performances, including strength, and dissipated energy, increased with increasing loading rate. The EPU meta-materials demonstrate their suitability for dynamic applications where high energy dissipation is crucial for reducing transmitted loads.

Theoretical solution for bond-slip behavior of composite structures consisting of H-shape beam and concrete based on experiment, numerical simulation, and theoretical derivation

This paper presents a theoretical solution for bond-slip behavior of SRC interface based on experiment, numerical simulation, and theoretical derivation. Push-out tests of five specimens were firstly carried out, based on which, the simplified bond-slip model was proposed. The specimen was fabricated using the Q345H-shape steel (400 mm × 200 mm×13 mm × 8 mm) and C50 concrete (350 mm × 600 mm) with a bonding length of 500 mm. A FE model was established based on the bond-slip model to analyze the nonlinear bonding stress that are difficult to obtain from the experiment. The research shows that although the bonding stress in the elastic stage is not uniformly distributed, it is basically uniformly distributed when the ultimate bearing capacity is reached. It proves that assuming the average interfacial bonding stress under ultimate load to replace the maximum bonding stress is reasonable, which is crucial for designer to estimate interface bearing capacity and maximum bonding stress. Moreover, five different stages were obtained by the FE model analysis and the theoretical equations of bonding stress nonlinear distribution were obtained based on the boundary condition of each stage. The theoretical solution clearly showed the relationship between interface nonlinear stress distribution and influencing parameters including the bond area, elasticity modulus of material, section area of specimen, bond-slip constitutive model, and the external load. Based on the theoretical model, the influences of different parameters can be directly obtained without conducting further experiments and establishing different FE models.

Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems

In order to explore the problems of load transfer and anchorage mechanisms of tensile anchors under pull-out load for geotechnical anchoring systems, a step-wise mathematical model is established which considers the linear–nonlinear shear stress and shear displacement of the anchorage segment, using an elasto-plastic constitutive model. The displacement, axial force, and shear stress of the anchorage interface in different stages (elastic, plastic, and debonding) are analyzed and solutions are derived. And the theoretical solutions for the ultimate pull-out load of the anchor at each stage are also presented. Two in situ pull-out tests are used to verify and apply these findings in engineering. The results show that the stepwise composite model could reflect the bonding, softening and residual characteristics of the anchoring interface. In the process of the pull-out load increasing, the pulling end of the anchor initially enters the plastic stage and the debonding stage, respectively, and the failure of the anchor occurs at the pulling end, and as the axial force transfers down deeper, the damage gradually spreads deeper. The axial force distribution of the anchorage section is a monotonically decreasing curve, and the peak point of the shear stress gradually moves deeper. The calculation results of the axial force distribution curve and load–displacement curve of the anchor are in good agreement with the measured values, which verifies the rationality and reliability of the theoretical prediction method. This method can provide a theoretical reference for the load transfer analysis and design of tension anchors for geotechnical anchoring systems.