Nonlinear Load-displacement Relationships Research Articles

Fatigue characteristics of rubber isolation layers used in high speed railway ballastless tracks: A theoretical and experimental study

In order to enhance operational conditions for high-speed trains on long-span bridges, the utilization of rubber under-ballast mats were taken as a novel isolation layer in ballastless tracks. Due to the increased train axle load, it is imperative to conduct investigation into the fatigue properties of the rubber isolation layer, so as to ensure the smoothness of the railways. In this work, a series of fatigue tests were performed on the rubber isolation layer to obtain the influence of stress amplitude and loading cycles on its fatigue performance. The results demonstrate a pronounced non-linear load–displacement relationship of the rubber isolation layer with increasing load, while the fatigue endurance significantly decreases with escalating the stress amplitude. Moreover, considering the hyperelastic properties of rubber material, the Mooney-Rivlin constitutive model was employed in finite element simulations. Subsequently, the stress distributions of the rubber isolation layer were analyzed, and the fatigue life was evaluated based on Luo stress method. The rubber isolation layer exhibits localized damage that does not affect the overall performance after 32 years of normal service condition. Therefore, the fatigue performance of the rubber isolation layer remains reliability, providing assurance for the application in ballastless tracks on high-speed railway long-span bridges.

Bending deformation and self-restoration of submicron-sized graphite cantilevers

Van der Waals (vdW)-stacked materials exhibit unique mechanical properties owing to their strong in-plane atomic bonds and weak interlayer vdW interactions. In this study, we experimentally demonstrate that submicron-sized graphite cantilevers can accommodate large out-of-plane deformations without fracturing and undergo self-restoration upon unloading. Submicron-sized cantilever beam specimens were fabricated from highly oriented pyrolytic graphite using the focused ion beam method and subjected to in situ transmission electron microscopy-based bending tests. The cantilevers exhibited a nonlinear load–displacement relationship, and their deformation was almost completely restored upon unloading, even though a large hysteresis loop was observed during the loading–unloading process. Moreover, similar loading–unloading curves were obtained during the repeated loading of the same specimen, indicating that the mechanical properties were restored by unloading even in the case of large nonlinear deformations. For larger deformations, plastic deformation occurred, and the self-restoration property was lost. However, no clear fracture was observed in the cantilevers even when they were deformed to a deflection angle of 62°. Even in this case, the deformation could be restored by electron-beam irradiation. Based on the obtained results, we propose a mechanism to explain the nonlinear and fully reversible deformation behavior of the specimens.

A modified strain gradient beam constraint model

The objective of the present paper is to develop a modified strain gradient beam constraint model (MSGBCM) to improve the modeling accuracy of small‐scale compliant mechanisms. First, a simple nano/micro flexure beam under the effect of end loads is considered. The virtual work principle is employed to formulate the nonlinear load–displacement behavior of the system, based on the modified strain gradient theory. It is observed that as the size of the structure becomes smaller, the elements of the elastic stiffness and load stiffening matrices severely deviate from their corresponding values in the beam constraint model (BCM). A closed‐form nonlinear strain energy expression is also derived for nano/micro flexure beams in terms of their tip displacements. This energy expression is utilized to model the nonlinear load–displacement relationships of small‐scale parallelogram (P) flexures. Moreover, analytical formulas are derived for the axial, transverse, and rotational stiffness values of P‐flexures. The most important observation is that in the small‐scale P‐flexures, the axial stiffness loss of small‐scale P‐flexures resulted from the movement of the stage in the transverse direction, may be seriously overestimated by BCM. The developed MSGBCM can be easily extended for investigating the static and dynamic behavior of more complicated micron and submicron size flexures.

A Calculation Method of Bearing Capacity of Single Squeezed Branch Pile Based on Load Transfer Method

A calculation method of the bearing capacity of single squeezed branch pile is established based on the load transfer method. In the method, the hyperbolic model is used to describe the nonlinear load-displacement relationship of pile-soil interaction at the pile tip, the pile skin, and the squeezed branch, and the theoretical expression of six load transfer coefficients of squeezed branch pile is given. The correctness of the method is proved by a small-scale model test in homogeneous soil and a large-scale model test in stratified soil. The results show that the calculation based on the load transfer method is applicable to predict the ultimate load in engineering application.

Yield interaction model of jack-up considering nonlinear load-displacement relationship before yield

Offshore structures with jack-up systems can be operated at depths up to 150 m and are used not only as drilling rigs and production rigs but also as support and accommodation units. Jack-up operation is carried out under environmental loads such as wind and wave, for which it is essential to understand jack-up behavior and structural response. As the boundary condition, the foundation model of offshore structures affects the vibration mode of a structure and, consequently, the behavioral and structural analysis results as well. Typical simple foundation models such as pinned and linear spring do not reflect soil-structure interaction (SSI) in jack-up analysis. As an alternative, the International Organization for Standardization (ISO) guidelines have suggested this SSI model as a simple secant model, a yield interaction model, and a time-consuming but accurate soil continuum model. In the present study, a structural analysis of jack-up is performed using the yield interaction model and the soil continuum model. The yield interaction model, Model B for clay, derived for consideration of the nonlinear behavior of soil, has been studied and continuously improved until recently. The existing model has generally assumed a linear load-displacement relationship in the elastic region; but this relationship may overestimate the load, as soil plasticity occurs gradually in practice. In this paper, the Hyperbolic Model B is proposed, based on which the horizontal and rotational load-displacement material curves in the region before yield are assumed to have a hyperbolic relationship. The regression equation for the initial stiffness accompanying the model is also suggested for single clay and soft-over-stiff clay. A fully coupled SSI analysis with a soil continuum model is performed to validate the proposed model. Large soil deformation accompanied by deep penetration is considered simultaneously in the structural analysis of jack-up. As a result, inside the yield envelope, the existing model may overestimate the moment acting on the soil, thereby underestimating the bending moment at the hull-leg joint. The Model B developed and proposed in this paper well predicts the soil response and bending moment distribution of the leg, which results are validated in comparison with those of the soil continuum model. The proposed model can also simulate the hysteresis soil response for a sinusoidal load like in the soil continuum model. These effects make that behavior of soil and structure before yield is well simulated, therefore, efficient design is available in terms of the fatigue strength assessment.

Reviewing Complex Static-Dynamic Concepts of Spine Stability: Does the Spine Care Only to Be Stiff to Be Stable?

Background Changes in the load-displacement relationship in spine segments suggesting alterations in biomechanical stiffness may not yield significant clinical information. Changes in Spine stiffness may arise secondary to neuro-muscular adjustments in the para-spinal muscles and may not be associated with physical anatomical laxity or motion restrictions at segmental articulations. Segmental stiffness may vary dynamically at different zones within the range-of-motion, suggesting a non-linear load-displacement relationship during motion. There is no linear, mechanistic relationship between spine pain and biomechanical markers of spine instability. Objective To review diagnostic assessment approaches of spine instability based on palpatory techniques, end-of-range radiography and imaging in light of our current understanding of biomechanical spine stability. Method The Medline and PubMed databases were screened for primary medical and engineering research articles and reviews on spine stability. Information related to bio-mechanical concepts and clinical decision-making were extracted and synthesized. Spine stability was described in two classical forms, the structural (anatomical) and the functional (physiological), the implications of static and dynamic instability was described in terms of biomechanical and mathematical models used to understand etiology of non-specific back pain. Results Evidence supports the view that dynamic adaptations in the load-displacement relationship of the spine may be resistive or assistive, depending on task-specific movements. Diagnosis of instability is based on structural and functional integrity of the segments in a static or dynamic context. Conclusion Development of specific criteria to define clinical spine stability, compatible with system-based biomechanical concept of spine stiffness, is an ongoing topic in clinical and basic science research.

Nonlinear stability analysis of rotationally-restrained imperfect doubly-curved composite shallow shells

The semi-analytical solution for nonlinear stability analysis of imperfect doubly-curved laminated composite shallow shells with rotationally-restrained edges and under in-plane loading is presented. The nonlinear governing equations are established using the Galerkin method, and the arc-length and quadratic control method is implemented to capture the snapping phenomenon of the doubly-curved composite shells. The nonlinear load-displacement relationships of four special curvature radii of doubly-curved shell structures are obtained, and they are compared and validated with the numerical finite element solutions. A parametric study is conducted to evaluate the effects of the initial imperfection, edge rotationally-restrained spring stiffness, various load parameters, and curvature radius on the nonlinear stability behavior of doubly-curved shells. Finally, the computational efficiency and capability of the semi-analytical solution are demonstrated in comparison with the finite element analysis. The present semi-analytical solution can be effectively and efficiently used in simplified nonlinear stability analysis of complex doubly-curved composite shallow shells with periodic and restrained boundary conditions.

Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part I: Theoretical background

Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of known geometry with postulated contact interaction laws including friction. The code consists of two entities: the contact model and the solid mechanics module. Part I of the paper concerns the modeling of internal contacts (called cracks for brevity), while part II is related to the integration of the developed contact model into a solid mechanics module that allows the description of wave propagation processes. The contact model is used to produce normal and tangential load-displacement relationships, which in turn are used by the solid mechanics module as boundary conditions at the internal contacts. Due to friction, the tangential reaction curve is hysteretic and memory-dependent. In addition, it depends on the normal reaction curve. An essential feature of the proposed contact model is that it takes into account the roughness of the contact faces. On one hand, accounting for roughness makes the contact model more complicated since it gives rise to a partial slip regime when some parts on the contact area experience slip and some do not. On the other hand, as we will show, the concept of contact surfaces covered by asperities receding under load makes it possible to formulate a consistent contact model that provides nonlinear load-displacement relationships for any value of the drive displacements and their histories. This is a strong advantage, since this way, the displacement-driven model allows for a simple explicit procedure of data exchange with the solid mechanics module, while more traditional flat-surface contacts driven by loads generate a complex iterative procedure. More specifically, the proposed contact model is based on the previously developed method of memory diagrams that allows one to automatically obtain memory-dependent solutions to frictional contact problems in the particular case of partial slip. Here we extend the solution onto cases of total sliding and contact loss which is possible while using the displacement-driven formulation. The method requires the knowledge of the normal contact response obtained in our case as a result of statistical consideration of roughness of contact faces.

A new method to approximate load–displacement relationships of spinal motion segments for patient-specific multi-body models of scoliotic spine

Load-displacement relationships of spinal motion segments are crucial factors in characterizing the stiffness of scoliotic spine models to mimic the spine responses to loads. Although nonlinear approach to approximation of the relationships can be superior to linear ones, little mention has been made to deriving personalized nonlinear load-displacement relationships in previous studies. A method is developed for nonlinear approximation of load-displacement relationships of spinal motion segments to assist characterizing in vivo the stiffness of spine models. We propose approximation by tangent functions and focus on rotational displacements in lateral direction. The tangent functions are characterized using lateral bending test. A multi-body model was characterized to 18 patients and utilized to simulate four spine positions; right bending, left bending, neutral, and traction. The same was done using linear functions to assess the performance of the proposed tangent function in comparison with the linear function. Root-mean-square error (RMSE) of the displacements estimated by the tangent functions was 44% smaller than the linear functions. This shows the ability of our tangent function in approximation of the relationships for a range of infinitesimal to large displacements involved in the spine movement to the four positions. In addition, the models based on the tangent functions yielded 67, 55, and 39% smaller RMSEs of Ferguson angles, locations of vertebrae, and orientations of vertebrae, respectively, implying better estimates of spine responses to loads. Overall, it can be concluded that our method for approximating load-displacement relationships of spinal motion segments can offer good estimates of scoliotic spine stiffness.

Kinematic and inertial forces in pile foundations under seismic loading

Under seismic excitation, a pile undergoes stresses stemming from both the vibration of the surrounding soil (kinematic interaction) and that of the superstructure (inertial interaction). The relative contributions of kinematic and inertial interaction on the magnitude of seismic forces induced in pile foundations are investigated by means of nonlinear two-dimensional (2D) finite element (FE) analyses. The interaction between the pile and the surrounding soil in three-dimensional (3D) type was idealized in the 2D analysis using soil–pile interaction springs with hysteretic nonlinear load displacement relationships. Parametric analyses are carried out, focusing on normalized seismic forces in the piles as affected by typical characteristics of structures and piles as well as ground nonlinearity. The results of the study provide a new interpretation of the interplay between pile kinematic and inertial seismic forces. The numerical results showed that for relatively short/stiff piles or where structure’s natural frequency is lower than that of the ground, the kinematic interaction can be the prime contributor to the seismic forces in the pile provided that the excitation frequency is not close to the natural frequency of the coupled soil–pile–structure system, fSSI. In a frequency band approaching fSSI, the inertial interaction generally prevails imposing large bending moments at or near the pile head. The results showed also that maximum kinematic seismic force does not always occur at the fundamental frequency of the deposit. For certain relative soil–pile stiffness and excitation amplitudes, the largest peak of the kinematic seismic forces in piles can be occurring at the second mode.

Modeling viscoelastic behavior of periodontal ligament with nonlinear finite element analysis

Background/purposeThe function and role of periodontal ligament (PDL) are of interest in a number of fields. Due to the complexity of the geometry and the impossibility of obtaining specimens, the mechanical properties of PDL remain uncertain. The object of this study was to investigate the viscoelastic behavior of PDL by means of nonlinear finite element analysis. Materials and methodsIn order to consider nonlinear force-displacement behavior, we proposed a finite strain viscoelasticity material model to represent PDL tissue. The strain energy consists of two kinds. The first is nonlinear in time, called viscoelasticity, and the other is geometrically nonlinear, called hyperelasticity. We obtained the material properties by retrograde calculation based on Ross's experiments. ResultsThe results demonstrate that viscoelastic properties mainly result from volumetric changes. The sources of volumetric change are fluids and the vascular system. Second, by combining the nonlinear geometric properties, we can simulate nonlinear load–displacement relationships. We found that the finite strain viscoelastic model can model both nonlinear load–displacement relationships and nonlinear time–displacement relationships. ConclusionThe nonlinear finite strain viscoelastic model can simulate both the creeping behavior and nonlinear load displacement behavior of teeth, and may be the best-fitting model for understanding the mechanical properties of PDL.

Vertical loads effect on the lateral pile group resistance in sand

Vertical loads effect on the lateral response of a 3×5 pile group embedded in sand is studied through a two-dimensional finite element analysis. The soil-pile interaction in three-dimensional type is idealized in the two-dimensional analysis using soil-pile interaction springs with a hysteretic nonlinear load displacement relationship. Vertical loads inducing a vertical pile head displacement of 0.1-pile diameter increase the lateral resistance of the single pile at a 60 mm lateral deflection by 8%. Vertical loads inducing the same vertical displacement applied to a pile group spaced at 3.92-pile diameter increase the overall lateral resistance by 9%. The effect on individual piles, however, depends on the pile position. The vertical load decreases the lateral resistance of the leading pile (pile 1) by 10% and increases the lateral resistances of piles 2, 3, 4, and 5 by 9%, 14%, 17%, and 35%, respectively. Vertical loads applied to the pile group increase the confining pressures in the sand deposit confined by the piles but the rate of increase in those outside the group is relatively small, resulting in the difference in a balance of lateral soil pressures acting at the back of and in front of the individual pile.

Soil–pile separation effect on the performance of a pile group under static and dynamic lateral loads

The effect of soil–pile separation is studied with respect to the performance of a laterally loaded pile group. Full-scale tests, which consist of a combination of a single and a 3 × 5 group pile under static and dynamic lateral loads, present a unique opportunity and allow a rigorous study without arbitrary parameter back-fitting. The coupled soil–pile system is idealized through two-dimensional finite elements with soil models idealized by a hyperbolic-type multiple shear mechanism. Nonlinear spring elements are used to idealize the soil–pile interaction through a hysteretic nonlinear load–displacement relationship. Joint elements with a separation–contact mechanism are used to idealize the separation effect at the soil–pile interface. Ignoring soil–pile separation in static tests overestimates the ultimate lateral load–carrying capacity by 43% for a single pile and 73% for the trailing pile in a closely spaced pile group. Moreover, neglecting soil–pile separation in dynamic tests overestimates the total group load–deflection curve in both the loading and unloading phases.

Novel re-entrant porous composite structure: a potential for orthopaedic applications

It was hypothesized that the nonlinear load-displacement relationship displayed by bone could be conferred on an implant by tailoring its structure, yielding an enhanced mechanical stimulation of the tissues. Composite structures would feature piezoelectric properties that could also stimulate osteogenesis. Preliminary mechanical and electromechanical investigations of such porous structures are presented. Initial trial bowtie specimens with various aspect ratii were made from Nickel powder via a solid free form process and from stainless steel shim stocks. Poled Barium Titanate plates were sandwiched between stainless steel bowtie cells to create composite structures.Results: Under quasi-static compression, the Nickel structures displayed a nonlinear mechanical behavior at small strains and an overall strain-stress relationship similar to bone. Under cyclic compressive tests to 0.6 percent strain, all structures presented a repeatable nonlinear strain-stress behavior. The curves were fitted by a second-order polynomial whose coefficients are function of the relative density of the structure to a power n. Composite stainless steel/BaTiO3 bowtie structures confirmed that their electromechanical properties can be tailored.Discussion: Certain patients present metabolic degeneration that hamper bone healing. A ductile and tough structural material with piezoelectric properties such as the new composite structures in development presents the potential to overcome those limitations. They could have the advantages of existing devices without some of the drawbacks. Those porous implants may reduce the needs, costs, and risks linked to the additional use and implementation of an electrical stimulator and BMPs. Furthermore, the solid free form technique gives control over the mechanical properties of the structure. Thus, the mechanotransduction activity of biologic cells can be fully exploited to trigger a faster implant-tissue bonding, which could lead to reduction of surgical cost and time.

Simplified design methods for pipelines subject to transverse and longitudinal soil movements: Discussion

The simple method for analysing the behaviour of buried pipeline subjected to transverse and longitudinal soil displacement, and proposed by the original authors, was critiqued. It was argued that in the transverse movement case, the assumptions of elastic-perfectly plastic soil and elastic pipe may have oversimplified the actual behaviour of the soil and the pipe, and that the use of a single value of ks was insufficient to model the nonlinear load displacement relationship. While the solution proposed by the original authors was simple and easy to use , it may be usable only in preliminary design. For detailed design a more rigorous solution has to be used, such as for example a beam on elastic foundation program that takes into account the plastic behaviour of the pipe and non-linear load-displacement relationship of the soil. It was pointed out that in actual practice most pipelines are laid in an excavated trench and then backfilled with other material and the backfill is usually weaker than the natural ground. This fact of two different materials can affect the value of the resistance factor. The reviewer (1994) derived a number of design charts to determine correction factors which have been checked against the results of a full-scale pipe loading test, and found excellent agreement. 3 refs.

Crack propagation in thin plates under double torsion

A method of analysis for characterizing non-linear crack propagation in thin plates under double torsion is presented. Quasistatic energy method is used to derive the fracture toughness and to study the stability conditions of crack propagation within the subcritical crack growth range.Non-linear relationship between the torque and angular deflection of a thin rectangular bar is well known. A thin rectangular plate containing a central longitudinal crack and acted upon by a system of four-point loading (two opposite forces on each side of the crack) causes non-linear structural response to this loading mode.The stability of crack propagation in thin plates under double torsion is examined for both the linear (thin plates) and non-linear (thick plates) cases. It is shown that improved stability can be attained if the specimen geometry and loading mode display a non-linear load-displacement relationship at constant crack area, as compared with the corresponding linear case.A series of experiments is conducted to measure the fracture toughness of PMMA material of different thicknesses. It is observed that the fracture toughness values evaluated analytically using the quasi-static energy method compare favourably with those measured experimentally.For the range of PMMA plate thicknesses considered, the effect of thickness on the fracture toughness of PMMA material is negligible. Finally, the validity of employing the double torsion specimen for fracture studies is examined by comparing the fracture toughness of thin plates of different thicknesses loaded separately by double torsion and by tension. No significant difference between the two sets of results is observed.

Quasistatic energy analysis of a slender D.C.B. specimen

This paper presents a method of quasistatic energy analysis for characterizing the fracture behaviour of a slender D.C.B. specimen having two parallel beam arms of low bending stiffness. Examples chosen are in-plane tension and anti-plane shear problems, which exhibit linear and nonlinear load-displacement relationships, respectively. The results derived analytically agree well with those obtained experimentally for both examples, thus successfully demonstrating the versatility of the quasistatic energy analysis in fracture studies. Stability of crack propagation for the geometries considered is analysed and found to be favourable unless there is a large negative value of d G/d a. Negative d G/d a was attained by contaminating portions of an uncracked specimen, and unstable crack propagation was observed experimentally in the contaminated regions.

Crack propagation with quasistatic energy analysis

In the application of quasistatic energy analysis, the P-u-A diagram has been found a valuable supplement in illustrating the characteristics of crack propagation. The diagram is useful not only for evaluating fracture toughness under quasistatic conditions, but also for representing important fracture parameters such as stability criteria, crack speed, strain rate, fatigue, environmental effects, etc. A general method of characterizing these parameters in the form of a P-u-A diagram is presented for both linear and non-linear load-displacement relationships at constant crack area. This was achieved by polynomial representation of the diagram. The analysis was extended to cover crack propagation under multiple-loading conditions. By introducing a complementary strain-energy function, equivalent load and compliance, a method of solution has been devised to avoid the laborious computations which may otherwise be required if the loadings become numerous and complex. The computed values of fracture toughness from the numerically represented P-u-A diagram, and the displacements evaluated with the equivalent compliance concept, compare well with those obtained experimentally.

Stability Conditions in Quasi-Static Crack Propagation for Constant Strain Energy Release Rate

Measurement of valid fracture toughness of engineering materials and design assurance of quasi-static crack growth in structures before the onset of fracture instability are the important factors in the study of fracture mechanics. Generalized equations of the stability conditions are presented not only for linear but also for nonlinear load-displacement relationships at constant crack area. Examples are used to illustrate the applications of the proposed equations, and stability conditions of different crack geometries and loadings commonly observed in engineering structures are discussed. Experiments conducted to verify the stability conditions are described and found to be satisfactory.

Piles in Cohesionless Soil Subject to Oblique Pull

Laboratory test data on rigid piles embedded or driven at various angles in cohesionless soils and subjected to oblique pull, 60 from the horizontal, are presented. The vertical or a slight out-batter pile position gives the greatest resistance, the ultimate loads for a 30 out-batter being less than 65% of the peak resistance. The pile roughness becomes a significant factor when the relative axial load component reaches a certain limit. A method for analyzing nonlinear load-displacement relationship of a rigid pile is presented.