Force-elongation Curve Research Articles

Achilles Tendon Stiffness: Influence of Measurement Methodology

ObjectiveMechanical stiffness derived from force-elongation curves is fundamentally different from shear wave (SW) elastography-based tissue properties. We compared these techniques, with a total of five methods of assessing Achilles tendon (AT) stiffness. MethodsSeventeen participants (12 male and 5 female) with unilateral AT rupture performed submaximal contractions at 30% and 10% maximal isometric contraction torque of the un-injured limb. SW velocity was acquired at rest. Force-elongation curves were assessed from the free AT and the medial gastrocnemius (MG) tendon. Mechanical stiffness was determined near the end of the linear region of the force-elongation curve and from the toe region. Bivariate correlations between mechanical stiffness and SW velocity, as well as pairwise t-tests between limbs, were computed. ResultsIn the injured limb, SW velocity correlated with MG tendon and free AT toe-region stiffness during 10% (r = 0.59, p = 0.020 and r = 0.60, p = 0.011, respectively) and 30% of submaximal contractions (r = 0.56, p = 0.018 and r = 0.67, p = 0.004, respectively). The un-injured limb showed no associations. In both limbs pooled together, SW velocity correlated with MG tendon toe-region stiffness in 30% of submaximal contractions (r = 0.43, p = 0.012). Free tendon mechanical stiffness was lower in the injured limb, with a mean difference of 148.5 Nmm⁻¹ (95% CI: 35.6–261.3, p = 0.013), while SW velocity was higher in the injured limb (1.67 m × s⁻¹, 95% CI; -2.4 to -0.9, p < 0.001). ConclusionSW elastography may reflect AT viscoelastic properties at the initial slope of the force-length curve with strains <1% but cannot offer insight into AT mechanics at higher loads. Extended toe regions in the injured limb could have caused the association between mechanical stiffness and SW-based stiffness.

Extraction of Equivalent Stress Versus Equivalent Plastic Strain Curve of Necking Material in Tensile Test Without Assuming Constitutive Model

Abstract An algorithm for extracting the equivalent stress versus equivalent plastic strain curve of a necking material in tensile test is proposed (the curve is called here the equivalent-plastic (EP) stress–strain (SS) curve). The presented algorithm traces the force–elongation curve via iterative finite element (FE) simulations without assuming a constitutive model and is suitable for a general-purpose FE solver available to a general audience. In the FE simulation and experiment, a slightly tapered geometry was employed around the specimen center to stably initiate necking there. The proposed algorithm and mentioned necking initiation method have been applied to extracting the EP SS curve of a high-strength steel material using an axisymmetric specimen. Because necking was initiated stably in simulation at the specimen center for a range of mesh sizes, the convergence of the extracted EP SS curve with the mesh size could be successfully verified. The EP SS curve of the tested material was extracted up to an EP strain of 0.85 with average strain intervals of approximately 2.5 mili-strain. The error values were less than 0.2 and 0.1% after four and seven FE simulations, respectively. The presented algorithm and necking initiation method in simulation can also be used for simultaneously quantifying the fracture EP strain of a necking material in the tensile test.

Strain inhibition of bacterial collagenase is consistent with a collagen fibril uncrimping mechanism in rat tail tendons

Mechanical strain inhibits bacterial collagenase from cleaving collagen. Additionally, the toe region of a soft tissue’s force–elongation curve arises from sequentially engaging collagen fibrils as the tissue lengthens. Together, these phenomena suggest that mechanical strain may gradually inhibit collagenase activity through a soft tissue’s toe region. Therefore, this investigation sought to test this hypothesis.92 rat tail tendon fascicles from 3 female sentinel animals underwent preliminary stiffness tests, and their force–elongation curves were fit to a collagen distribution model. This distribution-based model calculated the force magnitude corresponding to p% of collagen fibril engagement. Specimens were separated into one of five levels of p, and that level of force was maintained for two hours while being exposed to 0.054 U/mL of bacterial collagenase from C. histolyticum. The specimens were strained to failure following the creep test, and the relative reduction in stiffness was quantified to estimate the fraction of digested fibrils.Every 10% additional collagen engagement corresponded to a 6.3% (97% highest density interval: 4.3 – 8.4%) retention of stiffness, which indicated collagenase inhibition.The results of this investigation were consistent with a strain-inhibition hypothesis along with the established uncrimping mechanism in the toe region. These results support an interaction between mechanical strain and collagenolysis, which may be valuable for disease prevention or treatment.

The acute effects of higher versus lower load duration and intensity on morphological and mechanical properties of the healthy Achilles tendon: a randomized crossover trial.

ABSTRACTThe Achilles tendon (AT) exhibits volume changes related to fluid flow under acute load which may be linked to changes in stiffness. Fluid flow provides a mechanical signal for cellular activity and may be one mechanism that facilitates tendon adaptation. This study aimed to investigate whether isometric intervention involving a high level of load duration and intensity could maximize the immediate reduction in AT volume and stiffness compared with interventions involving a lower level of load duration and intensity. Sixteen healthy participants (12 males, 4 females; age 24.4±9.4 years, body mass 70.9±16.1 kg, height 1.7±0.1 m) performed three isometric interventions of varying levels of load duration (2 s and 8 s) and intensity (35% and 75% maximal voluntary isometric contraction) over a 3 week period. Freehand 3D ultrasound was used to measure free AT volume (at rest) and length (at 35%, 55% and 75% of maximum plantarflexion force) pre- and post-interventions. The slope of the force–elongation curve over these force levels represented individual stiffness (N mm−1). Large reductions in free AT volume and stiffness resulted in response to long-duration high-intensity loading whilst less reduction was produced with a lower load intensity. In contrast, no change in free AT volume and a small increase in AT stiffness occurred with lower load duration. These findings suggest that the applied load on the AT must be heavy and sustained for a long duration to maximize immediate volume reduction, which might be an acute response that enables optimal long-term tendon adaptation via mechanotransduction pathways.

Humidity influence on mechanics of paper materials: joint numerical and experimental study on fiber and fiber network scale

Paper materials are well-known to be hydrophilic unless chemical and mechanical processing treatments are undertaken. The relative humidity impacts the fiber elasticity, the interfiber joint behavior and the failure mechanism. In this work, we present a comprehensive experimental and computational study on mechanical properties of the fiber and the fiber network under humidity influence. The manually extracted cellulose fiber is exposed to different levels of humidity, and then mechanically characterized using atomic force microscopy, which delivers the humidity dependent longitudinal Young’s modulus. We describe the relation and calibrate the data into an exponential function, and the obtained relationship allows calculation of fiber elastic modulus at any humidity level. Moreover, by using confoncal laser scanning microscopy, the coefficient of hygroscopic expansion of the fibers is determined. We further present a finite element model to simulate the deformation and the failure of the fiber network. The model includes the fiber anisotropy and the hygroscopic expansion using the experimentally determined constants, and further considers interfiber behavior and debonding by using a humidity dependent cohesive zone interface model. Simulations on exemplary fiber network samples are performed to demonstrate the influence of different aspects including relative humidity and fiber-fiber bonding parameters on the mechanical features, such as force-elongation curve, strength and extensibility. Finally, we provide computational insights for interfiber bond damage pattern with respect to different humidity level as further outlook.

Reliability of human Achilles tendon stiffness measures using freehand 3-d ultrasound

Introduction: The Achilles tendon (AT) is the largest and strongest tendon in the human body and is subject to high forces during weight-bearing activities such as walking and running. Tendon stiffness, which is defined as resistance to elongation under load, is a key determinant of the efficiency of the triceps surae muscle output during human locomotion. Freehand three-dimensional ultrasound (3DUS) is a promising method for measuring stiffness of the Achilles tendon, and particularly the free AT (2-6 cm proximal to calcaneus), which is commonly injured. This study aimed to investigate the test-retest reliability and the smallest detectable change (SDC) of freehand 3DUS in measuring free AT stiffness in humans. Methods: The free Achilles tendon length of healthy participants (n=10) was scanned on the same day on two consecutive occasions (1 h apart) during rest and isometric planterflexion (PF) contractions at 20%, 40%, and 60% of maximum force. The slope of force-elongation curve over these force levels represented individual stiffness (N/mm). Relative reliability was assessed using the intra-class correlation coefficient (ICC) and absolute reliability was estimated with standard error of measurement (SEM), and smallest detectable change (SDC). Systematic bias in stiffness measures was explored by comparing test and retest distributions and Bland-Altman plots. Results: The test-retest reliability for free AT stiffness using freehand 3DUS was excellent [ICC= 0.994 (95% CI 0.978 to 0.999)]. The mean difference (SD) was -3.15 (± 26.6) N/mm, SEM was 18.82 N/mm, and SDC was 52.14 N/mm. The P-value for the difference between the means was = 0.72. The Bland and Altman plot showed no systematic bias within stiffness measures as data points were equally scattered around the mean difference (95% CI: -22.18-15.88) and within the 95% limits of agreement (LOA) = mean + (SD × 1.96). Discussion: The current study has established the reliability of freehand three-dimensional ultrasound for determining free AT stiffness in healthy humans. Freehand 3DUS could be implemented in research or clinical practice to measure and monitor changes in free AT stiffness that may occur with loading or as a consequence of tendon injury or pathology. Conflict of interest disclosure: The authors declare that they have no competing interests.

Reliability of Human Achilles Tendon Stiffness Measures Using Freehand 3-D Ultrasound

Achilles tendon (AT) stiffness is an important property of both human locomotor performance and injury mechanics. Freehand 3-D ultrasound (3-DUS) is a promising method for measuring stiffness of the Achilles tendon, particularly the free AT (2–6 cm proximal to calcaneus), which is commonly injured. The aim of this study was to investigate the test–retest reliability of freehand 3-DUS in measuring free AT stiffness in humans. The free Achilles tendon length of healthy participants (n = 10) was scanned on the same day on two consecutive occasions (1 h apart) during rest and isometric plantar flexion contractions at 20%, 40% and 60% of maximum force. The slope of the force–elongation curve over these force levels represented individual stiffness (N/mm). Relative reliability was assessed using the intra-class correlation coefficient (ICC), and absolute reliability was estimated with the standard error of measurement (SEM) and smallest detectable change. Systematic bias in stiffness measures was explored by comparing test and retest distributions and Bland–Altman plots. The test–retest reliability of free AT stiffness measured using freehand 3-DUS was excellent [ICC = 0.994, 95% confidence interval [CI]: 0.978–0.999)]. The mean stiffness values at test (361.83 N/mm [170.77]) and retest (364.98 N/mm [168.57]) did not significantly differ (p = 0.72), and the smallest detectable change was 52.14 N/mm. The Bland–Altman plot indicated the absence of systematic bias (95% CI: –22.18 to 15.88). Freehand 3-DUS provides reliable and precise measures of tendon stiffness and can be used to detect small changes in free AT stiffness in response to load or tendon pathology.

Performance Characterization and Pressure Prediction of Compression Socks

Objective of the current research is to develop the mathematical model for the prediction of compression pressure by incorporating some of new but missing variables; circumferential aspect of leg (Lc) and socks (Sc), true stress (σT), true/logarithm strain (eT), true modulus of elasticity (ET), deformed width (Wf) at b ankle position. Socks samples were purchased, hand washed, donned, marked, sliced to loops and then cut to rectangular strips to obtain force-elongation curve at different extensions context to practical elongation. Additionally, all samples were evaluated to analyze the influnce of multiple-machine washes as well as for identification of most influential fabric parameter on compresson pressure, statistically. Standard sized wooden leg (S9=24 cm) attached with pneumatic based sensors (Salzmann MST MK IV) to medial side was used. Force-extension diagram was obtained using Testometric tensile tester. Mathematical models were developed on the basis of principle of Laplace’s law. Developed two models were analyzed and comapred with two existing recently developed models statitistically using MINTAB 17 software.

Study of tensile properties of plain-woven fabrics in all-directional using energy method, Part I: theoretical study

In the first part of this series, a new mechanical parameter, that is, “Comprehensive Tensile Modulus (CTM)” is introduced and modeled to show the tensile behavior of plain-woven fabrics in the initial linear elastic region of the force-elongation curve subjected to tensile load and extended it simultaneously in all directions. Considering the initial load-extension behavior of fabrics, a mathematical-mechanical model is presented to predict the CTM of fabrics in the initial linear elastic region using Castigliano’s theorem. Based on the generated model, the initial sample length, dimension of the load imposed region, the geometrical shape created in the plain-woven fabric sample during tensile, the structural specification of these fabrics such as yarns sett, yarns crimp in fabric and mechanical properties of yarns such as bending rigidity in both warp and weft yarns affect in the comprehensive tensile modulus of fabric. In order to verify the conformity and accuracy of the model, a preliminary test was conducted on the prepared samples based on a novel tensile test method developed to measure load-extension curve of the fabric samples under the proposed loading condition. A reasonable agreement was found between theoretical and experimental results.

Force and deformation transmission characteristics of a compliant tendon-sheath actuation system based on Hill-type muscle model.

Artificial muscle is a kind of transmission actuator widely used in rehabilitation robots and wearable devices. However, there are some restrictions on the usage of these artificial muscles, including the short stroke length, complex structure, special power sources, and high nonlinear characteristics. Inspired by Hill-type muscle model, in this article, a new kind of artificial muscle using tendon-sheath and compliant springs is proposed to perform muscle-like characteristics. Force and deformation transmission models are proposed and validated by simulations and experiments. The experimental and simulation output results show nice goodness-of-fit and the R-square values are 0.9876 and 0.9046, respectively. Moreover, experiments are carried out in groups to analyze the transmission characteristics using different parameters, including variations of series springs, velocities, tendon diameters, and bending angles. The best R-square value of force-elongation curve and fitness curve could reach 0.9845, which indicates that the transmission model of the compliant artificial muscle can be used to express the transmission characteristics of the skeleton muscles.

Discontinuity in the In-plane to Out-of-plane Transition of Kirigami

Recently, we have shown that kirigami’s high extensibility emerges as a result of the transition in its deformation modes and that the transition observed through the force-elongation curve can be ...

Experimental study of the effect of filler on the ductility of filler-bitumen mastics

This research investigated the effect of filler on the ductility of filler-bitumen mastics, as a surrogate of the cohesive behaviour and the tensile properties. Various fillers (industrial products, mineral fillers or waste materials) were mixed with one paving grade bitumen, in various filler-to-bitumen ratios, and tested under the force-ductility test. The force-elongation curve obtained is affected by the constituents, but the peak force and total deformation energy are a mainly a function of the normalized filler concentration value. Moreover, the effect of the filler type and concentration on the specimen’s ductility is better perceived from the normalized post-peak deformation energy. Normalized filler concentration values close to 0.80 induce a clear reduction in ductility.

Low-cost variable stiffness joint design using translational variable radius pulleys

Robot joints are expected to be safe, compliant, compact, simple and low-cost. Gravity compensation, zero backlash, energy efficiency and stiffness adjustability are some desired features in the robotic joints. The variable radius pulleys (VRPs) provide a simple, compact and low-cost solution to the stiffness adjustment problem. VRP mechanisms maintain a preconfigured nonlinear force-elongation curve utilizing off-the-shelf torsional spring and pulley profile. In this paper, three synthesis algorithms are presented for VRP mechanisms to obtain desired force-elongation curve. In addition, a feasibility condition is proposed to determine the torsional spring coefficient. Using the synthesis methods and the feasibility condition, a variable stiffness mechanism is designed and manufactured which uses two VRPs in an antagonistic cable driven structure. Afterwards, the outputs of three synthesis methods are compared to force-elongation characteristics in the tensile testing experiment. A custom testbed is manufactured to measure the pulley rotation, cable elongation and tensile force at the same time. Using the experiment as the baseline, the best algorithm achieved to reproduce the desired curve with a root-mean-square (RMS) error of 13.3%. Furthermore, VRP-VSJ is implemented with a linear controller to reveal the performance of the mechanism in terms of position accuracy and stiffness adjustability.

A Finite Element Bendo-Tensegrity Model of Eukaryotic Cell.

Mechanical interaction of cell with extracellular environment affects its function. The mechanisms by which mechanical stimuli are sensed and transduced into biochemical responses are still not well understood. Considering this, two finite element (FE) bendo-tensegrity models of a cell in different states are proposed with the aim to characterize cell deformation under different mechanical loading conditions: a suspended cell model elucidating the global response of cell in tensile test simulation and an adherent cell model explicating its local response in atomic force microscopy (AFM) indentation simulation. The force-elongation curve obtained from tensile test simulation lies within the range of experimentally obtained characteristics of smooth muscle cells (SMCs) and illustrates a nonlinear increase in reaction force with cell stretching. The force-indentation curves obtained from indentation simulations lie within the range of experimentally obtained curves of embryonic stem cells (ESCs) and exhibit the influence of indentation site on the overall reaction force of cell. Simulation results have demonstrated that actin filaments (AFs) and microtubules (MTs) play a crucial role in the cell stiffness during stretching, whereas actin cortex (AC) along with actin bundles (ABs) and MTs are essential for the cell rigidity during indentation. The proposed models quantify the mechanical contribution of individual cytoskeletal components to cell mechanics and the deformation of nucleus under different mechanical loading conditions. These results can aid in better understanding of structure-function relationships in living cells.

Effects of tracking landmarks and tibial point of resistive force application on the assessment of patellar tendon mechanical properties in vivo

The different methods used to assess patellar tendon elongation in vivo may partly explain the large variation of mechanical properties reported in the literature. The present study investigated the effects of tracking landmark position and tibial point of resistive force application during leg extensions in a dynamometer.Nineteen adults performed isometric contractions with a proximal and distal dynamometer shank pad position. Knee joint moments were calculated employing an inverse dynamics approach. Tendon elongation was measured using the patellar apex and either the tibial tuberosity (T) or plateau (P) as tracking landmark.Using P for tracking introduced a bias towards greater values of tendon elongation at all force levels from 100 N to maximum tendon force (TFmax; p < 0.05). The differences between landmarks considering maximum tendon strain were greater at the proximal shank pad position (p < 0.05). Tendon stiffness was lower for P compared with T, but only in intervals up to 50% of TFmax (p < 0.05). The agreement between T and P for stiffness calculated between 50% and TFmax was acceptable with the distal, but poor with the proximal pad position.We demonstrated that using the tibia plateau and not the insertion as tracking landmark clearly affects the assessment of the force–elongation curve of the patellar tendon. However, using a distal point of resistive force application and calculating tendon stiffness between 50% and TFmax seems to yield an acceptable agreement between landmarks. These findings have important implications for the assessment of tendon properties in vivo and cross-study comparisons.

On Complete Solutions for the Problem of Diffuse Necking in Sheet Metal

Abstract The paper focuses on the assessment of so-called complete solutions for the problem of diffuse necking in sheet metal. The assessment is based on a virtual tensile test generated with the aid of a finite element model using a reference strain hardening behavior. The resulting displacement fields within the diffuse neck are then used to numerically deform a speckle pattern. Subsequently, the deformed images are post-processed using DIC software to acquire displacement fields and strain fields. The identification of the strain hardening behavior is based on the virtual force-elongation curve, the displacement fields and associated strain fields within the diffuse neck.

Achilles Tendon Adaptation During Transition to a Minimalist Running Style.

Achilles tendons (ATs) adapt to increased loading generated by long-term adoption of a minimalist shoe running style. There may be difference in the chronology and extent of adaptation between the sexes. To learn the chronology of AT adaptations in female and male runners who transitioned to a minimalist running style through a planned, progressive 12-wk transition program. Prospective cohort study of well-trained, traditionally shod runners who transitioned to minimalist shoe running. Repeated laboratory assessment at baseline and 3, 12, and 24 wk after initiating transition program. Fifteen women and 7 men (of 29 enrolled) completed the study. The authors used diagnostic ultrasound and isokinetic dynamometry to generate a force elongation curve and its derivatives at each time point. Greater adaptations were observed in men than in women, with men generating more force and having greater increases in CSA, stiffness, and Young's modulus and less elongation after 12 wk of training. Men demonstrated changes in AT properties that were consistent with increased loading of the triceps surae during exercise. The women demonstrated far smaller changes. Further investigation is warranted to understand when adaptations may occur in women and the implications of altered AT mechanical properties for performance and injury risk.

Achilles Tendon Properties of Minimalist and Traditionally Shod Runners

Tendon adapts to load through alterations in its composition and mechanical properties. Mechanical adaptation to increased load often involves increases in cross-sectional area (CSA), stiffness, and modulus. Runners exhibit these adaptations. To determine if runners wearing minimalist shoes had larger and stiffer Achilles tendons (AT) than traditionally shod runners. Cross-sectional study of well-trained, traditionally and minimally shod runners. Laboratory assessment of trained runners. 23 men (11 traditional, 12 minimalist) and 8 women (6 traditional, 2 minimalist). Runners wearing minimalist shoes had 4.2 ± 1.6 y of training experience in minimalist shoes. The authors used diagnostic ultrasound and isokinetic dynamometry to generate a force-elongation curve and its derivatives. Minimalist runners had a greater CSA: mean difference (MD) = 9.2 mm2, stiffness (MD = 268.1 N/mm), and modulus (MD = 202.9 MPa). ATs of minimalist runners experienced greater stress (MD 8.6 N/mm2) during maximal voluntary isometric contraction of the plantar-flexor muscles due to greater force of contraction (MD 798.9 N). The AT in minimalist runners adapts by increasing size, stiffness, and modulus, which is consistent with our understanding of mechanical adaptation of tendon to increased loading. Increased stress to the AT likely requires a slow transition to minimalist running to allow the AT to adapt without evidence of injury.

Ductile fracture of EN-AW 2024 aluminum alloy specimens with notches under biaxial loading. Part 1 – Experimental research

This paper presents the results of the experimental studies on ductile fractures of axisymmetric specimens with circumferential notches. A range of notch root radii is considered. The specimens underwent proportional biaxial loading (tensile–torsion) until failure. The load was forced onto the specimens by elongation as well as angular movement and was controlled by a device for the non-contact measurement. The EN-AW 2024 T351 aluminum alloy was used in this research. Special attention was paid to the change in shape of the fracture surface, changing the force-elongation curve and torsional moment–torsional angle curve, which are dependent on loading.

Patient-related factors influence stiffness of the soft tissue complex during intraoperative gap balancing in cruciate-retaining total knee arthroplasty.

How much force is needed to pre-tension the ligaments during total knee arthroplasty? The goal of this study was to determine this force for extension and flexion, and for both compartments, and to identify predicting patient-related factors. Eighty patients [55 females, mean age 71 (SD 9.7)] were recruited and had a navigated cruciate-retaining total knee arthroplasty. Distraction of the medial and lateral compartments of the extension and flexion gap (90°) with an instrumented bi-compartmental double-spring tensioner took place after finishing the bone cuts. Applied forces and resulting gap distances were recorded by the navigation system, resulting in a force-elongation curve. Lines were fitted with the intersection defined as the stiffness transition point. The slopes (N/mm) represented the stiffness of the ligamentous complex. Linear multiple regression analysis was performed to identify predicting factors. The amount of force at the stiffness transition point was on average 52.3 (CI95 50.7-53.9), 54.5 (CI95 52.7-56.3), 48.3 (CI95 46.2-50.2), and 59.3 (CI95 57.0-61.6)N for the medial and lateral extension and flexion gap, respectively, and varied considerably between patients. The force at the stiffness transition point was significantly different between extension and flexion and both compartments (P<0.05). Stiffness of the ligaments statistically significantly helped to predict the amount of force at the stiffness transition point, as well as body mass index, gender, and varus-valgus alignment. The amount of force at the stiffness transition point varies between 48 and 59N, depending on flexion/extension and compartment. Patient-related factors influence the stiffness transition point and can help predict the stiffness transition point. When forces higher than 60N are used for gap distraction, the ligamentous sleeve of the knee might be over-tensioned. Prognostic study, Level I-high-quality prospective cohort study with >80% follow-up, and all patients enrolled at same time point in disease.