Nonlinear viscoelastic characterization of bovine trabecular bone

The time-independent elastic properties of trabecular bone have been extensively investigated, and several stiffness–density relations have been proposed. Although it is recognized that trabecular bone exhibits time-dependent mechanical behaviour, a property of viscoelastic materials, the characterization of this behaviour has received limited attention. The objective of the present study was to investigate the time-dependent behaviour of bovine trabecular bone through a series of compressive creep–recovery experiments and to identify its nonlinear constitutive viscoelastic material parameters. Uniaxial compressive creep and recovery experiments at multiple loads were performed on cylindrical bovine trabecular bone samples (\(n = 19\)). Creep response was found to be significant and always comprised of recoverable and irrecoverable strains, even at low stress/strain levels. This response was also found to vary nonlinearly with applied stress. A systematic methodology was developed to separate recoverable (nonlinear viscoelastic) and irrecoverable (permanent) strains from the total experimental strain response. We found that Schapery’s nonlinear viscoelastic constitutive model describes the viscoelastic response of the trabecular bone, and parameters associated with this model were estimated from the multiple load creep–recovery (MLCR) experiments. Nonlinear viscoelastic recovery compliance was found to have a decreasing and then increasing trend with increasing stress level, indicating possible stiffening and softening behaviour of trabecular bone due to creep. The obtained parameters from MLCR tests, expressed as second-order polynomial functions of stress, showed a similar trend for all the samples, and also demonstrate stiffening–softening behaviour with increasing stress.     

Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone

Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples (\(\textit{n}\,{=}\,13\)) at loads corresponding to physiological strain level of 2000 \({\upmu }{\upvarepsilon }\). We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships (\(r^2\,{=}\,0.73,\ p\,{<}\,0.001\)) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV.     

The effect of damage on the viscoelastic behavior of human vertebral trabecular bone

The present study examines the viscoelastic behavior of cancellous bone at low strains and the effects of damage on this viscoelastic behavior. It provides experimental evidence of interaction between stress relaxation behavior and the effect of accumulated damage. The results suggest that damage is at least orthotropic in trabecular bone specimens under uniaxial loading. Simple linear models of viscoelasticity described the time-dependent stress-strain behavior at low strains before and after specimen damage, although better fits of these models were obtained prior to damage. Modeling the observed changes in relaxation times with damage accumulation appears necessary to successfully predict the post-damage viscoelastic response.     

Mechanical properties of the heart muscle in the passive state

The viscoelastic behaviour of the heart muscle (papillary muscle) in the passive unstimulated) state is studied by such methods as stress relaxation, creep, vibration and stress-strain testing. The tests are conducted on a newly developed electromechanical muscle testing device which is suitable for conducting active and passive tests on biological materials.     

An experimental and computational investigation of the post-yield behaviour of trabecular bone during vertebral device subsidence

Interbody fusion device subsidence has been reported clinically. An enhanced understanding of the mechanical behaviour of the surrounding bone would allow for accurate predictions of vertebral subsidence. The multiaxial inelastic behaviour of trabecular bone is investigated at a microscale and macroscale level. The post-yield behaviour of trabecular bone under hydrostatic and confined compression is investigated using microcomputed tomography-derived microstructural models, elucidating a mechanism of pressure-dependent yielding at the macroscopic level. Specifically, microstructural trabecular simulations predict a distinctive yield point in the apparent stress–strain curve under uniaxial, confined and hydrostatic compression. Such distinctive apparent stress–strain behaviour results from localised stress concentrations and material yielding in the trabecular microstructure. This phenomenon is shown to be independent of the plasticity formulation employed at a trabecular level. The distinctive response can be accurately captured by a continuum model using a crushable foam plasticity formulation in which pressure-dependent yielding occurs. Vertebral device subsidence experiments are also performed, providing measurements of the trabecular plastic zone. It is demonstrated that a pressure-dependent plasticity formulation must be used for continuum level macroscale models of trabecular bone in order to replicate the experimental observations, further supporting the microscale investigations. Using a crushable foam plasticity formulation in the simulation of vertebral subsidence, it is shown that the predicted subsidence force and plastic zone size correspond closely with the experimental measurements. In contrast, the use of von Mises, Drucker–Prager and Hill plasticity formulations for continuum trabecular bone models lead to over prediction of the subsidence force and plastic zone.     

Viscoelastic properties of passive skeletal muscle in compression—Cyclic behaviour

Skeletal muscle relaxation behaviour in compression has been previously reported, but the anisotropic behaviour at higher loading rates remains poorly understood. In this paper, uniaxial unconfined cyclic compression tests were performed on fresh porcine muscle samples at various fibre orientations to determine muscle viscoelastic behaviour. Mean compression level of 25% was applied and cycles of 2% and 10% amplitude were performed at 0.2–80 Hz. Under cycles of low frequency and amplitude, linear viscoelastic cyclic relaxation was observed. Fibre/cross-fibre results were qualitatively similar, but the cross-fibre direction was stiffer (ratio of 1.2). In higher amplitude tests nonlinear viscoelastic behaviour with a frequency dependent increase in the stress cycles amplitude was found (factor of 4.1 from 0.2 to 80 Hz).The predictive capability of an anisotropic quasi-linear viscoelastic model previously fitted to stress-relaxation data from similar tissue samples was investigated. Good qualitative results were obtained for low amplitude cycles but differences were observed in the stress cycle amplitudes (errors of 7.5% and 31.8%, respectively, in the fibre/cross-fibre directions). At higher amplitudes significant qualitative and quantitative differences were evident. A nonlinear model formulation was therefore developed which provided a good fit and predictions to high amplitude low frequency cyclic tests performed in the fibre/cross-fibre directions. However, this model gave a poorer fit to high frequency cyclic tests and to relaxation tests. Neither model adequately predicts the stiffness increase observed at frequencies above 5 Hz.Together with data previously presented, the experimental data presented here provide a unique dataset for validation of future constitutive models for skeletal muscle in compression.     

Viscoelastic properties of the human lumbodorsal fascia

The purpose of this study is to provide better understanding of the mechanical response of the lumbodorsal fascia to dynamic and static traction loadings. Since the fascia shows a viscoelastic behaviour, tests in which time is a variable were used, namely hysteresis and stress relaxation. Load-strain and load-time curves obtained from the hysteresis and stress-relaxation tests point out three different phenomena. First, an increase in stiffness is noticed when ligaments are successively stretched, i.e. strains produced by successive and identical loads decrease. Second, if a sufficient resting period is allowed between loadings, stiffening is reversed and strains tend to recover initial values. The third phenomenon, observed in stress-relaxation tests as time progresses, is ligament contraction in stretched and isometrically held samples. This third phenomenon may be explained by the possibility that muscle fibres capable of contracting spontaneously could be present in lumbodorsal fascia ligaments.     

In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen

Tendon is a hydrated multi-level fibre composite, in which time-dependent behaviour is well established. Studies indicate significant stress relaxation, considered important for optimising tissue stiffness. However, whilst this behaviour is well documented, the mechanisms associated with the response are largely unknown. This study investigates the sub-structural mechanisms occurring during stress relaxation at both the macro (fibre) and nano (fibril) levels of the tendon hierarchy. Stress relaxation followed a two-stage exponential behaviour, during which structural changes were visible at the fibre and fibril levels. Fibril relaxation and fibre sliding showed a double exponential response, while fibre sliding was clearly the largest contributor to relaxation. The amount of stress relaxation and sub-structural reorganisation increased with increasing load increments, but fibre sliding was consistently the largest contributor to stress relaxation. A simple model of tendon viscoelasticity at the fibril and fibre levels has been developed, capturing this behaviour by serially coupling a Voigt element (collagen fibril), with two Maxwell elements (non-collagenous matrix between fibrils and fibres). This multi-level analysis provides a first step towards understanding how sub-structural interactions contribute to viscoelastic behaviour. It indicates that nano- and micro-scale shearing are significant dissipative mechanisms, and the kinetics of relaxation follows a two-stage exponential decay, well fitted by serially coupled viscoelastic elements.     

Development of residual strains in human vertebral trabecular bone after prolonged static and cyclic loading at low load levels

Development of irreversible residual strains in trabecular bone may be a mechanism by which age-related non-traumatic vertebral fractures occur. To investigate this concept, static and cyclic loading tests were conducted at low loading levels for cylindrical cores of cadaveric vertebral trabecular bone. Stresses were applied equivalent to elastic strains of either 750 or 1,500 microstrain. Creep strains were measured during the tests, which lasted for 125,000 seconds (about 35 h), and for an additional 125,000 seconds after complete unloading. Emphasis was placed on the residual strains that developed, defined as the strain remaining at the end of the unloading phase. The results indicated that appreciable residual strains did develop, and were similar for static and cyclic loading. Irrespective of the applied load levels and loading modes, the residual strains that remained after the unloading phase were similar in magnitude to the originally applied elastic strain. Extrapolation of the observed residual strains to full recovery indicated that the time that would be required for full recovery was over 20 times longer than the duration of the applied loads. These results indicate that human vertebral trabecular bone does not creep in a linear viscoelastic fashion at low stress levels, and that creep mechanisms dominate the residual strains regardless of the loading mode. Taken together, these findings support the concept that non-traumatic vertebral fractures may be related to long-term creep effects because the trabecular bone does not have sufficient time to recover mechanically from creep deformations accumulated by prolonged static or cyclic loading.     

The mechanical characteristics of cancellous bone at the upper femoral region

Mechanical behaviour of trabecular bone at the upper femoral region of human bones has been studied by compression tests on trabecular bone specimens removed from normal femora obtained at autopsy. Compression tests were performed along three different axes of loading on wet specimens and high loading rates. Femoral head specimens proved to be the strongest for any axis of loading.

Large variation in compressive strength and modulus of elasticity is seen within and between femoral bone samples. Anisotropy and differences in anisotropy for the different regions have been observed. A significant correlation between mechanical properties (σ max − E) and bone mineral content of the specimen was found.

Tests on whole bone structures demonstrate that removal of the central part of the trabecular bone at the proximal femur reduces the strength for impact loading considerably (± 50%).     

A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour

Nearly all soft tissues, among which the vascular tissue is included, present a certain degree of viscoelastic response. This behaviour may be attributed in part to fluid transport within the solid matrix, and to the friction between its fluid and solid constituents. After being preconditioned, the tissue displays highly repetitive behaviour, so that it can be considered pseudo-elastic, that is, elastic but behaving differently in loading and unloading. Because of this reason, very few constitutive laws accounting for the viscoelastic behaviour of the tissue have been developed. Nevertheless, the consideration of this inelastic effect is of crucial importance in surgeries—like vascular angioplasty—where the mentioned preconditioning cannot be considered since non-physiological deformation is applied on the vessel which, in addition, can cause damage to the tissue. A new constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these two inelastic phenomena is presented here and used to fit experimental stress–stretch curves from simple tension loading–unloading tests and relaxation test on porcine and ovine vascular samples.     

Simulation of creep in non-homogenous samples of human cortical bone

Characterising the mechanisms causing viscoelastic mechanical properties of human cortical bone, as well as understanding sources of variation, is important in predicting response of the bone to creep and fatigue loads. Any better understanding, when incorporated into simulations including finite element analysis, would assist bioengineers, clinicians and biomedical scientists. In this study, we used an empirically verified model of creep strain accumulation, in a simulation of 10 non-homogeneous samples, which were created from micro-CT scans of human cortical bone of the femur midshaft obtained from a 74-year-old female cadaver. These non-homogeneous samples incorporate the presence of Haversian canals and resorption cavities. The influence of inhomogeneity on the response and variation in the samples in both creep and stress relaxation tests are examined. The relationship between steady-state creep rate, applied loads (stress relaxation and creep tests) and microstructure, that is bone apparent porosity, is obtained. These relations may provide insight into damage accumulation of whole human bones and be relevant to studies on osteoporosis.     

Simulation of creep in non-homogenous samples of human cortical bone

Characterising the mechanisms causing viscoelastic mechanical properties of human cortical bone, as well as understanding sources of variation, is important in predicting response of the bone to creep and fatigue loads. Any better understanding, when incorporated into simulations including finite element analysis, would assist bioengineers, clinicians and biomedical scientists. In this study, we used an empirically verified model of creep strain accumulation, in a simulation of 10 non-homogeneous samples, which were created from micro-CT scans of human cortical bone of the femur midshaft obtained from a 74-year-old female cadaver. These non-homogeneous samples incorporate the presence of Haversian canals and resorption cavities. The influence of inhomogeneity on the response and variation in the samples in both creep and stress relaxation tests are examined. The relationship between steady-state creep rate, applied loads (stress relaxation and creep tests) and microstructure, that is bone apparent porosity, is obtained. These relations may provide insight into damage accumulation of whole human bones and be relevant to studies on osteoporosis.     

On the viscoelastic character of liver tissue: experiments and modelling of the linear behaviour

This paper presents the results of recent investigations into the viscoelastic character of liver tissue, especially the linear response. The linear domain of liver tissue has been determined by a strain sweep oscillation test. The fluid-like viscoelastic behaviour of liver tissue was confirmed by stress relaxation experiments. To demonstrate the consistency and quality of the experimental data, the relaxation spectrum was used to reconstruct the relaxation modulus, as well as the storage and loss moduli. The generalised Maxwell model was successfully employed to model the mechanical behaviour in the linear viscoelastic domain.     

Dynamic Mechanical Properties of the Cell Walls of Some Green Algae

Stress relaxation and constant rate of strain measurements weremade on the cell walls of the green algae Nitella, Acetabularia,and Penicillus. These plants have, as theirmain structural polysaccharide,cellulose, mannan, and xylan respectively. The stress relaxationmeasurements were made in the temperature range 0 to 50 °Cfor the interval 0.02 to 10³ s. A limited number of constantrate of strain curves were also obtained for comparison withtheoretical curves deduced from the relaxation data. All three algae exhibit typical fibrous behaviour but Acetabulariacould not be mechanically conditioned to give reproducible results.The time-temperature correspondence of the data and its relevanceto the molecular organization of the cell wall are discussed.Structural changes with temperature, deduced in previous workfrom rigorous matching of the relaxation curves at differenttemperatures, are now thought not to occur. The matching procedureonly appears to work because the curves lack distinctive featureson which to base the superposition. Comparison of stress relaxationand constant rate of strain data show that the cell walls ofthese algae exhibit non-linear viscoelastic behaviour and apparentchanges in the quasi-static modulus with temperature can beexplained satisfactorily in terms of reaction rate theoriesfor viscoelastic processes.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Viscoelastic properties of isolated collagen fibrils

Understanding the viscoelastic behavior of collagenous tissues with complex hierarchical structures requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar level. Using a microelectromechanical systems platform, in vitro coupled creep and stress relaxation tests were performed on collagen fibrils isolated from the sea cucumber dermis. Stress-strain-time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell-Weichert model. The fibrils showed an elastic modulus of 123 ± 46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell-Weichert model with a fast time response of 7 ± 2 s and a slow time response of 102 ± 5 s. The fibrillar relaxation time was smaller than literature values for tissue-level relaxation time, suggesting that tissue relaxation is dominated by noncollagenous components (e.g., proteoglycans). Each specimen was tested three times, and the only statistically significant difference found was that the elastic modulus is larger in the first test than in the subsequent two tests, indicating that viscous properties of collagen fibrils are not sensitive to the history of previous tests.     

A chemo-mechanical constitutive model for transiently cross-linked actin networks and a theoretical assessment of their viscoelastic behaviour

Biological materials can undergo large deformations and also show viscoelastic behaviour. One such material is the network of actin filaments found in biological cells, giving the cell much of its mechanical stiffness. A theory for predicting the relaxation behaviour of actin networks cross-linked with the cross-linker α-actinin is proposed. The constitutive model is based on a continuum approach involving a neo-Hookean material model, modified in terms of concentration of chemically activated cross-links. The chemical model builds on work done by Spiros (Doctoral thesis, University of British Columbia, Vancouver, Canada, 1998) and has been modified to respond to mechanical stress experienced by the network. The deformation is split into a viscous and elastic part, and a thermodynamically motivated rate equation is assigned for the evolution of viscous deformation. The model predictions were evaluated for stress relaxation tests at different levels of strain and found to be in good agreement with experimental results for actin networks cross-linked with α-actinin.     

Three-axial strain controlled testing applied to bone specimens from the proximal tibial epiphysis

Reproducibility of the determination of Young's modulus and energy absorption along the three axes of trabecular bone cubes was analysed by non-destructive compression to 0.5% strain using different testing protocols. These protocols included testing with and without pre-conditioning to a viscoelastic steady state, and different orders of test directions. Reproducibility of conditioned tests was generally better than that of non-conditioned tests. No major effect of changing the order of the test direction was found. Three-axial conditioned testing of cubes from the proximal tibial epiphysis of five humans revealed a global transverse isotrophy while most cubes showed orthotropy. The ratio between stiffness along the long axis of the tibia and the stiffness in the transverse plane was 3.7 +/- 0.4 (mean +/- 2 SE). The corresponding ratios for elastic energy storage and viscoelastic energy dissipation were 2.5 +/- 0.2. There was no difference between the relative energy loss during a testing cycle (loss tangent) in the three axes.