Volumetric Compliance Research Articles

Fabrication, characterization and numerical validation of a novel thin-wall hydrogel vessel model for cardiovascular research based on a patient-specific stenotic carotid artery bifurcation

In vitro vascular models, primarily made of silicone, have been utilized for decades for studying hemodynamics and supporting the development of implants for catheter-based treatments of diseases such as stenoses and aneurysms. Hydrogels have emerged as prominent materials in tissue-engineering applications, offering distinct advantages over silicone models for fabricating vascular models owing to their viscoelasticity, low friction, and tunable mechanical properties. Our study evaluated the feasibility of fabricating thin-wall, anatomical vessel models made of polyvinyl alcohol hydrogel (PVA-H) based on a patient-specific carotid artery bifurcation using a combination of 3D printing and molding technologies. The model’s geometry, elastic modulus, volumetric compliance, and diameter distensibility were characterized experimentally and numerically simulated. Moreover, a comparison with silicone models with the same anatomy was performed. A PVA-H vessel model was integrated into a mock circulatory loop for a preliminary ultrasound-based assessment of fluid dynamics. The vascular model's geometry was successfully replicated, and the elastic moduli amounted to 0.31 ± 0.007 MPa and 0.29 ± 0.007 MPa for PVA-H and silicone, respectively. Both materials exhibited nearly identical volumetric compliance (0.346 and 0.342% mmHg−1), which was higher compared to numerical simulation (0.248 and 0.290% mmHg−1). The diameter distensibility ranged from 0.09 to 0.20% mmHg−1 in the experiments and between 0.10 and 0.18% mmHg−1 in the numerical model at different positions along the vessel model, highlighting the influence of vessel geometry on local deformation. In conclusion, our study presents a method and provides insights into the manufacturing and mechanical characterization of hydrogel-based thin-wall vessel models, potentially allowing for a combination of fluid dynamics and tissue engineering studies in future cardio- and neurovascular research.

Sensorless loaded test for machine tool 2D compliance characterization using a tool-mounted roller bearing

Machine tool compliance may affect a machined part's accuracy. Volumetric compliance is measured by a test apparatus, named ball-roller bearing device (BRBD). The volumetric compliance measurement procedure consists of a loaded circular test with a commanded radial interference displacement between a roller bearing mounted at the tool holder and a ball attached to the table causing a radial force to develop. The interference displacement and force are calculated from internal CNC data such as the encoder readings and the servo motor current. Disturbance from non-compliant effects is reduced by analyzing the data from at least two interference displacement levels. The estimated forces are validated using a dynamometer table while the estimated compliances are validated with a compliant rig made of a parallelogram flexure mechanism. At the maximum displacement, approximately 55 % and 45 % of volumetric compliance came from the machine and the BRBD, respectively. In the most compliant direction, the machine volumetric, machine volumetric with the rig, and predicted machine volumetric with rig compliance values are 0.12, 0.49, and 0.50 μm/N, respectively.

An intravascular optical coherence tomography technique for in vivo carotid intima-media thickness measurement and mechanical characterization

Arterial characteristic parameters including the intima-media thickness (IMT), strain, strain rate, elastic modulus, and compliance may be unevenly distributed due to local lesion growth. Thus, a combination of the full-field distribution of these parameters may provide more effective guidance for the diagnosis and treatment of vascular diseases than either one of them. Here, we develop an intravascular coherence tomography (IVOCT) based full-field measurement method to characterize the mechanical properties of carotid artery in vivo. Image processing methods are proposed based on U-Net model and centroid detection for accurate IMT measurement. The strain, strain rate, incremental elastic modulus and volumetric compliance are obtained based on the geometric changes of intima-media. Ex vivo silica gel phantom tests demonstrate the accuracy of the vascular wall segmentation by the U-Net based method is greater than 92%, and the error for elastic modulus measurement is less than 2%. A porcine carotid artery was imaged by IVOCT in vivo, and the IMT and the mechanical characteristics during cardiac cycle were measured. The proposed method facilitates the study of the uneven pathological evolution of the vascular wall and provides an effective measurement method for the study of vascular mechanical properties in vivo.

Modelling and indirect measurement of machine tool equivalent joint compliances

Machine tools require high geometric accuracy and stiffness. Both of these characteristics affect the radius of a radially loaded circular test. By using various loads, the machine's volumetric compliance can be studied as a function of position, orientation, and load. Further processing of the data using a kinematic and compliance model of the machine allows the equivalent joint compliances to be estimated. This model also allows to produce the characteristic patterns of loaded telescopic double ball bar readings associated with each compliance term. The compliance model contains numerous superfluous and confounded terms that are pruned from the model. The analytical model is then used to produce a numerical identification Jacobian that is further applied to estimate the compliances from test data gathered at various force levels. By using all force data at once global compliance values are estimated whereas using only adjacent force level data allows observing the change in compliance with force. The new nomenclature is introduced where each compliance term has three subscripts. The first subscript is the direction of the displacement, the second subscript is the applied force direction, and the third subscript is the relevant joint or axis. The dominant compliances are the X-axis on-axis compliance C XXX (confounded with the lateral compliance of the Y-axis C XXY ) and C YYY (confounded with C YYX ). It is observed that as the load increases from 76 to 706 N (by increments of 126 N), the dominant compliances increase by around 5%. Type A uncertainties of the calculated compliances are estimated from repeated measurements and are found to be relatively small. Some non-dominant compliances, such as the torsional compliance of the Y-axis C CCY account for deflection of less than 0.5% of that for the main compliances and has a negative value which is mechanically unexpected. It is explained in detail in the results and discussion section.

Compliance of Textile Vascular Prostheses Is a Fleeting Reality

Compliance is considered to be a major property influencing the long term performances of synthetic vascular substitutes that could play a role in anastomotic false aneurysm and intimal hyperplasia stenosis onset. Over the last decades, manufacturers have tried to develop substitutes that mechanically mimic arterial properties and avoid a compliance mismatch at the anastomoses in particular. However, data are missing about how initial compliance properties could change with time. The goal of this study was to evaluate how the compliance of vascular grafts evolves under cyclic loading conditions invitro. The compliance of three different models of commercially available textile polyethylene terephthalate (PET) grafts was evaluated. Tests were performed with and without their original coating. Compliance was assessed with a specific device dedicated to measure the deformations undergone by a graft under cyclic pressure loading conditions, using image analysis software. In each experiment, image analysis was performed under 60 and 140mmHg pressure loading conditions at loading start (H0) and after three, six, and 24h (H3, H6, H24) loading time. Average radial, longitudinal, and volumetric compliance was calculated from the obtained images. Twenty-four samples were tested. Results demonstrate that all values decreased significantly within only a few hours. On average, the loss of compliance after 3h of cyclic loading ranged on average from 35% for longitudinal compliance to 39% for radial compliance and 37% (p<.050) for volume compliance. After 24h, the loss of radial, longitudinal and volume compliance was respectively 63±3%, 60.5±2% and 61±7%. In this invitro model, PET graft compliance has already decreased significantly within 3h. The rapid loss of compliance identified in this experimental study helps explain the mismatch mentioned in clinical observations.

Determination of consolidation parameters based on the excess pore water pressure measurement using a newly developed U-oedometer

In this study, the U-oedometer, a novel modified oedometer cell equipped with tailor-made needle probes, is developed to easily and accurately measure the excess pore water pressure ( $$\Delta u$$ ) during 1D consolidation tests and to determine the coefficient of consolidation ( $$c_{\text{v}}$$ ). The 3D printing technique is applied to make simple yet robust modifications to the conventional oedometer cell for facilitating the installation of the needle probes. The tailor-made needle probes are designed in such a way that the volumetric compliance is lowered to avoid measurement bias. Subsequently, the $$\Delta u$$ -based method is proposed to determine $$c_{\text{v}}$$ , with the target of avoiding the intervention of human judgement and therefore minimizing the degree of subjectivity. The experimental results demonstrate that the measured $$\Delta u$$ matches the theoretical values of the Terzaghi 1D consolidation theory, showing that the estimated $$c_{\text{v}}$$ is sufficiently reliable. In addition to the determination of $$c_{\text{v}}$$ , the U-oedometer allows additional measurements of other soil properties during consolidation, including the coefficient of permeability ( $$k$$ ) and the coefficient of earth pressure at rest ( $$K_{0}$$ ). It is observed that k decreases with the reduction in void volume, due to the increase in the effective vertical stress ( $$\sigma_{\text{v}}^{'}$$ ). Further, the secondary compression seems to be a continuation of the primary consolidation, where the soil sample continues to deform at a relatively slower rate, associated with the slight decrease in $$k$$ . A constant value of $$K_{0}$$ is observed at any value of $$\sigma_{\text{v}}^{'}$$ in the loading path, while during secondary compression, $$K_{0}$$ slightly increases with time.

Integrated model control of brake–wheel system using bond graph method

Brake system is an important actuator of most active safety systems equipped on vehicles. It combines with the wheel to make vehicle decelerate and finally stop it. Moreover, brake system is an electronic, mechanical, and hydraulic hybrid system; it contains some highly nonlinear characters, which is a challenge to system control. In this article, an integrated model of brake system and single-wheel system using bond graph method is developed, in which the nonlinear characters of the volumetric compliance effect of brake fluid and the resistance effect of valves are taken into consideration. The accuracy and reliability of the brake system is verified by experiment. Nonlinear sliding-mode controller as well as sliding-mode observer is proposed. The controller is used to modulate inlet and outlet valves control signals according to the vehicle states, which will lead to cancel the usage of wheel cylinder pressure sensors. The controller is analyzed by different tire–road friction coefficient conditions. The results show that the proposed integrated bond graph model is accurate, and the nonlinear sliding-mode control is reliable on valves control signal regulation.

A biomechanically optimized knitted stent using a bio-inspired design approach

Vascular implants have always been a key area of research in medical textiles. Knitted structures have been proven to be suitable for stent applications on the basis of their looped mesh geometry, structural flexibility and ease of manufacturing. However, there are biomechanical constraints of plain knit constructions that can result in clinical complications after implantation and hence cannot be ignored. This study reports a new segmented knit design inspired by structural metamerism observed in the body design of some invertebrate animals. Metamerism is the phenomenon of having a linear series of body segments fundamentally similar in structure, but assigned to perform different functions. It was hypothesized that utilization of this simple and yet effective biological design approach in stent construction could improve the degree of control for optimizing stent biomechanical properties. The proposed segmented stent was constructed by incorporating an elastic filament component into a polyethylene terephthalate knitted stent at specific intervals along its length, also known as the ‘Plating Technique’. This technique generates a structure with alternately arranged stiff and elastic knitted sections which equip the stent with vital structural support and volumetric compliance properties, respectively. The stent design parameters (filament diameter, loop length, segmentation plan) were optimized to achieve significantly better biomechanical performance (bending flexibility, compression resistance, volumetric expansion, longitudinal extensibility) than a plain knit stent.

Continuous assessment of carotid intima-media thickness applied to estimate a volumetric compliance using B-mode ultrasound sequences

Recent reports have shown that the carotid artery wall had significant movements not only in the radial but also in the longitudinal direction during the cardiac cycle. Accordingly, the idea that longitudinal elongations could be systematically neglected for compliance estimations became controversial. Assuming a dynamic change in vessel length, the standard measurement of cross-sectional compliance can be revised. In this work, we propose to estimate a volumetric compliance based on continuous measurements of carotid diameter and intima-media thickness (IMT) from B-mode ultrasound sequences. Assuming the principle of conservation of the mass of wall volume (compressibility equals zero), a temporal longitudinal elongation can be calculated to estimate a volumetric compliance. Moreover, elongations can also be estimated allowing small compressibility factors to model some wall leakage. The cross-sectional and the volumetric compliance were estimated in 45 healthy volunteers and 19 asymptomatic patients. The standard measurement underestimated the volumetric compliance by 25% for young volunteers (p < 0.01) and 17% for patients (p < 0.05). When compressibility factors different from zero were allowed, volunteers and patients reached values of 9% and 4%, respectively. We conclude that a simultaneous assessment of carotid diameter and IMT can be employed to estimate a volumetric compliance incorporating a longitudinal elongation. The cross-sectional compliance, that neglects the change in vessel length, underestimates the volumetric compliance.

Uterine Artery Protein, DNA, RNA and Volumetric Compliance During Early and Late Ovine Pregnancy.

Uterine Artery Protein, DNA, RNA and Volumetric Compliance During Early and Late Ovine Pregnancy.

A development of a technique for measuring the compliance of the textile vascular prostheses

The objective of this study is to develop a technique for measuring the compliance of the textile vascular prostheses without membrane. The principle of this test is to investigate the dimensional changes of prostheses, using imaging techniques, submitted to internal pressure. The internal compliance is broken into three categories: the radial compliance, the longitudinal compliance and the volumetric compliance. The results have shown a significant difference in compliance between the polyethylene terephthalate (PET) vascular grafts and the healthy host arteries.

Biomechanics of brain tissue

The dynamic behavior of porcine brain tissue, obtained from a series of in vitro observations and experiments, is analyzed and described here with the aid of a large strain, nonlinear, viscoelastic constitutive model. Mixed gray and white matter samples excised from the superior cortex were tested in unconfined uniaxial compression within 15h post mortem. The test sequence consisted of three successive load-unload segments at strain rates of 1, 0.1 and 0.01 s⁻¹, followed by stress relaxation (n=25). The volumetric compliance of the tissue was assessed for a subset of specimens (n=7) using video extensometry techniques. The tissue response exhibited moderate compressibility, substantial nonlinearity, hysteresis, conditioning and rate dependence. A large strain kinematics nonlinear viscoelastic model was developed to account for the essential features of the tissue response over the entire deformation history. The corresponding material parameters were obtained by fitting the model to the measured conditioned response (axial and volumetric) via a numerical optimization scheme. The model successfully captures the observed complexities of the material response in loading, unloading and relaxation over the entire range of strain rates. The accuracy of the model was further verified by comparing model predictions with the tissue response in unconfined compression at higher strain rate (10 s⁻¹) and with literature data in uniaxial tension. The proposed constitutive framework was also found to be adequate to model the loading response of brain tissue in uniaxial compression over a wider range of strain rates (0.01-3000 s⁻¹), thereby providing a valuable tool for simulations of dynamic transients (impact, blast/shock wave propagation) leading to traumatic brain injury.

Analytical studies on creep of sealed concrete under multiaxial stresses using a microplane model

The Poisson's ratio resulting from multiaxial creep of concrete, which has been reported by some experimental works, is controversial. When calculated from measured strains, Poisson's ratio is very sensitive to small experimental error. This sensitivity makes it difficult to determine whether the ratio varies with time or remains constant, and whether it has a different value with stress states. To understand precisely the properties of multiaxial creep of concrete, a new approach is needed. In this study, microplane model is applied to optimally fitting test data extracted from experimental studies on multiaxial creep of concrete. Double-power law is used as a model to represent volumetric and deviatoric creep evolutions on a microplane. Six parameters representing the volumetric and deviatoric compliance functions are determined from regression analysis, and the optimum fits accurately simulate the test data. Poisson's ratio is calculated from the optimum fits and its value varies with time. Regression analysis is also performed on the assumption that multiaxial creep of concrete is isotropic. Four parameters are determined for this condition, and the error between the optimum fits and the test data is slightly larger than that for the six-parameter regression results. The constant Poisson's ratio with time is obtained from the results of the four-parameter analysis, and the constant value can be used in practice without serious error.

Effects of Cryopreservation on the Viscoelastic Properties of Human Arteries

The purpose of this study was to use our newly developed mock circulation loop to determine the effects of cryopreservation on the common carotid artery (CCA) and the superficial femoral artery (SFA). Fourteen healthy arteries (7 CCA and 7 SFA) harvested from multiple organ donors between the ages of 18 and 35 years were tested before and after cryopreservation at -140 degrees C using dimethyl sulfoxide and the vapor phase of liquid nitrogen. Mean storage time was 4.2 months. The mock pulse rate was 60 beats/min and the following four systolic/diastolic pressures settings were used: 50/110, 80/140, 110/170, and 140/200 mm Hg. Simultaneous measurements of intra-arterial pressure and external arterial diameter were made using an intra-arterial pressure sensor and external piezoelectric sensors. Measured data were used to calculate pulsatility, volumetric compliance, stiffness, midwall radial arterial stress, Young's modulus, and the incremental modulus. After SFA cryopreservation, no significant changes were observed. Conversely, CCA cryopreservation led to a significant decrease in compliance and pulsatility and a significant increase in stiffness. Young's modulus, the incremental modulus, and midwall radial arterial stress did not change significantly. A clearcut decrease in hysteresis was observed after cryopreservation in the CCA. No evidence of structural changes was detected on light and scanning electron microscopy. Baseline findings in this study were consistent with classification of the CCA as an elastic artery and the SFA as a muscular artery. Cryopreservation had no effect on the viscoelastic properties of muscular arteries (SFA). Cryopreservation affected only values related to the cylindrical shape of the elastic arteries (CCA). It had no effect on values related to wall structure.

Biaxial Mechanical Properties of Carotid Arteries From Normotensive and Hypertensive Rats

Hypertension is known to decrease arterial distensibility and volumetric compliance, as reported from pressure-volume experiments and ring and strip studies of human and animal large arteries. However, recent data using noninvasive in vivo recording of vessel diameter suggest that the cross-sectional compliance of large arteries can be unchanged or even increased in hypertensive subjects. The present study was performed to test the hypothesis that differences between volumetric and cross-sectional compliance could be related to differences in biaxial mechanical properties in normotensive and hypertensive rats. In normotensive (Wistar-Kyoto [WKY]) rats and spontaneously hypertensive rats (SHR) we measured the simultaneous changes in length and diameter of in situ isolated carotid arteries submitted to static pressures (50 to 200 mm Hg by steps of 25 mm Hg each). Carotid artery diameters and lengths were determined by video microscopy and computer-assisted image analysis. At low transmural pressure (50 mm Hg), carotid artery diameter was 710 +/- 41 microns in WKY rats and 980 +/- 31 microns in SHR (P < .01). In response to pressure increases, the carotid diameter increased by 91 +/- 6% in WKY rats and by 41 +/- 4% in SHR (P < .01). In parallel, the percent increase in carotid length was much larger in WKY rats than in SHR (31 +/- 2% versus 7 +/- 1%, respectively; P < .01). In WKY rats, longitudinal distensibility causes significantly larger volumetric values than cross-sectional compliance values; in contrast, because of the very small longitudinal distensibility, volumetric and cross-sectional compliances are almost identical in SHR.(ABSTRACT TRUNCATED AT 250 WORDS)

Hyperhomocysteinemia-Induced Vascular Damage in the Minipig

Previous attempts in animals failed to reproduce the metabolic, pathological, and clinical situations encountered in homocystinuric patients. Minipigs on a methionine-rich caseinate-based diet, however, have a special long-lasting postprandial plasma accumulation of methionine, the metabolic precursor of homocysteine. We hypothesized that these minipigs develop hyperhomocysteinemia in the long term. Angiotensin-converting enzyme (ACE) inhibition prevents atherogenic alteration of viscoelastic functions of arterial pulsatility and compliance and reduces fragmentation of vascular elastic laminae in the minipigs. We consequently analyzed the therapeutic effects of the captopril-hydrochlorothiazide combination against the typical hyperhomocysteinemia-induced alterations of vascular elastic features. Thirty-two Götingen minipigs were randomized as control diet-fed (C), captopril (25 mg/d)/hydrochlorothiazide (12.5 mg/d)-treated C (C+Cp), caseinate-based diet-fed (M), and M+Cp minipigs. After 4 months, M and M+Cp animals had hyperhomocysteinemia (9.64 +/- 4.10 mumol/L, n = 16) compared with C and C+Cp minipigs (5.67 +/- 1.14 mumol/L, n = 16) (P < .05). In the M group, one minipig died from thromboembolic syndrome, and one had pulmonary infarction. M minipigs presented with systolic-diastolic hypertension and extended reactive hyperemia, as well as a mega-artery syndrome in hyperpulsatile arteries due to expanded volumetric compliance, curtailed stiffness, strengthened vascular tension, and prevalence of the viscous wall component. In their arterial tree, hypertrophic endothelial cells covered a thickened subendothelial space. Major elastic lamina dislocations were observed, as well as hypertrophy and reorientation of smooth muscle cells, resulting in the settlement of spreading pathways for medial cells between muscular laminae. In C+Cp and M+Cp animals, serum and lung ACE activity were inhibited by 74% and 40%, respectively. Although the treatment with captopril-hydrochlorothiazide did not modify the hyperhomocysteinemia per se, the therapeutic effects of the drug combination are made evident by the absence of death and ischemic diseases in the M+Cp group. Specifically, the drug combination prevented diastolic hypertension and improved aortic blood flow by normalizing peripheral resistances, abolished the vascular hyperpulsatile characters, and restrained the fragmentation and the splitting of elastic fibers in capacitance arteries. In contrast, the drugs slightly prevented systolic and mean hypertension. In addition, the aortic stiffness and stress response remained altered and vascular smooth muscle cell hypertrophy was still observed in the M+Cp group. In minipigs, the present methionine-rich caseinate-based diet induced hyperhomocysteinemia, which reproduces the metabolic and histopathological situation found in homocysteic patients. Our results show that hyperhomocysteinemia-induced vascular alterations favor the viscous component of the wall rheology to the detriment of the elastic component. Furthermore, they extend to hyperhomocysteinemia the therapeutic effects characteristically shared by ACE inhibitors in association with hydrochlorothiazide against the atherogenic activation of elastinolytic processes.

Elimination of membrane compliance in undrained triaxial testing.II. Mitigation by injection compensation

A computer-controlled injection-correction system has been developed to continuously and completely mitigate the adverse effects of membrane compliance for undrained testing of granular soils. The system was developed with the explicit idea that it could be quickly and easily adapted to most any up to date triaxial testing apparatus. A relationship between effective confining stress and volumetric compliance can be predetermined, as volumetric-compliance errors have been shown to be a direct and repeatable function of effective confining stresses for a given material. The predetermination of volumetric-compliance errors represents a suitable basis for control of injection–removal compensation for membrane compliance during undrained testing. A closed-loop computer-controlled system continuously monitors changes in sample effective confining stresses such that precalculated volumetric-compliance errors can be continuously offset by injecting or removing volumes of water equal to those errors throughout the duration of each test. The validity of the injection system was verified by comparison of the compensated tests to results of large-scale (300 mm diameter) tests of the same materials, as compliance effects for large-scale tests of this material were negligible. Key words : membrane, penetration, compliance, undrained testing, triaxial, compensation, liquefaction.

A Compliant Biological Vascular Prosthesis

One requirement of the mechanical parameters for an acceptable vascular prosthesis is compliance. The compliance of a vascular prosthesis is defined as the fractional change in luminal volume per unit change in applied pressure. A compliant prosthesis has been correlated to prosthesis patency and long-term efficacy in an animal study. However, there have been very few reports on how to manufacture a compliant prosthesis. It is the objective of this study to research the processing methods to manufacture a reasonably compliant vascular prosthesis. A new fixative, polyepoxy compound, was used to fix an artery. The arteries were fixed under different degrees of longitudinal retraction. By locking in the collagen micro-structure at an overly relaxed state and then crosslinking said collagen, the resulting biological prosthesis exhibited extreme compliance and pliability. A prosthesis matching its arterial origin in tensile modulus was achieved by crosslinking an artery at its 45% retraction longitudinally. This flexible prosthesis showed a volumetric compliance index of 18.4 +/- 0.9 % delta/100 mmHg and a longitudinal tensile modulus of 942 grams/cm2. Our current study indicated that a prosthesis fixed with polyepoxy compounds has shown more pliability than that with glutaraldehyde. Further animal study to correlate prostheses patency to different degrees of compliance is needed to confirm this proposed manufacturing approach.

An Injection-Correction System to Mitigate Membrane Compliance

Abstract A computer-controlled injection correction system has been developed to continuously mitigate the adverse effects of membrane compliance for undrained testing of granular soils. The system was developed with the explicit idea that it could be quickly and easily adapted to almost any up-to-date triaxial testing apparatus. Methodology of the system is based on the direct and repeatable relationship between effective confining stress and volumetric compliance. A closed loop computer-controlled system continuously monitors changes in specimen effective confining stresses such that precalculated volumetric compliance errors may be continuously offset by injecting or removing volumes of water equal to those errors throughout the duration of each test. Implementation of the injection-correction system was demonstrated for a uniformly graded medium sand, exemplifying the potential usefulness of such a system.

Dilatancy hysteresis in rocks: a generalized ramberg-osgood approach

A review and interpretation of representative experimental data is given on the cyclic properties of rocks. Emphasis is placed on data concerning the development of hysteresis of volumetric strain. Volumetric strain occuring during stress cycles well below the peak stress value exhibits a remarkable hysteresis. The strain is either compactive or dilative. The hysteresis varies both in form and in width depending on the state of prior rock damage. Specifically, the volumetric compliance may either be higher in loading than in reloading or vice-versa, depending on the state of rock damage prior to the cyclic loading. Finally, the hysteresis form significantly varies with the number of cycles. A model is developed, which is a generalization to three dimensional cyclic loading of the well known Ramberg-Osgood equation for amplitudes. The model is based on an earlier idea of discretized kinematic hardening plasticity with a piecewise path independent stress - difference/strain - difference law. A particular anisotropic form of such a law is discussed in the light of the collected phenomenological data. The law is valid by pieces and is expressed through a non-linear hyperelastic relationship for amplitudes. The pieces are defined by characteristic points of the stress trajectory, like stress rate reversals or turns. To ensure the incremental continuity of such a law at reversals in non-radial cycles, the constitutive tensor of initial hysteresis moduli is assumed as a function of the reversal intensity. All irreversible strain effects of the material behaviour are lumped in the discrete points of the stress rate reversals, where material property parameters are adjusted. A special form of the constitutive functions is presented in the companion paper [12].