Instantaneous Stiffness Research Articles

Thermal Expansion of Elastic-Plastic Composite Materials

Exact relationships are derived between instantaneous overall thermal stress or strain vectors and instantaneous overall mechanical stiffness or compliance, for two binary composite systems in which one of the phases may deform plastically. Also, the local instantaneous thermal strain and stress concentration factors are related in an exact way to the corresponding mechanical concentration factors. The results depend on instantaneous thermoelastic constants and volume fractions of the phases. They are found for fibrous composites with two distinct elastically isotropic or transversely isotropic phases, and for any binary composite with elastically isotropic phases. The results indicate that in the plastic range the thermal and mechanical loading effects are coupled even if the phase properties do not depend on changes in temperature. The derivation is based on a novel decomposition procedure which shows that spatially uniform elastic strain fields can be created in certain heterogeneous media by superposition of uniform phase eigenstrains with local strains, caused by piecewise uniform stress fields which are in equilibrium with prescribed surface tractions. The method is extended to discretized microstructures, and also to the analysis of moisture absorption and phase transformation effects on overall response and on local fields in the two composite materials.

Biomechanical identification of injury to an intervertebral joint

Experimental and analytical finite element analyses were conducted to determine the effects of external mechanical load which produce spinal injury. Experimental studies on human thoracolumbar vertebra-disc-vertebra structures (functional unit) under axial compressive loading demonstrate four different phases in the load-deflection response. They are the initial ambient or preload, physiological, traumatic and post-traumatic loading phases. Initiations of trauma seen in the form of microfailures was defined as the point at which the instantaneous tangent stiffness of the structure begins to decrease for the first time. For both normal and degenerate segments, load at this level (microfailure load) was found to be about 80% (22.5%) of the conventional traumatic failure load which is defined as the point at which the stiffness of the structure suddenly drops to zero. However, the microfailure and conventional traumatic failure loads in degenerated discs were significantly lower than normal discs. Degenerate spines were more flexible than normal spines. Hypotheses regarding the presence of these microfailures in the spinal segment. which were not obvious on radiographs. were confirmed by morphological cryosections. Analysis of data with regard to energy absorbing characteristics indicated that the energy absorbed by normal discs between microfailure and traumatic failure was 43% of total energy required for traumatic failure. In degenerate discs this value was 28%. This may be an important index in determining the reserve strength in the spinal structure after the initiation of injury. A planar axisymmetric finite element model was constructed using geometry from experimental records. Material and geometrical nonlinearities were handled

Tension transients during steady shortening of frog muscle fibres.

Single intact fibres from frog muscle at 0-1 degrees C were stimulated to produce isometric tetani at a sarcomere length of about 2.25 micron, using a spot-follower apparatus to control the length of the central part of a fibre. When the plateau of the tetanus was reached the fibre was forced to shorten by applying a step and ramp length change in an approximation to an isotonic release. When tension had reached a steady level, Ti, during shortening, tension transients were elicited by applying step changes of length, complete within 0.2 ms, ranging from a stretch of 1.5 nm per half-sarcomere to a release of 6 nm per half-sarcomere. The tension transients recorded during shortening were qualitatively similar to those previously recorded in isometric tetani. There were four phases: phase 1, the change of tension during the step; phase 2, a rapid partial recovery of tension; phase 3, a delay or reversal of recovery; phase 4, a slower recovery of tension to the level before the step was applied. Measurements were made of the extreme tension, T1, attained during a step, and the level, T2, to which tension recovers in phase 2. The excursion of tension, [T1-Ti], during a small step of given size, fell with increase of shortening velocity, reaching about 40% of the isometric value near the maximum velocity of shortening. T2 fell as shortening velocity was increased and the fraction of steady tension recovered, T2/Ti, also decreased, so that the proportion of tension recovery in phase 4 increased. All the recovery phases became progressively more rapid with increase of shortening velocity. The early tension response was matched with a delay-line simulator so as to estimate the value of the instantaneous stiffness. Stiffness during shortening was found to decrease approximately linearly with tension, reaching about 35% of the isometric value as tension approached zero. It was impossible to match the early tension response in a rapidly shortening fibre without assuming decreased stiffness. The decline of stiffness is interpreted as due largely to reduced number of attached cross-bridges, but quantitative estimates would be affected by possible filament compliance and non-linearity of cross-bridge stiffness. The decrease in T2 also suggests fewer cross-bridges are attached as shortening velocity increases, but uncertainties about the processes determining phase 2 during shortening do not permit a precise estimate of stiffness to be made.(ABSTRACT TRUNCATED AT 400 WORDS)

Strain-space plasticity analysis of fibrous composites

Abstract A strain-space formulation of plasticity theory for metal matrix fibrous composites is discussed. Specific results are obtained for a composite system in which the fibers are elastic until failure, while the matrix is of the Mises type, with kinematic hardening. The material model of the composite is based on the assumption that the fiber diameter is vanishingly small, and the fiber volume fraction is finite. Explicit expressions are obtained for instantaneous strain concentration factors in the phases, for the instantaneous overall stiffness, and for the overall loading surface and hardening rule during mechanical straining along a prescribed path.

A maximum-dissipation principle in generalized plasticity

It is shown that in large-deformation generalized plasticity a local maximum-dissipation postulate is equivalent to the condition that the plastic strain rate (in the sense of Rice) cannot oppose the total strain rate, when strain space is regarded as a Riemannian manifold with the instantaneous Lagrangian tangent elastic stiffness as the metric tensor. From this condition, normality conditions in strain space (in this sense) and in the space of the second Piola-Kirchhoff stress (in the usual sense) are derived. With the additive decomposition of strain, the loading surface has essentially the same properties as in infinitesimal-strain plasticity. For the multiplicative decomposition, approximate normality rules are derived.

Creep trajectories for beech during moisture changes under load

Tensile creep experiments during concurrently changing humidities have been performed on Beech. There appears to be a “threshold compliance” level below which any change of moisture content, whether sorption or desorption, causes an increase in creep. Above this compliance level the creep appears to follow creep trajectories which form a shallow trough shape, for either humidification or dehumidification, when plotted with compliance as ordinate and moisture content as abscissa. The level of the threshold appears to vary with the instantaneous stiffness level of the material, measured at uniform moisture content. The mechano-sorptive creep below the threshold compliance could be explained in terms of the stress bias during hydrogen-bond breaking and remaking.

Force-velocity relation and stiffness in frog single muscle fibres during the rise of tension in an isometric tetanus.

The force-velocity (T-V) relation and the force-extension (T1) relation are determined at present times and at increasing isometric tensions during a tetanic contraction in frog single muscle fibres in which the passive compliance in series with the sarcomeres was made very small. The slope of the instantaneous T1 relation, the fibre stiffness, increases roughly in proportion to the level of the rising isometric tension at which the measurements were made. The value of V0 (the velocity of shortening under zero load) is time-independent, whereas the force T exerted during shortening at any velocity V lower than V0 increases gradually with time after the beginning of the tetanus volley and attains its steady state level before the isometric tension has attained the tetanus plateau and the fibre stiffness its final value. It is concluded that the delay of the development of the isometric tension and of the fibre stiffness with respect to the development of the T-V relation is determined by a specific factor of the contractile process. It is interesting to note that in a cross-bridge model of contraction, in which the value of the rate constant for cross-bridge formation is moderate, the recruitment of actin sites which is measured by the characteristics of the instantaneous T-V relation, is expected to lead significantly the actual cross-bridge formation, which is measured both by the instantaneous isometric tension and by the instantaneous stiffness.

Inelastic Tangential Stiffness for 2‐D Frames

A general instantaneous or tangential stiffness matrix for a plane frame member is presented. Expressions in the form of integral equations are developed for members having axial forces and arbitrary material properties, distributed loads and moment-axial force-curvature relations. The expressions are used for w sections of elasto-plastic material and several examples of plane frames are loaded to failure. Experimental results reported in the literature for plane frames tested to inelastic failure are closely matched by the analytical technique.

Muscular contraction: kinetics of crossbridge attachment studied by high-frequency stiffness measurements.

Instantaneous stiffness of frog skeletal muscle, an indication of the proportion of attached crossbridges, was determined drug the tetanus rise and after a step length change imposed during the tetanus plateau. During the onset of contraction as well as after a step, the ratio of stiffness to force differed from that determined during the tetanus plateau. The data after a step are predicted by the Huxley-Simmons model of muscular contraction, but the results during the rise suggest that a long-lived state may exist between crossbridge attachment and force generation.

Isometric contractile properties and instantaneous stiffness of amphibian skeletal muscle in the temperature range from 0 to 20 degrees C.

The isometric contractile properties of frog (Rana pipiens) and toad (Bufo bufo) sartorii have been studied over the temperature range from 0 to 20 degrees C. The isometric twitch tension was found to vary considerably between these two species and between muscles in the same species. Between 0 and 4 degrees C there was very little change in maximum isometric twitch tension. Between 4 and 12 degrees C several muscles from frog or toad showed a potentiation of twitch tension whereas others showed a decline. Over this temperature range the toad sartorii consistently demonstrated a greater potentiation. By 12 degrees C a steady decline in twitch tension in both muscles was seen as the temperature range the toad sartorii consistently demonstrated a greater potentiation. By 12 degrees C a steady decline in twitch tension in both muscles was seen as the temperature approached 20 degrees C. The maximum isometric tetanic tension recorded between 18 and 20 degrees C increased fractionally to an average of 1.504 +/- 0.029 (n = 4) for frog sartorii and to 1.377 +/- 0.008 (n = 5) for toad sartorii. The time to peak twitch tension and the half-relaxation time decreased markedly with an increase in temperature. Moreover, the half-relaxation time was reduced by a greater proportion than the time to peak twitch tension. Measurements of instantaneous stiffness by controlled velocity releases from the plateau of isometric tetani revealed that the large increase in isometric tetanus tension as the muscle was warmed was not accompanied by a corresponding increase in the total number of active cross-bridges. The possibility that a decreased availability of intracellular Ca2+ ions at the contractile sites contributing to the fall of isometric twitch tension at elevated temperatures is discussed. The possibility exists that at elevated temperatures a change inthe intrinsic contractile ability of the muscle occurs which produces an increased tension per cross-bridge.

The relation between stiffness and filament overlap in stimulated frog muscle fibres.

1. Tension transients were recorded at sarcomere lengths from 2.0 to 3.2 mum in isolated fibres from the tibialis anterior muscle of frogs during tetanic stimulation at 0-1 degrees C. 2. The length of a selected portion of the fibre was controlled by feed-back from a spot-follower device. The step was complete in 0.2 ms and the natural frequency of the force transducer was 10.8 kHz. 3. The transients were analysed by comparing the tension record with the output of an analogue circuit (delay line) which contained components representing (a) force transducer response, (b) fibre inertia, (c) viscosity and inertia of surrounding fluid, (d) passive stiffness and viscosity of the fibre, (e) tendon compliance and (f) stiffness and early tension recovery of the contractile apparatus. 4. In releases at different sarcomere lengths, the instantaneous stiffness and the early tension recovery attributed to the contractile apparatus varied almost exactly in proportion to the developed tension. In the later phases of the transient there were minor deviations from proportionality. 5. The results confirm that the entire transient represents events in the cross-bridges. 6. At full overlap, the compliance attributable to the cross-bridges is at least 80%, and probably well over 90% of the measured instantaneous compliance of the fibre. Stiffness can therefore be used as a measure of the number of attached cross-bridges. 7. The amount of instantaneous sliding movement of thick relative to thin filaments required to bring tension in a cross-bridge from the isometric value to zero is about 3.9 nm if filament and Z-line compliance are negligible, as suggested by the results. It is not however excluded that filament compliance, though small, may be sufficient to reduce this figure to 3.5 nm or possibly 3.1 nm. 8. The responses to quick stretch, unlike those to release, could not be satisfactorily matched with the delay line. The deviations suggest that the instantaneous elasticity is non-linear in stretches. 9. In resting fibres at all sarcomere lengths, the first peak of the tension response was determined chiefly by fibre inertia and viscosity, rather than elasticity.

Activation of the contractile system of insect fibrillar muscle at very low concentrations of Mg2+ and Ca2+.

The mechanical properties of single fibres and fibre bundles of glycerinated dorsal longitudinal muscle from lethocerus maximus were investigated in ATP-salt solutions containing only trace concentrations of free Ca2+ (pCa>9). A reduction in the magnesium concentration (pMg ∼ 7) resulted in an increase in the instantaneous stiffness of glycerinated insect flight muscle fibres, though very little accompanying tension was developed. Stiffness was measured either using small amplitude sinusoidal length changes of high frequency (1 kHz) or rapid rectangular form length changes. The ratio of stiffness to tension in solutions free of added magnesium and calcium was equal to or greater than that obtained from the tissue in the rigor state, and much larger than that obtained in the presence of both magnesium and calcium. Extrapolation of the linear part of the change-tension relationship (obtained during rapid length changes completed within 0.3 ms) back to zero tension indicated that the elastic elements of attached crossbridges were less extended under conditions of Mg2+-deprivation than during Ca-activation in Mg2+-rich solution. Following a quick stretch a delayed tension development similar to that obtained in the presence of magnesium and calcium ions was observed. The rise in tension was delayed with respect to the accompanying rise in stiffness and reached a peak value after about 2 s. Similar tension transients followed a subsequent release. The possibility that an unusually slow corss-bridge cycle might be responsible for these slow transients was suggested by the finding that the fibres showed a very low ATPase activity under these conditions which could be slightly activated by stretches. On increasing the free Ca2+ concentration during magnesium deprivation, the time course of the stretch induced tension transients became faster, while stiffness and the steady state tension rose to reach a ‘high tension state’ at aboutpCa 6.5.

Nonlinear analysis of cavities in rock salt

The paper covers some material and computational aspects of the rock mechanics of leached cavities in salt. A material model is presented in which the instantaneous stiffness of the salt is obtained by interpolation between the unloaded state and a relevant failure state. The model enables prediction of short term triaxial behaviour from uniaxial stress-strain curves. Key results from a nonlinear finite element calculation of a gas-filled cavity are given, and the general features are related to a simple nonlinear method of stress evaluation.

Isometric contractile properties of extensor digitorum longus muscle of the dystrophic hamster BIO 40.54

The contractile properties, including instantaneous stiffness, of normal and the genetically dystrophic BIO 40.54 hamster extensor digitorum longus muscle were compared at 60, 120, 170, 220, and 320 days of age. A general impairment in tension-generating ability of the dystrophic muscles was observed which correlated well with a corresponding change in the measured stiffness. A decrease in the mechanical half-relaxation time was observed at all age groups except 120 and 170 days of age. The data presented are discussed in terms of a defect in the Ca 2+ regulatory functions and/or energy production and utilization in the dystrophic skeletal muscle cell.

Isometric tension and instantaneous stiffness in amphibian skeletal muscle exposed to solutions of increased tonicity

The instantaneous elasticity and maximum isometric tetanic tension of isolated frog and toad sartorii have been measured in hypertonic Ringer solution. Although the mechanical response of contraction muscle continued to decrease as the tonicity of the bathing solution was increased to 1.26 X R, 1.52 X R, and 2.04 X R, a similar change in the instantaneous stiffness could not be shown. This finding was not expected on the basis of our current model of the cross-bridge mechanism which predicts that the instantaneous stiffness is a measure of the total number of tension-generating cross-bridges formed in a stimulated muscle. The compatability of our findings with an electrostatic theory of the cross-bridge mechanism proposed by Iwazumi (1970) is discussed.

Cross bridges as the major source of compliance in contracting skeletal muscle.

IT has been established that the elastic properties of contracting skeletal muscle are dominated by structures within the sarcomeres. We have demonstrated1 a decrease in the instantaneous stiffness of a contracting toad sartorius at lengths greater than body length (l0) coinciding directly with the decline in maximum isometric tetanic tension. This finding suggested that the cross bridges between actin and myosin filaments may be the major site of compliance in an active muscle, a conclusion in accordance with that reached by Huxley and Simmons2 on the basis of a study of tension transients following rapid releases in single fibres of frog semitendinosus. Also in agreement with this conclusion is the work of Cleworth and Edmond3 who used a light diffraction technique to demonstrate that any change in the length of the sarcomere during isometric contraction was less than 50 A, or less than 0.1 %. These studies suggest that the cross links could be the major site of elasticity within the sarcomere, but it is possible that significant compliance resides in the unbonded thin filaments, although X-ray diffraction studies of contracting muscle4 indicate that the I-filament length does not increase by more than 0.02%. The experiments reported here were carried out to extend previous observations of muscle stiffness into the range of lengths below the resting length.

Left ventricular function during systole and diastole in mitral incompetence

Left ventricular function during systole and diastole was studied in 17 young patients with mitral incompetence and sinus rhythm; 13 had established chronic mitral incompetence, 3 had prolapsing posterior leaflet syndrome and 1 had ruptured chordae tendineae. In chronic mitral incompetence and prolapsing posterior leaflet syndrome, clinical disability was related to a low forward stroke index, a large regurgitant volume and tall left atrial V wave. The defect was mechanical, and the ventricle had adapted by compensatory dilatation so that overall systolic function was normal, as measured by peak rate of rise of left ventricular pressure ( dP dt ) and ejection fraction. The dilated ventricle operated at a normal or slightly increased end-diastolic pressure, a consequence of normal resting tension (O point) and a more elastic myocardium with a low rate of change of instantaneous stiffness (m) and a flattened diastolic pressure-volume curve. In acute mitral incompetence, systolic function was normal but the sudden volume overload caused the unprepared ventricle with an almost normal modulus of elasticity and slope (m) to ascend its pressure-volume curve in diastole. End-diastolic pressure was increased.

Haemodynamic effects of beta blockade in hypertrophic cardiomyopathy using Sectral (acebutolol; M & B 17803A)

The immediate haemodynamic effects of Sectral (acebutolol; M&B 17803A) were measured in eight patients with hypertrophic cardiomyopathy (HOCUM) after intravenous administration of 10 mg and then an additional 20 mg of the drug. Resting heart rate decreased (-8%) and stroke and cardiac index were reduced (-20%) after the 30-mg dose. Peak LVdp/dt, Vmax, and ejection fraction were slightly depressed. Sectral reduced left ventricular outflow obstruction in systole and the early- and end-diastolic pressures. End-systolic vlueme increased and ventricular distensibility improved. The rate of change of instantaneous stiffness of the left ventricle (slope of the diastolic pressure-volume curve) remained constant.