Hook's Law Research Articles

Etude experimentale de la conductivité thermique de fluides non-Newtoniens sous cisaillement application aux solutions de Carbopol 940

Polymers exhibit physical properties intermediate to those of elastic solids which follow Hook's law and those of viscous fluids which behave according to Newtonian shear law, therefore, they are designated non-Newtonian fluids. Under a steady shear, non-Newtonian fluids exhibit not only the shear rate dependent viscosity but also their thermal conductivity changes with shear. It is the purpose of this paper to report on our investigation on the dependence of thermal conductivity on shear rate. Aqueous solutions of Carbopol 940 have been used as test fluids to investigate the effects of temperature and shear rate on thermal conductivity.

A method of measuring pupil-blocking force in the human eye.

A new method of measuring pupil-blocking force in the human eye is reported. The anterior segment of the eye was photographed using a slit-lamp apparatus and the color photographs then analyzed with a digitizer for converting the optical image of the anterior eye segment into the true image using a computer. This technique permits precise estimation of the relative positions of the anterior chamber, such as pupil margins, iris root insertions, and central anteroposterior pupillary axis. Pupil-blocking force was calculated from these data according to Hook's law. There was a statistically significant difference in pupil-blocking force between normal subjects (mean +/- SD, -0.027 +/- 0.029 lambda; lambda = Young's module) and patients with primary angle-closure glaucoma (mean +/- SD, +0.134 +/- 0.068 lambda) (P less than 0.001). In patients with primary angle-closure glaucoma, during an acute attack the pupil-blocking force was markedly high. This technique for measuring pupil-blocking force will provide valuable data for the analysis of the mechanism of primary angle-closure glaucoma.

Contraction and metabolic activity of electrically stimulated cardiac myocytes from adult rats.

An experimental setup has been developed, which allows electrical stimulation of cardiac myocytes and simultaneous measurement of oxygen consumption, lactate production, extent of shortening and of substrate uptake. In resting cells and in cells stimulated with 120 to 480/min the oxygen consumption ranged from 25 to 100 microliter/min X gww, with a linear relationship between rate of stimulation and oxygen consumption (VO2). When using 5 mM glucose plus 2 mM pyruvate or 10 mM lactate as substrates, isoproterenol (8 X 10(-8) M) augmented contraction and VO2 at all rates of stimulation. Assuming Hook's law for passive elastic behavior for the contracting myocytes over the length change observed, a good correlation exists between the degree of cell shortening calculated from VO2 per beat and the degree of contraction measured. This correlation can be used as a measure of the economy of O2 utilization.

Effect of pharmacologically active agents upon the human myometrium in vivo. Theoretical and technical aspects.

Smooth muscle is capable of contracting in response to mechanical distension. Even though the structural basis for this phenomenon is not completely clear, a method has been designed by which the contractile response to mechanical distension of the human uterine muscle can be determined in vivo. The distension is enforced sinusoidally within the frequency range 0.006-1 Hz, by filling and emptying a rubber balloon introduced into the uterine cavity. A tonometric index of the myometrium is defined using Hook's law after the approximate evaluation of a contratile modulus and the tangential stress and strain in the innermost layer of the uterine wall. The regression lines of tonometric indices upon frequency are determined and intercepts of the equations for basic conditions are compared with those obtained during the administration of pharmacologically active agents. In a pilot study, this system, now modified for the evaluation of drug effects during early pregnancy, has been tested and found reliable.

The strain-energy density function of the urinary bladder.

In previous articles we proposed a finite deformation theory of the urinary bladder. The present paper is the third article in a series of these works. Our aim is to study more carefully the strain-energy density function W of the urinary bladder, because the determination of this function W is one of the central problems in the finite deformation theory of a hyperelastic continuum. We found that, except for an infinitesimally small deformation region where Hook's law is valid, the third term in W of the Valanis-Landel type takes much smaller values than the other two terms, so that this term can be neglected. Hence, the formula giving the true stress t1 for the uniaxial extension test is reduced to the same one as the true stress t for the equal-biaxial extension test. This may be useful in practical applications.

Calculation of the mechanical stresses in a high field magnet by means of a layer model

A new model suitable for the calculation of azimuthal stresses σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</inf> well as the radial stress α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</inf> in different kind of layers (non current carrying and current carrying) is developed. For the calculation the validity of Hook's law is assumed. The equilibrium condition for a layer volume element leads to a differential equation for the radial component u of the displacement vector. The solution of the differential equation can be used to obtain a system of linear equations for the displacements of the boundaries of the layers. Examples are given where layer models are compared with models averaging the elastic constants of the used materials. In the case of intermediate layers with relatively small E-modul both models give approximately the same results for the stresses in the current carrying layers. In the case of a relatively large E-modul of the intermediate layers the layer model yields more. realistic results for the stresses in both types of layers. Finally the calculations shown for the layer model can be easely extended when additional reinforcing layers are considered.

DERIVATION OF CONSTITUTIVE EQUATIONS FOR IDEAL PLASTIC MATERIAL FROM CLAUSIUS-DUHEM INEQUALITY

The constitutive equations for an ideal plastic material are derived naturally from the Clausius-Duhem inequality and the yield function. An admissible thermodynamic process must satisfy the inequality. If there is no restriction on the stress and temperature rates in the elastic state, then we obtain the elastic constitutive equations. However, if they are related in the yield state by means of the yield criterion, then the constitutive equation may contain a term that corresponds to the plastic flow of the ideal plastic material. The isotropic case is considered, and we obtain Hook's law, the thermal isotropic expansion, and the Levy-St. Venant flow rule.

In vivo determination of the stress-strain relation of the human myometrium.

Smooth muscle is capable of responding by contraction to mechanical distension. Neither the structural basis for this phenomenon nor the theories of smooth muscle behavior are completely clear. A method is now described by which the contractile response to mechanical distension of the human uterine muscle in vivo is determined. The distension is enforced sinusoidally within the frequency range 0.005-2 Hz, by filling and emptying a rubber balloon introduced into the uterine cavity. A tonometric index of the myometrium is defined using Hook's law after the approximate evaluation of a contractile modulus and the tangential stress and strain in the inner layer of the uterine wall. The regression line of indices upon frequency is calculated and intercept and regression coefficient of the equation for basic conditions are compared with those obtained during administration of pharmacologically active agents. It is suggested that the differences in intercepts and regression coefficients be used as a measure of drug effect.

Behaviour of egg shell under compression in relation to deformation measurements1

1. The egg does not completely obey Hook's law when compressed between flat surfaces. 2. Deformation was not completely reversible due to crumbling at the points of contact with the force. 3. There are three components of non‐destructive deformation: (a) local deformation at the points of contact, (b) local surface crumbling and (c) deformation of the over‐all shell structure. 4. The latter two are uncontrolled, contributing to errors in the prediction of fracture force and depending on the non‐destructive force applied. 5. These effects should be considered in test selection and interpretation. 6. The optimum non‐destructive force was found to be 1 kg.

A geometric generalization of Hooke's law

A geometrical version of Hooke's law for prestressed materials is established using techniques suggested by general relativistic elasticity theory. The concept of dragged (D) tensor fields is used to present a simple coordinate independent form of Hook's law for prestressed materials: P̄ij = D Pij + Cijkl ηkl, where Pij is the initial stress, Cijkl are the second order elastic coefficients, and ηkl is the strain. The effects of initial stresses on Hooke's law have recently been given empirical significance by Wallace and the work here agrees with and simplifies his formalism. The elastic behavior of neutron stars, which requires a general relativistic treatment, is a case in which large prestresses of the type considered here are as important as the elastic modulii themselves. The version of Hooke's law developed here is the classical limit of the general relativistic theory of stellar elasticity. A simple derivation of the seismic response of a Newtonian self-gravitating body to elastic perturbations is presented.

An exact effective stress law for elastic deformation of rock with fluids

The exact expressions for the effective stress 〈σij〉 and, in particular, pressure 〈P〉 that cause elastic strain of material with pore fluids are, assuming only that Hook's law is valid, 〈σij〉 = σij - αPδij and 〈P〉 = Pc - αPp, where α = 1 - (K/Ks), Pc and Pp are confining and pore pressures, and K and Ks are the bulk moduli of the rock and grain, respectively. The equation for 〈P〉 was first suggested by Geertsma (1957) and by Skempton (1960) on empirical grounds. The expression does not depend directly on porosity, but when pores vanish the effective pressure 〈P〉 equals the confining pressure Pc, because then K=Ks. Thus the strain of a porous solid with pore pressure can be completely determined from the elastic modulus of the solid without pore pressure, if the effective stress law in the equation for 〈σij〉 is used. The exact expression for the effective stress describes quite accurately the measured strains in sandstone and granite samples at confining and pore pressures to 2.5 kb. The results are not applicable to inelastic processes, such as fracture, or elastic processes other than strain.

Elastic constants of the human lens capsule.

1. A technique is described whereby the elasticity of the human lens capsule has been determined at birth and throughout life. This technique requires three separate determinations: (a) thickness; (b) stress and strain; (c) Poisson's ratio; (a) the capsule was clamped between accurately perforated ground glass plates and its thickness determined by noting the change in depth of focus between Latex spherules adhering to its upper and lower surfaces; (b) the undisturbed capsule was then placed in a specially designed glass distension apparatus and the relationship between volume and pressure recorded when it was distended with isotonic saline. The permeability of the capsule was also measured; (c) in some cases Poisson's ratio was determined by measuring the change of thickness of the capsule and the height to which it rose when distended with isotonic saline at different pressures. An apparatus was designed for this purpose.2. The average thickness of the anterior capsule increases from birth until about the 60th year but thereafter it decreases slightly.3. Poisson's ratio was about 0.47 for both cat and human capsule, and no significant variations with age in human capsule could be detected.4. Corrected volume pressure curves obeyed Hook's law almost to the point of capsule rupture.5. In childhood Young's Modulus of elasticity is about 6 x 10(7) dyn/cm(2) and decreases to 3 x 10(7) dyn/cm(2) at 60 and 1.5 x 10(7) dyn/cm(2) in extreme old age.6. The ultimate tensile stress was 2.3 x 10(7) dyn/cm(2) in young capsules and 0.7 x 10(7) dyn/cm(2) in old ones. The maximum percentage elongation was 29 per cent and independent of age.7. The implications of these findings are discussed in relation to(a) the mechanical properties of the lens capsule;(b) the ageing of the lens capsule and basement membranes; and(c) the decrease in elasticity of the lens capsule as a cause of presbyopia.

Mechanical properties of paddy stem with reference to lodging bending moment-curvature diagram

Abstract The mechanical properties of cereal crops as related to lodging, breaking strength, rigidity, elasticity, buckling load etc. have been studied by various workers (1∼8). These properties were analyzed based on the theory of elastic body conforming to Hook's law. In the previous paper (9) the authors pointed out chat in the process of deformation of paddy culm, starting from slightest bending up to the point of break of the culm, two properties were observed, namely, elasticity and viscoelasticity. These properties can be clearly defined and investigated by the analysis of bending moment-curvature diagram (9) of stem. This paper deals with this diagram in relation to differences in varieties and fertilizer treatments.

Dynamic visco-elastic properties of glycerol-extracted muscle fibers.

1. The dynamic visco-elastic properties of glycerol-extracted muscle were investigated.2. The dynamic stiffness increases during contraction and decreases during relaxasion. The stiffness-tension relation is linear in most parts of either contraction or relaxation processes, and its tangent is almost equal to each other.3. When the sinusoidal changes in length (peak to peak) are less than 1.5 per cent of muscle length, the fiber's obey HOOK's law.4. The complex modulus measured at the frequencies of 20 or 40c/sec of contracted muscle is 7000 or 10000gwt·cm-2. The modulus is decreased to 800 or 1200gwt·cm-2 if the muscle is relaxed in 100 mM pyrophosphate-Na solution or 1 M KC1 solution, and to 2000gwt·cm-2 in 50 mM pyroposphate-Na solution containing 2mM MgCl2.5. Elastic modulus and viscosity are calculated based on the rheological analysis. The retardation time is 0.8 or 2.0 msec.6. When the muscle is stretched, the stiffness is always increased.7. From the time course of decay of tension after quick stretch, it is suggested that there would be another system possessed of time constant of a few seconds than the system detected by sinusoidal analysis.8. Visco-elastic properties and their changes in contraction process of glycerolextracted muscle are similar to those of living muscle.

フックの法則に従はない圓柱の挫屈計算に關する一考察 : 中實圓柱の場合

フックの法則に従はない圓柱の挫屈計算に關する一考察 : 中實圓柱の場合

フックの法則に從はざる材料の円〓の捩り剪斷内力を求める計算式に就いて

捩り試驗の結果より表題の如き剪斷内力を計算する爲には從來次の式が存在するのみである。[numerical formula]此の式は中空円筒に對してはτ_1τ_0の何れをも決定し得ないものであり且此の式の成立する爲には極限された假定が滿足されなければならないものである。本文は先づ理論的にτ_1τ_0の兩者を決定する計算式を導き、更に捩り速度の影響を受ける場合に對する同様な計算式を作り、最後に制限のはるかに寛大な條件の元にかなり一般的に成立する計算式として次の式を與へるものである。[numerical formula]