In a previous paper the variation method was developed to find the solution for the equation of motion of three dimensional viscoelastic materials, on the basis of Hamilton's principle of mechanics. Since the stress in that case was not conservative force, we could not get the variation function in a closed form. In the present paper it is shown that if the suitable representation is taken for the variation of strain-rate tensor and a somewhat different form is assumed for Hamilton's principle, we can obtain the variation function in a closed form as a matter of form. The Hamilton's principle is here applied to the variation of displacement-rate instead of that of displacement of the material point. In our case, the variation corresponds to the extremum of work done by the external force.
With attention to the equation of motion in the Euler-system (E-system), we put
ρξ=Divσ_E+ρK_E (1)
The Hamilton's principle is then assumed to be given in the form
∫_t∫_V[ρξ-Divσ_E-ρK_E]·δξdVdt=0 (2)
where ξ is the coordinate of the mass point in the material which is placed at x in the natural state, σ_E is stress tensor referring to E-system, K_E is external force acting on the unit mass of the sample and ρ is the material density. Making use of the displacement tensor a: Δξ=a·Δx where Δ denotes the difference of the coordinates of neighboring two points in the sample, we define the strain tensor e_L=a⁺·a/2 as well as the strain-rate tensor e_L=(a⁺·a+a⁺·a)/2 in the Lagrange system (L-system). The variations of these tensors are assumed to be δe_L=(δa⁺·a+a⁺·δa)/2 and δe_L=(δa⁺·a+a⁺·δa)/2, and these tensors are transformed to those in E-system by A_E=a^+-1·A_L·a^-1. In viscoelastic materials there exist two mechanisms; one of them is elastic, stored mechanism (1) and the other is viscous, dissipative mechanism (2). It is assumed that each of them is characterized by the corresponding stress tensor and strain tensor σ_i and e_i, i=1, 2. The elastic stored strain energy density and the viscous dissipative energy density are considered to be the functions of e₁ and e₂, respectively: w=w(e₁) and ε=ε(e₂). After suitable mathematical calculation, we can obtain the variation functions in both E-system and L-system as
I_E=∫_V1/2ρξ²dV+∫_Vw_E(a₁)dV+∫_V∫_tε_E(e_E2)dtdV+∫_V∫_ξρK_E·dξdV-_??__S∫_ξF_E·dξdS (3)
I_L=∫_V01/2ρ₀ξ²dV₀+∫_V0w_L(e_L1)dV₀+∫_V0∫_tε_L(e_L2)dtdV₀+∫_V0∫_ξρ₀K_L·dξdV₀-∫_S0∫_ξF_L·dξdS₀ (4)

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The purpose of this paper is to present a theory of the steady flow of a non-Newtonian liquid through a conical nozzle. Then the apparent viscosity μ is a function of the velocity gradient. Considering that μ is a function of the coordinates of the liquid particle, we have obtained the exact equations of motion of a non-Newtonian liquid in the conical nozzle. We have also derived the differential equation of the velocity distribution for a special non-Newtonian liquid with a power law flow curve. The equation of motion has been obtained on the following assumptions; i) the liquid is incompressible; ii) the motion of the liquid is not turbulent; iii) the motion is steady; iv) no body force acts on the liquid; v) the motion has an axial symmetry; vi) there is no slip at the wall; vii) the stream lines are the straight lines passing through the vertex of the cone, that is, the end effect is neglected; viii) the motion is so slow that the inertia term can be neglected.
We have taken a spherical coordinate system r, θ, and φ whose origin is at the vertex of the cone. Then the stress components are given by
τ_rr=2μ∂v_r/∂_r (1) τ_rθ=μ/r∂v_r/∂θ (4)
τ_θθ=2μv_r/r (2) τ_θφ=0 (5)
τ_φφ=2μv_r/r (3) τ_φr=0, (6)
where v_r is the velocity component.
From the assumptions i)∼v), vii), and viii), the equations of motion are
0=-∂p/∂r+{1/r²∂/∂r(r²τ_rr)+1/rsinθ∂/∂θ(τ_rθsinθ)-τ_θθ+τ_φφ/r} (7)
0=-1/r∂p/∂θ+{1/r²∂/∂r(r²τ_rθ)+1/rsinθ∂/∂θ(τ_θθsinθ)+τ_rθ/r-cotθ/rτ_φφ}, (8)
where p is the pressure and the equation of continuity is
1/r²∂/∂r(r²v_r)=0 (9)
Substitution of Eqs. (1), (2), (3) and (4) into Eqs. (7) and (8), with the help of Eq. (9), yields
∂p/∂r=2∂μ/∂r∂v_r/∂r+1/r₂∂μ/∂θ∂v_r/∂θ+μ/r²∂²v_r/∂θ²+μ/r²∂v_r/∂θcotθ (10)
∂p/∂θ=2μ/r∂v_r/∂θ+∂μ/∂r∂v_r/∂θ+2∂μ/∂θv_r/r (11)
The velocity gradient D is given by
D=-1/r∂v_r/∂θ (12)
We shall treat the special case of a non-Newtonian liquid obeying the power law
D=kτⁿ, (13)
where k and n are constants. Then the apparent viscosity μ is given by
μ=τ/D=KD^-(1-1/n), (14)
where
K=k^-1/n (15)
From Eq. (9), we get
v_r=f(θ)/r², (16)
where f(θ) is a function of θ alone.
From Eqs. (12) and (16) we obtain
D=r^-3{-f'(θ)} (17)
Thus the apparent viscosity μ is expressed as a function of ^r and θ as follows:
μ=Kr^3(1-1/n){-f'(θ)}^-(1-1/n) (18)
Substitution Eq. (18) into Eqs. (10) and (11) yields
∂p/∂r=-12K(1-1/n)r^-1-3/nf(θ){-f'(θ)}^-(1-1/n)
+K1/nr^-1-3/n{-f'(θ)}^-(1-1/n)f"(θ)-Kr^-1-3/n{-f'(θ)}^1/ncotθ (19)

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The Doolittle free volume equation
η=A exp (Bv₀/v_f), (1)
which was first proposed for the viscosity of n-alkanes, is widely accepted as the viscosity equation for various kinds of liquids and polymeric substances. However, Bueche modified it according to a normal coordinate computation for the thermal motion of a simple array of molecules connected by Hookian springs, and derived the following equation
η=A exp{B(v₀/v_f)²}. (2)
The purpose of the present study is to clarify the theoretical background for the free volume representation of the liquid viscosity from the view-point of the thermal fluctuation theory.
The probability that the thermal fluctuations of energy and volume will amount to ΔE and ΔV respectively in a small, but macroscopic system which is a part of a large macroscopic system, is denoted by prob. (ΔE, ΔV). By a simple thermodynamic computation, it is given by
Prob. (ΔE, ΔV)∝exp[-1/2kT₀{1/C_vT(ΔE)²+(p²/C_vT-2αp/C_vκT
+C_p/C_vV_κT)(ΔV)²+(2p/C_vT-2α/C_vκT)(ΔE)(ΔV)}]. (3)
in which C_v and C_p are the specific heats at constant volume and constant pressure respectively, α the thermal expansion coefficient, and κ_T the isothermal compressibility. Then, the probability that the volume fluctuation is larger than a certain critical value ΔV^* and the energy fluctuation is within the range of average root mean square is
Prob. (ΔV>ΔV^*, ΔE_??_√(ΔE)²)≅1/√2πγ√(ΔV)²_e/ΔV^*-γ/2(ΔV^*)²/(ΔV)², (4)
where γ is the ratio of the specific heat.
We consider a liquid system which is composed of N rigid sphere molecules in the potential field of square-well type, χ(V)=-(V₀/V)χ₀, where V₀ is an occupied volume and χ₀=-χ(V₀). According to a simple statistical mechanical computation the mean square value (ΔV)² will run as:
(ΔV)²=-kT(∂V/∂p)_T≅1/N(V_f)². (5)
in which V_f is the free volume, V-V₀. After further computation the following viscosity-free volume equation is obtained:
γ/2(Δv^*/v_f)² γ/2(v_v/v_f)²
η=Ae =Ae, (6)
where the critical volume required for the shift of a molecule, Δv^*, is assumed to be nearly equal to the occupied volume per molecule, v₀.
Equation (6) shows that logη is proportional to the square of the reciprocal of the free volume fraction, and this relation is the same as that obtained from Bueche's normal coordinate theory. The value of the parameter B in equation (2) is (γ/2) in the author's equation, and differs from the value of Bueche, (9/π).
In conclusion, the modified Doolittle equation, η=A exp{B(v₀/v_f)²}derived from the thermal fluctuation theory is in accordance with Bueche's result obtained from the normal coordinate theory, and against Cohen and Turnbull's theory.

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We have previously reported a new capillary viscometer in which two syringe needles are connected to top and bottom ends of a graduated glass tube and negative pressure is introduced by compressing a rubber ball. The viscosity of liquid was read by the maximum height of liquid sucked into the glass tube. In this device the gravitational force due to the weight of liquid worked against the negative pressure in the tube. This required a correction in calibration curves for the scale of tube and viscosity, and necessitated watching the instant when the liquid went up to the highest level.
In the present modification, the needle at the lower end is brought to the upper side of the tube, and the bottom of the tube is closed. The liquid is drawn in the tube by pressing a bellows made of phosphor bronze plates, and kept at the bottom of the tube. The reading of the height of liquid is easily made, and the correction for gravity is not necessary in the calibration curve.
The relation between the viscosity of liquid η_l and the volume V_l of liquid sucked is derived from the Poiseuille law in the form:
η_l=R_l⁴L_a/R_a⁴L_lη_a(V/V_l-1)
where η_a is the viscosity of air, V the volume evacuated by a bellows, R_l, R_a and L_l, L_a the radii and lengths of two needles, where suffixes l and a mean the liquid and the air respectively.
This relation was verified to hold well by measurements of the viscosities of standard oils. However, for the viscosity range lower than several centipoises, it was found that the correction for kinetic energy of flow should be added to the equation mentioned above.
The measurable range of viscosity can be adjusted by the adequate combination of dimensions of two needles and cover from 0.5 to 150 centipoises. The shear rate is altered in the range 10sec^-1∼10³sec^-1 by replacing a bellows by a motor driven pressure reduction device. The volume of liquid needed is less than 3ml and the time required is less than several seconds.
The examples illustrate the intrinsic viscosity of aqueous solution of polyvinylalcohol, the temperature variation of viscosity of oils and the temperature and concentration dependence of viscosity of milk.

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A high speed tensile tester was designed to study the tensile properties of thin polymer films using a flywheel. The tester consisted of a variable speed motor (1-HP) coupled to a flywheel, tooled from steel disc, through pulleys. It provided tensile speed over the range from 0.1 to 10m/sec. The force arising while testing the sample was detected by means of a semiconductor strain gauge whose output signal was fed to vertical deflection plate of an oscilloscope. The sweep of the oscilloscope was triggered when the light beam to a phototube was traversed by striking-fork mounted on the rim of the flywheel just before stretching the sample. The amplifier of the oscilloscope was flat from DC to 100kc. The resonance frequency of the strain gauge with a clamp for the sample was 1.1kc and its sensitivity was 2.5mV/kg if the carrier voltage was 1V.
The response of the high speed tensile tester to the force arising in stretching the sample was examined by means of an analogue computer. The response relates to the true stress, strain, the strain rate, resonance frequency and the damping of the gauge and upper and lower cut-off frequencies of the amplifier recorder system. The motion of the gauge should be represented by the following equation:
x+2lx+ω²x=F(t)/M
where x is displacement, M is the effective mass of the moving part including the mass of clamp. F(t) is considered stress function which shows load-time relation of the sample. The difference of the observed load-time curve from the true under 1m/sec of tensile speed is shown for various bandwidth of amplifier and resonance frequencies of the gauge. From the results, it is found that the observed load-time curve at 1m/sec of tensile speed is in accord with the true curve if the upper cut-off frequency of an amplifier is 5kc, the lower one is DC and resonance frequency of the gauge is 2kc, and at higher speeds these values increase proportionally except the lower cut-off frequency.
Considering the above analysis, stress-strain curves of the commercial film of cellulose-triacetate are indicated at the rates of strain from 200 to 8000%/sec. The results are compared with those of the tests carried out on the same materials with an Instron type tester at the rates of the strain of 0.17, 1.7 and 16.7%/sec.

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Kamaboko is a heat-coagulated fish paste containing some salt, sugar, starch and flavor, and is one of the favorites of Japanese people. The rupture properties of Kamaboko, such as the breaking strength and the breaking elongation, are considered as one of the most important factors in evaluating the commercial quality of this product. The quality of Kamaboko has been estimated in terms of pastelike compactness which in Japanese is known as“Ashi”. This in reality embodies rheological properties. The results of ranking Kamaboko on market by“Ashi”or compactness have shown that this is closely related to its being hard, but not so much so as to be tough, and also to its being fit for chewing chear cut. The organoleptic scores for the Kamaboko, how hard it is, were found to be related to the gel strength, the elastic energy stored in the specimen before breaking. On the contrary, the evaluation of Kamaboko being fit for chewing clear cut has not yet been established.
The data for the creep in a simple shear, the stress relaxation in extension and the damped free oscillation in torsion of commercial Kamaboko have shown that this is considered to be a thermorheologically simple material and a slightly crosslinked rubbery material which exhibits no viscous flow.
The results of stress relaxation of fresh fish meat paste were expressed as:
f/f₁₀=a-klogt,
where f is the stress at time t, f₁₀ the value of f at t=10sec, a and k the constants, and t the time, respectively. The stress relaxation measurements were made for the meat paste heated at various temperatures and heating times. It was found that the values of f₁₀ increase and those of k decrease with increase of temperature and heating time.
The measurements of the viscosity of the fish muscle extracts were made at various velocity gradients. The relation between the viscosity, η, and the velocity gradient, D, followed the de Waele-Ostwald law:
η=KDⁿ
where K and n are constants. Moreover, the viscosity, η, of fish muscle extracts at constant velocity gradient was found to be related to the concentration C as:
η_D=2=kC^m.

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Process cheese is an emulsion in a solid state consisting of protein denatured by enzyme, milk fat dispersed in the form of small particles and water containing salt.
In this experiment, a variation of viscoelastic property of process cheese with various moisture content and temperature was measured by means of three kinds of rheometer, i.e. a cone-plate viscometer, a compression tester and a dynamic viscoelastometer.
The static viscosity η measured by the cone-plate viscometer decreased logarithmically with increase of moisture content in the range from 35 to 42 percent and the activation energy ranging from 8.6 to 8.3Kcal/mol.
On the other hand, the elasticity and internal friction changed from 7×10⁶ to 2×10⁶ dyne/cm² and from 0.65 to 0.80 at 20°C respectively with the variation of moisture content.
When the compression-recovery cycles were repeated on the specimen, the elasticity of cheese increased gradually and reached a plateau region. Consequently, the process cheese has rheologically the behaviour of work hardening.
The dynamic viscoelasticity of casein networks in cheese texture with change of moisture content ranging from 36 to 42 percent was obtained; dynamic viscosity η_d=(1.8∼2)×10⁴ poise and dynamic elasticity E'=(8∼6.4)×10⁶ dyne/cm² respectively.
These values also change with the temperature of storage after processing of cheese.
Process cheese becomes harder by storage at lower temperature in the lower content of moisture. From the X-ray diffraction analysis of process cheese and paracasein, it was found that the variation of elasticity with moisture content was attributed to the change of network structure formed by casein micelles.

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The relations between the crystallization behaviour and the mechanical properties of fats were investigated. The process of crystallization was followed by a falling temperature differential thermal analysis. The physical state of solidified fats was modified to a great extent by varying both the crystallization temperature and the melt temperature. The hardness was measured by means of a micropenetrometer which was commonly used in fat industry. The dynamic elastic modulus E^* and loss factor tan δ were measured at 20c/s by a dynamic viscoelastometer. The static viscosity η was determined by a modified penetrometer which was also used as a parallel plate plastometer. Observation of the texture of crystal grains was made by a polarized microscope. The measurement of X-ray diffraction was also carried out.
The falling temperature DTA revealed the distinct variation in the crystallization behaviour among a variety of margarine and fat. Higher values in hardness was found to be accompanied with higher temperature required for initiation of crystallization, larger amount and more uniform distribution in latent heat, and a higher degree of crystallinity determined by X-ray diffraction analysis. This difference has mainly been caused by the difference in triglyceride components of the fats.
As a standard specimen of fat, refined tallow was employed. After melting at 80°C, the fat was crystallized at different temperatures from 30°to -30°C. With the decrease of crystallization temperature (T_c), the temperature initiating crystallization becomes lower and the latent heat becomes larger. The hardness measured by a micropenetrometer is maximum at T_c=20°C and minimum T_c=0°C. The hardness of T_c=-30°C is slightly higher than that of T_c=0°C. Gomparison of E^*, tan δ and η for specimens of T_c=20°, 0°, -30°C showed the same tendency. In the specimen crystallized at 20°C, which seemed to be the most favorable temperature for crystallization, large spherulitic crystals developed in the texture. And a rigid network connecting large crystal grains has been developed, hence the hardness is largest. In the specimen crystallized at 0°C, the size of crystal grain becomes smaller and irregular, and the network of crystal grains is intervened by the amorphous boundaries, resulting in the weaker network structure. The viscous flow takes place at grain boundaries and the structure is softest. In the specimen crystallized at -30°C, a number of small crystal grains are formed and packed closely, resulting again in a firm network structure of small crystal grains. Therefore the texture becomes more stable.
In order to investigate the effect of crystalline nucleus remaining in the molten fat, refined tallow was melted at various temperatures from 40°C to 200°C and crystallized at 0°C. With the increase of melt temperature (T_m), the temperature initiating crystallization and latent heat fall gradually and become nearly constant above T_m=80°C. The specimen of lower T_m is found to be softer. As crystallization takes place from the nucleus remaining in the molten state, the glycerides with higher melting point contribute preferentially to the growth of nucleus. As the result, the texture consists of crystallites with glycerides of high melting point, and of grain boundaries with glycerides of low melting point. Therefore the viscous flow in grain boundaries can take place more easily than in the specimen crystallized from completely molten state where the separation of glycerides of high and low melting point is not so clear during the crystallization process.

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Gotoh and Hirai¹⁾ found that the blocks of bentonite hydro-gels showed clear slip bands under critical pressure and the bands inclined at 45°in the direction of the force given. The slip patterns could not be observed with the specimens treated at above 800°C or with those dispersed in organic liquids. Gotoh, Hanai and Aida²⁾ investigated the viscoelastic behavior of the bentonite gels with the cone and plate viscometer and found that the gel slipped at certain critical shear stress.
In this report, the critical shear stress at which the slip bands appear was measured at various concentrations of Wyoming bentonite dispersed in water. The critical shear stress was observed with the following three kinds of method; (1) a compression method (Fig. 1), (2) a cone and plate viscometer method and (3) a two surface shearing box method. Fig. 2 shows a typical slip band observed by the compression method. Fig. 3 shows the viscoelastic behavior and slip of the bentonite gel observed at various shearing loads. Fig. 4 shows the relation between the critical stress and the concentrations. The results obtained by the three kinds of method fall on the same line which shows a characteristic curve analogous to the relation between the elasticity of gels of high polymers and their concentrations³⁾. It has been shown that the instantaneous elasticity of the bentonite gels shows also the same tendency with respect to their concentrations.
These results suggest that the bentonite gels has a network structure analogous to that of high polymers, but that the binding force is very weak in comparison with that of polymers. Further, it has been observed that a polarized light can pass through a thin film of the bentonite gel placed between two parallel glass plates at the instant when the plates are sheared (Fig. 6). This result suggests that the bentonite particles can be oriented under shearing stress, and such orientation may make the bentonite gels easy to slip.

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In this paper, a study has been made to obtain the macro deformation of clay skeleton as accumulation of micro displacements of clay segments with which the clay skeleton is composed. The joints of clay segments are classified in assumption into two types: the one is the joint where no relative sliding is thought to occur between the segments though variation is produced in the intercepting angle between the segments by external stress. In the other type of the joint, the segment slides on the surface of the adjacent segment holding the adsorbed water around the joint, when the force applied on the segment exceeds the frictional resistance between the segments. The probability of the sliding segment is calculated under the assumption that the applied force and the frictional resistance are distributed as certain Gaussian distribution functions. The resisting forces on the sliding segment are assumed to consist of the following 3 forces: the recovering force due to the imbalance of repulsive force and the attractive force existing between the segments, the viscous resistance of adsorbed water which is considered as Eyring's viscosity and the frictional resistance between the segments. From these considerations, the mechanical model has been obtained as shown in Fig. 3 which represents the rheological behaviour of clay skeleton accompanying neither consolidation nor fracture.

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Reiner observebed the so-called Ostwald flow patterns in the system of rubber-like materials in the solvent as toluene. This flow starts as the Newtonian flow in the range of low shearing rate, and the flow pattern is followed as non-Newtonian in the medium shearing rate until realtered as the second Newtonian at the high shearing rate.
This type of flow patterns are recognized as generalized Newtonian flow in the later age. This type of flow are observed by many invesigators as Philippoff.
This type of flow, however, is scarcely observed in the gel-type material or in the suspensions of powder-like material in the fluid. Especially, the flow patterns of these suspensions free from thixotropic effects, have not been observed until the present time. In this reports, the following items have been investigated, using three type viscometers known as the rotational viscometer in the low shearing rate ranging 0.4∼80sec^-1, the Kriger-Maron viscometer as the medium shearing rate ranging 30∼2000sec^-1, and BS type viscometer as the high shearing rate ranging 1000∼1000000sec^-1,
(1) The clay powder suspensions show the typical Ostwald pattern when the thixotropic effects are completely separated by adopting the procedure of the author's previous papers.⁹⁾¹⁰⁾
(2) The flow pattern of the suspensions are customarily recognized as simply pseudo-plastic or dilatant type. The Ostwald's flow pattern of these suspensions may be consistent with these traditional concept when the procedures illustrated in Figs. 7 and 8 are adopted.
(3) The silica gels showed also typical Ostwald's pattern in the 2∼8% concentration range. The dilatant flow obtained from the ZnO-water system are decided as Ostwald-like pattern as measured in the wide shearing rate range.

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The study has been made of the relations between the physical property of liquid and the thickness of a layer of liquid remaining on the wall, after draining the liquid and the thickness of the liquid layer coating a solid body drawn out of a liquid. The thickness h of coated layer was calculated following equation by the weight method.
h=W/2_ρS
W: corrected weight of the coated layer that is released of surface tension effect.
ρ: density of liquid
S: coated area
The relations between the thickness of coated gelatin solution and glycerin solution, the physical properties of those and the moving velocity of the solid are shown in Table I, which indicates the difference between the observed and the calculated value.
These differences are attributed to the temperature difference between the solid and the liquid, or between ambient air and the liquid which cause viscosity increase of liquid which remains on the solid. The experimental results which exclude temperature difference are shown in Fig. 6 and these results accord with the values calculated by equation (4) which is derived from neglecting the surface tension below 6.38cm/sec of coating velocity. In Fig. 8, equation (8) is obtained which combines the force F of lifting solid and the above mentioned equation (4).
As shown in Fig. 9, Fig. 10, the observed values accord with the values calculated by equation (8) below the coating velocity 2.22cm/sec within the values of viscosity ranging from 0.050 to 10 poise. The region where equation (8) is satisfied becomes narrow with an increase of the coating velocity. For example, the observed values accord with those calculated within the viscosity range 0.050-0.780 poise at coating velocity 6.38cm/sec. Therefore the thickness of coated layers can be calculated by measuring dF/dt and using equation (8) in the above designated region. The deviation of the observed thickness in the high viscosity region points to the appearance of non-Newtonian behavior.
Then the deviation in the low viscosity region as shown in Fig. 6, Fig. 8 and Fig. 10 is considered to be owing to the dynamic surface tension effect which appeared at high speed coating. The equation (5) containing the term of surface tension is derived by Deryagin⁶⁾. In order to obtain the required thickness 0.0176cm, the coating velocity was calculated 13.2cm/sec by the equation (5) and 8.02cm/sec by the equation (4) respectively. The expected thickness was obtained by coating gelatin solution with 8m/min.

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The dynamic viscosity η' and rigidity G' have been measured by means of the rotational cylinder type rheometer described elsewhere for seven types (A to G) of high-density polyethylene manufactured by three different processes. The measurements have been carried out in a frequency range from 0.001 to 0.5cps at five temperatures ranging from 180 to 260°C.
These materials do not necessarily have the same frequency dependence of η' and G', and can be classified into four groups. Polymers D and E have the same frequency dependence of η' and G' in the frequency range covered by the experiment and η' vs. ω or G' vs. ω curve (ω denotes angular frequency) can be superposed into a composite curve by shifting them along a straight line having the slope of -1 or the abscissa. This also holds for polymers F and G, but the η' and G' curves for these polymers intersect with those for polymers D and E. The polymers A and B have the same frequency dependences of η' and G' on each other but are different from the above materials.
The polymer C is quite different from others. The frequency dependence of η' for this polymer is much larger and that of G' is much smaller than the other materials.
In some cases, the difference in the frequency dependence of η' and G' for different materials may be ascribed to the presence and the nature of double bonds included in the polymer chains as verified from the infrared spectrum. In other cases, the difference may be due to the difference in the molecular weight distribution as elucidated by the fractionation with the column method and turbidity titration.
The frequency dependence of η' for the polymer having bimodal distribution is greater than for the polymer having normal distribution, and that of G' is reverse. The absolute values of η' and G' for the polymer having bimodal distribution tend to be higher than the normal for a given weight-average molecular weight. Moreover, the polymer having bimodal distribution have larger shift factor and lower fluidity than the normal, indicating that the molecular weight distribution is very important to the flow properties of polymer melts.

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As widely known, glass fiber reinforced plastics have very great value of strength weight ratio and many other excellent properties, being heat and water proof and having electric resistance among others. Hence glass fiber reinforced plastics are attracting the attention of the engineering world. The investigation about the strength of glass reinforced plastics is performed by many researchers from different standpoints, the summarizing consideration there of being required by many engineers. The present paper proposes to give such summarizing considerations about the strength of glass fiber reinforced plastics to meet the above mentioned requirement.

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It has been made very clear by recent works of many researchers that the distribution of mechanical relaxation time in the rubbery region of a linear amorphous polymer is related closely to the shape of the molecular weight distribution. Attempts have been made to express this relation quantitatively, from a phenomenological point of view.
These criteria have been considered helpful as indicators of the degree of polymer polydispersity. One is that the height of the relaxation spectrum increases with increasing sharpness of the molecular weight distribution, and this fact was previously discovered by Tobolsky and his co-workers and the indicator of the degree of polymer polydispersity in this case is here shown by E_m.
The other is that the relaxation spectrum approximates a“box”more closely with increasing sharpness of molecular weight distribution, and this fact was ascertained by Tobolsky and Murakami, and the indicator of the degree of polymer polydispersity in this case is here expressed by α, and α is satisfied by the equation α=τ_mE_m/η.
Some researchers have previouly studied the relation between the steady state compliance J_e and the structure of polymers. According to Leaderman, Ninomiya, and others, J_e appears to be dependent upon the molecular weight distribution rather than the molecular weight of polymers. The authors have succeeded in proving that J_e is dependent upon the molecular weight distribution by applying procedure X developed by Murakami and Tobolsky.
According to procedure X, relaxation modulus E_r (t) can be written by the following equation using the distribution of relaxation time in the rubbery region.
E_r(t)=E_me^-t/τm+E_m-1e^-t/τm-1+…… (1)
The steady state viscosity ηt is shown by
η_t=∫^∞₀E_r(t)dt (2)
Substituting equation (1) into equation (2), we obtain
η_t=E_mτ_m+E_m-1τ_m-1+…… (3)
The steady state compliance J_e is indicated by
J_e=1/η²t∫^∞₀∫^∞_t^*Er(t)dtdt^* (4)
Substituting equation (1) into equation (4), we obtain
J_e=1/η²_t(E_mτ²_m+E_m-1τ²_m-1+……) (5)
In the region of t>>τ_m, equation (3) is simplified into equation (6), and equation (5) is similarly modified into equation (7), neglecting the second and more terms.
η_t=E_mτ_m (6)
J_e=1/η²_tE_mτ²_m
=1/E_m (7)
In conclusion, equation (7) shows that the steady state compliance J_e is proportional to 1/E_m. Here E_m is just the indicator of the polydispersity of polymers as mentioned above.
To be more exact however, J_e becomes α²/E_m by using the other indicator α.
The data actually obtained and the logical conclusion is thus compared. It is discussed also whether the steady state compliance J_e can precisely be the parameter of the molecular weight distribution.

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The rheo-optical techniques such as dynamic birefringence are very useful for the study of the relation between mechanical properties and the structure of high polymers, especially crystalline polymers. As a method in this line, an attempt has been made to measure the change in infrared absorption simultaneously with stress and strain for polymer films under stretching. As described in the previous paper¹⁾, the double beam type infrared spectrometer has been modified in such a way as to allow one of the beams go through a specimen mounted on stretcher similar to the Instron type tensile tester in its principle. The spectrometer has also been equipped with a quick return device for the motor in order to record the absorption in a rather narrow range of wave length repeatedly in a short time. In the present paper, the infrared dichroism of low-density polyethylene has been measured by means of this spectrometer.
The sample film used in this study was made by the inflation method, and is the same as that used in the previous paper¹⁾.
The dichroic ratio, D=A_⊥/A_||, where A_⊥ and A_|| are the absorbance (optical density/thickness) for polarized radiations whose electric vectors are perpendicular and parallel to the stretching direction, has been determined for the 720 and the 730cm^-1 bands. For the 730cm^-1 band, D increases slowly up to about 10% elongation, very rapidly thereafter up to about 30% elongation, and then slowly again up to about 70% elongation. D for the 720cm^-1 band, on the other hand, decreases slowly first, and then rapidly until it goes through minimum at about 30% elongation. Then it gradually increases. The orientation function for the α-axis, F_α, has been determined from D for the 730cm^-1 band, and the function for the b-axis, F_β, from D for the 720cm^-1 band, assuming that this band is primarily due to the crystalline phase. It is interesting to note that the change of these orientation functions with strain corresponds very well to the stress-strain diagram for the same film.
If we employ the current folded chain theory for the structure of crystalline polymers, this correspondence enables us to interprete the stress-strain curve very reasonably.
In the earlier stage of stretching up to the elastic limit, the spherulites or superstructures of, lamellae may deform elastically without appreciable orientation or unfolding of folded chains. In the neighborhood of the yield point, such spherulites or superstructures may split into smaller domains, and they can be oriented in the direction of stretching without appreciable unfolding up to 30% elongtion. Above this elongation the unfolding of folded chains in the oriented domains occurs predominantly, and the unfolded chains and crystals reformed tend to be oriented again in the direction of stretching.

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A new method of simultaneous measurements of the changes in stress and infrared dichroism on elongation of polymer films was devised by using a double beam infrared Spectrometer. The film was installed in a stretching apparatus and was placed in front of the entrance slit of the spectrometer where both the sample and the reference beams came together. Two polarizers were used: one was placed in the sample beam and the other in the reference beam. Thus the sample and the reference beams were polarized to have the electric vectors in parallel and perpendicular to the stretching direction of the film, respectively. With this arrangement the spectrometer responded only to a difference in the transmittance of the two beams. Then, by setting the spectrometer at one of the wave numbers of the absorption maxima one could record the change in its dichroism continuously during the mechanical treatment of the film. The stress generated by the mechanical treatment was transformed to the electric signal by means of a strain gage pasted on the stretching apparatus and was recorded after having been amplified.
According to the theoretical considerations of this method, it is concluded that for the sample of unilateral orientation the quantity T recorded by the spectrometer is related to the orientation function F of the transition moment with respect to the stretching direction by the equation,
F=1/3A_od_o/d_nlog1/T,
where A_o is the absorbance of the unstretched film and d_o and d_n are the thicknesses of the unstretched and n% stretched films, respectively.
The method was applied to the stress relaxation phenomena of vulcanized natural rubber. The simultaneous measurements of stress and infrared dichroism were made at constant elongations less than 600% at room temperature. The absorption bands examined were those observed at 1664, 1380, 1361, 1129, and 844cm^-1, where the last one was a crystallization-sensitive band of natural rubber. When the film was elongated, all the bands gave rise to the instantaneous dichroism to some extent, which was followed by the gradual increase except for the band at 844cm^-1. These results show that there is some correspondence between the dichroic change and the stress relaxation. It is concluded that the stress relaxation observed is attributable to the molecular orientation in the amorphous region rather than the crystal orientation which is completed almost immediately after the elongation.

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As sequel of the previous paper relating the rheological properties of the mechanical mixture of two polymer components to the degree of mixing, the temperature dependence of tensile stress relaxation of the two types of mixed systems, polyvinyl acetate-polymethyle methacrylate system and polyvinyl acetate-lightly crosslinked polymethyle methacrylate system, was investigated over a temperature range covering enough the glass-transition temperatures of the two polymer components.
The procedure, the so-called “time-temperature superposition” was carried out in comparison with several parameters, such as fractional free volume and its thermal expansion coefficient which were temporarily determined on a basis of the free volume concept on the viscosity in relation to the William-Landel-Ferry's equation, with those of the individual polymer components.
Although the fractional free volume and its thermal expansion coefficient thus determined for the mixed systems were just of apparent, the results may, at least qualitatively, deny the simple additivity of the free volumes of the two polymer phases and suggest the existence of a sort of physical interaction between the phases, i. e., the internal pressure induced by one phase to the other due to the difference of thermal expansion coefficient between the phases.

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In the previous paper¹⁾, it was concluded from the NMR studies by means of X-rays of the crosslink density and the combined type of sulfur, that the rubber molecules of filled vulcanizates formed three different phases, the highly motional and uncrosslinked, the unmotional and collectively crosslinked, and the degenerated phase around the fillers. However, these phases were not apparently separated, but had a continuous distribution from the highly motional to the unmotional.
In this paper, the behaviours under cyclic deformation of these vulcanizates which had wide distribution of molecular mobility have been investigated by NMR method with respect to stresses by energy and entropy, density and crosslink density. Natural rubber vulcanizates loaded with white fillers, CaCo₃ (I), CaCO₃ (II), ZnO, MgCO₃ and clay and carbon blacks, HAF, MPC and FT were adopted as samples. These were cyclically extended to the extension ratio α=5 and the above properties were measured at every cycle.
The results are as follows:
1) From changes in density and crosslink density, it is deduced that the bond of rubber molecules to the white fillers is weak and cavities have appeared on the surface of the fillers in the direction of extension, but in case of carbon black loaded vulcanizates, the bond is strong and the degenerated phase around the fillers has increased by extension.
2) Stresses by entropy and energy of filled vulcanizates, have both increased, and the rubber molecules tend to separate into more motional and more unmotional phases. The greater the amount of the degenerative phase around the fillers, the greater these tendencies are.
3) In case of smaller extension (Strain ±20%) and multicyclic deformations, the case is the same as in larger extensions; the mobility of rubber molecules of filled vulcanizates also proceed to two different states, more ordered and more disordered. Then the phases become separated more apparently after the cyclic deformations.

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Single crystals or crystals like those of various polymers are prepared by slow cooling of dilute solution. The solvents used are as follows; cyclohexanol for polyoxymethylene (POM), ethyl alcohol for polyethylene oxide (PEO), xylene for branched polyethylene (b-PE) and for polypropylene (PP), decalin for polybutene-1 (PB-1), diethylene glycol for polyvinyl alcohol (PVA) and glycerol for nylon 6 (PA-6). The dynamic modulus E' and loss modulus E" are measured at frequencies 3.5, 11, 35, 100 and 138c/s over a temperature range from -180°C to the softening temperature by using the direct-reading dynamic viscoelastometers, Vibron Model DDV-I and II. Measurements have been made of the mat of solution-grown crystals and of the bulk-crystallized or solvent-cast film. The latter has been used for comparison.
It has been clarified that remarkable crystalline dispersion appears in all the solution-grown polymer crystals, even for the samples of PEO, b-PE, PB-1 and PA-6, in which the crystalline dispersions are scarcely observed in the bulk-crystallized state. This means that the solution-grown crystal has by far the higher crystallinity, and that there exists the intrinsic relaxation mechanism of the crystal phase for all the crystalline polymers. The crystalline dispersion is usually composed of the low temperature side with a low activation energy and the high temperature side with a high activation energy, as we have already reported on linear polyethylene. Two separate crystalline dispersions are observed for PEO (around 0°C and 50°C) and PA-6 (around 130°C and 200°C).
The activation energy for crystalline relaxation ΔH^* of various polymers is as follows: 25kcal/mole for PB-1, 30kcal/mole for PE, 48kcal/mole for PP and 58kcal/mole for POM. The empirical relationships between ΔH^* and the temperature of the E" maximum at 100c/s, T_c, are
ΔH^*=402 T_c-115000 (cal/mole),
and ΔH^*=407 T_m-132500 (cal/mole).
It is concluded that the structural factors contributing to the increase of T_m or T_c are at the same time to increase the value of ΔH^*. ΔH^* is related to the activation volume ΔV^* by the following equation.
ΔH^*=(α/β)·ΔV^*·T_c
where α is the volume expansion coefficient of the crystal and β is the compressibility of the crystal. By assuming that the value of α/β for linear PE is generally applicable to the other polymers, ΔV^* is evaluated as 520cc/mole for PB-1, 550cc/mole for PE, 820cc/mole for PP and 860cc/mole for POM. These values are converted to the corresponding number of main chain atoms, which are 18, 40, 37 and 87 for PB-1, PE, PP and POM respectively. These values decrease at the low temperature side of T_c and increase at the high temperature side. At the temperature higher than T_c by 10 or 20°C the value of ΔV^* almost agrees with that of the molecular segment corresponding to the lamella thickness. In such circumstances the segment is considered to be possible of diffusion within the lamella with resulting change on the lamella in shape like that of an amoeba. This makes possible the initiation of lamella thickening by nucleation mechanism, as presented by Hirai.

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The measurements have been made of the sonic moduli (pulse propagation moduli) and the tensile moduli simultaneously on the filaments produced from various polymers to investigate the orientation characteristics of molecules in the drawing process. The tensile moduli have been measured dynamically by giving the sample sinusoidal strain, whose amplitude is about 0.4% and frequency is 0.12cps, and by stress relaxation statically. These three moduli gives different information each on the behavior of polymer molecules during the elongation. The variance of pulse propagation time (ΔT) has also been measured when the sample is given dynamic strain (Δγ) or stress (ΔS) of the low frequency of 0.12cps. The physical meanings of the coefficients ΔT/Δγ and ΔT/ΔS are discussed from the point of molecular orientation.
The measuring apparatus consists of pulse propagation viscoelastic meter, strain meter, stress meter, bias current part, drawing device and X-Y recorder. The dynamic tensile moduli have been measured by Lissajous method. Only the change of pulse propagation time has been amplified after the electric current corresponding to the propagation time of the sample, which was not dynamically strained, was cut off by the opposite bias current. The measurements have been made mainly on polypropylene filaments (1050d) at room temperature (about 30°C). The results are as follows:
(1) The dynamic tensile moduli (E') and pulse propagation moduli (E_s) do not change so much for polypropylene filaments of lower draw ratio, but increase much for that of higher draw ratio with the increase of static strain. E' increases monotonously with the increase of draw ratio. E_s, however, has a minimum at the draw ratio of about 1.5. The reason why the E_s curve passes the minimum seems to be found in the process of crystalline orientation which gives much influence upon the drawing process at the lower elongation. At the higher elongation, E' and E_s increase in accordance with the increase of amorphous orientation.
(2) The propagation moduli on polypropylene filaments were measured during the stress relaxation. The stress relaxation moduli (E_r) decrease monotonously within the measured time. Because of increase of molecular orientation E_s increases to about 100sec and reaches the constant value which corresponds to the strain. R. S. Stein et al. have reported that crystalline orientation is completed within 1sec from the observation of dynamic birefriengences and dynamic X-ray diffraction. Taking accounts of these results, it is considered that the change of E_s within 100sec has been occasioned by the relaxation of the orientation of the amorphous parts though the orientation is usually complete within 100sec. The change of E_s which is derived from crystalline orientation seems to be too fast to observe.
(3) The change of pulse propagation time (ΔT) when a filament received dynamic strain (Δγ) or dynamic stress (ΔS) are recorded on the X-Y recorder with either of them. The coefficients ΔT/Δγ or ΔT/ΔS are defined as“strain-pulse propagation coefficient”or“stress-pulse propagation time coefficient”respectively. The strain-pulse propagation time coefficient is useful as a measure showing the degree of the allowance of molecular strain. The coefficients of polypropylene are positive for lower elongated filaments, negative for moderately elongated filaments, and zero for highly elongated filaments. The change of the coefficient with the increase of the static strain also shows this tendency.

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The mechanisms of wear phenomena for fibrous materials are classified in four kinds under various rubbing conditions as follows; (1) frictional wear, (2) ploughing or abrasive wear, (3) shaving wear and (4) fatigue wear. In this experiment, the steel roller (18mmφ) with sharp bezels on surface is used for the abradant, in order to carry on the experiment under the shaving wear condition.
The samples used are monofilaments of polypropylene (417^d) and polyethylene (402^d).
The following results are obtained;
(1) The dead weight added on the fiber during wear test, S_B, is expressed as follows;
S_B=C_oN_B^-K_s
where N_B is the number of rubbing cycles up to rupture, C_o and K_s are constants.
(2) The temperature dependence of dead weight to break at N_B=1000, S_B·1000, shows two maxima for both the polymers.
(3) In the case of polypropylene, the maxima of S_B·1000 appear at about 48°C and 98°C. In the region 20∼50°C, there is but little change in tensile strength, elongation and total frictional force, therefore the maximum of S_B·1000 at about 48°C seems to be due to the variations of wear-resistance and internal friction with temperature. On the other hand, the maximum of S_B·1000 at about 98°C may be considered as derived from the temperature dependence of mechanical properties.
(4) In the case of polyethylene, the maximums of S_B·1000 appear at lower temperature than on the case of polypropylene. But the temperature dependence of S_B·1000 in both the polymers shows very similar behavior with each other against the reduced temperature, T-Tα_a, where Tα_a is the temperature at which the main dispersion appears in the dynamic viscoelastic behavior of the polymers.
Consequently, the temperature depndence of wear-resistance seems to be dependent on polymer properties, especially molecular motion of polymer.
(5) The relationship between the wear resistance and the mechanical properties is described qualitatively as follows;
amount of wear loss ∞total frictional force/modulus×tensile strength×elongation.

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This paper is concerned with the formulation of a thermodynamic strain-time function for a 2-phase material when subjected to a constant stress in an isothermal temperature field.
It is shown on basis of irreversible thermodynamics that the stress-deformation relation of such systems is of the same mathematical form as the relation obtained from a simple rheological model described in an earlier publication. A correspondence is established between the phenomenological coefficients contained in the thermodynamic function and the structural parameters employed in the mechanical model.

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