Rheology: Concepts, Methods and ApplicationsChemTec Publishing, 2006 - 474 There are few comprehensive books on the market on the subject of Rheology -- the complex science dealing with flow and deformation of matter -- and these are several years old. At least now there is a book that explains the meaning of a science that many scientists need to use but only a few can fully grasp. It does so by striking the balance between oversimplification and overload of theory in a very compelling and readable manner. The authors' systematic presentation enables the authors to include all components of Rheology in one volume. The first four chapters of this book discuss various aspects of theoretical Rheology and, by examples of many studies, show how particular theory, model, or equation can be used in solving different problems. The main emphasis is on liquids, but solid materials are discussed in one full chapter as well. Methods of measurement and raw data treatment are included in one large chapter which constitutes more than one quarter of the book. Eight groups of methods are discussed giving many choices for experimentation and guidance on where and how to use them properly. The final chapter shows how to use rheological methods in different groups of products and methods of their manufacture. Usefulness of chemorheological (rheokinetical) measurements is also emphasized. This chapter continues with examples of purposeful applications in practical matters. |
Spis treści
Rheology as an independent branch of natural sciences emerged more than 70 years ago It origin | 1 |
Superposition of liquidlike and solidlike features in behavior of technological materials is di | 2 |
The definition of rheology as a branch of natural science and the subject of rheological studies | 3 |
In its origin the term rheology applies to flowing media since the main root of the word mean | 4 |
In the framework of solidstate mechanics such effects as a longterm behavior engineering prop | 5 |
properties of material may change in space due to the gradient of concentration in multicompo | 6 |
Press Cambridge 2000 | 7 |
Publisher Dordrecht 2001 | 8 |
Example | 236 |
4542 Thixotropy | 237 |
demonstration of the secondorder effects in the mechanics of continuum | 238 |
A shaft is twisted with a torque T as in Question 43 However the torque is high enough to prod | 239 |
This part of the book is devoted to selection and evaluation of modern experimental methods of rh | 241 |
Relative methods of viscosity measurement are based on comparison of properties of fluid under in | 242 |
velocity is not very high and flow is laminar | 244 |
529 | 246 |
Rheology is a science dealing with deformation and flow of matter Relationships between stresses | 9 |
In our case we do not consider force distribution because we have selected a small surface area | 10 |
113 | 11 |
Let us consider a plane section of a unit cube in Fig 113 The section is shown in Fig 114 | 12 |
normal stress sE | 13 |
The threshold effects on the material behavior can be treated in an unambiguous manner using th | 14 |
Any combination of invariants I1 I2 and I3 is also invariant with respect to the orientation of | 15 |
1110 | 16 |
This stress tensor shows that shear stresses are absent at any direction in space The tensor exp | 17 |
1117 | 18 |
Analysis of uniaxial loading can be useful in solving many practical problems For example let u | 19 |
Then writing the sum of projections of all forces stresses are multiplied by the unit areas of | 20 |
1120 | 21 |
The change of distances between points inside a body can be monitored by following changes of ver | 22 |
With neglecting the terms of higher orders than dx it is easy to calculate the difference ds2 | 23 |
As shown in section 111 any tensor can be defined by two vectors It is similar with deformati | 24 |
The sense of this difference or the meaning of the second the socalled antisymmetrical part of | 25 |
Pure geometrical analysis demonstrates that the diagonal components of tensor dij expressed by Eq | 26 |
1213 | 27 |
In the first case let l0 1 and Dl 01 The socalled engineering measure of deformation is | 28 |
If elongations are small Dl1 1 and Dl2 1 the difference between eItotal and is negligib | 29 |
Another tensor of large deformations is also frequently used This is the inverse or reciprocal | 30 |
1227 | 31 |
The schemes of twodimensional plane simple shear for an element of a body in small deformation | 32 |
The results obtained from Eqs 1235 indicate that in simple shear no volume change occurs becau | 33 |
Motion of a body is characterized by velocity which is a vector If velocity at any given point | 34 |
135 | 35 |
The physical meaning of substantial time derivative DDt requires that the derivative is calcul | 36 |
This scale must be large enough to distinguish individual molecules or their segments The charac | 37 |
Stress is a measure of forces acting on a point and it is defined as a relative force or a force | 38 |
the spherical part which represents volume | 39 |
1962 or EC Young Vector and Tensor Analysis Marcel Dekker New York 1993 | 40 |
Calculate stresses acting on a thread being suspended by its end and stretched by its own weight | 41 |
Two simple and easy to grasp concepts or models describing mechanical properties of materials o | 43 |
These relationships appear to be very simple and in fact many real materials behave as predict | 44 |
However many other materials have different behavior It is possible to note a slow decay of for | 45 |
A large number of empirical equations were proposed to describe st functions during relaxation | 46 |
Experimental linear relaxation and creep functions are demonstrated here by their spectral repres | 47 |
The creep function by its physical meaning is an increasing function of time having a definite | 48 |
The discrete approximation helps us to understand the meaning of spectral representation of the c | 49 |
In order to find rheological characteristics of material it is preferable to carry out an experim | 50 |
This equation is equivalent of Eq 221 with the same meaning of members The new factor here is | 51 |
The last central equation of the theory of periodic oscillation in studies of material properties | 52 |
Another useful graphical interpretation of dynamic experiment in measurement of viscoelastic prop | 53 |
Then sind is expressed by the ratio of the ellipse surface area to the surface area of a circums | 54 |
The above formulated parameters are used to describe viscoelastic effects and characterize proper | 55 |
In order to compare this equation with the standard formulation of Hookes law X can be treated | 56 |
The last expression is the analogue of Eq 2211 The ratio kG has the meaning of a retardati | 57 |
Let us assume that in a multicomponent The Kelvin Voigt model one partial modulus equals zero | 58 |
The Maxwell and the KelvinVoigt models can be joined in parallel in series or combinations the | 59 |
where C1 and C2 are constant expressed via ki or four rheological parameters of the Burgers mode | 60 |
Q CU | 61 |
241 | 62 |
The hereditary part of Eq 245 is | 63 |
Memory effects become complicated and sometimes rather unexpected when temperature changes during | 64 |
2414 | 66 |
Eq 2417 shows that the relaxation and creep functions are not independent but related to each | 67 |
Equilibrium value of the coefficient of the first difference of normal stresses18 corresponding t | 68 |
More complicated modes of deformation can also be studied on the basis of the general relationshi | 69 |
the pair of dynamic functions either J and J or G and G are in fact not independent bu | 70 |
Double lines in the upper part of the scheme indicate the relationship between relaxation and ret | 71 |
2512 | 72 |
second no experiment gives the absolutely correct measured value but only within some experim | 74 |
In this approach a relaxation spectrum is not a pure mathematical image but any mode of a spectru | 75 |
at 2524 | 76 |
The result of the computeraided calculations is also a set of discrete modes The set of relaxat | 77 |
2528 | 78 |
A chain contains N + 1 identical beads and N identical springs Resistance to the displacement | 79 |
There is only a single independent retardation time and all others can be expressed by it The n | 80 |
and all other times can be calculated from a spectrum using Eq 266 Eq 2612 relates relaxat | 81 |
2613 Model of a rotating coil47 | 82 |
Later the idea of entanglements was developed on the basis of a model of a single freedraining | 83 |
Other versions of the same approach were also discussed based on different distribution of the fr | 84 |
It is important that the values Lab and qab are not specified in the original theory because it | 85 |
It was also shown60 that the theory permits calculation of characteristic relaxation time qd us | 86 |
it is very desirable that the theory correctly predicts the frequency dependencies of G and G | 87 |
This equation is assumed to be valid for highmolecular mass polymers where the whole MMD lies a | 88 |
The mixing rules for polydisperse polymers discussed in modern literature69 are based on some mol | 89 |
2624 | 90 |
Based on Eq 2621 the following analytical equation was obtained6973 which helps to determine | 91 |
It was noticed that the time or frequency dependencies of viscoelastic properties measured at | 92 |
This master curve is related to the reduction temperature which is the largest of the examined i | 93 |
all relaxatio | 94 |
It is interesting to notice that Tg is the glass transition temperature then for many polymer | 95 |
Analogous experimental data were observed and published for various polymers and they form the ba | 96 |
Combining experimental data of these figures one can see the full range of relaxation covering a | 97 |
This conclusion is very important for numerous applications of viscoelastic polymeric materials | 98 |
Concerning monodisperse polymers the general solution was proposed for relaxation behavior in th | 99 |
The theoretical and applied meaning of nonNewtonian behavior of liquids is complex Therefore a | 100 |
up to | 101 |
The most well known example of instability is the formation of a neck the effect of necking Th | 102 |
Based on these and many other analogous experimental data it is reasonable to suppose that the | 103 |
These equations are valid for the steadystate flow demonstrating that nonNewtonian behavior is | 104 |
It is also possible to decompose the response nonlinear signal into several harmonics at differe | 105 |
An interesting phenomenon was elucidated from comparison of linear and nonlinear viscoelastic pr | 106 |
This is directly opposite to linear behavior for which all relaxation modes are independent and | 107 |
it is desirable that the structure of a constitutive equation be convenient for applied calcula | 108 |
It was shown that the KBKZ model correctly describes many special rheological effects in various | 109 |
can be considered as a very special case of a factorable KBKZ model where | 110 |
2810 | 111 |
2814 | 112 |
Initial information necessary for constructing a nonlinear constitutive equation is the correct | 113 |
2816 | 114 |
A more complicated experimental technique can also be used and sometimes is used for constructing | 115 |
2 R Hooke 16351703 outstanding English experimentalist physicist and architect He invente | 116 |
1977 A I Isayev C M Wong J Polym Sci Polym Phys 26 2303 1988 | 117 |
17 1990 Rheol Acta 28 65 1993 C Elster J Honerkamp J Weese Rheol Acta 31 161 1 | 118 |
873 1999 | 119 |
LJ Zapas J Nat Bur Stand 68B 103 1964 The constitutive equation is called by the name | 120 |
Additional question | 121 |
Additional question 1 | 122 |
Rheology deals with materials in liquid and solid states However as can be derived from its nam | 123 |
In a simple shear | 124 |
It is also useful to formulate the rheological equation of state of Newtonian liquid in an invari | 125 |
3110 | 126 |
If the shear stress is proportional to the shear rate Newtonian liquid its graphic representat | 127 |
The increase of polymer concentration in solution shift from upper left corner of Fig 321 dow | 128 |
Qualitative explanation of nonNewtonian flow is based on analysis of the interrelation between t | 129 |
Here there is no gradual decrease in viscosity but a rapid drop in viscosity by several decimal | 130 |
Flow of viscoplastic materials below the yield stress may occur and viscosity in this range is on | 131 |
Structured systems are nonNewtonian viscoplastic liquids Their viscosity below the yield stress | 132 |
There are two main cases of anisotropy First there are liquid crystals11 which initially have | 134 |
Concentration dependence of viscosity of LC polymers will be discussed in more detail in section | 135 |
There are several general and commonly used approaches for fitting experimental data obtained in | 136 |
Eqs 331 satisfy the main conditions which are necessary for fitting experimental data of non | 137 |
The yield stress sY is one of their fundamental parameters The most popular and simple equatio | 138 |
3313 | 139 |
In general the viscosity of materials depends on their properties This section is limited to de | 140 |
Chemical bonds between chains decrease the average length of chain between neighboring entangleme | 141 |
where the constant h is introduced as a coefficient of the firstorder term A more rigorous de | 142 |
The structure of equations commonly used for hc dependence shows that they can be presented in | 143 |
The difference in the concentration dependence of viscosity for different polymers is determined | 144 |
3322 | 145 |
3343 Viscosity of suspensions | 146 |
This equation is converted to the Einstein equation at low concentrations It satisfies numerous | 147 |
3328 | 148 |
The structure of Eq 3329 demonstrates that nonNewtonian behavior results from polymer polydis | 149 |
Rubbery elasticity is characterized by the rubbery modulus Ge or by the reciprocal value equi | 150 |
These values are only estimates because of difficulties in preparation of monodisperse polymers | 151 |
If a rod is rotating inside a rheological liquid such liquid instead of being displaced out o | 152 |
3422 First normal stress difference quantitative approach | 153 |
The direct relationship between the first normal stress difference N1 and the shear stress s | 154 |
347 | 155 |
The common agreement that the normal stresses are related to elasticity of liquid is quantitative | 156 |
These equations demonstrate that the normal stresses are secondorder effects with respect to the | 157 |
This equation permits estimation of stored elastic energy in nonlinear regimes of steady flow | 158 |
The typical example of die swell or postextrusion swelling is shown in Fig 3414 for a polym | 159 |
Both reasons may exist simultaneously | 160 |
The existence of a very sharp maximum on the grt curves suggests that the maximum on the s+t | 161 |
The rheological behavior of this material is generalized in Fig 355 where the dependencies of | 162 |
Systems such as greases toothpaste and other analogous viscoplastic materials are not rubbery a | 163 |
57 | 164 |
The deformation process in some systems may lead not to a decrease but to an increase in viscosi | 165 |
The same material may exhibit either thixotropic or antithixotropic effect and the observed rheo | 166 |
Examination of the transient behavior in shearing gives even more pronounced effects if one measu | 167 |
Volume effects dilatancy caused by shear are also possible in elastic bodies as well as in vis | 168 |
Finally it is interesting to report that structure effects in shearing occur in a self oscillat | 169 |
The general result of thermodynamic studies of deformationinduced phase transitions is a theory | 170 |
Two dotted lines are drawn in Fig 3515 They show that the apparent transition temperature beg | 171 |
The influence of shear flow on kinetics of crystallization is shown in Fig 3517 where the kine | 172 |
The conformational transition gives us a possibility to obtain material with fully extended chain | 173 |
The term limits of shear flow in the title of this section should be understood as the limit of | 174 |
where Q is the flow rate Dp is the pressure drop h is viscosity R and L are the radius and the | 175 |
The Blasius rule is an empirical generalization of numerous experimental data obtained by differe | 176 |
In concentrated solutions and polymer melts other types of instabilities occur before the onset | 177 |
Stability of stream along the whole channel in spite of the appearance of sharkskin related to | 178 |
Existence of wall slip is accompanied by surface charge formation tribological effect that is | 179 |
The stickslip phenomenon is directly related to the existence of multivalued flow curves or hy | 180 |
The hysteresis effect accompanied by regular periodic oscillations of the instant output rate is | 181 |
It is interesting to mention that breaks occur along the channel axis where shear stresses are a | 182 |
Adhesive and cohesive strengths of polymeric materials are similar Hence whether there is an ex | 183 |
Regular surface defects or regular variations of the form of a stream formally does not correspon | 184 |
Uniaxial extension is not easy to induce in some liquids It is difficult to maintain the shape o | 185 |
At first glance the experimental results presented in Fig 372 seem to contradict the Trouton | 186 |
The increase in the elongational viscosity at high deformation rates is sometimes considered as a | 187 |
If the deformation rate is increased the steady flow regime cannot be reached because the sample | 188 |
However the extension cannot be carried any further An increase in the deformation rate leads t | 189 |
The experimental data presented in Fig 376 constitute only a part of the characteristics of th | 190 |
The transition through the domains corresponds in polymer physics to wellknown transitions throu | 191 |
The increase in stress was discussed by the authors of original publications in terms of strain h | 192 |
3731 Multiaxial elongation | 193 |
The tubeless siphon effect is schematically presented in Fig 3710a The fluid from a large fre | 194 |
where a is a constant of the order of one M is the molecular mass of macromolecule h is the i | 195 |
3710 | 196 |
The stability in relation to the draw resonance phenomenon depends on whether fluid has extension | 197 |
perhaps during the period of observation or an experiment unrecoverable deformations are so s | 198 |
All real liquids are complex liquids and the following principal effects are more or less pronoun | 199 |
Nobel prize in chemistry 1936 | 200 |
27 EO Kraemer Industr Engng Chem 30 1200 1938 | 201 |
49 AG Dodson P Townsend K Walters Comput Fluids 2 317 1974 | 202 |
76 BA Wolf Macromol Chem Rapid Commun 1 231 1980 Macromolecules 17 615 1984 | 203 |
102 VI Brizitsky GV Vinogradov AI Isayev YuYa Podolsky J Appl Polymer Sci 20 25 | 204 |
127 Extensional flow of polymer materials was the subject of numerous investigations One can fin | 205 |
27 1984 | 206 |
Additional question | 207 |
What temperature rise is expected? | 208 |
The concept of solid is an idealization of real behavior of numerous materials Some of them are | 209 |
Formulating the rheological model of an elastic solid ie writing its rheological equation of | 210 |
the equation is written for onedimensional deformations extension | 211 |
Let us assume that the linear relationships between spherical and deviatoric parts of both tensor | 212 |
Hydrostatic pressure p in uniaxial extension is | 213 |
Then it is evident that E 0 and this inequality can be fulfilled only if | 214 |
431 | 216 |
433 | 217 |
or | 219 |
Therefore the general solution in determining principal stresses in a threedimensional deformat | 220 |
Materials obeying Eq 4416 are sometimes called neoHookean because this equation is the most | 221 |
Elastic potential in the form of Eq 447 was formulated as a consequence of the molecular kine | 222 |
4420 | 223 |
Treating experimental data in terms of Eq 4424 assumes that they ought to be presented in the | 224 |
If linear functions Eq 447 or Eq 4420 are not sufficient and they are not for threedime | 225 |
The general form of the dependence of shear stress on deformation for simple shear its measure i | 226 |
The last remark in this section regards time effects Time must not be mentioned in this section | 227 |
This approach leads to nonlinear dependencies in deformation of any arbitrary type The potentia | 228 |
The limit of elasticity is a result of standard experiments in uniaxial extension of solid sample | 229 |
The elastictoplastic transition as well as fracture can be considered as some critical event | 230 |
values of stresses in the domain of plastic flow do not exceed 106 Pa depending on viscosity a | 231 |
452 | 232 |
In the deformation of elastic solids the work done is stored in the form of elastic potential en | 233 |
It is reasonable to suggest that in reality both effects are superimposed and the dominating mech | 234 |
A general criterion of limiting states must reflect the influence of both shear and normal stress | 235 |
where K pR48L is the shape factor or formfactor determined from known dimensions of the me | 247 |
5216 | 248 |
Such computation method of determining a flow curve is sometimes examined within the framework of | 249 |
where as earlier is the mean output based velocity r is density and a is the coefficient re | 250 |
For lowviscosity inelastic liquids the socalled Couette correction plays the dominant role in | 251 |
with mk varying with the flow rate or pressure | 252 |
where l is the heat transfer coefficient and k is a coefficient of temperature dependence of vis | 253 |
Slip may not necessary occur continuously it may alternate with adhesion This phenomenon which | 254 |
5227 Adsorption on a channel surface | 255 |
Velocity measurement of meniscus curving of an initially strictly cylindrical sample is a special | 256 |
The upper limit of shear stress is determined by the appearance of instability excessive thermal | 257 |
The schematic diagram of such an instrument is shown in Fig 526 This instrument basically con | 259 |
Gas viscometers are instruments typically used for research purposes For realization of their gr | 260 |
52623 Viscometers with varying load | 261 |
These instruments seem primitive and obsolete but they play an important role in standardization | 262 |
Glass viscometers are manufactured with different sizes of measuring reservoirs and capillaries | 263 |
All this makes such testing machines an excellent base for manufacturing viscometers of constant | 264 |
This instrument makes it possible to conduct viscosity measurements in the range from 25 to 104 | 265 |
During material testing by rotational rheometry different regimes of deformation are possible T | 266 |
532 | 267 |
scanning regimes of tests when torque changes according to a predetermined program | 268 |
538 | 269 |
Eq 5313 according to its structure and physical meaning is identical to Eq 5213 in the th | 270 |
5320 | 271 |
Torque with respect to the vertical axis generates shear stresses s From the equilibrium condit | 272 |
The instruments with small angles d are of basic practical interest The shear rate field as wel | 273 |
5324 Disk viscometers | 274 |
5338 | 275 |
Calculation of apparent viscosity measured in a spherical viscometer is also possible for nonN | 276 |
The device with guarding cylinders In this device the outer cylinder revolves Above and below t | 277 |
5328 Temperature effects | 278 |
Thus viscosity measurements constitute the right choice only in the region of laminar flow The | 279 |
At the same time such instruments are not always suitable for studying new phenomena and refined | 280 |
The schematic diagram that is shown in Fig 537 can be realized in the existing rotational visc | 281 |
The same company manufactures other high precision elastoviscometers known under the names RDA I | 282 |
The torque in this instrument is measured on the drive shaft The measured shear stress ranges fr | 283 |
The most common instrument of such a type is the Brookfield viscometer Different modifications o | 284 |
The automatic torque control systems | 285 |
In one of the modifications of this instrument the size of clearance between the cylinders is bro | 286 |
The electromagnetic method of imposing torque for viscosity measurements in the region of low she | 287 |
53452 The Mooney viscometer | 288 |
53453 The Goettfert instrument | 289 |
Repeating these measurements at different shear stresses one can obtain the dependence | 290 |
5351 | 291 |
The subscript i shows that N2 determined by this equation is related to the values at the surface | 292 |
a moving plate is | 293 |
In these instruments a sample as in the preceding case is placed between two parallel disks b | 294 |
Then by plotting the dependence of h4t versus Ft it is not difficult to find viscosity from | 295 |
Hence the viscosity of liquid being investigated is determined from the measured dependence of F | 296 |
where rl and rs are respectively the density of liquid being investigated and the material fro | 297 |
5410 | 298 |
In some technical applications an indentor prepared in the form of cone is used This is espec | 299 |
the wall effect on a sphere motion is neglected ie the sphere radius is much smaller than t | 300 |
557 | 301 |
A cylindrical tube can be used under a preselected angle in one of the variations of method T | 302 |
Production of instruments with a falling sphere is simple It is sufficient to have a glass tube | 303 |
In the laminar motion of any body in a viscous Newtonian liquid a force resisting motion F is | 304 |
Viscometers with a falling cylinder are very simple to construct but their use in practice is li | 305 |
5621 The simplest measuring schemes | 306 |
The end of extrudate in Fig 561 is moved with velocity of vt which is varied with time in s | 307 |
5623 Tubeless siphon instruments | 308 |
5625 High strain rate methods | 309 |
An axisymmetric and planar stagnation flow involves impinging two melt streams through lubricated | 310 |
Let a uniform isotropic sample be placed between two parallel plates A and B Fig 571 The ga | 311 |
573 | 312 |
The results of measurements in the vicinity of w0 are unreliable because even a slight error in m | 313 |
The solution of Eq 5713 with these boundary conditions gives the function xz t and its part | 314 |
An equation describing torsional oscillations of cylindrical sample caused by twisting of one of | 315 |
5719 | 316 |
5726 | 317 |
5732 | 318 |
The resonance method is applicable for measuring G and G at a single resonance frequency for lo | 319 |
5745 | 320 |
Samples of other geometrical forms can also be used in the damping oscillation experiment utilizi | 321 |
Since the solution of Eq 5755 is proportional to eiwt one obtains | 322 |
5772 Longitudinal waves | 323 |
5767 | 324 |
5775 | 325 |
5778 | 326 |
5782 | 327 |
5787 | 328 |
Parallel disks with shifted axes Small radial shift of axes leads to periodic changes of velocit | 329 |
Many hundreds of experimental devices have been constructed for measuring viscoelastic properties | 330 |
Electromagnetic excitation is also widely used in the vibratingreed method shown in Fig 577 | 331 |
upper | 332 |
581 | 333 |
However the main interest here is application of optical techniques to study the flow of polymer | 334 |
583 | 335 |
As an example Fig 582 shows the dependence of birefringence Dn on maximum tangential stress | 336 |
In addition the stressoptical rule is not applicable to filled polymers even in cases when they | 337 |
The rheooptical method can also be used for measuring quasiequilibrium compliance phase angl | 338 |
587 | 339 |
Instruments in which birefringence during flow is measured are called dynamooptimeters or rheo | 340 |
A combination of viscometric and optical schemes of measurements is especially effective for obse | 341 |
The quantity N1c in the linear deformation domain equals where g0 is the amplitude of deformat | 342 |
This approach called velocimetry is successfully realized by utilizing several physical phenome | 343 |
1 Flow of viscous liquid is always accompanied by heat output because work must be done and dissi | 344 |
36 This was done in earlier publications devoted to the Weissenberg effect See eg K Weissenb | 345 |
72 The fourroll method was first suggested by A Keller and described in series of papers for e | 346 |
J Appl Polym Sci 7 685 1963 and widely used by JK Gillham and coauthors | 347 |
diameter mm | 348 |
Prove that Eq 5739 is valid for materials exhibiting low losses | 349 |
Rheological measurements provide us with properties of materials Fundamental rheological theorie | 351 |
The most important correlations based on vast experimental data are dependence of viscosity on av | 353 |
As was shown in section 34 the dependence of coefficient of normal stresses Y0 on molecular m | 354 |
Comparison of linear and branched polymers in uniaxial extension see section 353 and Fig 37 | 355 |
Melt flow index MFI see section 526 is a characteristic of viscosity directly related to mo | 356 |
It is important to stress two terms always present in description of any method specified condi | 357 |
Measurement of gelation temperature helps to estimate the wax content but also data similar to th | 358 |
The rheological behavior of viscoplastic materials was discussed in section 322 The main pecul | 359 |
In real practice measurements of viscoelastic properties can be carried out in a limited experim | 360 |
WernerBratzler Shear | 361 |
Properties of chocolate mass19 are characterized by its rheological parameters The recommended m | 362 |
Elasticity is not essential for application of these materials although viscoelastic properties | 363 |
Another problem in estimation of blood properties is its instability caused by temperature and c | 364 |
Application of rheological analysis was found useful for other biological materials such as bone | 365 |
The yield stress in concentrated suspensions in a low viscosity matrix depends on the concentrati | 366 |
From an applied point of view existence of a critical concentration at which strong shear thicke | 367 |
A structure formed after applying an electrical or magnetic field has some strength due to intera | 368 |
An example illustrating magneto rheological effect is shown in Fig 6216 where the dependence | 369 |
Rheology operates with different measures of resistance to deformation primarily determined by v | 370 |
Rheokinetics of linear polymerization depends on the process chemistry andor the chain growth me | 371 |
Calculation of time dependence of viscosity based on pure chemical and kinetic arguments is illus | 372 |
The effects of phase separation in polymerization processes regardless of cause is always manif | 373 |
calculation of viscosity change during technological process with application to design of equi | 374 |
Two matters are important for technology estimation of the geltime t and formulation of the | 375 |
Very rapid increase in viscosity suggests the use of exponential formulas of various types A num | 376 |
Let the induction period when oligomer can be treated as lowviscosity liquid be t0 at low shea | 377 |
6333 Curing after gelpoint | 378 |
Diagrams such as those presented in Fig 6310 can be constructed for different curing systems | 379 |
Incomplete curing is shown in Fig 6312 In this case the limiting values of elastic modulus a | 380 |
6314 | 381 |
Rheokinetic study is a convenient and sensitive method of monitoring chemical reactions of polyme | 382 |
Therefore a crucial step is listed on the right side of the second line This is the checking of | 383 |
Principle of material objectivity expresses the idea that behavior of material must be written | 384 |
where vi are the components of velocity vector a is the thermal diffusivity r is density and cp | 385 |
The simplest case is flow through a tube of a circular crosssection | 386 |
where sR is the shear stress at wall which is expressed in usual manner as and sY K and n are rh | 387 |
Equilateral triangle | 388 |
6410 | 389 |
A pumping screw is often used in different technological processes In particular various extrus | 390 |
where A and B are constants N is the rotational speed of the screw and m is an exponent in a pow | 391 |
Calculation of profile evolution after leaving a forming calibrating die is a purely rheologica | 392 |
6412 | 393 |
Extension of the calendering theory to flow of nonNewtonian liquids is made according to the sam | 394 |
6420 | 395 |
6434 Molding technologies | 396 |
6435 Compression molding | 397 |
Let us assume that in the case of a strip fluid is confined between two sides in the width direc | 399 |
Injection molding of thermosets and reactive fluids which are able to form infusible crosslinked | 400 |
The injection molding cycle can be divided into three stages These include cavity filling packi | 401 |
Pressure required to fill a tubular mold cavity is reciprocally proportional to its radius to the | 402 |
1 The number of publications devoted to investigation of rheological properties of polymeric subs | 403 |
29 Ch Ancey H Jorrot J Rheol 45 297 2001 | 404 |
P Slatter K Wilson Rheol Acta 43 2004 | 405 |
Prove Eq 646 for the Bingham liquid Explain the necessary conditions required for movement of | 406 |
A intensity of dissipation in flow | 407 |
DR draw ratio | 408 |
G0 instantaneous shear modulus | 409 |
k ratio of radii | 410 |
Pv part of the pressure drop responsible for resistance of a channel | 411 |
U0 initial voltage | 412 |
Z rigidity of spring | 413 |
deformation in periodic oscillation | 414 |
l retardation time in a discrete spectrum with different indices | 415 |
s+ shear growth stress function | 416 |
What is the equilibrium state of a liquid and a solid in the absence of stresses? | 417 |
Calculate stresses acting in a thread being suspended by its end and stretched by its own weight | 418 |
where g is the gravitational acceleration | 419 |
For Maxwellian liquid with a relaxation time q what is the residual stress in comparison with t | 420 |
where 0 is the instantaneous modulus see Eq 224 The function Gq ie a relaxation spe | 421 |
Eq 2311 and its solution show that the Burgers model describes the behavior of a material with | 422 |
Answer | 423 |
Relaxation of shear stresses after sudden cessation of steady flow is described by the formula s | 424 |
Analyze the movement of the weight after the force is ceased Is it possible to find the componen | 425 |
Can viscosity be negative? Explain the answer | 426 |
Calculate shear stresses in flow of liquid through a straight tube if flow is created by the pres | 427 |
The last expression is known as the Poiseuille equation | 428 |
This equation shows that for a powerlaw type liquid with arbitrary value of the exponent n not | 429 |
which is valid at r rY At r rY velocity is constant and equals to urY | 430 |
An experimenter measured viscous properties of material at different shear rates and obtained a f | 431 |
Normal stresses in shear appear as a secondorder effect However at high shear rates they excee | 432 |
Comment | 433 |
ie the intrinsic viscosity of suspension of solid spheres always equals 25 | 434 |
Answer | 435 |
The elongation ratio l is | 438 |
and in compression at l 2 | 439 |
Answer | 440 |
Calculate the velocity profile in flow of Newtonian liquid through an annulus produced by two coa | 441 |
Answer | 442 |
Prove that Eq 5739 is valid for materials exhibiting low losses | 443 |
Answer | 444 |
Compare sensitivity of different rheological methods to estimation of molecular mass | 445 |
Prove Eq 646 for a Bingham liquid Explain the necessary conditions required for movement of B | 447 |
Answer | 448 |
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wide frequency range 92 | 474 |
Inne wydania - Wyświetl wszystko
Rheology: Concepts, Methods and Applications Aleksandr I︠A︡kovlevich Malkin,Avraam I. Isayev Widok fragmentu - 2006 |
Rheology: Concepts, Methods, and Applications Alexander Ya. Malkin,Avraam I. Isayev Podgląd niedostępny - 2011 |
Kluczowe wyrazy i wyrażenia
A.Ya Acta amplitude apparent viscosity appears application birefringence calculated capillary chain changes characteristic coefficient components concentration constant constitutive equation cylinder deformation rate dependence deviatoric dynamic dynamic modulus effect elastic elastic modulus elongational example experimental data expressed extrudate Figure flow curve Fluid Mech force formulation frequency function G.V. Vinogradov initial instrument large deformations limit linear low shear macromolecules Malkin measurements mechanical method modulus molding molecular mass Newtonian liquid non-linear non-Newtonian normal stresses obtained oscillations parameters phase Phys plastic polymer polymer melts polymeric possible pressure problem properties of material radius range ratio relationship relaxation spectrum Rheol rheological equation rheological properties rotation rubber sample shear flow shear rate shear stress shown in Fig simple shear solid solution stress tensor structure surface temperature theory thixotropic tion torque transition tube uniaxial extension velocity viscoelastic viscometer viscometry yield stress