Plastics Engineering 3E Episode 15 docx

If this is repeated for all the plies then we get The limiting condition is thus ax = 583 MN/m2 as predicted by Tsai-Hill... The shear rate in the metering zone will be given by... Hence

(3.17) This is a non-symmetric laminate The Q matrices are given in the text and the

A , B and D matrices are determined from

3

A = CQf (hf - hf-1)

f = 1

- E x - - a11 a12 ai6 bii b12 b i b - - N , -

CY a21 a22 a26 b21 bn b26 N y

N x y

YXY - ab1 a62 a66 bbi b6z b M

Kx - ~ I Ib12 bla dii d12 d16 M , where N x = 30 N/mm

K y hl a2 h b d21 d22 d2b M y

- K x y - - b b l b62 BM d61 db2 d a - - M x y -

4.65 x lob3 -2.68 x low3 0.036 -8.00 x 0.031

(3.18) (a) Maximum Stress Criterion

The local stresses are given by

The Tsai-Hill equation gives

Therefore failure is predicted

(3.19) (i) For an applied stress of a, = 1 MN/m2, the stresses in the 1-2 direction are

given by

0 1 0 2 r12

- = 1867, - = 200, - = 92.4 Hence an applied stress of a, = 92 MN/m2 would initiate a shear failure before tensile

or compressive failures occurred

(ii) Maximum Strain Criterion

From the data given

(iii) Tsai-Hill Criterion

Letting X be the multiplier for a,

( x z ) 2 - (x+)? + ( z)2 + ( F)2 = 1

Solving for X gives X = 83.8

Hence the limiting condition is a, = 83 MN/m2

This predicts a stress of ax = 583 MN/m2

If this is repeated for all the plies then we get

The limiting condition is thus ax = 583 MN/m2 as predicted by Tsai-Hill

(3.21) K , = ẵa,)'/~ (E tan (s)) 'I2

The shear rate in the metering zone will be given by

Hence the characteristics shown may be drawn

The operating point is at the intersection of the two characteristics

2cos24 = 2cosz4 -+ 3(1 - 2sin24) = 2(1 - sin24)

thus

which can be solved to give 4 = 30"

C.S.U = d + D2 tan 10" = D2(1 + tan 10")

s.a = 20 + 2D/ cos 10 + 2 0 tan 10"

Since the dome formed is a free surface it m a y be assumed to have a constant thick- ness, t

When sheet just touches the base

OTD film thickness x (blow-up)*

6qVL(2h - H ) H3

42.0

2.85 10-3 6.25 x 10-3 1.28 x

3.68 x

8.5 x 10-2

0.21

1.31 x 105 1.2 x 105 1.17 x 105

1.02 x 105 8.8 x 104 7.13 x 104

r = 1 x lo4 N/m2 and q = 8 x 104 Ns/m2

(5.2) For a Power Law fluid

y a (r)""

or l o g e o< l / n log AP

So the slope of a graph of (log Q) against (log A P ) will be a straight line of slope l / n

if the Power Law is obeyed From the attached graph l/n = 1.268 SO n = 0.7886 Consider the point, Q = 0.8 kglmin and AP = 5.2 MN/m2

(5.3) Cone and Plate

0.062 x 180 True shear rate, i, = - = = 1.18 S-'

3 x 1.18 2n x (37.5 x 10-3)3

r = - 3T -

2nR3 Shear Stress,

Therefore the true shear rate on the cone and plate is equivalent to a shear rate of 0.69(1.18) = 0.817 on the ram extruder

At i, = 0.817 r = 11.98(0.817)0.362 = 11.13 kN/m2

11.13 - 10.7

% difference = = 4.06%

10.7 (5.4) If the material is assumed Newtonian and the shear rates are equated then

6Q2 4Ql

T H z nR3

- = - 4TH2Qi

6nR3

If the material is Non-Newtonian then the true shear rates should be equated

So for n = 0.37 correction factor = 0.9094

so Natural time = - 'I = 1.3 x 103 = 4.3 x s

v n(3 10-3)2 x 37 10-3 Dwell time = - =

A P 1.5 x 106

A T = - = = 0.45"C

pc, 3 3 x 106 (5.9) Flow rate = 3 d m i n

From flow curves, at 9 = 150

G = 3.05 x 104 N/m2

From graphs for swell ratio BST = 1.41, BSH = 1.99

Thus approximate dimensions of extrudate would be 42.3 mm deep x 21.2 mm wide x

R = 2.83 mm (5.15) Since the power law equations are going to be used rather than the flow curves,

it is necessary to use the true shear rates

From Fig 5.3, qo = 3 x 104 NdmZ (ie q at 1; = 1 s-I) and n = 0.33

So from illustrative example (5.3)

Total Pressure loss = 22.1 MN/rn2

At exit, i, = 1910 So t = 3.6 x lo5 N/mZ, G = 5 x lo4 N/m2

36

.5

so yR = - = 7.2 -+ BH = 2.43, BT = 1.56

So wall thickness = 1.21 mm, diameter = 62.4 m m

(5.19) Assuming no swelling at the die exit

6V 6 x 15 x

y = - = = 120 s-L

H 0.75 x 10-3 From the flow curves, t = 1.3 x Id N/mz

2.08 New H = = 0.36 x lod3 m New V = 15 x 2.08 x 1.455 x = 45.39 x mls

6 x 45.39 0.36

And so on until the iteration converges, at which point

H = 0.405 x m, D = 29.2 m m

Total pressure loss = 58 MN/m2

(5.22) (a) As shown in the text, the flow rate of a power law fluid in a circular section

(5.23) From the equations in the previous example

For a Newtonian fluid n = 1 so .tf/t = 0.316

For a Non-Newtonian fluid, n = 0.3f!,/e = 0.424

(5.24) (a) As shown in the previous examples, if the pressure is held constant

but

So substituting for .t gives

1 / 1 3

140 x 106 7.53 x 10-5

(b) As shown in the text for the flow of a power law fluid in a capillary

but

3n + 1 Qdt = r R 2 d e SO d e = ( Q d t ) / Z R 2 ) 2q0@ = (m) nn 3 d P nR2 R3"+' 2q0Q"dt = (e) ltR3n+3 dP

which m a y be integrated to give

0.3

2 ioq7 10-511.3 n(5 x 10-3)3.9 (6%)

= (29.36t) MN/mz

The variations of Q and P in each case are shown in the attached graph

(5.25) Volume of cavity = 50 x 10 x 3 x

So time to fill this = 1.5 seconds

Gate: volume = 2 x 4 x 0.6 x lo-' m3

So time to pass through = 4.8 x

Total time to fill = 1 + 1 + 1.5 = 3.5 seconds

Now, as shown in previous example, for a constant flow rate situation the pressure build-up in the machine is given by

For PE, n = 0.33 and qo = 3 x 104 (from flow curves)

Therefore after 3.5 seconds P = 4.7 MN/m2

But Pressures losses = (0.6 + 0.6 + 0.13) = 1.33 MN/m2

0.29 x (1.5 x

1 x 10-7

Since the haul-off speed is 0.4 m/s, the water bath would need to be at least (0.4 x 6) m long ie 2.4 m

(5.27) Using the equation derived in the text:

Using the data given, the flow ratios will be

Material Flow Ratio (LIH)

LDPE

polypropylene

PVC

POM acrylic polycarbonate nylon 66 ABS

From the flow curves the value of shear stress to give r / G = 5.5 is 7 = 2.25 x

1 6 N/m2 (also G = 4.1 x 104 N/m*) From the flow curves at T = 2.25 x 1 6 N/m2,

6Q

9 = 620 = - + Q = 52 x m’/s = 147 kghour

TH2 2uh 2 x 2.5 x 4 x lo6

So characteristic time = A/E = 4.95

For acrylic, ,,IqR = 10[(28.32~-40)+(9.54~-50)110-~ = 4 53

Similarly for the others V / q R = 0.95, 1.415,2.03 and 1.28,

Also for q / q R = 1, A A T -BAP

For acrylic - = - - A ' 28'32 - 2.97 MN/m2"C

AT 9.54 Similarly for the others, ( A P ) / ( A T ) = 1.17, 1.87,3.07 and 1.93

Celluloid, 2 Cellulose, 1 Centrifugal casting, 337 Charpy impact, 152, 155, 157

Chemical resistance, 5

Chopped fibres, 329 Chopped strand mat, 329 Clamping forces, 293, 326, 401 Clamping systems, 284 Coefficient of friction, 28, 72 Coefficient of thermal conductivity, 31 Coefficient of thermal expansion, 33, 61 Co-extrusion, 275

COI, 34 Cold drawing, 44

Cold press moulding, 332 Complex modulus, 11 1 Compliance, 52 Compliance matrix, 183 Composites, 8, 168, 327 Compression moulding, 323, 334 Compression zone, 246

Compressive modulus, 57, 68 Concentric cylinder viscometer, 370 Cone and plate viscometer, 369 Coni-cylindrical dies, 357

192, 228, 233, 328

50 1

Critical fibre length, 227

Critical flaw size, 133

Critical oxygen index, 34

Critical volume fraction, 176

Equivalent section, 67 ESC, 27

EVA, 11, 26 Extenders, 3 Extensional flow, 359 Extensional stiffness matrix, 196, 205, Extensional strain rate, 345, 388 Extrudeddie characteristics, 257 Extrusion, 246, 377

Extrusion blow moulding, 268 Extrusion coating, 273 Extrusion stretch blow moulding, 272

209

Fatigue, 5, 26, 138 Fatigue limit, 141 Fatigue of composites, 238 Feed zone, 246

Fibres, 3, 25, 168 Filament, 328 Filament winding, 330, 337 Fillers, 3, 169

Film blowing, 265, 379 Filter pack, 250 Fire retardant, 34 Flame retardants, 3 Flammability, 34 Flashing, 293 Flexural modulus, 10, 22, 45, 57 Flow coefficient, 260

Flow defects, 375 Fluorocarbons, 5 Foam, 9,26, 68, 298 Folded chain theory, 421 Fourier number, 391 Fractional recovery, 104 Fracture, 1 19, 131 Fracture mechanics, 121 145, 154 Freeze-off time, 397

Friction 28 Fringed micelle model, 421

Gas injection moulding, 299 Gates, 286

Gelcoat, 330 Glass fibres, 25, 28, 39, 168, 173, 231,

233

Glass transition temperature, 30, 117

Hot press moulding, 333

Hot runner moulds, 290

Hydrolysis, 26, 283

Imaginary modulus, 112

Impact, 147

Impact behaviour of composites, 240

Injection blow moulding, 303

Injection moulding, 278, 330, 338, 377,

Injection stretch blow moulding, 273

Insulated runner moulds, 291

Interfacial shear strength, 227, 231

Latent heat of fusion, 405

Lateral strain ratio, 58

LCP, 12

LDPE, 12, 26, 33, 132, 135, 308

Leakage flow, 255

LEFM, 121, 145, 154

Linear elastic fracture mechanics, 127

Liquid crystal polymers, 12

Matched die forming, 309 Material selection, 22 Maximum strain criterion, 233 Maximum stress criterion, 233 Maxwell model, 85, 112 Mean effective pressure, 294, 401 Mean stress, 143

Melt flow index, 373 Melt flow rate, 373 Melt fracture, 375 Metallocene, 13 Metering zone, 246 Mixing zones, 248 Modulus, 20, 39, 41, 47, 59, 67, 283 Molar gas constant, 136

Monomers, 2 Mould clamping force, 293 Moulds, 285

Muenstedt polynomial, 353 Multi-daylight moulds, 290 Multi-layer materials, 218 Multi-layer mouldings, 66

Natural time, 368 Neck ring process, 272 Negative forming, 306 Newtonian flow, 346, 367 Nip gap, 313

Non-Newtonian flow, 35 1 Non-symmetric laminates, 223 Nozzles, 283

Nylon, 4, 13, 26, 28, 34, 64, 135, 149, 171,228, 261, 283, 295, 418

Olefinics, 16 Optical properties, 34 Orientation, 424 Oxidation, 26, 27

Parison, 268 Parkesine, 2 PBT, 8, 11 15 PEEK, 7, 8, 15, 27, 28, 174, 177, 181,

189 328

Permeability, 35 Perspex, 2

Plate constitutive equations, 197, 210

Plug assisted forming, 307

Plunger injection moulding, 280 376

Refractive index, 34 Reinforcement, 3 Relaxation, 42, 87, 89, 93 Relaxation modulus, 49 Relaxation time, 87, 367 Relaxed modulus, 51 Residence time, 367 Residual strain, 109 Resin injection, 335 Resistance, 32 Retardation time, 88 Rheological models, 35 1 Rib design, 74

RIM, 302 Rotational moulding, 3 18 Rotational viscometer, 369 Roving, 328

RRIM, 302 Rule of mixtures, 173, 226 Runners, 287 377

Safe working stresses, 135

S A N , 34

Sandwich mouldings, 66,298

SCORIM, 301 Screws, 282 Secant modulus, 21 55 Semi-crystalline, 4, 11 Sharkskin, 375 Shear modulus, 57, 59, 68 180, 183, 217,345

Shear rate, 344 Shear stress, 345 Shear viscosity, 345 Shellac, 1

Shift factor, 116 Short fibre composites, 226

Stress concentration factor, 121, 148

Stress intensity factor, 127, 146

TPR, 10 Tracking, 32 Transfer moulding, 326 Transverse modulus, 179

Triaxial stresses, 149,427 Tsai-Hill criterion, 233 Twin screw extruders, 262

Uni-directional composites, 182, 188 Unrelaxed modulus, 51

Urea formaldehyde, 17 uni-directional plies, 202

Vacuum bag moulding, 331 Vacuum injection, 336

Van der Waals forces, 3 Vent wne, 249

Venting, 288 Viscoelasticity, 25,42, 84

Viscosity, 42, 344 Voigt model, 87 Volume fraction, 171 Vulcanisation, 10 vacuum forming, 306

Waveform, 142 Wear, 5,28 Weathering 27 Wedge shaped die, 362 Weight fraction, 171 Williams Landel and Ferry, 117

W L F equation, 117

XLPE, 13

Yam, 328 Yield locus, 132 Young’s modulus, 20, 426

Zener model, 92 Ziegler Natta catalysts, 13