PREDICTING THE TENSILE STRENGTH OF SHORT GLASS FIBER

REINFORCED INJECTION MOLDED PLASTICS

John F. O’Gara, Glen E. Novak and M. G. Wyzgoski*

All Formerly of Delphi Research Labs

*American Chemistry Council Consultant

Abstract

The tensile strength of a composite is dependent on the properties of the fiber, the

properties of the matrix resin, the fiber content, the geometry and orientation of the fibers, and

the interfacial strength between the fiber and the matrix. Plaques and tensile bars were injection

molded of 30 wt% glass filled polybutylene terephthalate (PBT), 33 wt% glass filled nylon-66

(PA), and 15 and 30 wt% glass filled polycarbonate (PC). A modified Kelly-Tyson theory was

examined for predicting tensile strength from the known microstructure. We have found we can

successfully model the strength with knowledge of the fiber length distribution, the average

through-thickness fiber orientation, and the stress-strain curve for the unfilled resin. Surprisingly

accurate strength predictions within 10% have been validated for both flow and cross-flow

directions for PBT, PC, and PA, although slightly larger errors have been observed for some PA

samples. The approach proposed in this work has been validated using data determined on

composites whose fiber contents and lengths are typical of injection molding. A key finding of

this work is that in injection molded glass fiber reinforced thermoplastics most of the fibers

(~98% for PBT and PC; ~70% for PA) are shorter than the theoretically calculated critical fiber

length. This can greatly simplify the analysis and allows for a quick estimate of the strength

values of any reinforced plastic using material data that is generally available.

Purpose

The purpose of this study was to develop and validate a predictive capability for tensile

strength in glass fiber reinforced thermoplastic materials to enable strength predictions in

injection molded components.

2

Introduction

Glass fibers are commonly added to thermoplastic resins to improve the stiffness,

strength, thermal stability, and shrinkage behavior of the material. In injection molding

processes, short-glass fibers are the most widely used reinforcement due to their relatively low

cost and ease of manufacturing. Short-glass fiber composites have found extensive

applications in the automotive environment, including intake manifolds, radiator tanks, gears,

rack and pinion housings, etc. During the injection molding of short-glass reinforced plastic

parts, material flow produces in-plane and through-thickness variations of the fiber orientation,

which results in anisotropic mechanical properties. These properties depend not only on fiber

orientation distributions with respect to the loads that are applied to the composite, but also on

the glass fiber lengths and concentration.

The stiffness (elastic modulus) and strength (tensile strength) of these composites are

two of the most basic mechanical properties and their prediction is essential in the design of

these parts. Previously the authors participated in a Thermoplastic Engineering Design (TED)

venture between General Motors and General Electric (currently Sabic Innovative Plastics), a

Department of Commerce sponsored Advanced Technical Program (ATP) administered by the

National Institute of Standards and Technology. In the TED program [1-3], stiffness models,

which take into account the fiber orientation and lengths, have been refined to give predictions

to within 10% for tensile bars and plaques. Much of this work has found its way into commercial

mold-filling packages. Thus, for a part designer, there exists a link between processing

simulations and the resultant stiffness of a part. However, commercial mold-filling packages do

not predict the tensile strength of a part. While the modeling of strength has received much

attention in the literature [4-10], there has been little validation due to the paucity of experimental

data related to many of the parameters in these models.

The mechanical properties of any composite material depend on the details of the

microstructure. This paper presents an analytical method that considers the effects of fiber

length, diameter, and orientation for predicting the tensile strength of short glass fiber reinforced

thermoplastics. We have gathered extensive microstructural information on a series of tensile

bars and plaques that allow us to critically assess the modeling of strength. We have not

attempted to validate all the current theories, but have focused on the Kelly-Tyson strength

model that has received considerable attention over the years since its introduction [4]. A

modified version of the Kelly-Tyson strength model is tested and a procedure for estimating

many of the necessary parameters is developed.

3

Experimental

Materials and Mechanical Testing:

Standard ASTM type I tensile bars were molded from two different lots of 30 wt% glass

filled polybutylene terephthalate, (PBT), which are simply designated as PBT Lot A and Lot B,

unfilled PBT, 33 wt% glass filled nylon-66 (PA), unfilled PA, unfilled polycarbonate (PC) and 15

and 30 wt% glass filled PC. To establish strength data that are more general with respect to flow

orientation than data from injection molded tensile bars, plaques were also molded from these

materials in geometries of 76.2 x 279.4 x 2.92 mm, 152 x 381 x 3 mm, and 152 x 203 x 6.35

mm. The material supplier's recommendations were followed for all molding conditions.

For strength measurements on the above end gated plaques, small ISO Type 1BA

tensile bars were cut from both flow and cross-flow orientations as shown in Fig. 1. Strength

Traditional

Method

Specimens Cut From

Edge-Gated Plaques

Gate

Highly axial

aligned glass

ASTM Type I

molded tensile

specimen

Flow Crossflow

Figure 1. Schematic for the cutting of samples from an edge-gated plaque for assessing anisotropic properties.

values for the injection molded ASTM Type I tensile bars were measured on with a 25 kN load

cell. Strain was measured using an extensometer with a 50 mm gage length. The crosshead

rate was 5mm/min with a grip separation of 135mm. Load and strain data were recorded for the

smaller ISO Type 1BA tensile bars cut from the end gated plaques with a 10kN load cell and

strain was measured using an extensometer with a 25.4 mm gage length. Rather than test at a

constant crosshead speed, all samples were tested at the same standard strain rate of 0.1

mm/mm/min.

4

mfff

ff

fww

wv

/)1()/(

/

Fiber Content:

Fiber content is usually given by the material suppliers in terms of a weight percentage.

However, composite properties are better related to fiber volume fraction than weight fraction.

Knowing fiber and matrix densities (ρf and ρm), one can easily convert weight fraction (wf) to

volume fraction (vf) as

Using the density of 2.54 g/cc for glass, fiber volume fractions for glass-fiber-reinforced PBT,

PA, and PC are calculated and shown in Table I.

Table I. Fiber Volume Fraction

Material Matrix density (ρm) wf vf

PBT 1.31 0.30 0.181

PA 1.14 0.33 0.181

PC 1.20 0.30 0.168

PC 1.20 0.15 0.077

Fiber Length Measurements:

A 12.7 mm square piece was cut from the center of each plaque or tensile bar. The

glass was subsequently separated from the matrix by burning off the resin using a microwave

muffle furnace. The PC and PA resins were removed using a single stage burn off procedure

(30 minutes at 475°C). The PBT resin was removed using a 3-stage burn off procedure (275°

for 90 minutes, 375°C for 30 minutes and 525°C for 30 minutes). The staged approach was

found necessary with the PBT to get clean burn off with a minimal level of fiber distortion. After

resin removal, the fibers were dispersed in 2-propanol and analyzed using image analysis as

previously described [3]. The fiber length distributions have also been previously reported [3].

The average fiber lengths for each tensile bar and plaque are given in Table II.

(1)

5

Fiber Orientation Measurements:

Details of the fiber orientation measurement procedure have also been previously

described [1, 3]. A similar method was employed for this study. A one inch long through-

thickness section was cut from the center of each tensile bar and plaque. The samples were

subsequently mounted in epoxy, polished, and photographed under the microscope. The

individual fibers appear as an ellipse and the orientation tensor components are extracted from

the measured major and minor axes. For a given tensile bar or plaque, a common convention is

to describe “1” as the flow direction, “2” as the cross-flow direction, and “3” as the through-

thickness direction. Orientation is described using the diagonal elements of 2nd order tensors,

a11, a22, and a33 (where a11+a22+a33 = 1), which are usually reported or plotted as a function of a

normalized thickness. Assuming the principal flow direction is in the “1” direction, these diagonal

tensor elements (a11, a22, and a33) provide a quantitative description of the orientation state in

each of the directions. For example, values of a11= 0, a22= 0, or a33= 0 correspond to no

alignment in the 1, 2, or 3 directions, respectively, while values of a11= 1, a22= 1, or a33= 1

correspond to perfect alignment in the 1, 2, or 3 directions, respectively. Orientation profiles for

the tensile bars and plaques studied in this work were previously reported [3]. For this work, we

make use of the average orientation values across the thickness of the plaques as shown in

Table II.

6

Table II. Average orientation across the thickness and average fiber length for each tensile bar and plaque.

Material avg. a11 avg. a22 avg. a33 avg. l (µm)

PBT Lot A Tensile Bar

0.83 0.14 0.03 155

PBT Lot A 76x279 Plaque 0.77 0.19 0.04 168

PBT Lot B Tensile Bar 0.84 0.13 0.03 224

PBT Lot B 76x279 Plaque 0.81 0.17 0.02 212

PBT Lot B 152x381 Plaque 0.74 0.23 0.03 221

PC +30 Tensile Bar 0.81 0.16 0.03

150

PC +30 76x279 Plaque 0.76 0.22 0.02 170

PC+15 152x381 Plaque 0.60 0.38 0.02 262

PC +30 152x203 Plaque 0.63 0.32 0.05

167

PA Tensile Bar 0.85 0.12 0.03 210

PA 76x279 Plaque 0.80 0.17 0.03 227

PA 152x381 Plaque 0.71 0.26 0.03 204

Fiber Diameter Measurements:

The reinforcing efficiency of a glass fiber in a composite relates to not only the length but

also the diameter of the glass. The measurement of fiber diameters is a byproduct of the

orientation measurement techniques. As noted above, fibers appear as ellipses on a polished

cross-section and the measurement of the minor axes allows one to extract the fiber diameter

distribution for each of the materials. Fiber diameter distributions for PBT, PA, and PC are

summarized in Table III. It is apparent that PBT and PC materials have higher average

diameters with broader distributions than the PA material.

Table III. Fiber diameter statistics for each material

Material Min. (µm) Max. (µm) Mean (µm) Standard Deviation

PBT Lot A 8.0 18.2 12.6 1.8

PBT Lot B 7.7 19.0 13.0 2.0

PA 6.1 14.0 9.5 1.2

PC 6.6 18.3 12.3 1.7

7

Theory

Modified Kelly-Tyson Strength Model:

For composites containing continuous, unidirectional fibers, the tensile strength can be

predicted by the simple rule of mixtures,

σc = σf vf + (1-vf) σm (2)

where σc is the composite tensile strength, σf is the tensile strength of the fiber, vf is the volume

fraction of fibers, and σm is the stress carried by the matrix at the failure strain. We define the

failure strain as the strain at the maximum stress. In this approach, the basic assumptions used

are that there is a perfect bond between fibers and matrix and that the strain in the fibers is

equal to the strain in the matrix.

However, in the case of discontinuous, unidirectional fibers, there is no longer equal

strain in the fibers and the matrix since the fibers are separated by matrix material at the fiber

ends. Since the matrix has a lower modulus than the fiber, the longitudinal strain will be higher

than in the adjacent fibers. One approach to dealing with this problem was proposed by Kelly

and Tyson [4] using the concept of a critical fiber length. In this model, if a perfect bond between

the matrix and fiber exists, the difference in longitudinal strains causes a shear stress

distribution across the matrix/fiber interface. In addition, the stress along the fibers is not uniform

for short fibers. Assuming that both the matrix and fibers behave as linearly elastic materials,

the stress in the fibers builds up in a linear manner from 0 at the fiber end to the maximum

stress that the fiber can be loaded, σf, as illustrated in Fig. 2.

A critical minimum fiber length is needed to build up sufficient stress to fracture the fiber.

This critical length, lc, is given by

lc = (σfd/2τ) (3)

where σf is the ultimate tensile strength of the fiber, d is the fiber diameter and τ( note this is the

small Greek letter tau, which shows up in cursive form in some equations) is the interfacial

shear strength between the fiber and the matrix or the shear strength of the matrix , whichever

is less. The critical fiber length is defined, as the minimum fiber length required for the maximum

fiber stress to equal the ultimate fiber strength at its midlength.

8

lf < lc lf = lc lf > lc

Ultimate fiber tensile strength (fu)stress

lc/2 lc/2 lc/2 lc/2

Figure 2. Shows how the stress varies along the length of a fiber (lf) when the fiber is shorter than the critical length

(lc) and longer than the critical length. For the case, lf > lc, a finite length (lc/2) at each end of the fiber is required for

stress in the fiber to reach a maximum.

For aligned short fibers where the length is shorter than lc, the maximum fiber stress is

not reached. If internal stress effects between adjacent fibers are ignored, fiber failure does not

occur. Either the matrix/fiber interfacial bond fails, and fiber pull-out is observed, or the matrix

itself fails. For this situation, the composite failure strength is given by the simple addition of the

contribution from each component.

σc = vf σ (l/d) + (1-vf) σm for (l < lc) (4)

“fiber” + “matrix”

For aligned short fibers where the length is greater than lc, the composite failure will be

mainly accompanied by fiber breakage. The stress is assumed to increase linearly from the fiber

end until it reaches the ultimate fiber strength at a distance ½ lc from the fiber end, as shown in

Fig. 2. The average fiber strength is as follows,

avg. σf = σf (1- lc/2l) (5)

9

For this situation, the composite failure strength is again given by the simple addition of the

contribution from each component.

σc = vf σf (1- (lc/2l) + (1- vf) σ’m for (l > lc) (6)

“fiber” + “matrix”

These models were developed for a single length, unidirectional fiber. However, one

must account for both the fiber orientation and fiber length distributions in real composites. A

modified rule of mixtures has been proposed, which involves a summation over the fiber length

distribution and the incorporation of an orientation factor to the fiber contributions [5-7]. This

model takes the final form as follows:

'12

12

max

min

imf

llj

lcljj

cfjf

lcli

llic

ffic v

l

lvn

l

lvnD

(7)

“sub-critical fiber” + “super-critical fiber” + “matrix”

where D is an orientation factor, lc is the critical fiber length, li is the fiber length below lc, lj is the

fiber length equal and above lc, ni or nj are the percentage of fibers with lengths li or lj,

respectively, vf is the volume fraction of fiber, σf is the strength of the fiber, and σm’ is the matrix

strength at the composite failure. The failure of a composite according to this equation is

characterized by either fiber pull-out for the sub-critical fibers or fiber breakage for the super-

critical fibers combined with matrix fracture. Depending on the fiber length distribution and fiber

volume fraction, if there is substantial fiber breakage, then σm’ should correspond to the matrix

strength at the fiber failure instead of the ultimate matrix strength, σm [10].

10

Results and Discussion

There are numerous studies in the literature, which make use of some form of the

modified Kelly-Tyson model. Templeton [5] successfully used this model to predict the tensile

strength of glass filled nylon, polypropylene, and polybutylene terephthalate. However, the

average fiber lengths were four to ten times longer (800 to 2500 µm) than the lengths observed

in this study (~200 µm). In addition, Templeton added another term, defined as B, to equation

(7) (simply multiply the right side of the equation by B), where B is a measure of the bonding

efficiency between the fiber and the matrix. This parameter cannot be measured directly and

was used as a “fitting” parameter to make successful polypropylene strength predictions. This

parameter was not needed in the predictions of the nylon and polybutylene terephthalate

materials. Eriksson et al. [6, 7] followed the degradation of glass fibers in a nylon recycling

study, and were able to model the relative decline in tensile strength using this model. Although

length measurements were made, no quantitative orientation measurements were attempted.

Estimates were made for the parameters, D, τ, lc, and σm from literature values on similar

materials but different plaques. Van Hattum and Bernardo [9] have also used this model

successfully to model the strength behavior of tensile bars of carbon fiber polypropylene

composites. They have expanded the analysis to include fiber strength distributions, probability

density functions describing the fiber length distributions, and to account for arbitrary fiber

orientation relative to the direction of loading. Despite the greater utility of their approach, their

analysis is limited by the fact that they only have tensile bars with fiber lengths much lower than

the calculated critical fiber length.

The relative success in the aforementioned studies suggests that a detailed analysis of

our samples is warranted using the modified Kelly-Tyson equation. As noted previously, we

have extensive microstructural information, including fiber length, diameter, and orientation, on

the actual plaques and tensile bars that were mechanically tested. This information allows us to

critically evaluate the parameters in the model. A discussion of how we approach the

determination of each of the parameters follows.

σf

It is impossible to measure the strength of the glass fiber present in a particular lot of

material. The strength is dependent on the elemental composition of the glass, as well as the

processing conditions during its manufacture and its incorporation into the composite. These

latter processes may introduce flaws on the fiber surface that influence the ultimate strength. A

11

fiber with a larger diameter or longer length has a greater surface area and, thus, potentially

more flaws. Fiber strengths measured as a function of gauge length show higher strengths with

smaller gauge lengths [5,9]. Literature values for the strength of commercial E-glass range from

2.2 to 3.45 GPa. Templeton [5] measured the strength of one commercially available E-glass

fiber as a broad distribution, ranging from 1.6 to 3.45 GPa, with an average around 2.7 GPa.

Ericksson, et al., [6,7] used a value of 2.2 GPa in their modeling. For the modeling in this work,

we have used a single value of 2.4 GPa for the strength of the glass fiber. No attempt has been

made to take into account the differences in diameters that were measured for PBT and PC,

which are greater than PA.

σm

The strength of each resin was determined by molding tensile bars of the unfilled resins

and testing under the same conditions as with the filled materials. The stress-strain curve for

each is shown in Figs. 3-5. The ultimate matrix strength refers to the yield stress, which is taken

at the maximum inflection point and is included on each of Fig.3-5.

Given that we have collected the whole stress-strain curve, the matrix stress at the fiber failure

strain can also be determined. Assuming a fiber strength of 2.4 GPa and a fiber modulus of

72.5 GPa, a failure strain of 0.033 is calculated. The values for the matrix strength, σm’, at the

fiber failure strain are also included in Figs. 3-5. For the PBT sample, the matrix fails at

approximately the same strain as the fiber.

lc and τ

The critical fiber length lc and the interfacial shear strength, τ, are related to one another

through equation (3) using the measured diameters and the ultimate glass strength. Knowledge

of either lc or τ is needed for the model. Unfortunately, each is difficult to reliably measure.

Several experimental techniques have been developed to measure the interfacial strength

between the matrix and the fiber, including, the fiber fragmentation test, the protruding fiber

length test and the microindentation test, to name a few [5, 11-13]. However, a round-robin test

[12] revealed that it is not possible to reliably measure the interfacial strength or the critical fiber

length. Thus, as an approximation, we assume that there is good adhesion between the matrix

12

Figure 3. Stress-strain curve for unfilled PA with maximum stress identified as well as stress at the fiber failure

strain of 0.033.

and the fiber, and thus, the limiting shear strength of the matrix can be used. The shear strength

of the matrix is itself difficult to measure and very little data exists in the literature [14]. Assuming

an isotropic matrix, the shear strength can be estimated by the von Mises criterion from the

tensile strength of the unfilled matrix [15],

3mm (8)

The calculated matrix shear strength, τm, and the resultant critical fiber length for each

material are given in Table IV using a fiber strength of 2.4 GPa and the average diameters in

Table III. The critical fiber length for PA is much lower than PC and PBT. This is a result of the

lower glass fiber diameter in this material as well as the higher matrix shear strength. A

comparison of the fiber length distributions to the calculated critical fiber lengths reveals that for

PBT and PC most of the fibers (~98%) are found below lc, while for PA ~70% are below lc , as

shown in Table V.

0

10

20

30

40

50

60

70

80

90

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

strain

80.85 MPa @ 0.047 strain (max matrix stress)

76.1 MPa @ 0.033 fiber failure strain

stress (MPa)

13

Figure 4. Stress-strain curve for unfilled PC with maximum stress identified as well as stress at the fiber failure

strain of 0.033.

Figure 5. Stress-strain curve for unfilled PBT with maximum stress identified. For this sample, the matrix fails at

approximately the same strain as the fiber failure strain.

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

strain

61.54 MPa @ 0.058 strain (max matrix stress)

55.3 MPa @ 0.033 fiber failure strain

0

10

20

30

40

50

60

0 0.01 0.02 0.03 0.04 0.05 0.06

strain

53 MPa @ 0.032 strain

(max matrix stress)

53 MPa @ 0.033 fiber failure strain

stress(MPa)

stress(MPa)

14

Table IV. Matrix tensile strength, calculated von Mises shear strength,

and calculated critical fiber length

Material σm (MPa) τm (MPa) lc (µ)

PBT 53.0 30.6 494 Lot A

510 Lot B

PC 61.5 35.5 415

PA 80.8 46.9 244

Table V. Average fiber length for each tensile bar and plaque and the percentage

of fibers below and above the critical fiber length lc

Material avg. l (µ) % l < lc % l > lc

PBT Lot A tensile bar

155 99 1

PBT Lot A 3x11” plaque 168 98 2

PBT Lot B tensile bar 224 96 4

PBT Lot B 3x11” plaque 212 97 3

PBT Lot B 6x15” plaque 221 97 3

PC tensile bar 150 98 2

PC 3x11” plaque 170 96 4

PC+15, 6x15” plaque 262 81 19

PC 6x8” plaque 167 96 4

PA tensile bar 210 71 29

PA 3x11” plaque 227 64 36

PA 6x15” plaque 204 72 28

D

The orientation coefficient, D, was used as a scaling factor by McNally et al. [9] and

Templeton [6]. Our approach is to utilize the average through-thickness value for a11 and a22 to

represent the degree of orientation in the flow and cross-flow directions, respectively. These

second order tensor values range from 0 to 1, where 1 corresponds to complete alignment in a

given direction and 0 corresponds to no alignment. The use of a11 and a22 will allow for

predictions of the flow and cross-flow strength properties of the plaques, respectively.

15

Modeling the Composite Strength

The necessary material parameters have been established and we now continue by

modeling the composite strength for the tensile bars and plaques. We have taken three

approaches to predicting the strength data. First, we predict the strength with equation 7 using

the length distribution data previously reported [4] and the matrix stress at the fiber failure strain,

σm’. We refer to this as Model 1. Second, we predict the strength with equation 7 using the

length distribution and with the ultimate matrix stress, σm, instead of σm’. We refer to this as

Model 2. Finally, we take a much simpler approach and predict the strength using a single

average length and the ultimate matrix stress, σm. For this model, we make use of only the sub-

critical fiber term and the matrix term in equation 7. We refer to this as Model 3. Summaries of

the predicted vs. experimental strength values for all three models are shown in Figs. 6-8.

The relative predictions of strength for the flow orientation, including tensile bars and

plaques are shown in Fig. 9, while the relative predictions of strength for the x-flow orientation of

the plaques are shown in Fig. 10. All three models are within 10% for the PC and PBT materials,

other than the 6x15” PC plaque of 15-wt% glass-fiber. Since the majority of fibers in these

samples are below lc, Model 3 is sufficient to predict the strength. The ultimate strength of the

matrix is used since there will be minimal fiber breakage. Thus, the strength of most PC and

PBT samples can be modeled very simply with only knowledge of the average through-

thickness orientation, the average fiber length, and the ultimate strength of the unfilled matrix.

The 6x15” PC plaque of 15-wt% glass fiber is more similar to the results for the PA

samples. These plaques have 20-30% of the fibers greater than lc, so Model 1 appears to be

best. For these samples, we need to account for the overall distribution of fibers, and the

composite failure may be more related to fiber breakage. Thus, using the matrix stress at the

fiber failure in the model is more appropriate. Therefore the calculation of strength for these

materials is a little more involved in that both a length distribution and the matrix stress at fiber

failure are required. Nevertheless, other than the 3x11” PA sample, it should be noted that the

simple approach of Model 3 is within 15% of the experimental values. Replacing the full-length

distribution with the average length will affect the final strength predictions by only an additional

~6%.

16

Plaques and tensile bars:

Predicted vs. Experimental Strength (flow)

75

100

125

150

175

200

225

75 100 125 150 175 200 225

Experimental strength (MPa)

Pre

dic

ted

str

en

gth

(M

Pa

)

10% range

model 1

Plaques:

Predicted vs. Experimental Strength (X-flow)

40

50

60

70

80

90

100

110

120

60 70 80 90 100

Experimental strength (MPa)

Pre

dic

ted

str

eng

th (

MP

a)

10% range

model 1

Figure 6. Absolute strength values: predicted vs. experimental. Model 1 corresponds to use of σm’ (matrix stress at

fiber failure strain) and complete fiber length distribution.

17

Plaques and tensile bars:

Predicted vs. Experimental Strength (flow)

75

100

125

150

175

200

225

75 100 125 150 175 200 225

Experimental strength (MPa)

Pre

dic

ted s

tren

gth

(M

Pa)

10% range

model 2

Plaques:

Predicted vs. Experimental Strength (X-flow)

40

50

60

70

80

90

100

110

120

60 70 80 90 100

Experimental strength (MPa)

Pre

dic

ted s

tren

gth

(M

Pa)

10% range

model 2

Figure 7. Absolute strength values: predicted vs. experimental. Model 2 corresponds to use of σm (ultimate

matrix stress) and complete fiber length distribution.

18

Plaques and tensile bars:

Predicted vs. Experimental Strength (flow)

75

100

125

150

175

200

225

75 100 125 150 175 200 225

Experimental strength (MPa)

Pre

dic

ted

str

eng

th (

MP

a)

10% range

model 3

Plaques:

Predicted vs. Experimental Strength (X-flow)

40

50

60

70

80

90

100

110

120

60 70 80 90 100

Experimental strength (MPa)

Pre

dic

ted

str

eng

th (

MP

a)

10% range

model 3

Figure 8. Absolute strength values: predicted vs. experimental. Model 3 corresponds to use of σm (ultimate matrix

stress) and single average fiber length.

19

tensile bars +plaques (flow): strength simulations

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

PC

tb

PC

3x11"

PC

6x15"

15%

PC

6x8"

PB

T A

tb

PB

T A

3x1

1"

PB

T B

tb

PB

T B

3x1

1"

PB

T B

6x1

5"

PA

tb

PA

3x11"

PA

6x15"

material

dif

fere

nce

betw

een

exp

eri

men

tal

an

d p

red

icte

d s

tren

gth

(%

)

model 1

model 2

model 3

Figure 9. Relative predictions of strength for flow orientations (% difference between experimental and

predicted). Model 1: corresponds to use of σm’ (matrix stress at fiber failure strain) and complete fiber length

distribution. Model 2 corresponds to use of σm (ultimate matrix stress) and complete fiber length distribution. Model 3

corresponds to use of σm (ultimate matrix stress) and single average fiber length.

20

plaques (x-flow): strength simulations

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

PC

3x11"

PC

6x15"

15%

PC

6x8"

PB

T A

3x1

1"

PB

T B

3x1

1"

PB

T B

6x1

5"

PA

3x11"

PA

6x15"

material

dif

fere

nce

betw

een

exp

eri

men

tal

an

d p

red

icte

d s

tren

gth

(%

)

model 1

model 2

model 3

Figure 10. Relative predictions of strength x-flow orientations (% difference between experimental and predicted).

Model 1: corresponds to use of σm’ (matrix stress at fiber failure strain) and complete fiber length distribution. Model

2 corresponds to use of σm (ultimate matrix stress) and complete fiber length distribution. Model 3 corresponds to

use of σm (ultimate matrix stress) and single average fiber length.

21

It is somewhat surprising that despite the many assumptions made in the specification of

the parameters that go into this model, the strength can be predicted to within 10- 15%. Overall,

the relative strength predictions for the different materials reveal systematic trends dependent

on the material; PA has larger positive errors than PC, while PBT has negative errors. These

systematic errors may be related to the simple approach taken and reflect shortcomings in the

values used for some parameters.

Although the current models are accurate enough for predicting component strength

properties, it is possible that further improvements can be made for each material. It is therefore

of value to question some of the model assumptions. For example, we assume that the matrix

strength is the same for the unfilled and filled materials and is not influenced by processing or

the presence of the concentrated fibers. The matrix strength will be affected by changes in

molecular weight, the presence of moisture, and for semi-crystalline materials, the morphology.

It is most likely that for the 3x11” PA plaque, where the flow direction predictions are off by 25%,

that there has been a change in the level of crystallinity or morphology. Similar over-predictions

of the moduli have been noted with these plaques, despite careful drying of the samples. This

suggests that the matrix properties used in the model may be wrong. Recent glass fiber

reinforced PA composite modeling by Foss [18] suggest that a PA stress-strain curve having a

reduced yield stress must be “backed out” from the model calculations in order to accurately

capture the composite properties. More accurate strength predictions may be anticipated if

micromechanics calculations are utilized to define “effective” matrix stress-strain relationships.

We also assume that the fiber/matrix interfacial strength is equal to the calculated shear

strength for the matrix. We further assume that the interfacial strength between the glass fibers

and the matrix is not influenced by the processing. The model will over-predict the tensile

strength of a composite where the glass fiber matrix interfacial strength is compromised.

Surprisingly, the work of Ericksson et al. [7,8] in their analysis of the recycling of PA composites,

suggests this may not be a major problem with the model. Despite multiple compounding of

their materials, they saw no evidence to suggest degradation in the fiber/matrix interface or the

matrix itself.

22

Concluding Remarks

All the parameters in this analysis of the modified Kelly-Tyson equation are measured or

estimated from measured properties, except for the strength of the glass fiber. One could make

this an adjustable parameter for a given lot of material, however, the overall success of the

modeling with a value of 2.4 GPa precludes making that next step at this point. As a first

approximation for any material, Model 3 seems appropriate for predicting the strength. For the

most part, predictions are within 10 to 15% for PBT, PC, and PA. The strength can be modeled

very simply with only knowledge of the average through-thickness orientation, the average fiber

length, and the ultimate strength of the unfilled matrix. Given that the orientation can be

predicted in commercially available mold-filling software, strength predictions could be easily

implemented similar to predictions that are currently done for stiffness.

Acknowledgments

This research was part of a Thermoplastic Engineering Design (TED) venture between

General Motors and General Electric (currently Sabic Innovative Plastics), a Department of

Commerce sponsored Advanced Technical Program (ATP) administered by the National

Institute of Standards and Technology. The authors would like to thank Charles C. Mentzer

(retired), Peter H. Foss, and Richard P. Schuler (retired) of the GM R&D Center for molding the

samples, James P. Harris (retired) of the GM R&D Center for fiber length measurements, and

Peter H. Foss of GM R&D Center and Louis P. Inzinna of GE CR&D for assistance in the

testing. Finally, the assistance of Falgun Rathod and Vauhini Telikapalli, formerly graduate

students at Wayne State University, in making fiber orientation measurements is greatly

appreciated.

23

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