This document discusses numerical modeling of the behavior of uneven thermoplastic polymers using experimental stress-strain data and pressure dependent yield criteria to improve design practices. It presents the following: 1) Four polymers (PA-66, PA-6, PP, HDPE) were tested under tension and compression to determine their unevenness levels. PA-6 and PP showed significant unevenness, with compressive yield strengths 25-70% higher than tensile strengths. 2) Pressure dependent yield criteria that account for unevenness, such as conical and parabolic modified von Mises models, were reviewed. These criteria incorporate the effect of hydrostatic pressure on yield strength. 3) Experimental stress-strain data was

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ICM11
Numerical modeling of uneven thermoplastic polymers
behaviour using experimental stress-strain data and pressure
dependent von Mises yield criteria to improve design practices.
Gustavo H. B. Donatoa*
, Marcos Bianchib
a
Professor of the Mechanical Engineering Department, Ignatian Educational Foundation (FEI), Humberto de Alencar Castelo
Branco Av., 3972, São Bernardo do Campo, São Paulo, ZIP 09850-901, Brazil.
b Master in Mechanical Engineering, Ignatian Educational Foundation (FEI), Humberto de Alencar Castelo Branco Av., 3972, São
Bernardo do Campo, São Paulo, ZIP 09850-901, Brazil.
Abstract
Classical plasticity theories and yield criteria for ductile materials, such as Tresca and von Mises original
formulations predict that yielding is independent on the hydrostatic stress state (pressure), which means that tensile
and compressive stress-strain behaviours are considered equal and are equally treated. This approach is reasonable for
ductile metallic materials but sometimes inaccurate for polymers, which commonly present larger compressive yield
strength, therefore being characterized as uneven. Polymer unevenness can be of great interest for mechanical
structural design since components that present regions operating under compression can be optimized taking this
phenomenon into account. Compressive stress-strain data for polymers are very scarce in the literature, and as a step
in this direction this work presents three key-activities: i) four selected polymers were tested under tension and
compression to identify unevenness and assess its levels; ii) pressure dependent yield criteria applicable for polymers
were briefly reviewed; iii) experimental results were incorporated in adapted design practices using modified yield
criteria implemented in optimization and finite element computations. Results show that taking unevenness into
account and implementing modified criteria in the numerical techniques for structural calculation can provide mass
reductions up to ~ 38% even with simple geometric changes, while keeping original safety and stiffness levels.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICM11
Keywords: Uneven polymers; Pressure dependent yield criteria; Experimental testing; Structural improvement.
1. Introduction
Classical plasticity theories and yield criteria for ductile materials, such as Tresca and von Mises
original formulations [1], include several assumptions, such as: i) the material is isotropic and
homogeneous; ii) deformation proceeds under constant volume; iii) tensile and compressive yield
* Corresponding author. Tel.: +55-11-9688-8917; fax: +55-11-2347-3166.
E-mail address: gdonato@fei.edu.br.
1877-7058 © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11
doi:10.1016/j.proeng.2011.04.311
Procedia Engineering 10 (2011) 1871–1876

2. 1872 Gustavo H. B. Donato and Marcos Bianchi / Procedia Engineering 10 (2011) 1871–1876
Nomenclature
E elastic modulus
I1 first stress invariant
J2 second invariant of the deviatoric stress tensor
k critical value of the von Mises yield criterion
me,t unevenness level in terms of yield strength considering (e) engineering and (t) true data
1,2,3 principal normal stresses
h hydrostatic stress
uts ultimate tensile stress
vM von Mises equivalent stress
vM-C,P modified von Mises equivalent stress considering (C) conical and (P) parabolic Mises models
ys-c,t yield strength under (c) compression and (t) tension
ys-off-x yield strength evaluated using the offset method with offset level = x%
ys-max yield strength evaluated using the first point on the stress-strain-curve were d /d = 0
strengths are equal; iiii) yielding phenomenon is uninfluenced by the hydrostatic component of the stress
state (pressure) [2]. The last two assumptions mean that tensile and compressive stress-strain behaviours
are identically treated in terms of structural integrity, which is reasonable for ductile metallic materials,
but most times inaccurate for polymers, ceramics and even brittle metals [1]. Engineering ductile
thermoplastic polymers, which are focused here, usually present larger compressive yield strength,
therefore being characterized as uneven. This is a direct result of chains arrangement and
deformation/interaction micromechanism, which are dependent on the hydrostatic stress level [2]-[3].
Under tensile loading, a direct tendency of chains alignment takes place due to bending and torsion of the
covalent bondings (including tie-chains from the amorphous region and regularly oriented chains from the
cristallytes), which characterizes elastic stiffness under tension and causes a preliminary orientation of
chains. Yielding begins when secondary bonds (relatively weak, but responsible for connecting chains)
are broken and relative slip between chains occurs. The existence of a positive hydrostatic stress state
reduces entanglement and promotes separation between chains, favoring this phenomenon. Under
compression (negative hydrostatic stress state), on the other hand, chains get closer, entanglement
increases and chains alignment is indirect (based on shear and transversal deformation, not aligned to the
loading axis). Consequently, the occurrence of yielding demands energy to surpass entanglement, align
chains and break secondary bonds. This extra energy requirement is sometimes significant and explains
the larger compressive yield strength presented by some polymers. Additional details will not be given
here due to space limitations, but can be found in Bower [2], Krevelen [3], Ward & Sweeney [4] and
Mascarenhas et al. [5]. Unevenness level in terms of yield strength is denoted “m” and is defined here as
m = ys-c/ ys-t . (1)
Experimental results including tensile and compressive stress-strain data for polymers are scarce in
the literature until the present days, however, some available data reveal that unevenness for engineering
polymers usually presents levels between 20 % and 30 % [4]-[7]. However, a polypropylene (PP) studied
by Jerabek et al. presented m 1.50 [8] and an investigation conducted by the authors using the materials
database of CES EDUPACK 2009 software [9] revealed that, for the available 198 unfilled thermoplastic
polymers, the unevenness ranges from m 0.60 to m 7.00, being for most cases 1.00 m 2.00.
Unfortunately, these uneven mechanical properties are in general not considered by current design and
integrity assessment practices, which are based on protocols developed during several decades for
metallic materials with great success. However, current market competitive context demands the use of
new materials and nowadays the gains based on the simple substitution of metallic materials for polymers
are becoming saturated and more laborious. In this context, a better understanding of the mechanical

3. Gustavo H. B. Donato and Marcos Bianchi / Procedia Engineering 10 (2011) 1871–1876 1873
behaviour of emerging materials (such as polymers) supported by the use of pressure dependent yield
criteria and numerical optimization and calculation techniques represent potential opportunities for
structural improvement by taking advantage of the larger compressive yield strength of polymers.
As a step in this direction, this work evaluates the effects of implementing pressure dependent yield
criteria on design practices for components with regions working under compression. First, four polymers
were uniaxially tested under tension/compression to obtain unevenness levels. In the sequence,
calculations including optimization and finite element procedures were developed to incorporate the
different criteria in the design process and assess stiffness, mass, stresses and safety factors of an example
component. The exploratory results show that the use of pressure dependent criteria with experimental
data can reduce mass up to 38 % keeping original stiffness and safety against yielding.
2. Analytical background - pressure dependent yield criteria
Original von Mises yield criterion proposes that yielding occurs when the second invariant of the
deviatoric stress tensor (J2) reaches a critical value (k2
) [10], as stated by Eq. (2). Consequently, yielding
occurs if Mises equivalent stress ( vM – presented by same equation) is greater than the tensile yield
strength ( ys-t). Since the hydrostatic stress ( h) can be written in terms of the first stress invariant (I1) as
presented by Eq. (3), it can be realized that there is no effect of h on this failure prediction (the locus for
this criterion is the well known cylindrical tube aligned with the 1 = 2 = 3 axis) [1].
(2)
(3)
To consider uneveness and include the pressure dependency on Mises original yield criterion, Hu and
Pae [11] proposed an expansion of Eq. (2) as a polynomial in I1 (Eq. (4)). Following the procedures
presented by Ehrenstein and Erhard [12] and Miller [13], N = 1 and N = 2 conduct, respectively, to
conically and parabolically modified Mises criteria presented by Eqs. (5,6). It can be realized that in the
conical model (Eq. (5)) the effect of I1 is linear, providing the yield surface shown by Fig. 1(a). In the
parabolic model, in its turn, Eq. (6) reveals that the effect of I1 is quadratic, providing the yield surface of
Fig. 1(b). Yielding occurs if the equivalent stress ( vM-C or vM-P) is larger than tensile yield strength ( ys-t).
Fig. 1(c) compares original and modified models for different m under plane stress conditions. It can be
realized that modified theories deviate yield loci to the compressive-compressive regime, being conic
model more sensitive to m and h due to the linear effect of I1. The parabolic model is considered more
representative of experiments [4]-[7], which is exemplified by Fig. 1(d) for selected uneven polymers.
(4)
(5)
(6)
3. Experimental procedures and results
Four thermoplastic polymers (PA-66, PA-6, PP and HDPE) were tested under tension and
compression. One 3 meters long rod with 1 inch diameter was purchased for each material and 10
specimens (5 tensile and 5 compressive) were machined using CNC equipments and small passes from
the bar centerline according to ASTM D638 [14] for tension (rectangular cross section with thickness =
7.0 mm and width = 13.0 mm) and ASTM D695 [15] for compression (cylindrical specimen with length
( ) ( ) ( )2
13
2
32
2
21
2
2
2
1
σσσσσσσ −+−+−== vMkJ
32113 σσσσ ++== Ih
=
⋅+=
N
i
i
i IkJ
0
1
2
2 α
( ) ( ) ( ) ( ) ( ) ( )[ ]−+−+−++⋅−⋅=−
2
13
2
32
2
211
2
1
11
2
1
σσσσσσσ mIm
m
CvM
( ) ( ) ( ) ( ) ( )[ ]2
13
2
32
2
21
2
11
2
1
2
1
2
1
σσσσσσσ −+−+−+⋅
−
+⋅
−
=−
m
I
m
m
I
m
m
PvM

4. 1874 Gustavo H. B. Donato and Marcos Bianchi / Procedia Engineering 10 (2011) 1871–1876
= 25.4 mm and diameter = 12.7 mm). The specimens were kept and tested at 21 ºC and 60 % relative
humidity, using the same strain rate for tensile and compressive testing (0.051 min-1
), in order to obtain
comparable results. Tensile tests were conducted using a 30 kN electromechanical INSTRON machine
(model 5567) and compressive tests using a 250 kN servohydraulic MTS machine (model 810). Were
determined for all specimens: i) elastic modulus (E); ii) offset yield strength ( ys-off-x) for x = 0.2, 0.5, 1.0
and 2.0 % plastic strain offsets; iii) maximum yield strength ( ys-max) based on the point where d /d = 0.
Due to the excellent agreement between the five specimens tested under tension and compression for
each material, Fig. 2 presents selected curves (one under tension and one under compression) for each
material in order to illustrate unevenness. It can be realized that PA-6 and PP are clearly uneven
considering yielding. Table 1 presents results for elastic moduli and yield strength for 2.0 % offset and
maximum methods. Results for other offsets were very close and are not presented here. This table
includes unevenness considering engineering (me) and true data (mt). Results reveal that PA-6 and PP
present relevant unevenness, between 25% and 70% for engineering data and between 9% and 27% for
true data depending on the yield point definition. True data should be considered more realistic and safe.
(a)
σ1
σ2
σ3
σ1 = σ2 = σ3
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
A / o
B / o
von Mises - original
m = 1,3 - conic
m = 1,3 - parabolic
m = 2,0 - conic
m = 2,0 - parabolic
Plane Stress ( C = 0)
3.1=m
m
PVC_1 1.33
PC 1.20
PS 1.30
x PVC_2 1.30
+ PMMA 1.30
σ σ
σ σ
− σ σ
− σ σ
(b)
σ1
σ2
σ3
σ1 = σ2 = σ3
(c) (d)
Fig. 1. Illustrative yield surfaces plotted relative to the three principal axes considering (a) conically modified and (b)
parabolically modified von Mises criteria. (c) Yield loci for original Mises criterion compared to conically and
parabolically models for m = 1,3 and m = 2,0 and (d) experiments compared to parabolic model prediction [6].
(a) (b)
(c) (d)
Fig. 2. Engineering stress-strain curves for (a) PA-66, (b) PA-6, (c) PP and (d) HDPE. Legends indicate specimens
under tension (ST) and compression (SC).

5. Gustavo H. B. Donato and Marcos Bianchi / Procedia Engineering 10 (2011) 1871–1876 1875
Table 1. Results under tension (ST) and compression (SC). Unevenness levels were calculated for engineering (me)
and true (mt) stress-strain data. All moduli and stresses values are MPa, while m values are dimensionless.
PA-66 PA-6 PP HDPE
STeng SCeng me STeng SCeng me STeng SCeng me STeng SCeng me
E 2766±270 1984±90 --- 2871±209 2340±61 --- 1778±112 1682±91 --- 1650±93 932±86 ---
ys-2.0 64.9±1.0 64.9±0.4 1.00±0.02 60.4±0.9 75.3±0.8 1.25±0.02 31.7±0.5 44.1±0.6 1.39±0.02 20.0±2.3 21.6±0.4 1.08±0.12
ys-max 64.6±1.1 69.5±3.3 1.08±0.05 63.3±0.8 107.5±0.7 1.70±0.01 36.1±0.2 55.9±0.1 1.55±0.01 23.5±0.4 30.9±0.9 1.31±0.03
STtrue SCtrue mt STtrue SCtrue mt STtrue SCtrue mt STtrue SCtrue mt
ys-2.0 67.0±1.0 61.1±0.7 0.91±0.02 63.4±0.9 69.2±0.6 1.09±0.02 33.0±0.8 40.9±0.5 1.24±0.03 19.9±0.4 20.2±0.4 1.02±0.03
ys-max 67.2±1.1 62.9±1.1 0.94±0.02 68.7±0.9 87.6±1.2 1.27±0.02 39.2±0.3 46.5±0.9 1.19±0.02 26.0±0.8 26.8±0.3 1.03±0.03
4. Exploratory design application and discussion
To illustrate the potential of considering uneven polymers for structural design, this section contains a
brief exploratory application. In order to take advantage of this phenomenon, the component being
designed must present regions loaded predominantly and permanently under compression. One such
example is the mechanical joining system called snap-fit (Fig. 3a). Considering that the snap-fit beam
presents L = 25 mm and is loaded by a force F = 66 N, bending ( ) and shear ( ) stresses are maximum in
the ABC plane (clamped region) and can be analytically calculated neglecting stress intensity factors
using Bernoulli’s and elasticity theories [16] or computed using refined finite element models. Based on
and stresses in each point, equivalent stresses and respective safety factors can be computed based on
Eqs. (2,5,6) for conventional and modified theories. Due to technological interest, PP was selected for the
exploratory investigation. Considering true stress-strain data and 2.0 % offset results (highly
representative of the average behavior of PP), average elastic modulus for tension and compression is E =
1730 MPa, ys-2.0-t = 33.0 MPa, ys-2.0-c = 40.9 MPa and m = 1.24.
Based on the original (usual in practice) rectangular cross section presented by Fig. 3(b) (named
section #1), it is well known that maximum and minimum stresses occur respectively at the top and
bottom fibers. For a good description of stress fields, refined 3D solid finite element models were
developed using ~ 130000 8-node isoparametric hexahedric elements. MD Patran/Nastran 2010 finite
element code was used for the computations, however, Mises parabolic model was implemented using
Fields and PCL functions since it was not available for use. Conventional Mises equivalent stresses are
equal at the top and bottom of section # 1, leading to approximately unitary safety factors (S.F.-up,bottom =
1.03) as presented by Table 2. However, adopting parabolically modified Mises criterion for this case
study (Eq. 6), a different safety factor emerges for the bottom fibers of section # 1 (S.F.-bottom = 1.31). This
occurrence demonstrates that in this position there is 31 % extra safety that was not accounted for by
original Mises criterion and can be optimized, as in the sequence. Looking for shifting the neutral axis to
a higher position, and consequently increasing stresses at the bottom in order to balance safety factors
accounting for unevenness, a routine based on the Generic Reduced Gradient Nonlinear Optimization
Code (GRG2) was implemented. Using analytical stress calculations, the code enforced original stiffness
(by keeping inertia I = 125 mm4
) and unitary safety factors at the top and bottom considering Mises
parabolic model and changing geometric features for the optimized cross sections. Two interesting results
using one and two opposite trapezoids are presented in Figs. 3(c,d). These sections were then
implemented in finite element models similar to the one for section # 1and provided the additional results
shown by Table 2. It can be realized that taking unevenness level into account in the optimization
algorithm, cross sections not approved by conventional von Mises criterion (S.F. < 1.00) are considered
safe using the parabolically modified theory (S.F. 1.00), while keeping original stiffness and reaching
area (mass) reductions up to 38%. Figure 4 illustrates results for equivalent Mises stress fields in the
clamped region of section #2 not considering (a,b) and considering (c,d) the 24% unevenness of PP.
5. Concluding remarks
• All tested materials presented stiffness reduction under compression (lower elastic modulus E).
• Considering deviation, PA-66 and HDPE presented even yield strength. PA-6 and PP presented
unevenness levels between 25% and 70% for engineering data and between 9% and 27% for true data.

6. 1876 Gustavo H. B. Donato and Marcos Bianchi / Procedia Engineering 10 (2011) 1871–1876
• As strains are significant when evaluating yield strength, considering true stress-strain data is
recommended for realism/safety.
• The incorporation of pressure dependent criteria in optimization and finite element codes provided mass
reductions up to 38 % while keeping original stiffness and safety factors, which encourages future
developments in the field and implementation of modified criteria for polymers in commercial codes.
12.00
5.00
2.50
12.14
5.60
5.72
3.10
13.84
3.102.00
8.52
5.60
R1
(a) (b) Section # 1 (c) Section # 2 (d) Section # 3
Fig. 3. (a) Cantilever beam snap-fit, (b) usual rectangular cross section, (c) proposed trapezoidal and (d) proposed double trapezoidal
optimized cross sections. Neutral axes are indicated by the dashed lines.
MPavM 86.32+=σ
MPavM 55.39−=σ
MPaPvM 24.33+=−σ
MPaPvM 60.31−=−σ
(a) (b) (c) (d)
Fig. 4. Finite element models for section # 2 considering (a,b) original von Mises yield criterion and (c,d) parabolic von Mises yield
criterion incorporating PP unevenness. Color ranges are the same for all figures and stresses are pointed in the ABC plane (clamp).
Table 2. Results for the three evaluated cross sections presented by Fig. 4. In each case, stresses and safety factors
were computed using conventional (vM) and parabolically modified (vM-P) von Mises yield criteria.
Section # 1 Section # 2 Section # 3
vM vM-P vM vM-P vM vM-P
eq-up (MPa) 32.00 32.69 32.86 33.24 33.45 33.46
eq-bottom (MPa) 32.00 25.25 39.55 31.60 40.74 32.63
S.F.up 1.03 1.00 1.00 1.00 0.99 0.99
S.F.bottom 1.03 1.31 0.83 1.05 0.82 1.01
I (mm4
) / A (mm2
) 125.00 (ref.) / 60.00 (ref.) 125.00 (-0.0%) / 50.00 (-16.7%) 125.20 (+0.2%) / 37.19 (-38.0%)
Acknowledgements
This work is supported by Ignatian Educational Foundation (FEI, Brazil).
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