Mechanical characterization and modeling of poly...

Mechanical Characterization and Modeling of Poly(b-hydroxybutyrate)-co-poly(b-hydroxyvalerate)–AlfaFiber-Reinforced Composites

Ridha Ben Cheikh,1 Aaron Michel,2 Sarah Billington2

1Universit�e Tunis El Manar, Ecole Nationale d’Ing�enieurs de Tunis, Campus El Manar, 1002 Tunis, Tunisia

2Department of Civil and Environmental Engineering, Stanford University, Stanford, California

The mechanical properties of biobased composites ofpoly(b-hydroxybutyrate)-co-poly(b-hydroxyvalerate) bio-polymer continuously reinforced with unidirectionalAlfa fibers are investigated via tensile testing of ori-ented composite laminates. Simple mechanical modelsfor the elastic stiffness, strength, and nonlinear hard-ening of the biobased composites are proposed withan emphasis on techniques that only require the inde-pendent properties of the fiber and matrix to facilitatecomposite design. Rule of mixtures (ROM) approachesare found to effectively predict the elastic properties ofthe composites but generally overestimate strength.Modified ROM approaches that discount the contribu-tion of the matrix in the fiber loading direction and thecontribution of the fiber in the transverse loadingdirection provide the most accurate strength predic-tions. Apparent elastic properties for composites withvarying fiber orientations are predicted using a modi-fied orthotropic laminate plate method which wasfound to overestimate composite stiffness in off-axisloading directions. Postyield nonlinear hardening ismodeled using a calibrated continuum yield and plas-ticity model and demonstrated to provide a closematch of the experimental results. POLYM. COMPOS.,00:000–000, 2014. VC 2014 Society of Plastics Engineers

INTRODUCTION

Motivation and Background

Synthetic plastics and composite materials are integral

to a variety of engineering applications, but increasing

social concern over the negative impact of primarily

petroleum-derived plastics on the environment has moti-

vated the development of biobased alternatives. Although

partially biobased materials, such as wood plastic compo-

sites, are already widely used [1, 2], fully biobased plas-

tics and composites are still emerging technologies. An

interesting subset of biobased materials is comprised of

rapidly renewable composites made from bacterially syn-

thesized polyhydroxyalkanoates and natural fibers. These

composites have been proposed within a closed-loop life-

cycle in which the degradation of the biobased materials

produces a biogas which can be collected and used as a

fuel source or feedstock to grow more biopolymer [3].

Engineered Biobased Composites

Biobased composites with continuous unidirectional

fiber reinforcement have been broadly investigated for

use in construction, automotive, and industrial applications

[4–6]. Continuously reinforced composites typically exhibit

superior mechanical properties when compared with those

reinforced with discontinuous or short fibers, especially in

the direction of the continuous reinforcing [7]. Continu-

ously reinforced uniaxial composites are most commonly

used when high strength or stiffness is required in one of

the principal loading orientations of a structure.

Proposed Applications

Biobased composites can be designed to have stiffness

and strength properties suitable for many engineering

applications, including transportation, biomedical devices,

packaging, and construction [8]. Construction is a particu-

larly attractive market for biobased materials because of

high turnover rates to landfills [9, 10]. The variety of

potential uses for biobased composites requires the opti-

mization of their mechanical material properties for spe-

cific design applications. To this end, the development of

practical design tools and modeling techniques is essential

for predicting the performance of biobased materials and

facilitating their acceptance in industry.

Correspondence to: Aaron Michel; e-mail: [email protected]

Contract grant sponsor: The United States EPA STAR and Fulbright

Fellowship Programs.

DOI 10.1002/pc.22829

Published online in Wiley Online Library (wileyonlinelibrary.com).

VC 2014 Society of Plastics Engineers

POLYMER COMPOSITES—2014

In this research, the mechanical properties of compo-

sites of poly(b-hydroxybutyrate)-co-poly(b-hydroxyvaler-

ate) (PHBV) continuously reinforced with Alfa fibers are

investigated. Elastic stiffness and ultimate strength models

are proposed with an emphasis on simple predictive tech-

niques which use the independent mechanical properties

of the fibers and matrix.

MATERIALS

Alfa (Esparto) Fibers

Alfa, commonly known as esparto grass or Stipa tena-

cissima plant, is widely cultivated in the dry climates of

northern Africa and the southern Iberian Peninsula. In

Tunisia, for example, it covers approximately 3,500 km2

and has an annual production of 60,000 kg [11]. Alfa is a

heterogeneous polymer composed principally of cellulose

(45%), hemicelluloses (25%), lignin (23%), and wax

(5%) [12] and is primarily used in the production of high-

quality papers, ropes, and carpets because of its advanta-

geously high strength and stiffness.

Alfa fibers are extracted from the stem using a process

which consists of boiling the plant in sodium hydroxide

(NaOH) and subsequently bleaching the fiber in sodium

hypochlorite (NaClO). The duration of each processing

step is known to impact the mechanical integrity of the

resulting fibers. For example, extended heating and

bleaching treatment times yield short and fine fibrils,

whereas shorter processing intervals result in long and

light fibers with good mechanical properties [13]. The

Alfa fibers used in this work were obtained using an opti-

mized extraction process consisting of boiling the stem

for 2 h in a 3 N NaOH solution at 100�C and atmospheric

pressure, and then bleaching the resulting fibers in a 40%

NaClO aqueous solution for 1 h [13–15]. The resulting

fibers were approximately 350 mm long. The directional

mechanical properties of the fibers are reported in Table

1 and correspond to the material orientation defined in

Fig. 1.

PHBV

Poly(b-hydroxybutyrate) (PHB) is an isotactic aliphatic

polyester synthesized by bacteria as a carbon storage

mechanism. The intracellular polymer can be harvested

and processed for engineering applications, such as com-

modity plastics, biomedical devices, and composites.

PHBV is synthesized by copolymerization of hydroxybu-

tyrate with hydroxyvalerate, which is useful for increasing

the flexibility, processability, and thermal stability of neat

PHB [16].

Pelletized PHBV polymer was obtained from TiananVR

(Enmat-Y1000P); reported tensile properties for Y1000P

are provided in Table 1. The Y1000P was blended with

an unreported amount of proprietary nucleating agents by

the manufacturer to improve melt processing [17].

EXPERIMENTAL METHODS AND RESULTS

Experimental Methods

Composite Manufacture. Pelletized PHBV was proc-

essed using a Brabender Plasti-CorderVR

extruder operated

FIG. 1. Schematic of the composite fiber–matrix ply layup sequence and representative tensile response for

PHBV–Alfa fiber composites loaded in the x, xy, and y directions.

TABLE 1. Ranges of elastic properties for PHBV [19] and Alfa fiber

[13].

Property PHBV Alfa fiber

Ex (MPa) 2,800–3,500 21,500–23,300

Ey (MPa) 2,800–3,500 5,300–6,200

Gxy (MPa) 1,070–1,380 3,590–3,860

txy 0.27–0.30 0.34

rx,u (MPa) 31–34 187–306

ry,u (MPa) 31–34 29–31

ex,u (%) 2.5–4.0 1.5–2.4

ey,u (%) 2.5–4.0 0.26–0.34

Ei 5 tensile modulus, Gij 5 shear modulus, mij 5 Poisson’s ratio,

ri,u 5 ultimate tensile stress, and ei,u 5 ultimate tensile strain.

2 POLYMER COMPOSITES—2014 DOI 10.1002/pc

at 25 rpm with a heating profile ranging from 160�C at

the hopper to 150�C at the die. Extruded 50.8 3 1 mm

films were combined with unidirectional Alfa fibers via

compression molding on a 267-kN PHIVR

hot press. The

Alfa fibers were oriented relative to their respective load-

ing direction and held using double-sided tape before

melt compounding. The composites were then hot pressed

at 207 kPa for 4–6 min, whereas both top and

bottom heating platens were maintained at 180�C. After

compression molding, the composites were cooled at

room temperature under a 15-kg weight for 24 h. The

resulting two-ply PHBV–Alfa fiber composite sheets

had thicknesses, which ranged between 3.2 and 4.0 mm,

and fiber volume fractions, which ranged between

20 and 27%. Tensile specimens consistent with ASTM

D638 [18] were cut from the composite sheets using a

band saw.

Fiber Volume Fraction Measurement. Fiber weight

fractions were calculated using the mass of the fibers and

matrix before composite processing. The weight fraction

was confirmed after processing by dissolving the compos-

ite samples in chloroform at 80�C and massing the

remaining insoluble Alfa fibers. Fiber weight fractions

were converted to volume fractions using the known den-

sities of the Alfa fiber (1.4 g/cm3 [14]) and PHBV poly-

mer (1.25 g/cm3 [19]).

Mechanical Testing and Measurement. Uniaxial

mechanical properties were determined via tension testing

using an MTS 858 table-top hydraulic actuator. Tensile

stresses were calculated using a 14.7-kN load cell and

average cross-sectional areas; tensile strains were meas-

ured directly with a 50.8 mm extensometer (MTS, Model

Number 634.12F-24). The force resolution of the load

cell is 60.5% of the applied load, and the strain resolu-

tion of the extensometer is 60.18% of the applied dis-

placement. Engineering stress and strain are reported in

place of true stress and strain with negligible loss of

accuracy due to the brittle nature of the composites (i.e.,

<5% elongation at break).

Experimental Results. Herein, the x–y plane illustrated

in Fig. 1 defines the consistent material coordinate sys-

tem, where x is oriented parallel to the fiber axis and y is

oriented perpendicular to the fiber axis. The 1-2 plane

defines the tensile loading coordinate system, which

rotates clockwise about the z-axis by the angle h with

respect to the x-axis. Tensile loading is coincident with

the 1 direction.

Average elastic and ultimate strength constants, and

their corresponding coefficients of variation, are provided

in Table 2 for the PHBV–Alfa fiber composites. Average

composite volume fractions were 27, 20, and 20% for the

x (h 5 0�), y (h 5 90�), and xy (h 5 45�) oriented com-

posites, respectively. Differences in the composite volume

fractions are attributed to manufacturing variances; how-

ever, volume fraction variation is accounted for in the

mechanical analysis of all composites. A total of five rep-

licate composite specimens were tested in each fiber

orientation.

The tensile modulus is calculated as the chord between

points defined at 5 and 15% of the ultimate composite

strength and is indicated on each stress–strain curve in

Fig. 1. Approximate yield values of 36.3, 10.6, and 10.0

MPa for the x, y, and xy specimens, respectively, were

defined at the point where the instantaneous tensile modu-

lus deviated from the initial elastic modulus by 5%. A

mixture of fiber breakage and pullout (Fig. 2) character-

ized the ultimate failure of the composites, indicating

nonuniform fiber–matrix mechanical bond compatibility

at break. The composites also exhibited considerable post-

yield nonlinear hardening before failure.

TABLE 2. Experimentally determined composite material properties.

Material property Mean CV (%)

Ex 8,629 MPa 6.4

Ey 2,990 MPa 14.2

Gxy 1,234 MPa 3.8

txy 0.315 8.5

rx,u 64.3 MPa 8.2

ry,u 14.0 MPa 9.9

sxy,u 9.80 MPa 10.1

Ex,u 0.015 7.4

Ey,u 0.006 20.1

cxy,u 0.020 27.8

Ei 5 tensile modulus, Gij 5 shear modulus, mij 5 Poisson’s ratio, ri,u 5

ultimate tensile stress, sij,u 5 ultimate shear stress, ei,u 5 tensile strain at

ultimate, and cij,u 5 shear strain at ultimate.

FIG. 2. Typical failure modes for composites with alfa fibers oriented

in the x (u 5 0�), y (u 5 90�), and xy (u 5 45�) directions. [Color fig-

ure can be viewed in the online issue, which is available at wileyonline-

library.com.]

DOI 10.1002/pc POLYMER COMPOSITES—2014 3

MODELING METHODS AND RESULTS

Modeling Methods

Several modeling approaches were used to calculate

the elastic and ultimate strength properties of the compo-

sites. A focus on rule of mixtures (ROM) methods

reflects the objective of using predictive models that rely

only on the properties of the polymer matrix and reinforc-

ing fibers. Nonlinear hardening properties were also mod-

eled using a calibrated continuum method.

Elasticity Models

The underlying isostrain and isostress assumptions of

elastic laminated composite analysis are shown in Fig. 3. In

the isostrain state (Fig. 3a), displacement compatibility is

enforced between each ply of a composite material,

whereas the stress in each layer is allowed to vary. In the

isostress state (Fig. 3b), force compatibility is enforced

between each ply, whereas the strain between layers is

allowed to vary. Common mechanical analogs for isostrain

and isostress are springs in parallel and series, respectively.

ROM

The ROM method is a basic geometric approach for

estimating the elastic and strength properties of a compos-

ite material using known constituent volume fractions and

mechanical constants. The ROM principle is shown in

Fig. 3 and is provided explicitly for the elastic modulus

(Ei, Gij) in Eqs. 1–3 [20].

ECx 5EF

x VF1EMð12VFÞ (1)

ECy 5

VF

EFy

1ð12VFÞ

EM

" #21

(2)

GCxy5

VF

GFxy

1ð12VFÞ

GM

" #21

(3)

Here, the subscripts represent the material orientation

(see Fig. 1) and the superscripts C, F, and M correspond

to the composite, fiber, and matrix, respectively.

Oriented Orthotropic Elasticity

Composite materials generally exhibit orthotropic elas-

ticity, which, for thin planar materials, can be character-

ized using the following plane stress, transversely

isotropic constitutive relationship:

rx

ry

rxy

8>><>>:

9>>=>>;5Ce

eex

eey

cexy

8>><>>:

9>>=>>; (4)

Ce5

E2x

Ex2Eym2xy

ExEymxy

Ex2Eym2xy

0

ExEymxy

Ex2Eym2xy

ExEy

Ex2Eym2xy

0

0 0 Gxy

266666664

377777775

(5)

Here, rx, ry, and rxy and eex, ee

y, and cexy are the ply

level elastic stresses and strains, respectively, in the x, yand xy directions. Ce represents the ply level stiffness

matrix, which is populated with the elastic constants of

each composite ply. The ply level properties can then be

transformed using a suitable rotation matrix to predict the

apparent stiffness, �Ce, of the composite subjected to load-

ing in any direction h as follows:

�Ce5T21CeðT21ÞT (6)

Where:

T5

cos 2h sin 2h 2sin hcos h

sin 2h cos 2h 22sin hcos h

2sin hcos h sin hcos h cos 2h2sin 2h

2664

3775 (7)

The apparent composite stiffness can then be deter-

mined using the stiffness of each oriented composite ply

and the laminate sequence. This procedure is commonly

referred to as classical laminate plate theory (CLPT)

[20]. The implementation of CLPT in this research is

similar to the isostrain ROM approach, in which the indi-

vidual elastic properties of the fiber and matrix are

assigned to alternating plies and the ply sequence is

determined by the known composite geometry and fiber

volume fraction.

Eq. 6 can be solved directly to obtain useful equations

for relating the oriented ply level material properties to

the apparent elastic properties of a composite material.

Expressions for the apparent stiffness and Poisson’s ratio

in any loading direction h are given by:

E1ðhÞ5cos 4h

Ex1

sin 4hEy

1sin 2hcos 2h1

Gxy2

2mxy

Ex

� �� 21

(8)

FIG. 3. (a) Isostrain and (b) isostress models for layered composite

materials.


m12ðhÞ5E1ðhÞmxy

Exðcos 4h1sin 4hÞ

�

2cos 2hsin 2h1

Ex1

1

Ey2

1

Gxy

� �� (9)

Ultimate Strength and Nonlinear Hardening Models

Five strength prediction models are evaluated for

PHBV–Alfa fiber composites. These methods range from

ROM models to techniques that incorporate the statistical

distribution of fiber strength and the impact of progres-

sive damage and fiber load redistribution. The investi-

gated models estimate strength using the material

properties of the matrix and reinforcing fibers alone. A

calibrated orthotropic nonlinear hardening model and

associated failure criterion are also demonstrated.

ROM and Modified ROM

The isostress and isostrain assumptions of the ROM

model established in section Elasticity Models can be

modified to predict composite ultimate tensile strength

(ri,u) as follows:

rCx;u5rF

x;uVF1rMu ð12VFÞ (10)

rCy;u5

VF

rFy;u

1ð12VFÞ

rMu

" #21

(11)

The ROM model for strength assumes that the matrix

and fiber maintain bond compatibility up to failure and

discounts the contribution of matrix cracking. Therefore,

the ROM model is considered an upper bound estimate of

composite strength.

A simplified modified rule of mixtures (MROM)

model for strength is also proposed. In the MROM model,

composite failure in the fiber (x) direction is controlled

by the volume fraction and strength of the fiber alone,

whereas in the transverse (y) direction strength is con-

trolled by the volume fraction and ultimate strength of

the matrix alone, as defined in Eqs. 12 and 13.

rCx;u � rF

x;uVF (12)

rCy;u � rM

u ð12VFÞ (13)

The MROM model assumes poor bond compatibility

in each loading direction and effectively establishes a

lower bound strength prediction.

Global Load-Sharing and Simultaneous Fiber Failure

The strength of unidirectional continuously reinforced

composites can alternatively be estimated using a global

load-sharing (GLS) principle. GLS is a statistical method

that accounts for progressive fiber failure and subsequent

fiber load redistribution by shear lag through the compos-

ite matrix [21]. In the GLS model, the strength of a com-

posite loaded in the fiber direction is controlled by the

strength of the reinforcing fibers, which are statistically

characterized by the Weibull distribution [21, 22]. Wei-

bull parameters r0, L0, and m, and the fiber-matrix inter-

facial shear strength, s, and the average fiber diameter, D,

are used to estimate the composite strength as follows:

rCx;u5VF3rc

m11

m12

� �2

m12

� �1=m11

(14a)

Where rc is considered the critical fiber stress and is

given by:

rc52rm

0 sL0

D

� �1=m11

(14b)

The GLS model, which is known to overestimate the

strength of many composites that exhibit damage localiza-

tion, was modified to account for simultaneous fiber fail-

ure (SFF) initiated by a weak fiber within a group [22].

The SFF model uses the interaction between the strengths

of the fiber, matrix, and fiber–matrix interface to quantify

a number of SFFs within an arbitrary region of the com-

posite. The SFF method can be summarized by the fol-

lowing modified composite strength equation:

rCx;u5VF3rc

m11

m12

� �2

m12

1

n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVF1n21

VF

r� �1=ðm11Þ

(15a)

where,

ln ðnÞ5alnsffiffiffiffiffiffiffiffiffiffiffi

r0rMu

p !

1b (15b)

The parameter n is defined as the number of SFFs. It

has been suggested that n can be approximated as a loga-

rithmic function of the geometric average of the fiber–

matrix interface strength and fitting parameters a and b[22]. Other statistical models, such as local load-sharing

[23], have also been demonstrated to have robust strength

prediction capabilities but require more rigorous analysis

and empirical calibration and are not investigated here.

Cooper and Kelly Transverse Model

Transverse composite strength can alternatively be pre-

dicted using a model developed by Cooper and Kelly

(C–K) for low fiber volume fraction composites with

weak transverse matrix–fiber bonding [24]. The C–K

model predicts failure at points of minimum matrix thick-

ness, which are calculated using a simple volumetric rela-

tionship that assumes an average fiber spacing based on

an estimated cylindrical fiber packing density derived


from the fiber volume fraction. The transverse composite

strength can then be estimated as:

rCy;u5rM

u 12

ffiffiffiffiffiffiffiffi4VF

p

r !(16)

Orthotropic Nonlinear Hardening and Failure Crite-

rion. Nonlinear behavior in composites is generally an

artifact of the progressive failure of elastic plies throughout

the thickness of a laminate, which results in gradual stiff-

ness degradation before ultimate failure. This phenomenon

is commonly modeled by defining a suitable criterion that

dictates the failure of an individual ply, such as those pro-

posed by Hill, Hoffman, Tsai-Wu, and Hashin [25, 26].

Damage and nonlinearity is captured by updating the com-

posite stiffness matrix as the failure state of each ply is

exceeded. For composites that exhibit nonlinear fiber and

matrix constitutive behavior, damage-based strength mod-

els are insufficient for capturing composite nonlinearity.

Continuum plasticity techniques use a yield surface and

appropriate hardening model to define the evolution of the

yield surface. Although plasticity models can provide accu-

rate performance predictions, they are potentially limiting

because of the need for significant empirical calibration.

The simplified plasticity formulation used in this work

is proposed by Sun and Chen [27] and requires a single

shear parameter to define the anisotropic yield surface

and plastic hardening behavior. The Sun–Chen model

assumes linear elasticity in the fiber direction and uses

the following yield condition:

2f 5r2y12a66r

2xy (17)

where 2f is the effective yield condition and a66 is a shear

parameter. Equation 17 leaves a single coefficient, a66, to

be determined using a minimum of two off-axis tension

tests. Experimental stresses, r1, and plastic strains, ep1, in the

loading direction are converted to effective stresses, �r, and

plastic strains, �ep, using an associative flow rule as follows:

�r5hðhÞr1 and �ep ep1

hðhÞ (18)

where h(h) is given by:

hðhÞ5 3

2ðsin 4h12a66sin 2hcos 2hÞ

� �1=2

(19)

The relationship between the effective stresses and plas-

tic strains is then described by a power law of the form:

�ep5Að�rÞn (20)

where A and n are exponential terms determined by linear

regression of the logarithmic effective plastic strain ver-

sus effective stress curve. The parameter a66 is iterated by

either trial-and-error or a linear optimization procedure

until the values of A and n converge for a minimum of

two off-axis tensile tests. After A, n, and a66 are deter-

mined, the tensile constitutive response of a composite

with any fiber orientation h can be predicted as follows:

e15r1

E1

1½hðhÞ�n11Arn1 (21)

An extension of the plasticity model is a failure crite-

rion proposed by Sun, Chung, and Chang [28], which is

also defined by a single shear parameter, a�66. The failure

criterion, which is similarly calibrated using the results of

two off-axis tension tests, takes the form:

k2cr5r2

y12a�66r2xy; where kcr5ry;u (22)

Modeling Results

Elastic Constants. Elastic properties for the PHBV

polymer and Alfa fibers obtained from literature are

reported in Table 1. Although the longitudinal elastic

modulus for a continuous fiber can be measured directly

from a tension test, the transverse and shear moduli for a

fiber are typically determined indirectly by composite

tests. Therefore, the elastic constants for the Alfa fibers

were derived from the test results of Ben Brahim and Ben

Cheikh on unsaturated polyester Alfa fiber composites

[13] using CLPT. The range of calculated longitudinal

fiber moduli agreed well with measured properties for

Alfa fiber bundles tested in direct tension reported by

Paiva et al. [14]. Unless otherwise noted, the reported

ranges of physical and mechanical properties are consid-

ered in subsequent composite calculations and are illus-

trated in each figure with error bars.

Elasticity Predictions. The predicted elastic moduli in

the x, y, and xy directions are compared with the experi-

mentally determined values in Fig. 4. The elastic proper-

ties predicted by the ROM models in each loading

direction closely matched the experimental results, indi-

cating good initial fiber–matrix compatibility. As

expected, ROM and CLPT yielded precisely the same

results for loading in the fiber direction. The off-axis

ROM and CLPT predictions differed however as a result

of the discrepancy between the isostrain and isostress

assumptions of the CLPT and off-axis ROM methods,

respectively.

A range of apparent moduli for PHBV–Alfa compo-

sites with fiber volume fractions between 10 and 30%

and fibers oriented between 0 and 90� with respect to the

tensile loading direction are presented in Fig. 5a. The

elastic modulus of the 0� aligned composite, which is the

most sensitive to fiber volume fraction variation, was

accurately captured by the 30% volume fraction curve

(actual VF 5 27%). Composites with loading angles


greater than 45�, which are expected to exhibit isostress

dominated behavior, are overestimated using CLPT but

are closely matched using the ROM method. Here, the

utility of CLPT is demonstrated for predicting the appa-

rent tensile modulus of unidirectional biobased compo-

sites with varying loading orientations, but, in the case of

off-axis specimens, isostrain CLPT predictions tend to

overestimate the experimental responses.

Poisson’s ratios calculated by Eq. 9 for composites

with orientations between 0 and 90� are compared with

experimental results in the x, y, and xy directions in Fig.

5b. Here, Poisson’s ratio was determined directly in the

0� orientation and predicted using the elastic composite

constants in the 45 and 90� orientations. Although the

method is not predictive using independent fiber and

matrix properties, it does provide a reasonable and there-

fore useful approximation of the relationship between

Poisson’s ratio and fiber orientation with only minimal

material characterization.

Ultimate Strength Predictions. Predictions using

ROM, or any elastic technique that is predicated on per-

fect fiber–matrix bond compatibility up to failure, tend to

overestimate composite strength. Ultimate strength predic-

tions using ROM and MROM strength models are com-

pared with experimental results in Fig. 6 in the x and yloading orientations. The unmodified ROM model signifi-

cantly overestimates the tensile strength in both the x

and y loading directions, suggesting fiber–matrix bond

degradation before failure, premature fiber rupture, or a

combination of the two. The MROM models presented in

Fig. 6 differ for the two loading directions. In the aligned

fiber (x) direction, the MROM discounts the contribution

of the matrix at ultimate and bases composite failure on

the strength of the fiber alone. In the transverse (y) direc-

tion, the MROM treats the fibers as voids in the matrix

and failure is controlled by the volume fraction and ulti-

mate strength of the matrix alone. These simplified

MROM models predict the strength of the aligned compo-

sites well but overestimate the strength of the transverse

composites. It has been suggested that transversely

aligned fibers in composites can weaken matrix–fiber

interfaces and act to initiate and propagate matrix fracture

[24, 29]. Such phenomena could lead to an ultimate

FIG. 4. Experimental (EXP) results compared with rule of mixtures

(ROM) and classical laminate plate theory (CLPT) predictions.

FIG. 5. (a) Longitudinal modulus calculated using CLPT for variable

fiber volume fractions (VF) and fiber ply orientations compared with

experimental and ROM results at 0, 45, and 90� and (b) variation of the

theoretical and experimental Poisson’s ratio with respect to fiber ply

orientation.

FIG. 6. Ultimate strength predictions using the rule of mixtures

(ROM), simplified modified rule of mixtures (MROM), simultaneous

fiber failure (SFF) (rx only), and Cooper–Kelly (C–K) (ry only) models

compared with experimental (EXP) results for x and y loading

orientations.


transverse composite strength below that of the polymer

matrix alone.

The GLS strength prediction in the aligned fiber direc-

tion is calculated using Weibull parameters (L0 5 1 mm,

r0 5 1078 MPa, m 5 2.74) and average fiber properties

(D 5 80 lm, s 5 14 MPa) established in an investigation

of the same Alfa fibers as those used herein [14]. The

GLS model predictions by Eq. 14a significantly overesti-

mated the strength of the composite (results not shown),

suggesting that fiber load redistribution is limited in the

PHBV–Alfa fiber composites. This discrepancy is also an

indication that the relatively weak matrix is unable to

transfer shear lag stresses as the fibers progressively frac-

ture, leading to failure localization.

The SFF strength prediction in the aligned fiber direc-

tion is calculated using the same parameters as the GLS

model and the additional term, n. The number of SFFs, n,

is calculated per Eq. 15b with a 5 4.2 and b 5 12.6, as

suggested in Ref. [22] for the range of investigated com-

posites, which included several that exhibited similar ulti-

mate strengths to the Alfa–fiber composites. The SFF

result also overestimated the composite strength (Fig. 6),

which suggests that either (1) a group fiber failure initi-

ated by a single weak fiber was not the primary failure

mechanism or (2) the proposed a and b parameters do not

accurately reflect the number of SFFs exhibited by the

composites tested here. The strength prediction from the

SFF analysis did however provide a better approximation

than the GLS model and was within the error bounds of

the experimental results.

Results from the C–K model for the transverse compo-

sites presented in Fig. 6 provide a close match to the exper-

imental data. The accuracy of this model validates the

assumption that the transverse interfacial fiber–matrix bond

between the Alfa fibers and PHBV polymer provides an

ineffective stress transfer mechanism and that the strength

is instead controlled by the minimum matrix cross section.

The results of the fifth strength prediction method are

shown in Fig. 7a. The one parameter model was cali-

brated using the results of the 45 and 90� off-axis tension

tests. Therefore, the resulting a�66 coefficient of 0.475 pro-

vided exact strength predictions for each experimental

value because these values also served as calibration

points. Although the utility of a simple strength model for

composites with any arbitrary fiber orientation is clear,

further testing of off-axis specimens is necessary to accu-

rately validate this method.

Nonlinear Hardening. The one parameter plasticity

model was calibrated using the stress–strain results of off-

axis tests in the 45 and 90� loading directions. Linear

optimization of the effective stress and plastic strain data

for the off-axis specimens yielded an a66 coefficient of

1.05. The resulting stress–strain predictions are presented

in Fig. 7b. As expected, the nonlinearity of the 0� was

not captured because the Sun–Chen model neglects plas-

ticity in the fiber direction. In contrast, the calculated

constitutive response of the 45 and 90� composites is rep-

resentative of the experimental response for both the yield

stress and nonlinear hardening modulus.

The Sun–Chen model was selected for its simplicity,

ease of implementation, and accuracy for oriented com-

posite materials. Although this model is unable to capture

the plasticity observed in the fiber direction of the Alfa–

PHBV composites, it provides good predictions of off-

axis nonlinearity and strength. Further calibration of this

model using off-axis tensile specimens is expected to pro-

vide a more robust estimation of the composite constitu-

tive behavior. This technique is nonetheless useful for

determining the postyield composite response, especially

when coupled with a strength prediction model.

CONCLUSION

The applicability of elastic stiffness and ultimate

strength prediction models has been demonstrated for bio-

based composites made from PHBV polymer and Alfa

fibers. In general, reinforcing PHBV with unidirectional

bundles of Alfa fibers resulted in improved elastic

FIG. 7. (a) Comparison of experimental and theoretical ultimate

strengths using the Sun et al. model calibrated at u 5 45 and 90� with

a�66 5 0.475 and (b) comparison of experimental and theoretical consti-

tutive responses using the Sun–Chen model calibrated with a66 5 1.05.


stiffness and ultimate strength and reduced elongation at

break when compared with neat PHBV.

Isostress and isostrain ROM models provided suitable

estimates of initial composite stiffness in the 0, 45, and

90� fiber orientations. CLPT estimates of composite stiff-

ness as a function of reinforcing fiber orientation pro-

vided a good match to the experimental data for aligned

composites, but overestimated stiffness in the off-axis ori-

entations. This discrepancy demonstrates the limitations

of CLPT for unidirectional composites and highlights the

known importance of isostrain models for off-axis stiff-

ness predictions. The CLPT method did however provide

reasonable estimates of Poisson’s ratio in the 0, 45, and

90� loading orientations.

ROM strength predictions tended to overestimate ulti-

mate strength, suggesting that the perfect fiber–matrix bond

assumptions of the ROM method were violated before fail-

ure. Strength estimates using a proposed modified ROM

model closely matched the experimental results in the

aligned and transverse fiber orientations. A GLS model

developed for unidirectional synthetic composites overesti-

mated the biobased composite strength in the fiber direc-

tion, suggesting that failure of PHBV–Alfa fiber composites

is generally nonprogressive and does not exhibit significant

fiber load redistribution. A modified GLS model that con-

siders a group failure mechanism initiated by a weak fiber

also overestimated the strength in the fiber direction, but

provided a more reasonable match of the experimental

results. Predictions of transverse composite strength with

the C–K model closely matched the experimental results,

further suggesting poor interfacial stress transfer between

the fibers and matrix. The strength prediction findings have

broad implications for biobased composite design because

they suggest that accurate estimates of composite strength

can be obtained with limited knowledge about the compo-

site’s constituent materials, namely the fiber volume frac-

tion and the ultimate strength of the fiber and matrix.

A calibrated nonlinear hardening continuum model was

demonstrated to accurately capture the postyield response

of off-axis composites. An extension of this model was also

demonstrated to calculate the ultimate strength of off-axis

composites, but was limited by the availability of off-axis

test data to fully validate the model. These techniques are

useful for demonstrating the complete characterization and

modeling of a biobased composite in the elastic, nonlinear,

and ultimate strength regimes.

ACKNOWLEDGEMENTS

The opinions expressed in this article are those of the

authors and do not necessarily reflect the views of the

sponsors.

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