Brittle Failure of Dry Spaghetti

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Brittle failure of dry spaghetti

ARTICLE in ENGINEERING FAILURE ANALYSIS · OCTOBER 2004

Impact Factor: 1.03 · DOI: 10.1016/j.engfailanal.2003.10.006

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Gustavo Guinea

Universidad Politécnica de Madrid

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Universidad Politécnica de Madrid

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Brittle failure of dry spaghetti

G.V. Guinea *, F.J. Rojo, M. Elices

Departamento de Ciencia de Materiales, E.T.S.I. Caminos, Canales y Puertos, Universidad Politecnica de Madrid, c/Profesor Aranguren,

Ciudad Universitaria s/n, 28040 Madrid, Spain

Received 5 September 2003; accepted 1 October 2003

Available online 13 February 2004

Abstract

This paper investigates the tensile properties and brittle fracture of dry durum semolina fibers (spaghetti), and

provides quantitative values for the strength and toughness of this material. Tensile tests on spaghetti of different

lengths were performed, and the results correlated with the micrographic observation of fracture surfaces and flaw

distribution. The tests were analyzed according to two widely-used failure theories for brittle materials: those of weakest

link statistics and linear elastic fracture mechanics, pointing out their applicability and limitations for this material.

2004 Elsevier Ltd. All rights reserved.

Keywords: Tensile properties; Fracture toughness; Weibull statistics; Brittle fracture; Food technology

1. Introduction

Durum wheat semolina is the base material for spaghetti, fusilli, and other pasta products. Semolina is

processed by adding water, extruding the dough into the desired shape – which gives it its characteristic

flavor – and drying it under well controlled conditions to prevent the development of cracking. Dry pasta is

basically made of starch granules uniformly dispersed in a continuous protein phase known as gluten.

When pasta is extruded in long cylindrical fibers with a diameter between 1 and 2 mm it is given the

commercial name of spaghetti.

The mechanical strength of dry pasta is ordinarily used as a standard of quality control because it isclosely related to the semolina properties (mainly gluten content [1,2]) and to the pasta processing, specially

to the drying step (which has proved critical to the quality of the final product [3]). In addition, mechanical

measurements are simple and can be easily integrated in the production plant, and provide useful infor-

mation for the design of packing and shipping operations.

To assess the mechanical performance of dry spaghetti, flexural tests and compression tests (where the

final collapse is due to fiber buckling) are usually performed. A nominal rupture strength is obtained from

* Corresponding author. Tel.: +34-91-336-66-79; fax: +34-91-336-66-80.

E-mail address: [email protected] (G.V. Guinea).

1350-6307/$ - see front matter 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfailanal.2003.10.006

Engineering Failure Analysis 11 (2004) 705–714

www.elsevier.com/locate/engfailanal

https://www.researchgate.net/publication/285237659_Statistical_evaluation_of_tests_for_assessing_spaghetti-making_quality_of_durum_wheat?el=1_x_8&enrichId=rgreq-7d96539d-e6df-4564-ab62-a4c3e20e998b&enrichSource=Y292ZXJQYWdlOzI0NTE2MTI3MTtBUzozMDIxMzY5NzIyNTExMzdAMTQ0OTA0NjQ3NTM3NA==http://mail%20to:%[email protected]/https://www.researchgate.net/publication/285237659_Statistical_evaluation_of_tests_for_assessing_spaghetti-making_quality_of_durum_wheat?el=1_x_8&enrichId=rgreq-7d96539d-e6df-4564-ab62-a4c3e20e998b&enrichSource=Y292ZXJQYWdlOzI0NTE2MTI3MTtBUzozMDIxMzY5NzIyNTExMzdAMTQ0OTA0NjQ3NTM3NA==https://www.researchgate.net/publication/284534199_Standardization_of_cooking_quality_analysis_in_macaroni_and_pasta_products?el=1_x_8&enrichId=rgreq-7d96539d-e6df-4564-ab62-a4c3e20e998b&enrichSource=Y292ZXJQYWdlOzI0NTE2MTI3MTtBUzozMDIxMzY5NzIyNTExMzdAMTQ0OTA0NjQ3NTM3NA==http://mail%20to:%[email protected]/


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these tests by dividing the maximum load recorded by the cross sectional area of the specimen. Although

useful for comparative purposes, nominal strengths do not measure the intrinsic properties of the material,

and are dependent on the geometry and shape of the specimen.

A better understanding and modeling of the mechanical behavior of dry pasta needs to be supported byappropriate knowledge of the material properties which must be insensitive – by definition – to the specific

experimental procedure by which they are determined.

This work aims at characterizing the tensile behavior of dry spaghetti, and at providing values for the

tensile strength and fracture toughness of this material. Four tensile test series were performed on speci-

mens of different lengths, analyzing also their fracture surfaces. The results show that dry spaghetti fibers

are close to the ideal linear-elastic behavior, and can be characterized by a definite value of fracture

toughness. The paper also demonstrates that statistical models based on the weakest-link do not explain

satisfactorily the influence of size on tensile properties.

The next section introduces the material and the experimental methods used in this work. Section 3

discusses the results of the tensile tests by applying both the weakest link model – routinely used to

evaluate fracture of brittle fibers – and the Linear Elastic Fracture Mechanics (LEFM) theory,

and examines their applicability to pasta fibers. The paper closes with the main conclusions and the

references.

2. Materials and methods

2.1. Material and specimen geometry

Dry commercial semolina fibers with a diameter of 1.65 mm (Barilla spaghetti) were used for this work.

All the fibers were obtained from the same package (300 U) to ensure homogeneity, and were stored andtested under well controlled temperature and humidity conditions to minimize the effect of hydration/de-

hydration processes. The nominal storage and testing conditions were 20 2 C and 40 5% relativehumidity (RH).

Spaghetti dry matter was composed of starch (76%) and gluten (13%). The moisture content of the fibers,

measured by the weight loss after heating at 105 C for 4 h, was estimated as 4.7%.

Fibers were nominally 500 mm long. They were cut in samples of 400, 250, 175 and 135 mm corre-

sponding to the four tensile test series planned. The tested fiber lengths (free length between the upper and

lower grip) were set equal to L ¼ 300, 150, 75 and 35 mm, respectively. All the specimens were obtainedfrom fresh fibers from the package.

2.2. Tensile tests

Fibers were tested with a universal testing machine (Instron 4111) driven at 1 mm/min elongation rate.The fibers were clamped between flat jaws, placing 2 mm-thick pieces of silicone rubber between the fiber

and the metallic grips to avoid damage and a premature failure. The length of the fiber between the grips

was set to the specified value (300, 150, 75 or 35 mm) within 0.5 mm, the anchoring length being roughly50 mm for all the specimens.

Tests were carried out at 20 C and 40% RH. Maximum loads were recorded with an Instron load cell of

0.5N accuracy, rejecting tests in which the fibers broke in the grips. To measure the entire stress–straincurve – and not only the maximum load – an extensometer with 50 mm gage length (INSTRON 2620-602)

was attached to some of the fibers.

The fibers were appropriately identified after testing, and stored under controlled conditions (20 C and

40% RH).

706 G.V. Guinea et al. / Engineering Failure Analysis 11 (2004) 705–714


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2.3. Fracture surfaces

Fracture surfaces were analyzed by optical microscopy (Zeiss Axiovert 100A). A planar, quasi-circular

flaw, perpendicular to the fiber axis, was systematically observed in all the specimens. The defect wascharacterized by its radius, r , and ligament, b, as shown in Fig. 1. The fiber diameter and flaw size were

measured by means of a calibrated microscope to 4 lm accuracy. The average radius of the fibers, R, was0.824 0.004 mm (95% confidence interval).

In addition to optical measurements, some selected samples were metallized (10 nm Au–Pd, Energy

Beam Sciences Ultra-Spec 90) and observed by scanning electron microscopy (SEM) with a JEOL JSM-

6300 microscope (observation conditions:10 kV and 1010 A). The micrograph of a fractured section isshown in Fig. 2.

3. Results and discussion

A total of 232 fibers were tensile tested, which yields an average of 58 for each fiber length. The tensile

strength, r, was computed as the ratio between the maximum load of the test and the fiber cross-sectional

area at the fracture plane.

A remarkably linear-elastic behavior until rupture was noticed in all the fibers, as illustrated in Fig. 3.

Inelastic deformations at rupture were under 2% of total specimen deformation. The modulus of elasticity

was 5.0 GPa.

3.1. Statistical analysis

It is customarily assumed that the maximum stress that brittle materials can withstand varies unpre-

dictably, even if a set of nominally identical specimens are tested under the same conditions, so statistical

theories are drawn on to describe the strength of these materials, and the weakest link is one of the mostpopular models adopted, specifically when fibers are the concern [4,5].

In its most general formulation, the weakest link model states that the cumulative probability of failure

P f of a brittle fiber of length, L, subjected to a load, r, is given by [6]:

P f ¼ 1 exp½C ðrÞ L; ð1Þ

Fig. 1. Optical micrograph of a fractured surface. The fiber diameter is 1.63 mm.

G.V. Guinea et al. / Engineering Failure Analysis 11 (2004) 705–714 707


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where C ðrÞ is the concentration function, which represents the number of defects per unit length having astrength lower than r. The derivation of Eq. (1) assumes that defects are dilute – non-interacting – and

randomly distributed.

Weibulls classical form for C ðrÞ is the potential equation [7]:

C

ðr

Þ ¼ 1

L0

rrthr

0

m

if r > rth;

¼ 0 if r6rth; ð2Þ

where L0 and r0 are reference values, and m is the Weibulls modulus. rth is the threshold strength below

which no failure will occur, and which is generally assumed to be equal to zero.

Eq. (1) together with the concentration function (2) gives the equation of a straight line of slope m when

represented in a ln½ lnð1 P f Þ vs lnðr rthÞ plot:

ln½ lnð1 P f Þ ¼ lnð LÞ lnð L0rm0 Þ þ m lnðr rthÞ; ð3Þ

which furnishes a simple way of evaluating m from a linear fitting of (3) to the experimental pairs (r; P f ).

Fig. 2. SEM micrograph of the surface shown in Fig. 1.

Fig. 3. Stress–strain curves of semolina fibers.



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In practice, P f values are not directly accessible from experiments, and a statistical estimator is used to

allocate the failure probability to each strength level. A useful procedure to evaluate P f is based on ar-

ranging strength values in ascending order and assigning them a probability proportional to their position, i

[8]:

P i ¼ i 0:5 N

; ð4Þ

N being the number of samples.

Fig. 4 plots ln½ lnð1 P f Þ as a function of lnðrÞ for the four series tested, each corresponding to adifferent fiber length. P f values were estimated by (4), and rth was set at zero. The figure shows how the

values corresponding to the same fiber length, L, lie with good approximation on a straight line, as stip-

ulated by (3). Nevertheless, the overall behavior does not match the expected dependence; Weibulls

modulus is not constant – it varies with L – and the straight line corresponding to each fiber length does not

translate to the right as L decreases.

Weibulls parameters can be directly determined without resort to probability estimators such as (4). Thebest estimate of these parameters is by the maximum likelihood method [9], which seeks the set of pa-

rameters that maximize the function that gives the probability of obtaining the set of experimental points

actually measured. Unfortunately, this method does not yield a better fit than (3) when applied to the

experimental data, as is shown in Table 1.

A third set of parameters obtained by a direct least square fitting of Eqs. (1) and (2) is also shown in

Table 1. Once more, Weibulls modulus – and the other parameters – are dependent on fiber length, and it is

not possible to find a set of parameters to characterize the tensile behavior of the fibers.

The influence of fiber length on the mean tensile strength is shown in Fig. 5. Error bars display the

standard deviation, and the number of tests is shown in brackets. The figure illustrates the wide scatter of

the results, in which for the same fiber length, extreme values can differ by 70% (e.g., from 19 up to 33 MPa

for L ¼ 150 mm).The mean strength of Weibulls distribution, r, is given by [4,6]:

r ¼ r0 L0 L

1=mCð1 þ 1=mÞ r0 L0

L

1=m0:63661=m; ð5Þ

where the error is within 0.5% for 5 < m < 50 [6].

Fig. 4. Logarithmic plot of ln½ lnð1 P f Þ vs lnðrÞ of the four series tested.



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The least square fitting to Eq. (5) of the experimental data in Fig. 5 gives the dashed curve plotted in the

figure, whose analytical expression is:

r ¼ 38:338 L1=16:5; ð6Þ

from which m ¼ 16:5 and r0 L1=m0 ¼ 39:4 mm1=m MPa are readily obtained. These values are comparable tothe averages shown in Table 1, but the correlation coefficient is very poor ( R ¼ 0:774).

As the Weibull concentration function (2) has been found to describe successfully the fracture of most

brittle materials, it was considered first in this work. However, the results suggest that Weibull s distribution

does not satisfactorily explain the tensile behavior of dry semolina fibers.

To explore the general shape of the concentration function for this material, C ðrÞ was estimated byð1= LÞ lnð1 P f Þ, as immediately derived from Eq. (1). The probability P f was obtained from Eq. (4) for

Table 1

Weibulls parameters obtained by different estimation methods

Method L (mm) m r0 L1=m0 (mm

1=m MPa) Correlation coefficient

Eq. (3) 300 15.2 42.7 0.988150 10.9 45.3 0.990

75 17.6 37.1 0.989

35 13.9 43.0 0.993

Average 14.4 42.0 0.853

Maximum likelihood 300 13.9 44.3 0.994

150 10.1 47.1 0.992

75 18.0 37.0 0.996

35 13.8 42.5 0.997

Average 14.0 42.7 0.849

Least square fitting of

Eqs. (1) and (2)

300 14.2 43.7 0.995

150 9.8 47.7 0.994

75 17.0 37.5 0.996

35 13.9 43.1 0.997

Average 13.7 43.0 0.848

Fig. 5. Variation of tensile strength with fiber length.



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every fiber length series. The estimated values of C ðrÞ are plotted in Fig. 6, which clearly shows that theexperimental data cannot be modeled by a single concentration function independent of the fiber length.

This result cast doubts on the applicability of the weakest link statistics to dry semolina fibers.

3.2. Flaw geometry

As stated in Section 2, a planar, internal crack, oriented perpendicularly to the fiber axis was present in

all the fractured sections. The shape of this crack with a good degree of approximation can be considered

circular (see Figs. 1 and 2). The crack size was practically constant in all the specimens, independently of the

fiber length. The mean crack radius, r , of all the specimens was 0.199 mm, which is about 1/4 of the mean

fiber radius, with a coefficient of variation (ratio of the standard deviation and mean) of 0.18.

The position of the cracks is presented in Table 2, which gives the values of the relative ligament b= R (seeFig. 1). The crack position does not seem to be related to fiber length; neither does its mean value nor its

standard deviation vary significantly with L. It is worth mentioning that in opposition to crack radius, the

relative ligament displays a large coefficient of variation – of the order of 0.51 – and b= R practically rangesover all the interval (0,0.8).

A close look at the fracture surfaces reveals that crack faces exhibit a characteristic morphology in which

starch granules are visible and partially detached from the gluten matrix, thus rendering a rough surface

(Figs. 2 and 7). On the contrary, the region outside the crack presents a smooth surface with morphological

characteristics similar to cleavage.

These observations suggest that internal flaws could have been developed by shrinkage or other anal-

ogous mechanisms during manufacture, when the fiber was wet and in a soft visco-plastic state. Once

Fig. 6. Concentration function of dry semolina fibers.

Table 2

Mean values and standard deviations of relative crack ligaments (b= R)

L (mm) b= R

Mean Standard deviation Coefficient of variation

300 0.347 0.178 0.513

150 0.326 0.155 0.475

75 0.328 0.171 0.521

35 0.320 0.179 0.559

All 0.332 0.169 0.509



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completely dried, flaws would remain inside the fibers and would trigger their brittle fracture when tensile

tested.

3.3. Fracture mechanics

Internal cracks have not received so much attention by researchers despite their interest for fiber fracture

analysis, and only recently has a report been published of stress intensity factors, K I for a planar, circular,

eccentrical crack, oriented perpendicularly to the fiber axis and remotely loaded [10]. For the crack ge-

ometry depicted in Fig. 1, K I reaches the maximum value at the point of the crack front closest to the fiber

surface, which can be expressed as [10]:

K I ¼ 2pr ffiffiffiffiffipr

p f ðr = R; b= RÞ; ð7Þ

r being the remote stress applied to the fiber and f ðr = R; b= RÞ a non-dimensional function of the relativecrack radius, r = R, and crack ligament, b= R, given by [10]:

f ðr = R; b= RÞ ¼ 1 þX5i¼1

C i0ðr = RÞð2iþ1Þ=2 þX3i¼1

Ln½1 þ ðr = RÞ2ifC i1Ln2½ðb= RÞðr = RÞ þ C i2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr = RÞðb= RÞp ; ð8Þ

where C ij coefficients are given in Table 3. The expression above is claimed to be valid for b= R > 0:005 y r = R < 0:6 with 1% accuracy [10].

Fig. 7. Fracture morphology of the crack surface and adjacent zones. The fiber diameter is 1.65 mm.

Table 3

C ij coefficients

i C i0 C i1 C i2

1 +0.01242 –0.3097 +1.185

2 –6.388 +1.547 –3.723

3 +16.89 –0.8769 +2.628

4 –9.838



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According to LEFM, the crack in the fibers will propagate when the maximum K I reaches K IC, the

fracture toughness which is assumed to be a material property. Fig. 8 shows the values of K IC computed

with Eq. (7) for all the fibers tested. A stable value K IC ¼ 0:478 MPa ffiffiffiffi

mp

is drawn from the figure, inde-

pendent of fiber length. The coefficient of variation is under 10%, and the scatter is similar in each group of

fiber length.

4. Conclusions

This paper analyzes the tensile behavior and strength of dry semolina fibers, commonly known as

spaghetti, and their dependence on fiber length.In the conditions investigated, 20 C and 40% RH, semolina fibers are brittle and show a striking linear-

elastic behavior with less than 2% inelastic deformation at rupture. The modulus of elasticity is 5.0 GPa.

Weibulls analysis of rupture loads, when applied to a set of fibers of the same length, works properly

yielding a modulus close to that of ceramics (m ¼ 14). Nevertheless, the model fails to explain the tensilebehavior of fibers of different lengths. A similar result is obtained when the weakest link model with a

general concentration function is considered.

The inapplicability of statistical theories to spaghetti fibers is probably related to the presence of a

planar, circular, internal flaw in all the fractured surfaces. The regular size observed, close to 1/4 of fiber

diameter, does not fulfil the requirement of randomly distributed defects prescribed by weakest link

statistics. The defects show a peculiar rough texture where the starch granules are removed from the

protein matrix. The rest of the broken surface is flat, and some patterns recall cleavage fracture. Themorphology suggests that flaws have been generated at the manufacturing stage, probably during drying,

by shrinkage when the fibers were wet and plastic. This is consistent with the uniform size observed in all

the defects.

The existence of a crack-like defect in combination with the linear-elastic behavior of the material makes

it possible to analyze the breaks by fracture mechanics. The authors, in a previous paper, developed an

expression for K I valid for internal, circular flaws which has been applied to the fibers in this work. The

fracture toughness of dry pasta, measured for the first time, shows a value of 0.478 MPa ffiffiffiffi

mp

.

The results show that LEFM is applicable to dry pasta, and could be a useful tool to model its me-

chanical behavior. This opens the possibility that LEFM parameters such as fracture toughness could be

used in the future to measure the quality of this product.

Fig. 8. K IC values of dry semolina fibers.



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Acknowledgements

The authors gratefully acknowledge financial support for this work by the Spanish Ministry for Science

and Technology under Grant No. MAT2000-1355.

References

[1] Matsuo RR, Dexter JE, Kosmolak FG, Leisle D. Statistical evaluation of tests for assessing spaghetti-making quality of durum

wheat. Cereal Chem 1982;59:222–8.

[2] DEgidio MG, DeStefanis E, Fortini S, Galterio G, Nardi S, Sgrulletta D, et al. Standardization of cooking quality analysis in

macaroni and pasta products. Cereal Foods World 1982;27:367–8.

[3] Donnelly BJ. Pasta: raw materials and processing. In: Lorenz KJ, Kulp K, editors. Handbook of cereal science and technology.

New York: Marcel Dekker; 1991 [chapter 19].

[4] Chawla KK. Fibrous materials. Cambridge: Cambridge University Press; 1998.

[5] Elices M, Llorca J, editors. Fiber fracture. Amsterdam: Elsevier; 2002.

[6] Bazant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. Boca Raton: CRC Press; 1998.[7] Weibull W. The phenomenon of rupture in solids. Ingeniors Vetenskaps Akademien Handlingar 1939;153:55.

[8] Advanced Technical Ceramics, EN843. European Standards: 1996.

[9] Lu C, Danzer R, Fischer FD. Fracture statistics of brittle materials: Weibull or normal distribution. Phys Rev E

2002;067102(65):1–4.

[10] Guinea GV, Rojo FJ, Elices M. Stress intensity factors for internal circular cracks in fibers under tensile loading. Eng Fract Mech

2003;71(3):365–77.


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