the use of full-field measurement methods in composite material

The use of full-field measurement methods in composite material

characterization: interest and limitations

Michel Grediac*

Laboratorie d’Etudes et de Recherches en Mecanique des Structures, Universite Blaise Pascal Clermont II, 24,

Avenue des Landais, BP 206, Aubiere Cedex 63174, France

Abstract

An overview of the use of full-field measurement techniques for composite material and structure characterization reported in the recent

literature is presented in this paper. The features of the main types of measurement techniques are first briefly described. Specific advantages

of using these techniques in a context of composite material characterization are then highlighted. Critical issues that require further research

and development are finally examined.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Identification; B. Mechanical properties; D. Mechanical testing; D. Physical methods of analysis

1. Introduction

During the two last decades, the improvement in image

processing with microcomputers has caused non-contact

measurement techniques to become more and more

popular in the experimental mechanics community. Some

full-field measurement techniques like moire, interfero-

metry or photoelasticimetry were known and used

beforehand. These techniques suffered however from the

non-automatic processing of the fringe patterns they

provided, leading to some heavy, boring and unreliable

by-hand manipulations before obtaining relevant infor-

mation in terms of displacement or strain. In the recent

past, thanks to the dramatic advances in microcomputer

and camera technology, many research groups devoted to

optics, experimental mechanics or data processing have

been developing suitable techniques based on the use of

optical devices, digital cameras, algorithms and softwares

which automatically process images. These techniques

directly provide displacement or strain contours onto

specimens under testing. Temperature fields are also

available thanks to infrared scanning cameras. Such

measurements constitute in fact a new type of tool for

researchers in mechanics of solids, which is especially

interesting in the field of composite material characteriz-

ation. Indeed, composites present some features like

heterogeneities at different scales which render such full-

field measurements very attractive.

It must be emphasized that the above techniques of

strain/displacement components deliver extensometric

informations of deformed surfaces. It is somewhat different

of similar approaches devoted to morphology characteriz-

ation of materials. In this last case, images are captured

directly or through a microscope, digitized and processed to

obtain various geometrical properties of the surface under

investigation. Percentage and orientation of fibers [1],

morphological features of braided composites [2], micro-

crack properties [3] are some examples of use of this type of

image analyses but this point is not addressed in this paper.

The aim here is to examine the interest and the

limitations of full-field measurement techniques in compo-

site materials characterization. The main techniques avail-

able and their main features are presented in the first part of

the paper. Their application in the field of composite

material characterization is addressed in the second part.

The numerical processing of these fields to identify

parameters of constitutive equations is eventually discussed

in the last part of the paper.

2. Full-field measurement techniques

Several types full-field techniques have been proposed

and used in composite material characterization in the

last decade. The nature of the measurements can be

1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesa.2004.01.019

Composites: Part A 35 (2004) 751–761

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* Tel.: þ33-4-73-40-75-29; fax: þ33-4-73-40-74-94.

E-mail address: [email protected] (M. Grediac).

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displacement, strain or temperature. Displacements are

measured with various techniques [4], for instance speckle

[5], speckle interferometry [6], geometric moire [7], moire

interferometry [8], holographic interferometry [9], image

correlation [10] or grid method [11]. Strain can be obtained

by numerical differentiation of the above displacement

fields with suitable algorithms [12] or directly, for instance

with shearography [13,14], speckle shearing photography

[15] or by moire fringes shifting [16]. These techniques can

be classified according to various criteria based on the

nature of the physical phenomenon involved. For instance,

non-interferometric (speckle, grid method) and interfero-

metric (speckle interferometry, moire interferometry)

techniques [17] are the two main categories of techniques

which are discussed below. Note that temperature fields can

also be measured with infrared cameras [18,19,20], but the

features of this technique are not recalled below.

2.1. Displacement field measurement techniques with

non-interferometric methods

2.1.1. Speckle photography

When a rough surface is illuminated with a coherent

light, the rays are scattered in all directions and at all

distances from the surface. The scattered waves distribution

displays a random spatial variation of intensity called

speckle pattern which contrast mainly depends on the

degree of coherence of the incident radiation and the surface

roughness. The scattered rays are usually collected with a

lens and focused on a screen. Speckle patterns are in fact

randomly coded patterns, which carry information about the

surface under investigation. Moreover, if the surface

undergoes changes, the position and the irradiance of the

pattern are changed. Techniques that monitor the positional

changes of the speckles are classified under speckle

photography. The displacement field is obtained by

calculating the so-called cross-correlation function

kI0ðx; yÞ; Iðx; yÞl; where I0ðx; yÞ and Iðx; yÞ are, respectively,

the intensity distribution before and after loading of the

specimen. Such calculations are carried out with suitable

algorithms after digitizing the image captured by a camera.

The main advantage of the speckle method is the ability to

measure large and out-of-plane displacements. Several

books or review articles are devoted to this technique

[21–24] for instance.

2.1.2. Image correlation

Generally, it does not matter how the speckles are

formed in the above method. Hence, incoherent light can

be used. Surfaces under investigation can therefore be

prepared with white painting and sprayed with a black

aerosol. This leads to a random structured aspect, which

can be observed with a digital camera. In this case, the

speckles can be considered as physically attached to the

surface unlike the above laser speckles. Since incoherent

white light is used, neither laser nor specific optical device

are required. Moreover, the preparation of the surface is

very simple and the displacements are easily obtained by

matching different zones of two images captured before

and after loading of the specimen. This image processing is

performed with the same type of algorithms as above. This

method is therefore easy to use and straightforward, as

illustrated by the literature on this subject [10,25–27] for

instance, but its sensitivity is lower than laser speckle

because of larger speckle size. This method is popular

thanks to its simplicity.

2.1.3. Geometric moire and grid method

Analyzing the evolution of a regular network of

equispaced and parallel lines which deform is the base of

the grid and moire methods. In the first case, the network is

physically attached to the surface of the specimen [28]

whereas it is obtained with some geometric methods in the

second one. For instance, in-plane moire fringes [29–31]

are obtained by superposing a grating applied to a surface

and a second one called reference grating. When the

specimen is loaded or moved, a fringe pattern is generated

since the location of the lines relative to the reference

grating changes. These fringes represent the component of

the displacement normal to the reference grating. Out-of-

plane displacements normal to the surface plane are

represented by the fringes of a shadow moire [32]. In this

case, fringes are obtained by superposing a grating placed in

front of the surface and its shadow. Such optical arrange-

ments are used in practice to determine the deflection field

of bent or vibrating plates. Finally, slopes are determined

with reflection moire [33]. In this case, fringes are produced

by superposing a grating placed in front of the surface and

its shadow.

2.2. Displacement field measurement techniques with

interferometric methods

Interferometric methods increase sensitivity of the above

methods by more than one order of magnitude. They are

however susceptible to environmental disturbances like

vibrations because of their high sensitivity. Moire inter-

ferometry and electronic speckle pattern interferometry

(ESPI) are the two main methods which have been

developed for displacement measurement.

2.2.1. Moire interferometry

The superposition of two coherent laser beams of light

provides interference and fringes which can be considered as

regular reference grids which frequency is usually greater

than 1000 lines/mm. Moire interferometry is based on the

same physical phenomenon as the above geometric moire

methods, but it can be observed that the grating frequency is

much greater. This method therefore extends the possibilities

of the moire method to the submicron level [8]. First, a

specimen grating has to be imprinted onto the surface of the

specimen. The grating is obtained by exposing a photographic

M. Grediac / Composites: Part A 35 (2004) 751–761752

plate to a two-beam interference pattern. The resulting fringes

overcoated with a thin layer of aluminum are then transferred

to the specimen with a layer of epoxy. The reference grating is

also obtained via two-beam interference. The superposition

of these two gratings provides moire fringes from which in-

plane displacement are deduced. Out-of-plane displacements

can also be measured [34]. In this case, the reference grating is

placed perpendicular to the surface. The superposition of the

first beam generated by reflection from the specimen surface

and the second one reflected from the reference grating

provides moire fringes since the first beam is distorted by the

out-of-plane deformation of the specimen. Consequently, the

fringes here are contours of the out-of-plane displacement.

2.2.2. Electronic speckle pattern interferometry

ESPI is based on the coherent addition of the scattered

light from the specimen surface and a reference laser beam

[5,35,36]. The phase changes within one speckle because of

the object displacement are coded by the reference beam

with the assumption that the microstructure is modified. The

displacement field is extracted by correlation of two speckle

patterns, one taken before and one taken after object

displacement. The method requires surfaces, which scatter

light. Hence neither grating nor smooth surface are

necessary.

2.3. Strain field measurement techniques with

interferometric methods

Only a few techniques directly provide strain fields

among which the shearing interferometry is the most

popular. The principle of shearing interferometry is to

interfere an optical wavefront with a shifted copy of itself

[37]. The resulting fringes are related to the gradient of the

optical phase from which the gradient of the displacement

are deduced. Strain components or local rotations are

therefore directly obtained from these measurements.

Various devices have been proposed in the literature to

create this shearing effect [14,38–42]. Ligtenberg proposed

a double-exposure method to measure curvatures of bent

plates [43]. These quantities are directly proportional to the

surface strain components thanks to the Love-Kirchhoff

assumption [44]. Various optical set-ups providing curva-

ture fields are described in Ref. [45].

2.4. Choosing a method

All the above methods have been used in various

applications of experimental mechanics. An important

question for the user is the optimal choice of the method

for a given problem. The answer is not easy to find for

different reasons:

† the published work is scattered over several journals,

making it difficult for a reader to have ready access to the

information.

† vocabulary and physical phenomena involved in these

techniques come from other fields than mechanics like

optics or data processing. This causes most of potential

users to feel uneasy when reading the description of the

optical arrangements and choosing a method becomes

therefore troublesome.

† the output from a full-field measurement system is

optical data like speckle or fringes in which the

information is encoded. These data must therefore be

processed by a programme to provide displacement or

strain patterns, giving rise to a black box in which the

different steps are often confuse and cannot always be

clearly distinguished.

† the metrological features of any system should be defined

and characterized but extensive data sheets from manu-

facturers are generally not available. Indeed, the word

‘accuracy’ is too general and some important notions

coming from metrological aspects should be clarified. For

instance, any data sheet should at least quantify the

following properties: sensitivity, uncertainty of the

measurements, repeatability, resolution, spatial resolution

[46]. This task is however somewhat difficult to perform

because most of the arrangements are in-house designed.

Moreover, some extrinsic parameters of the optical

methods themselves influence the results like the

displacement or strain gradients in the field or the

performance of the digital camera used for capturing

the fields. Some studies are devoted to the evaluation of

the accuracy of full-field measurement systems [47,48],

but such a concern is too seldom taken into account in the

literature. Standards in this field should have to be urgently

proposed to define some objective criteria, which would

be useful to calibrate and to compare the methods through

round robin confrontations or benchmarks for instance. It

should be pointed out that the Versailles Project on

Advanced Materials and Standards (VAMAS) which is

directly involved in pre-standards research activities has

launched a Technical Working Area called ‘TWA26 Full

Field Optical Stress and Strain Measurements’ which is

devoted to the development of standards for the use offull-

field stress and strain measurement techniques [49]. Some

documents dealing to the evaluation of non-contact

systems are already available [46–48,50,51]. These

papers show that the first problem is to properly define

the terminology, showing that the community of exper-

imental mechanics and composite materials has still much

effort to spend before correctly using calibrated systems.

† programmes which process the images provide very

attractive colorful displacement or strain patterns which

often fascinate the user, avoiding relevant questions

concerning the above issues.

2.5. Image processing

A common property of most of the above methods is

that they produce a fringe pattern as output. The quantity

M. Grediac / Composites: Part A 35 (2004) 751–761 753

of interest (in-plane displacement, deflection, strain) is

coded at the scale of the fringe period. These fringes must

therefore be analyzed to convert them into a continuous

phase map directly proportional to the quantity of interest.

For a more accurate evaluation than just counting the

fringes, the phase fringe pattern has first to be filtered and

processed. Since this pattern contains discontinuities due

to the wrapping of the phase modulo 2p, it must be

unwrapped and finally correctly scaled. The recent

connection between image digital processors, optical test

arrangements and microcomputers has opened new

possibilities for automatically processing these patterns.

Various algorithms are available and described in

Refs. [52,53] for instance. It must be emphasized that

the situation here is somewhat similar as the situation for

the systems themselves described above. Indeed, the

softwares provided with optical set-ups often appear like

‘black boxes’ for the users who are therefore unable to

correctly estimate their capabilities. For instance, the

influence of noisy data on the final quantity provided by

the softwares remains often unknown.

3. Application to the composite material and structure

characterization

3.1. Introduction

In the last volume of the international journal Exper-

imental Mechanics [54], 19 papers over 61 reported the use

of full-field measurement techniques, showing an important

interest of the community of experimental mechanics for

these methods. Full-field measurements are well suited to

analyze the specific mechanical properties of composite

materials because of their anisotropic and heterogeneous

nature. Papers found in the literature which report the use of

full-field measurement techniques for composite material

and structure characterization can be classified for instance

according to the link between measurements and modeling.

Seven groups were found applying this criterion. The link

between measurements and modeling presently increases

from one category to another:

1. non-destructive testing and inspection;

2. verification of boundary conditions applied to tested

coupons;

3. experimental evidence of local gradients due to material

heterogeneities;

4. cracking characterization;

5. verification of assumptions under which theoretical or

numerical models are built;

6. validation of models;

7. identification of constitutive parameters.

Some recent examples belonging to the above categories

are presented in the following sections.

3.2. Non-destructive testing and inspection

Full-field measurement techniques are used in non-

destructive testing and inspection since a defect induces as a

singularity in a field which can easily be detected [55]. In

composites materials, these defects may be due to the

manufacturing process (delaminations for instance) or to the

degradation of the structure during its live (damage,

impacts, macrocracks, etc.). Waldner [56] detected a

localized damage within the foam of a sandwich structure

due to an impact which did not damage the skin. The

specimen was slightly heated with an infrared lamp and the

deformation during cooling was detected with ESPI and

shearography, showing the location of the impact. Indeed,

the defect causes a strain concentration, which manifests

itself in the form of fringes lying close together. The same

procedure was used to evaluate the influence of removing

paint from a sandwich part with a strong pulsed laser [57]

and to detect a simulated flaw in an epoxy patch [56]. The

influence of the location of the defect on the size of the

disturbance of the field was investigated, showing that

defects closer to the surface induced a larger deformation

than defects located below [58,59]. The resolution of ESPI

allows the detection of damaged zones in terms of small

cracks due to fatigue for instance, as shown in Ref. [56].

Some compact set-ups are now available and make it

possible to use them in an industrial environment [60–63]

for instance.

The infrared thermography is also a very good tool for

damage analysis. The analysis of the thermal changes of the

specimens during a test allows a qualitative monitoring of

the damage evolution. Thermal maps were used for instance

in Ref. [64] to analyze the static notch sensitivity of various

GFRP specimens in terms of damage evolution. In Ref. [65],

typical defects like glue infiltration, water ingress and

disbonds were detected on the blade of a wind turbine made

of sandwich composite structure. The use of thermography

in order to detect defects in sandwich structures is illustrated

in various papers [66–68] for instance. Note finally that the

determination of the fatigue limit of various materials

among which composites can be performed with infrared

thermographic techniques [69–72].

3.3. Verification of boundary conditions applied to tested

coupons

Full-field measurements often reveal parasitic effects that

occur during testing composite materials: misalignments of

the grips, heterogeneous strain fields or local effects near the

loading zones are immediately detected. For instance, it is

well-known that designing a test which leads to a state of

pure shear stresses a difficult task. Concerning sandwich

structures, the shear properties are determined through some

usual tests like bending [73] or shear tests [74]. Some

parasitic effects occur however during these tests like the

indentation under the loading points [75], the bending


of the loading steel plates [76] or the influence of free edges

[77]. In Ref. [78], the grid method and its companion

software ‘Frangyne’ developed by Surrel [79] were used to

quantify these two last effects during an ASTM C273 shear

test [74]. The heterogeneity of the shear strain field was

clearly highlighted as well as the existence of local

transverse tensile stresses near the free edge, which initiate

early failure. A similar study has been carried on the off-axis

test on composites [80]. The heterogeneity of the shear

strain field due to the grips was highlighted and this parasitic

effect was reduced thanks to oblique tabs [81]. In the same

way, the correlation method was used to assess the

heterogeneity of the strain field in fabrics subjected to a

shear test [82].

3.4. Experimental evidence of local gradients due to

material heterogeneities

Composite materials are heterogeneous at different

scales. This leads to some local variations of strain

components around their average values. Only these

quantities are measurable with usual transducers like strain

gauges because of their too high spatial resolution. Evidence

of local variations of the strain field in a composite is

however important because they may initiate cracks for

instance. These variations have been observed in different

studies, for instance by Post and co-workers with moire

interferometry. In Ref. [83], the through-thickness longi-

tudinal displacement field in various bent specimens was

measured with moire interferometry. Striations and zigzag

fringes were clearly highlighted through the thickness of

unidirectional composites under five-point bending. This is

the consequence of high-localized shear strains in resin-rich

zones between adjacent plies. Shear strain through the

thickness of quasi-isotropic stacking sequences under three-

point bending were deduced from the longitudinal displace-

ment field, showing free-edge shear strains in region of

normal stresses, especially near the top and bottom of the

specimen. The shear strain variation due to the cyclic

variation of shear compliance of individual plies was also

clearly pointed out. The same technique was used to

measure the strain field onto thick composite coupons under

compression [8,84]. Very important variations were

observed. Note that curved surfaces can also be investigated

with the same technique, as shown in Refs. [85–87] where

the ply-by-ply deformation of composite plates at the

cylindrical surface of a hole in a specimen under tension/

compression is examined. Very local phenomena can also

be investigated with full-field measurement techniques. For

instance, the failure mechanisms in laminates with dropped

plies was studied in Ref. [88] with moire interferometry.

The strain gradients near a ply drop of an enlarged specimen

were measured. More recently, the tension/bending coup-

ling effect in the warp and fill yarns and the shear strain

concentration in resin-rich regions were observed thanks to

the grating shearography method [14,89]. Significant local

variations of the strains were detected. Similar conclusions

were obtained with ESPI [90].

3.5. Cracking characterization

Full-field measurements are very useful to detect cracks.

These cracks appear indeed as singularities in a displace-

ment field. The magnitude of the displacement jump along a

direction perpendicular to a crack provides its width.

Thanks to the resolution of most of the techniques described

in Section 2, crack widths of some micrometers only can be

measured. Using full-field measurements is especially

effective in the case of brittle materials where multicracking

appears since the location of the cracks is not known a

priori. For instance, the grid method [91] and image

intercorrelation [92] were used to measure the width and

the profile of cracks in a concrete structure reinforced with

composite materials. These results were used to build up

some relevant models describing the mechanical response

of such structures. Cement-based composites with steel

fibers were also investigated in Ref. [93]. In this study,

multicracking of specimens under three and four-point

bending was studied in terms of relationship between

loading and number of cracks, crack width and crack profile.

3.6. Verification of assumptions under which theoretical or

numerical models are built

Numerical models used for computing stresses in

composite materials are often built under some assumptions

concerning the displacement and strain fields. For instance,

the calculation of the transverse shear in laminated

structures is usually carried out under some assumptions

concerning the through-thickness displacement field. The

classical laminated theory assumes that straight lines

perpendicular to the mid-plane remain straight after loading

[94], Thimoshenko or Mindlin theories that these straight

lines remain straight but not perpendicular to the mid-plane,

more refined theories that a warping appears, leading to a

non-uniform through-thickness shear strain and stress

components. Reddy proposed a cubic horizontal displace-

ment distribution [95,96] whereas Touratier suggested a

sine repartition [97,98] to model this warping. This warping

is locked at the center of bent beams in any case [99]. The

relevance of these different assumptions was verified and

discussed by measuring the displacement field through the

thickness of bent composite beams, either with the grid

method [100] or with ESPI [101], highlighting the influence

of the span-to-depth ratio on their validity.

3.7. Validation of models

Full-field measurement techniques are efficient tools to

validate theoretical models when heterogeneities are

involved. For instance, a theoretical model describing

the post-buckling response of composite I-sections was


developed in Ref. [102]. This model was then verified by

measuring the out-of-plane deflection of web and flange

due to buckling during compression tests on columns

[103]. The shadow moire method was used in this study

to measure the deflection field due to buckling. In the

same way, the influence of joining techniques on the

response of assembled composite beams was investigated

with moire interferometry and compared with some

theoretical models in Ref. [104]. A meso-model based

on the anisotropic damage theory was proposed in Ref.

[105] to evaluate the degradation state in a laminate.

This model was then validated through full-field

measurements performed on bi-axial specimens with

intercorrelation.

3.8. Identification of constitutive parameters

Procedures which allow the identification of constitutive

parameters from heterogeneous strain fields are very

promising. They allow indeed the determination of several

constitutive parameters from a reduced number of tests.

Two main difficulties arise however. First, an amazing

amount of measurements must be processed since the input

data is a field, that is up to some hundreds of thousands of

measurements and the question is to manage this unusual

quantity of information. Second, no direct relationship is

available between measurements and unknown parameters.

Specific procedures have therefore been proposed in the

recent past. The main features of two of them are described

below.

3.9. Conclusion

Various applications of full-field measurement tech-

niques for composite material and structure characterization

have been described in this section. The link between

measurements and modeling increases from one case to

another as illustrated in Fig. 1, showing the interest of this

type of measurements for a better understanding of the

mechanical response of composites.

4. Identification

4.1. Introduction

The determination of parameters of constitutive

equations of materials is usually obtained thanks to

mechanical tests for which the loading conditions and

the specimen geometries are such that homogeneous

stress/strain fields take place [106]. In this case, a simple

relationship between applied loading and local strain

measurements directly provides the unknown parameters.

It is however well-known that such pure states of stress/strain

Fig. 1. Link between measurements and modeling.


are often difficult to obtain in composite materials because

of their anisotropy and their heterogeneity at different

scales. Moreover, the number of unknown parameters which

characterize these anisotropic materials—even in linear

elasticity—is much more important than in the case of

isotropy, leading therefore to a more important number of

mechanical tests to measure them. Finally, the manufactur-

ing conditions influence the final mechanical properties of

composite materials, but taking out test samples from large

industrial structures may often be unacceptable. For all

these reasons, an increasing interest is found in the literature

in methods allowing the identification of parameters of

constitutive equations from testing configurations leading to

heterogeneous stress/strain fields, either for composites or

other materials. From a practical point if view, the main

breakthrough is the relaxation in the demands in terms of

specimen geometry and loading conditions, leading to much

more freedom in the nature and in the location of the applied

loading as well as in the shape of the specimens to be tested.

For instance, such strategies allow the design of exper-

iments in which the conditions are much closer to those of

practical or industrial situations. From a theoretical point of

view, a much more important number of parameters

influence the heterogeneous stress/strain field, leading to

their possible identification if suitable procedures are

available. This leads to a reduced number of tests and

specimens to get the maximum number of constitutive

parameters. Finally, parameters can be determined in testing

configurations in which complex loading paths are intro-

duced. For all these reasons, this type approach is very

attractive but the main problem here is the lack of analytical

solutions for the stress/strain/displacement fields, leading to

the lack of relationship between measurements and

unknown parameters.

In the recent past, several strategies have been proposed

to extract constitutive parameters from heterogeneous strain

fields, either for composites or for other materials. Such a

problem is often referred as inverse problem in the

literature. The corresponding direct problem is the deter-

mination of the stress/strain/displacement fields when the

geometry, loading distribution f and constitutive equations

are known (see Table 1). In the inverse problem, the

parameter identification is carried out within the framework

of constitutive equations which are a priori assumed to be

relevant. The displacement or strain fields are known as well

as the geometry of the specimen geometry and the resulting

force F applied to the specimen. The aim here is to describe

the headlines, the advantages and the limitations of two

strategies found in the literature to solve this inverse

problem, namely updating finite element models and the

virtual fields method.

4.2. Updating finite element models

The finite element method provides displacement/-

strain/stress fields in almost any case of loading conditions,

specimen geometry and constitutive equations. It is there-

fore the most popular numerical solution of the direct

problem. It has been also used as a tool for solving

iteratively the inverse problem, leading often to so-called

mixed experimental/numerical methods. In this case, a first

calculation is performed with an initial set of constitutive

parameters. Nodal displacements are collected and com-

pared to their experimental counterparts. The difference is

quantified with an objective function, which is often the sum

of the squared differences between numerical and exper-

imental data. The idea is then to minimize iteratively this

estimator with respect to the constitutive parameters. Many

optimization algorithms are available for minimizing this

objective function. They are based on the numerical

calculation of a sensitivity matrix, which allows the step-

by-step determination of new sets of constitutive par-

ameters. The procedure stops when the objective function is

lower than a given threshold value. Such procedures have

been successfully simulated for instance by Hendricks [107]

and used for the determination of anisotropic bending

rigidities of wooden plates [108]. More recently, promising

results have been obtained by Meuwissen [109,110] for the

determination of the plastic response of some metals

(displacement fields were measured in this case with the

image correlation technique). Allix and Vidal [111] reduced

the number of iterations necessary to reach convergence and

improved the quality of each iteration using the so-called

LATIN method (Large Time INcrement method) [112,113].

It is clear that this type of approach can also be used for

anisotropic materials. In the same way, LeMagorou et al.

[114] optimized the loading conditions in terms of location

of the applied forces to determine the constitutive

parameters of the elastic/viscoelastic response of wooden

plates under bending.

The two main drawbacks of this type of approach are (i)

the fact that the procedure is a ‘numerical black box’ built up

with the suitable solution of the direct problem and an

objective function in which the physical meaning is not

sufficiently present [115], (ii) often the computing time since

iterative calculations are required, (iii) the fact that boundary

conditions in terms of tractions must often be known to

perform the calculations. Indeed, only the resulting force F

applied to tested specimens are measurable in practice

whereas the loading repartition f is required in the finite

element model apart in the cases of concentrated forces.

Table 1

Direct and inverse problems

Direct problem Inverse problem

Known Geometry Geometry

Loading distribution f Resulting force F

Type of constitutive equations Type of constitutive equations

Constitutive parameters u or e

Unknown u; e ; s Constitutive parameters


4.3. The virtual fields method

This approach has been proposed in Ref. [116]. It clearly

departs from the above ones since it is direct and does not

require any iterative calculation in many cases. The basic

idea is rather simple since it consists in writing the global

equilibrium of the tested specimen with the principle of

virtual work. Introducing the constitutive equations and

particular virtual fields leads to an equation where the

constitutive parameters only are unknown, provided that the

strain fields are measured, either directly or indirectly, by

numerical differentiation of measured displacement fields

for instance. The idea is then to write the principle of virtual

work with different and independent virtual fields. This

leads to a system of equations in which the constitutive

parameters are unknown. If at least as many virtual fields as

unknowns are chosen and if the constitutive parameters

write as polynomials of the strain, like in the cases of

elasticity and damage, the system becomes linear and the

unknowns are directly determined after inversion of the

system. This approach has been simulated and applied in

various cases of composite materials characterization in the

recent past [116–131] for instance. The key point here is the

choice of the virtual fields, since an infinity of virtual fields

verifying the principle of virtual work is available. A

method has been recently proposed to optimize the choice of

the virtual fields with respect to the sensitivity to noisy data

[132–135]. This dramatic improvement renders the virtual

fields method much more easy to implement and reliable.

The main advantages of this method are (i) its directness in

many cases and (ii) the fact that the influence of the above

problem posed by the boundary conditions can be avoided

by choosing virtual fields in which only the resulting forces

F virtually work. The main limitation is the reduced number

of results obtained in the non-linear case, but this point is

presently under progress [136] for instance.

5. Conclusion

Thanks to recent advances in the sensing technology and

image processing, various techniques are now available to

measure displacement, strain or temperature fields during

mechanical tests on composite materials and structures. The

major breakthrough here is the possibility to detect

heterogeneities which can be due to different phenomena:

defects or cracks within material, mechanical set-ups

causing parasitic effects, strain gradients due to distributed

mechanical properties for instance. Full-field measurement

techniques are very attractive and it is therefore expected

that they will increase their presence and acceptance in

mechanical characterization of composites in the near

future. Different technologies have emerged and found a

niche where they can offer specific performance but it

should be noted that quite a few have evolved into

commercial products till now. More research is still required

to calibrate the different techniques or image processing

softwares, to propose reliable black-box units requiring little

experience to operate and to develop robust procedures that

allow the determination of constitutive parameters from

these fields which constitute in fact a new type of data.

References

[1] Zak G, Park CB, Benhabib B. Estimation of the three-dimensional

fibre orientation distribution in short fibre composites by a two

section method. J Compos Mater 2001;35(4):316–39. Technomic

Publishing Co., Inc.

[2] Lian B, Jiang L, McGrath JJ, Jaranson J. Quantitative determi-

nation of morphological features of triaxially braided composites

by the use of machine vision. Compos Sci Technol 2000;60:

159–66. Elsevier.

[3] Ammouche A, Riss J, Breysse D, Marchand J. Image analysis for the

automated study of microcracks in concrete. Cement Concrete

Compos 2001;23:267–78.

[4] Kobayashi A, Handbook on experimental mechanics, Weinheim:

Society for Experimental Mechanics, VCH publishers, Inc.; 1999.

[5] Dainty JC, Laser speckle and related phenomenon, Berlin: Springer;

1984.

[6] Leendertz JA. Interferometric displacement measurement on

scattering surfaces utilizing speckle effect. J Phys E 1970;3:214–8.

[7] Theocaris PS, Moire fringes in strain analysis, Elmsford: Pergamon

press; 1969.

[8] Post94, High sensitivity moire: experimental analysis for mechanics

and materials, Berlin: Springer; 1994.

[9] Kreis T, Holographic interferometry: principles and methods, Berlin:

Wiley-VCH; 1996.

[10] Sutton MA, Wolters WJ, Perters WH, Ranson WF, McNeill SR.

Determination of displacements using an improved digital corre-

lation method. Image Vis Comput 1983;1(3):133–9.

[11] Sirkis JS. System response to automated grid method. Opt Engng

1990;29(12):1485–93.

[12] Surrel Y. Moire and grid methods: a signal-processing approach. In:

Pryputniewicz RJ, Stupnicki J, editors. Interferometry ‘94: Photo-

mechanics. Berlin: Springer; 1994. p. 213–20.

[13] Templeton DW. Computerization of carrier fringe data acquisition,

reduction and display. Exp Tech 1987;11(11):26–30. Society for

Experimental Mechanics.

[14] Bulhak J, Surrel Y. Grating shearography. In: Interferometry ‘99:

Techniques and Technologies, 20–23 September, Pultusk, Poland,

vol. SPIE 3744.; 1999. p. 506–15.

[15] Rastogi PK. Speckle shearing photography: a tool for direct

measurement of surface strains. Appl Opt 1998;37:1292–8. Optical

Society of America.

[16] Wasowski JJ, Wasowski LM. Computer-based optical differentiation

of fringe patterns. Exp Tech 1987;11(3):16–18. Society for

Experimental Mechanics.

[17] Surrel Y. Optique C1. www.cnam.fr/instrumesure, autres ressources

telechargeables, Optique C1.

[18] Bakis CE, Reifsnider KL. Nondestructive evaluation of fiber

composite laminates by thermoelastic emission. In: Thompson

DO, Chimenti DE, editors. Review of progress in quantitative NDE

plenum. 1987. p. 1109–16.

[19] Heller M, Dunn S, Williams JF, Jones R. Thermoanalysis of

composite specimens. Compos Struct 1988;11:309–24.

[20] Rauch BJ, Rowlands RE. Thermoelastic stress analysis. In:

Kobayashi A, editor. Handbook on experimental mechanics.

Weinheim: Society for Experimental Mechanics, VCH publishers,

Inc.; 1993. p. 581–99.


http://www.cnam.fr/instrumesure

[21] Joenathan C. Speckle photography, shearography and ESPI. In:

Rastogi PK, editor. Optical measurement techniques and appli-

cations, vol. 6. Boston: Artech House, Inc.; 1997. p. 151–82.

Chapter 6.

[22] Sihori RS, Speckle metrology, New York: Marcel Dekker; 1993.

[23] Sihori RS, Selected papers on speckle metrology, Washington, DC:

SPIE Optical Engineering Press; 1991.

[24] Gasvik KH, Optical metrology, New York: Wiley; 1987.

[25] Chiang FP, Asundi A. White light speckle method of experimental

strain analysis. Appl Opt 1979;18:409–11. Optical Society of

America.

[26] Chu TC, Ranson WF, Sutton MA, Petters WH. Application of digital

image correlation techniques to experimental mechanics. Exp Mech

1985;3:232–44. Society for Experimental Mechanics.

[27] Sutton MA, McNeill SR, Helm J, Chao YJ. Advances in two-

dimensional and three-dimensional computer vision. In: Rastogi PK,

editor. Photomechanics, topics applied physic 77. Berlin: Springer;

2000. p. 323–72.

[28] Parks VJ. Strain measurement using grids. Opt Engng 1982;21(4):

633–9.

[29] Dantu M. Recherches diverses d’extensometrie et de determination

des contraintes. Conference GAMAC; 1954.

[30] Burch JM, Forno C. High resolution moire photography. Opt Engng

1982;21(4):604–14.

[31] Boone PM, Vinckier AG, Denys RM, Sys WM, Deleu EN.

Application of specimen-grid moire techniques in large scale steel

testing. Opt Engng 1982;21(4):615–25.

[32] Pirodda L. Shadow and projection moire techniques for absolute or

relative mapping of surface shapes. Opt Engng 1982;21(4):640–9.

[33] Ritter R. Reflection moire methods for plate bending studies. Opt

Engng 1982;21(4):663–71.

[34] Asundi A, Cheung MT, Lee CS. Moire interferometry for

simultaneous measurement of u; v;w. Exp Mech 1989;29:258–60.

Society for Experimental Mechanics.

[35] Jones R, Wykes C, Holographic and speckle interferometry,

Cambridge, MA: Cambridge University Press; 2000.

[36] Robinson DW, Reid GT, Interferomgram analysis: digital fringe

pattern measurement techniques, Philadelphia, PA: IOP Publications

Ltd; 2001.

[37] Krishnaswamy S. Techniques for non-birefringent objects: coherent

shearing interferometry and caustics. In: Rastogi PK, editor.

Photomechanics, topics applied physic 77. Berlin: Springer; 2000.

p. 295–321.

[38] Ronchi V. Forty years of history of a grating interferometer. Appl

Opt 1964;3:437. Optical Society of America.

[39] Wyant JC. Double frequency grating lateral shear interferometer.

Appl Opt 1973;12:2057. Optical Society of America.

[40] Patorski K. Talbot interferometry with increased shear. Appl Opt

1985;24:4448. Optical Society of America.

[41] Tippur HV, Krishnaswamy S, Rosakis AJ. Optical mapping of crack

tip deformations using the methods of transmission and reflection

coherent gradient sensing: a study of crack tip K-dominance. Int J

Fract 1991;52:91. Kluwer Academic Publisher.

[42] Lee HS, Krishnaswamy S. A compact polariscope and shearing

interferometer for mapping stress fields in bimaterial systems. Exp

Mech 1996;36:404. Society for Experimental Mechanics.

[43] Ligtenberg FK. The moire method. In: Proceedings of SESA, vol.

12.; 1954. p. 83–98.

[44] Timoshenko S, Woinowsky-Krieger S, Theory of plates and shells,

New York: McGraw-Hill; 1958.

[45] Kao TY, Chiang FP. Family of grating techniques of slope and

curvature measurements for static and dynamic flexure of plates. Opt

Engng 1982;21(4):720–42.

[46] Surrel Y. Some metrological issues in optical full field techniques.

In: Proceedings of Interferometry XI: Techniques and Analysis.

CDROM, vol. SPIE 4777; 2002.

[47] Picard P. Evaluation of phase shifting speckle interferometry

accuracy. In: Jacquot P, Fournier JM, editors. Interferometry in

speckle light. Berlin: Springer; 2000. p. 431–8.

[48] Novak J. Error analysis of three-frame algorithms for evaluation of

deformations. In: Jacquot P, Fournier JM, editors. Interferometry in

speckle light. Berlin: Springer; 2000. p. 439–44.

[49] www.shef.ac.uk/mecheng/mechsol/exptmech/twa26.

[50] Sutton MA. Standard guide for evaluating non-contacting optical

strain measurement systems. Revised draft available at wwwshefa-

cuk/mecheng/mechsol/exptmech/twa26; 2001.

[51] Schreier HW, Sutton MA. Systematic errors in digital image

correlation due to undermatched subset shape functions. Exp Mech

2002;42(3):303–10. Society for Experimental Mechanics.

[52] Moritomo Y. Digital image processing. In: Kobayashi A, editor.

Handbook on experimental mechanics. Weinheim: Society for

Experimental Mechanics, VCH publishers, Inc.; 1993. p. 969–1029.

[53] Kreis T. Computer-aided evaluation of holographic interferograms.

In: Rastogi PK, editor. Holographic interferometry. Berlin: Spinger;

1994.

[54] Experimental Mechanics, vol. 41. Society for Experimental

Mechanics; 2001.

[55] Waldner S, Kress G. Nondestructive testing of composite materials

with electronic speckle pattern interferometry. In: Rastogi P, Inaudi

D, editors. Trends in optical non-destructive testing and inspection.

Amsterdam: Elsevier; 2000. p. 257–71.

[56] Waldner S. Removing the image-doubling in shearography theory

and application. In: Proceedings of SPIE 944; 1996. p. 247–55.

[57] Hertwig M. Macroscopic and microscopic surface topographies

evaluated by improved speckle contouring methods. In Proceedings

of SPIE 3098; 1997. p. 90–105.

[58] Anastasi RF, Serabian SM, Shuford RJ, Das-Gupta DK. Nondes-

tructive detection of simulated delamination in composite laminates

by laser-speckle shearography. Exp Tech 1987;11(6):28–31. Society

for Experimental Mechanics.

[59] Shang HM, Toh SL, Chau FS, Shim VPM, Tay CJ. Location and

sizing disbonds in glassfiber-reinforced plastic plates using shearo-

graphy. J Engng Mater Technol 1991;11(113):99–103.

[60] Newman JW. Production and field inspection of composite aero-

space structures with advanced shearography. In: Proceedings of the

22nd International SAMPE Technical Conference on Advanced

Materials; 1995. p. 1243–9.

[61] Ettermeyer A, Honlet M. Nondestructive testing with TV-hologra-

phy and shearography. In: VDI Berichte Nr 1118, IMECO

Symposium: Laser Metrology for Precision Measurement and

Inspection in Industry; 1994. p. 33–8.

[62] Ettermeyer A, Wang Z, Walz T. Application of 3D speckle

interferometry to material and component testing. In: Proceedings

of SPIE 1997; 1997. p. 188–94.

[63] Walz T, Ettermeyer A. Automatic shearography system for

helicopter rotor blades. In: Proceedings of SPIE 3397; 1998. p.

187–92.

[64] Quaresimin M, Guglielmino E. Static notch sensitivity of GFRP

composites. In: Proceedings of Experimental Techniques and Design

in Composite Materials 5, September 2000, Cagliari; 2000.

[65] Dattoma V, Marcuccio R, Pappalettere C, Smith GM. Thermo-

graphic investigation of sandwich structure made of composite

materials. NDTE Int 2001;34:515–20. Elsevier.

[66] Favro LD, Xiaoyan H, Thomas RL. Quantitative thermal wave

measurements of defects in composite aircrafts structures. In:

Proceedings of the 44th International SAMPE Symposium Exhibi-

tion, Long Beach, California, USA; 1999.

[67] Vickstrom M, Backlund J, Olsson KA. Non-destructive testing of

sandwich construction using thermography. Compos Struct 1989;13:

49–65. Elsevier.

[68] Vickstrom M. Thermography: a fast and reliable method for

detecting flaws in foam-cores sandwich structures. Infrared Observer

1989;5569:15–28.


http://www.shef.ac.uk/mecheng/mechsol/exptmech/twa26

http://wwwshefacuk/mecheng/mechsol/exptmech/twa26

http://wwwshefacuk/mecheng/mechsol/exptmech/twa26

[69] Luong MP. Fatigue limit evaluation of metals using an infrared

thermographic technique. Mech Mater 1998;28:155–63. Elsevier.

[70] Gudlielmino E, Gudlielmino ID, La Rosa G, Pasta A. Analysis

of laminate composite under cyclic testing using thermal

infrared imagery. In: Proceedings of International Conference on

Material Engineering, 1996, Galipoli-Leece, Italy; 1996. p. 899–

906.

[71] La Rosa G, Risitano A. Analysis of a multiaxial test on a C/C

composite by using digital image correlation and a damage model.

Int J Fatigue 2000;22:65–73. Elsevier.

[72] Quaresmin M. Fatigue of woven composite laminates under tensile

and compressive loading. In: Proceedings of the 10th European

Conference on Composite Materials (CDROM), ECCM10, Brugge,

Belgium; 1999.

[73] ASTM C373, Method of flexure tests of flat sandwich structures,

Philadelphia, PA: American Society for Testing and Materials; 1999.

[74] ASTM C273, Shear test in flatwise plane of flat sandwich

construction or sandwich cores, Philadelphia, PA: American Society

for Testing and Materials; 2000.

[75] Shuaeib FM, Soden PD. Indentation failure of composite sandwich

beams. Compos Sci Technol 1997;57(9):1249–59. Elsevier.

[76] O’Connor DJ. A comparison of test methods for shear properties of

the cores of sandwich construction. J Test Eval 1989;17(4):241–6.

American Society for Testing and Materials.

[77] O’Connor DJ. An evaluation of test methods for shear modulus of

sandwich cores. The Int J Cement Compos Lightweight Concrete

1984;6(1):3–12.

[78] Grediac M, Dufort L. Experimental evidence of parasitic effects in

the rail shear test on sandwich beams. Exp Mech 2002;42(2):

186–93. Society for Experimental Mechanics.

[79] Surrel Y. Fringe analysis. In: Rastogi PK, editor. Photomechanics,

topics applied physic 77, Berlin: Springer; 2000. p. 55–99.

[80] Pierron F, Alloba E, Surrel Y, Vautrin A. Whole-field assessment of

the effects of boundary conditions on the strain field in off-axis

tensile testing of unidirectional composites. Compos Sci Technol

1998;58(12):1939–47.

[81] Sun CT, Chung I. An oblique end-tab design for testing off-axis

composite specimens. Composites 1993;24:619–23. Elsevier.

[82] Launay J, Lahmar F, Boisse P, Vacher P. Strain measurement in tests

on fibre fabric by image correlation method. Adv Compos Lett 2002;

11(1):7–12. Adcotec Ltd.

[83] Post D, Czarnek R, Joh D, Wood JD. Deformation measurements of

composite multi-span beam shear specimens by moire interferome-

try. Technical Report, NASA Contractor Report; 1984.

[84] Guo Y, Post D, Han B. Thick composites in compression: an

experimental study of micromechanical behavior and smeared

engineering properties. J Compos Mater 1992;26:1930–92. Tech-

nomic Publishing Company, Inc.

[85] Mollenhauer DH. Interlaminar deformation at a hole in laminated

composites: a detailed experimental investigation using moire

interferometry. PhD Thesis, Virginia Polytechnic Institute and

State University; 1997.

[86] Mollenhauer D, Reifsnider K. Measurements of interlaminar

deformation along the cylindrical surface of a hole in laminated

composite materials by moire interferometry. Compos Sci Technol

2000;60:2375–88. Elsevier.

[87] Boeman RG. Interlaminar deformation on the cylindrical surface of a

hole in laminated composites: an experimental study. Report 91-07.

Technical Report, Virginia Polytechnic Institute and State Univer-

sity; 1991.

[88] Xing YM, Yun H, Dai FL. An experimental study of failure

mechanisms in laminates with dropped plies. Compos Sci Technol

1999;59:1527–31. Elsevier.

[89] Vautrin A, Lee JR, Molimard J, Surrel Y. Full-field optical

techniques: applications to strain measurement and mechanical

identification. In: Proceedings of the 10th European Conference on

Composite Materials (CDROM), ECCM10, Brugge, Belgium;

2002.

[90] Nishiyabu K, Uetsuji Y, Masaru Z, Prodromou A, Huysmans G,

Verpoest I. Strain fields on plain woven fabric composites under an

axis tensile load: comparison of experimental and numerical results.

In: Proceedings of ECCM9, Brighton (CDROM); 2000.

[91] Avril S, Vautrin A, Ferrier E, Hamelin P, Surrel Y. Experimental

analysis of reinforced-concrete beams repaired with CFRP sheets.

Compos Part A.

[92] Muller M, Toussaint E, Grediac M. Bridging cracks with composite

materials in concrete structures: experimental study and modeling.

Compos Part A.

[93] Destrebecq JF, Toussaint E, Grediac M. A refined study of the crack

propagation in cement-based composite beams under bending.

Cement Concrete Compos. Submitted for publication; 2003.

[94] Tsai SW, Hahn HT. Introduction Compos Mater 1980; Technomic.

[95] Reddy JN. A simple higher-order theory for laminated composite

shells. J Appl Mech 1984;51(3):745–52. Transaction of the

American Society of Mechanical Engineers.

[96] Reddy JN, Liu CF. A higher-theory shear deformation theory of

laminated elastic shells. Int J Solids Struct 1985;23(3):319–30.

Elsevier.

[97] Touratier M. An efficient standard plate theory. Int J Engng Sci 1991;

29(8):901–16. Elsevier.

[98] Touratier M. A refined theory of laminated shallow shells. Int J

Solids Struct 1992;29(11):1401–15. Elsevier.

[99] Dufort L, Drapier S, Grediac M. Closed-form solution for the cross-

section warping in short beams under three point bending. Compos

Struct 2001;52:233–46. Elsevier.

[100] Dufort L, Grediac M, Surrel Y. Experimental evidence of the cross-

section warping in short composite beams under three point bending.

Compos Struct 2001;51:37–47. Elsevier.

[101] Bosia F, Facchini M, Gmur T, Botsis J. Experimental analysis of

composite laminates subjected to bending. In: Proceedings of the

International Conference Interferometry in Speckle Light, 25–28

September 2000, Lausanne, Switzerland; 2000. p. 355–62.

[102] Papila M, Akun MA, Niu X, Ifju P. Post-buckling of composite I-

sections. Part 1. Theory. J Compos Mater 1999;35(9):774–96.

Technomic Publishing Company, Inc.

[103] Papila M, Akun MA, Niu X, Ifju P. Post-buckling of composite I-

sections. Part 2. Experimental validation. J Compos Mater 1999;

35(9):797–821. Technomic Publishing Company, Inc.

[104] Liu D, Shakour E. Joining forces in assembled composite beams.

J Compos Mater 1999;35(22):1985–2008. Technomic Publishing

Company, Inc.

[105] Perie JN, Calloch S, Cluzel C, Hild F. Analysis of a multiaxial test on

a C/C composite by using digital image correlation and a damage

model. Exp Mech 2002;42(3):318–28. Society for Experimental

Mechanics.

[106] Hodgkinson JM, Mechanical testing of advanced fibre composites,

Woodhead Publishing Limited; 2000.

[107] Hendricks MAN. Identification of the mechanical properties of solid

materials. Doctoral Dissertation, Eindhoven University of Technology;

1991.

[108] Foudjet A. Methodes de caracterisation de materiaux anisotropes.

Application au bois. Doctoral Dissertation, INSA Lyon; 1980.

[109] Meuwissen MHH. An inverse method for the mechanical character-

ization of metals. Doctoral Dissertation, Eindhoven University of

Technology; 1998.

[110] Meuwissen MHH, Oomens CWJ, Baaijens FPT, Petterson R,

Janssen JD. Determination of the elasto-plastic properties of

aluminium using a mixed numerical–experimental method. J Mater

Process Technol 1998;75:204–11.

[111] Allix O, Vidal P. A new multi-solution approach suitable for

structural identification problems. Comput Meth Appl Mech Engng

2002;191:2727–58. Elsevier.


[112] Boucard PA, Ladeveze P, Poss M, Rougee P. A non-incremental

approach for large displacement problems. Comput Struct 1997;

64(1–4):499–508. Elsevier.

[113] Champaney L, Cognard JY, Dureissex D, Ladeveze P. Large scale

applications on parallel computers of a mixed domain decomposition

method. Comput Mech 1997;19(4):253–63. Springer.

[114] LeMagorou L, Bos F, Rouger F. Identification of constitutive laws

for wood-based panels by means of an inverse method. Compos Sci

Technol 2002;62(4):591–6. Elsevier.

[115] Bonnet M, Bui HD, Constantinescu A. Principes variationnels et

exploitation de mesures de champs en elasticite/variational prin-

ciples and exploitation of field measurements in elasticity.

Mecanique et Industries 2003;4:687–97.

[116] Grediac M. Principe des travaux virtuels et identification/principle of

virtual work and identification. Comptes Rendus de l’Academie des

Sciences 1989;II(309):1–5. Gauthier-Villars, in French with

Abridged English Version.

[117] Grediac M. Mesure des rigidites de flexion de stratifies minces

anisotropes a l’aide d’essais sur plaques. Doctoral Dissertation,

Ecole Nationale Superieure des Mines de Saint-Etienne/Universite

Claude Bernard—Lyon I; 1989.

[118] Grediac M, Vautrin A. A new method for determination of bending

rigidities of thin anisotropic plates. J Appl Mech 1990;57:964–8.

Transactions of the American Society of Mechanical Engineers.

[119] Gauthier-Villars M, Vautrin A. Mechanical characterization of

anisotropic plates: experiments and results. Eur J Mech/A Solids

1993;12(6):819–38. Gauthier-Villars.

[120] Grediac M. Dossier de travaux d’habilitation a diriger des

recherches. Ecole Nationale Superieure des Mines de Saint-

Etienne/Universite Jean Monnet, Saint-Etienne; 1993.

[121] Grediac M, Pierron F, Vautrin A. The Iosipescu in-plane shear test: a

new approach based on displacement field processing. Compos Sci

Technol 1995;51:409–17. Elsevier.

[122] Grediac M. On the direct determination of invariant parameters

governing the bending of anisotropic plates. Int J Solids Struct 1996;

33(27):3969–82. Elsevier.

[123] Grediac M. The use of heterogeneous strain fields for the

characterization of composite materials. Compos Sci Technol

1996;56:841–6. Elsevier.

[124] Grediac M, Paris P-A. Direct identification of elastic constants of

anisotropic plates by modal analysis: theoretical and numerical

aspects. J Sound Vib 1996;195(3):401–15. Academic Press.

[125] Grediac M, Pierron F. A T-shaped specimen for the direct

characterization of orthotropic materials. Int J Numer Meth Engng

1998;41:293–309. Wiley.

[126] Grediac M, Fournier N, Paris P-A, Surrel Y. Direct identification of

elastic constants of anisotropic plates by modal analysis: experiments

and results. J Sound Vib 1998;210(5):645–59. Academic Press.

[127] Grediac M, Pierron F, Surrel Y. Novel procedure for complete in-

plane composite characterization using a T-shaped specimen. Exp

Mech 1999;39(2):142–9. Society for Experimental Mechanics.

[128] Grediac M, Fournier N, Paris P-A, Surrel Y. Direct measurement of

invariant parameters of composite plates. J Compos Mater 1999;

33(20):1939–65. Technomic.

[129] Pierron F, Grediac M. Identification of the through-thickness moduli

of thick composites from whole-field measurements using the

Iosipescu fixture: theory and simulations. Compos: Part A 2000;

31(4):309–18. Elsevier.

[130] Pierron F, Zhavaronok S, Grediac M. Identification of the through-

thickness properties of thick laminates using the virtual fields

method. Int J Solids Struct 2000;37(32):4437–53. Elsevier.

[131] Grediac M, Auslender F, Pierron F. Applying the virtual fields

method to determine the through-thickness moduli of thick

composites with a nonlinear shear response. Compos: Part A 2001;

32(12):1713–25. Elsevier.

[132] Grediac M, Toussaint E, Pierron F. Special virtual fields for the

direct determination of material parameters with the virtual fields

method. 1. Principle and definition. Int J Solids Struct 2002;39:

2691–705. Elsevier.



method. 2. Application to in-plane properties. Int J Solids Struct

2002;39:2707–30. Elsevier.

[134] Grediac M, Pierron F. Numerical issues in the virtual fields method.

Int J Numer Meth Engng; 2001. John Wiley and Sons, 2004;59:

1287–312.



method. 3. Application to the bending rigidities of anisotropic plates.

Int J Solids Struct 2003;40:2401–19.

[136] Giraudeau A, Pierron F. Simultaneous identification of stiffness and

damping properties of isotropic materials from forced vibrating

plates. Comptes-Rendus de l’Academie des Sciences; 2003. Elsevier.

2003;331:259–64.


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