The Power of Ultrasonic Characterisation for Completely Assessing the Elastic Properties of Materials

Jean-Marie MORVAN 1

1 CANOE, Cheminnov – ENSCBP, 16 avenue Pey Berland, 33600 Pessac - France Phone: +33 54000 3854; e-mail: morvan@plateforme-canoe.fr

Abstract From a single ultrasonic test, it is possible to provide materials developers and structures engineers every data on the elasticity of the material and thus limit the mechanical tests to selected samples only. However, despite the existence of an European standard, this method remains unknown. Moreover, the ultrasonic characterization is the only method that can fully monitor residual mechanical properties after damage or fatigue, or if it is used simultaneously with a standard tensile test, to survey the variation of the mechanical properties of the sample and thus provide real constitutive laws and data for FE simulations. The purpose of this contribution is to explain the fundamentals of the method, illustrate it with some results and compare them with standard mechanical tests. Keywords: Ultrasonic characterization, anisotropy, damage, stiffness tensor 1. Introduction With the continuous demand from internal or external customers to reduce the costs of composite materials while the numbers of reinforcements both in materials and directions increase, the problem arises of limiting the number of mechanical tests to be performed, especially at the critical stages of development and qualification. The aim of this paper is to show how, for more than twenty years now, it has been possible with ultrasonic characterisation to provide data on the elastic properties of materials and their variation when the samples are damaged whether by mechanical tests or environmental ageing. The first chapter of this paper will briefly discuss the limitations and drawbacks of conventional mechanical tests. Then, the ultrasonic characterisation will be explained. The last part of this paper will show some chosen results obtained with this method. 2. Limitations of the conventional mechanical tests Standard mechanical tests (traction, flexion) are pretty easy to perform properly on isotropic materials and even if the number of samples is commonly five as set by standards, the cost remains low. Nevertheless, when non linear behaviours are studied, difficulties start to arise. Of course, standards and internal procedures are helpful for explaining how to measure elastic properties with such data as secant or tangent moduli but it involves introducing “tailored” moduli compared to the intrinsic elastic modulus and sometimes tracing a simple slope can become more difficult than though at first sight, figure 1. Difficulties are increasing when damage deactivation phenomena are involved, hysteresis loops with different slopes, etc… For example, composite materials such as ceramic matrix composites exhibit, during cycle tests, damage deactivation phenomena with an opening/closing of the cracks created under tensile loading which is very different from plasticity hence conventional tests can mislead the understanding of the strain mechanisms of these materials.

11th European Conference on Non-Destructive Testing (ECNDT 2014), October 6-10, 2014, Prague, Czech Republic

Figure 1. Determining elastic modulus difficulties. Nowadays, the widespread of composite materials in every industrial sector is intensifying, and, with it, the use of anisotropic materials. This makes the number of tests to grow because the numbers of independent elastic properties increase with the lack of material symmetry. For example when an isotropic material is studied, only two independent elastic constants are necessary for modelling its behaviour but with an orthorhombic material, no less than nine independent constant are needed if an accurate and complete constitutive law has to be written. Equation 1 shows the general form of the stiffness tensor CIJ in the case of a material with three planes of symmetry :

CIJ =

C11 C12 C13 0 0 0C22 C23 0 0 0

C33 0 0 0C44 0 0

Sym. C55 0C66

!

"

########

$

%

&&&&&&&&

(1)

The number of tests will increase accordingly and with it the cost of characterisation (testing but also manufacturing and cutting samples especially off axis ones). From a simple plate of the material, some samples will have to be cut at various angles (most commonly 0°, 90° and 45°), tensile and flexural tests have to be prepared, … One must also keep in mind that some moduli are very difficult to measure : Young Modulus in the thickness direction, in-plane shear modulus… 3. Ultrasonic characterization of the elastic properties and strain partition under load 3.1 Ultrasonic characterization The main principles of ultrasonic evaluation have been given by Roux [1] for the elastic coefficients evaluation of homogeneous anisotropic materials. In order to identify the nine elastic constants Cij which fully describe the elastic behaviour of an orthotropic material, the

wave propagation velocities are collected in the two accessible principal planes (planes (1, 2) and (1, 3), figure 2) and in a non principal plane (plane (1, 45°) described by the bisectrix of axis 2 and 3, figure 2). The identification in plane (1, 2) allows to measure four elastic constants : C11, C22, C66 and C12 and three others are obtained in plane (1, 3) : C33, C55 and C13. The two remaining coefficients C23 and C44 are identified by propagation in the non principal plane (1, 45°). However, when the material exhibits a tetragonal symmetry, Plane (1, 45°) becomes a principal plane and it becomes impossible to measure independently these two stiffnesses [2]. The value of the in-plane shear modulus is then obtained from the phase velocity of a shear wave generated with a pair of contact transverse transducers. This value is used together with Plane (1, 45°) data to simplify and to improve optimisation.   The confidence interval associated to each identified constant is then estimated by a statistical analysis [3]. The ultrasonic device, figure 2, consists in an immersion tank associated to a tensile machine. It allows to study the complete stiffness tensor variation under load thus it is possible to know which coefficients are affected during a damage process.

Figure 2. Sample instrumented for ultrasonic characterisation Wave speed measurements are performed by using ultrasonic pulses which are refracted through the sample immersed in water. The measurement of the phase velocity of the pulses is done by a signal processing method using Hilbert transform [4]. The characterization of anisotropic materials using an ultrasonic method gives access to the purely elastic part of their behaviour. The complete determination of the stiffness tensor of a composite presenting an orthorhombic symmetry allows, by simply inverting the tensor, to pass on to a description in terms of compliances which authorized a reconstruction of the elastic hardening curve.  When people working with ultrasounds speak about stiffnesses, most of the time, other people argue that engineering constants are more commonly use for writing constitutive laws. Hopefully, relationships exist between stiffness tensor and engineering constants [5]. The accuracy and the reliability of determination of the complete stiffness tensor of a composite presenting an orthorhombic symmetry allows, by simply inverting the tensor, a description in terms of compliances which is necessary to write the results in engineering constant form :

SIJ =CIJ

−1 (2)

Ei =1Sii

(3)

ν ij =−SijSii

(4)

Gij =1Sijij

(5)

Where SIJ is the compliance tensor, Ei , the Young modulus in direction i, ν ij , the Poisson ratio between directions i and j and Gij , the Coulomb modulus between directions i and j. From these relationships, it is clear that if the material is anisotropic, the engineering constant can only be obtained once the complete upper part of the stiffness tensor has been evaluated. 3.2 Strain partition In order to study accurately the damage evolution in composites, the experimental device that couples an ultrasonic immersion tank to a tensile machine has been enhanced with the use of an extensometer [6] that record the total macroscopic strain. As the complete stiffness tensor is measured, the description in terms of compliances is known by simply inverting the tensor and the elastic strain value along the tensile axis is calculated by using the generalized Hooke's law :

ε elastic3          = S33σ 3 (6) where ε elastic3          is the elastic axial strain along tensile direction 3 and σ 3 is the stress along tensile direction 3. The total strain ε total3       measured with the extensometer is therefore written as being the sum of the elastic strain ε elastic3           assessed from ultrasonic measurements and of an inelastic strain ε inelastic3           which mainly finds its origin in damage of the material :

ε total3          = εelastic3         + ε

inelastic3           (7)

This allows not only to know what is the real ratio of elastic strain of the material beyond damage threshold but also to verify the accuracy of the ultrasonic characterisation. The strain can be calculated at every step of ultrasonic characterisation.

4. Chosen results In order to illustrate what kind of otherwise unobtainable results can be reached with the method described, this chapter will present some example of past works. 4.1 Accuracy of the strain partition This first illustration was obtained with a quite simple material : an high impact resistance PMMA with 25% core shells in which very little damage occurs. The sample was supplied by GRL (now part of Arkema group). It was submitted to tensile stress in direction 3. Figure 3 shows the variation of quasi longitudinal (QL) and quasi transversal (QT) waves velocities in Plane (1, 3) at zero stress and at nearly failure stress. Whatever the velocity considered, it hardly varies showing that little damage occurs.

Figure 3. Slowness variation in propagation plane (1,3) at 0 MPa and 35.5 MPa

Figure 4 shows the result the strain partition. One can see that the elastic slope issued from ultrasonic measurement fits very well the elastic slope that can be drawn from the extensometer data before plasticity and crazes appear leading to inelastic strain.

Figure 4. Comparison between total strain measured with an extensometer and elastic strain

calculated from data obtained by ultrasonic characterisation

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0 MPa35.5 MPa

Mode QT

Mode QL

(1)

(3)

0 0,2 0,4 0,6 0,8 1 1,2 1,40

5

10

15

20

25

30

35

40

%

MPa

ε εεanélastique totalélastique

!

4.2 Measurement of the 3D effects of damage 4.2.1 2D SiC-SiC under tensile loading The investigated material was a woven ceramic-ceramic composite made by SEP (now Herakles) [7]. Under tensile stress, this composite exhibits a non linear behaviour related to the matrix microcracking because the matrix has a lower failure strain than the fibres. The SiC clothes are balanced weaves, so directions 2 and 3 are symmetrically equivalent, and the composite presents a tetragonal symmetry with six independent stiffnesses. The 2D SiC-SiC specimen was submitted to tensile stress in the direction 3 which is parallel to one of the bundle direction. The slowness curves at various loads are gathered in figure 5.

Figure 5. Slowness curves at various stress levels for a 2D SiC-SiC (103 m/s)-1

Figure 6. Variations of the stiffness tensor coefficients (GPa) of a 2D SiC-SiC sample and their 90 % relative confidence interval versus applied stress in direction 3

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

210 MPa120 MPa 80 MPa 0 MPa

(2)

(1)

Q.S.

Q.L.

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

210 MPa120 MPa 80 MPa 0 MPa

(3)

(1)

Q.S.

Q.L.

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

210 MPa120 MPa 80 MPa 0 MPa

(45°)

(1)

Q.S. 2

Q.L.

Q.S. 1

80100120140160180

0 50 100 150 200 250

C11

(MPa) 0

150

300

450

600

0 50 100 150 200 250

C22

(MPa) 0

100

200

300

400

0 50 100 150 200 250

C33

(MPa)

0

25

50

75

100

0 50 100 150 200 250

C55

(MPa) 0

25

50

75

100

0 50 100 150 200 250

C13

(MPa) 20

45

70

95

120

0 50 100 150 200 250

C44

(MPa)

0

100

200

300

400

0 50 100 150 200 250

C23

(MPa) 50

75

100

125

150

0 50 100 150 200 250

C12

(MPa) 0

25

50

75

100

0 50 100 150 200 250

C66

(MPa)

Figure 6 shows the variations of the variation of the stiffness tensor coefficients of the sample and their 90 % relative confidence interval versus applied stress in direction 3. It is obvious that the greatest variation occurs on C33 with a nearly 60% drop. The stiffness along the tensile direction exhibit a three zones behaviour : linear before the damage threshold which occurs around 80 MPa and then two decrease zones : one with a great slope when matrix microcracking occurs at the bundle scale and one with a slighter slope after around 120 MPa when microcracking occurs at the fibre scale inside the bundles. The complete stiffness tensor being known, the elastic axial strain associated to the tensile stress can be calculated for every stress step as a function of the S33 variation. This strain, together with the total strain measured with an extensometer, and the resulting inelastic strain are plotted on figure 7. The dots represent the steps of ultrasonic characterisation. The non linearity of elasticity is clearly highlighted after the damage threshold but if the inter-bundles cracking greatly affects the non linearity, the intra-bundles one seems to be more moderate. The particular kinetic of the test allows to see that an important part of the inelastic strains is generated by fibre matrix sliding.

Figure 7. Strain partition under load of a 2D SiC-SiC

4.1.2 2D C-SiC under cyclic loading The non-linear mechanical behaviour of a 2D C-SiC ceramic matrix composite was investigated under cyclic loading [8]. The sample has a tetragonal symmetry and was submitted to tensile stress in the direction 3 parallel to one of the bundle direction. The sample was supplied by SEP (now Herakles). Figure 8 shows the stress strain curve obtained with the extensometer. No linear elasticity domain can be identified, thus measuring the Young Modulus in a conventional manner is a problem and this material exhibits a strange behaviour with loading-unloading cycles similar to elastoplastic materials which is a nonsense for such case of brittle material. The ultrasonic characterisation has been of great help for understanding the damage mechanism of this material. After a similar test to the one described before, compliance S33 has been calculated. Figure 9 shows its variation during the cycles. The variation is non linear but compliance at a given stress point has the same given value either when loading or unloading. This shows that no further damage occurs during a given cycle.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

50

100

150

200

250

Strain (%)Δεbundle sliding

Stre

ss (

MPa

)

elasticε fibre slidingεεelastic εtotal+

Figure 8. Stress Strain curve of a 2D C-SiC

Figure 9. Variations for each cycle performed of the compliance along the tensile with its 90

% relative confidence interval versus applied stress in direction 3 Figure 10 shows what happen during the cycles from the strain point of view. The elastic strain has been calculated with compliance obtained from ultrasonic measurements. Surprisingly the linear variation that occurs from the macroscopic point of view of the extensometer is the sum of two non linear mechanism : a non linear elastic one, that can only be highlighted by ultrasonic wave propagation, and another non linear inelastic strain mechanism finding its origin in the opening closing of the transverse microcracks created by increasing load. So considering as it is usually done that the permanent strain is the inelastic part of the strain is a mistake. Ultrasonic characterisation is the only mean of avoiding it.

Figure 10. Figure 7. Strain partition under load of a 2D C-SiC of the last cycle performed

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

50

100

150

200

250

300

350

Total Strain (%)

Stre

ss (M

Pa)

9

10

11

12

13

14

15

16

17S33

Stress(MPa)

300300300300300 3003000 00 0 0 0 0 3000

(1000.GPa)-1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

50

100

150

200

250

300

350

Strain (%)

Stre

ss (M

Pa)

ε εεinelastic elastic total

5. Conclusion The aim of this paper was to briefly explain the basics and advantages of ultrasonic characterisation. This method is described more precisely in European standard EN14186 [9]. Of course, the method still have some drawbacks such as the necessity of immersing the sample in a fluid, the duration of the ultrasonic test, usually an average of 30 minutes for recording the data in the three planes, which increase greatly the duration of the test when strain partition under load is needed and the necessity to be skilled with the evaluation procedure. Nevertheless, the method allows for most materials the measurement of the complete stiffness tensor at every stress step and only one sample is needed. When one is developing new materials, mechanical tests can only be limited to measurement of ultimate stress and strain. Moreover, the strain recorded will validate the accuracy of ultrasonic characterisation on the elastic part of the curve. The knowledge of the complete stiffness tensor, or engineering constants tensor whatever the description chosen, diminishes the risk of misunderstanding the strain mechanisms and leads to accurate constitutive laws and will enhance FEA predictions. On the analytic models side, the use of these data has been used so far for crack patterns and characteristic lengths identification, crack densities variations or interfacial sliding stress measurement [10]. Acknowledgments The author would like to express gratitude to Pr. Stéphane Baste from I2M laboratory in Bordeaux, France, for pleasant joint research, and COFREND, the French Confederation for the Non-Destructive Testing, for sponsoring part of this communication. References 1. J. Roux, "Elastic wave propagation in anisotropic materials," IEEE 1990 Ultrasonics

Symposium, Honolulu, Dec. 1991, pp 1065-1073, 1990. 2. B. Hosten, "Stiffness matrix invariants to validate the characterization of composite-

materials with ultrasonic methods", Ultrasonics, 30, n°6, pp 365-371, 1992. 3 B. Audoin, S. Baste S and B. Castagnede, "Estimation de l’intervalle de confiance des

constantes d’élasticité identifiées à partir des vitesses de propagation ultrasonores", CR. Acad Sci Paris; t.312:679–86, 1991.

4 B. Audoin and J. Roux, "An innovative application of the Hilbert transform to time delay estimation of overlapped ultrasonic echoes", Ultrasonics, 34, pp 25–33, 1996.

5 SW Tsaï, Composites Design, Think Composites 1987. 6 S. Baste and J-M. Morvan, "Under load strain partition of a ceramic matrix composite

using an ultrasonic method", Exp Mech 6;36: pp 148–54, 1996. 7 J.-M.Morvan and S. Baste, "Effects of two-scale transverse crack systems on the non-

linear behaviour of a 2D SiC-SiC composite", Materials Science and Engineering, A250:231-240, 1998.

8 J.-M. Morvan and S. Baste, "Effects of The Opening/Closure of Microcracks on the Non-Linear Behavior of a 2D C-SiC Composite under Cyclic Loading", International Journal of Damage Mechanics, vol. 7 no. 4, pp 381-402, October 1998.

9 EN 14186, "Advanced technical ceramics - Mechanical properties of ceramic composites at room temperature - Determination of elastic properties by an ultrasonic technique", 2008.

10 J.-M.Morvan and S. Baste, "Evaluation of the Interfacial Sliding Stress of Ceramic Matrix Composites Under Tensile Loading", Review of Progress in Quantitative

Nondestructive Evaluation, pp 1193-1200, 1998.