Bending elastic stiffness constants of the tested anisotropic plate and the computed errors.
Nowadays, the demand and the necessity for the use of materials with specificcharacteristics have increased in many engineering fields. Due to this necessity of making new materials, composite materials have been an alternative, or maybe the unique option, to attempt a large number of design requirements suchas high strength-to-weight ratio,high resistance to mechanical shocks, chemical attacks, corrosion, and fatigue, that cannot be obtained only from the commonly used structural materials (metals, ceramics, polymers and wood). Because of this, their applications are present in the main industries such as aerospace, automotive, marine, and sportive
Thanks to their flexibility characteristicsthere are many combinations and arrangements and, consequently, constitutive properties, that are possible to be achieved. This particularity represents one of the main advantages of these materials. Nevertheless, some factors related to arrangements as the number of layers and the orientation of the fibers can introduce abehavior called anisotropy that, in the most of the cases, is not required. The anisotropy makes the structural analysis more complex due to increasing the number of independent variables, as for example, the number of bending and extensional elastic stiffness constants.
Recently, wide part of works presented by scientific literature whose goal is to identify constitutive parameters of materials (these being composites or not) is based in the called “inverse problems”. Experimental data such as geometry, resultant forces and strain (or stress) fields are used as input data, and the unknown variables are the required constitutive parameters. In general, the solution is basically associated to two methods: iterative (or also called indirect methods) and non-iterative (or also called of direct methods). The first one is related to optimization problems where the design variables are constitutive parameters and the objective function represents, in general, a residue (or error) between experimental and numerical (generally obtained by finite elements) data. For the numerical simulations, it is considered structures that have the same geometrical characteristics and boundary conditions of the real ones. For each step, required parameters are checked out, and, the optimum represents the iteration whose residue (objective function) has its lowest value. Unlike indirect methods, the direct methods are ones where the required parameters are computed from the solution of constitutive equation(s) that are functions of these parameters.
In general, according to the type of experimental test, it is possible to separate the methods of elastic property identification in two categories: static (destructives and non-destructives) and dynamic methods (non-destructive), as shown in Fig.1. A large number of identification techniques that use data from these categories of tests have been proposed, especially ones dedicated to composite materials. It is possible to say that these techniques identify effective properties of the entire material. The way as each formulation is built, and, the adopted procedures and devices are the main differences among the many proposed methodologies.
Static tests with monotonic load are experimental tests that were more commonly used in the last years, and maybe the simplestones, for this material property identification. Despite the simplicity of these tests, some aspects render them less attractive than dynamic tests, such as the fact of requiring a number of samples with fiber orientations according to standard norms, for example, American Society for Testing and Materials (ASTM), which, in the most of cases, are not in accordance to the real characteristics of the required material. Furthermore, some variables difficult of controlling during the tests can contribute to worsen the experimental results, e.g., the presence of non-uniform stress fields near the ends of the sample from the clamped boundary conditions. For these reasons, dynamic tests have been considered an interesting alternative. In general, they are tests that combine experimental data with numerical methods, and allow the identification of elastic constants from only one unique sample or even from composite material part. Sample is usually thin plate (that reflect Kirchhoff’s hypotheses), cylindrical shell, or beam. In many cases, input data of the numerical methods are natural frequencies and/or mode shapes.
Many authors have proposed to identifythe elastic constants by iterative procedures adopting Rayleigh-Ritz and finite element as numerical methods. The difference among the identification techniques based on these iterative procedures is basically in the way as the optimization problem is formulated, for example, the type of adopted search method to find the minimal, the boundary conditions, the geometric characteristics of the samples, the type of anisotropy of the test material, the type of experimental devices, and the numerical method used to compute the mode shapes (or operational modes) with their respective frequencies (Deobald& Gibson, 1988; Pedersen &Frederiksen, 1992; Lai & Lau, 1993; Ayorinde& Gibson, 1995; Rikards&Chate, 1998; Ayorinde& Yu, 1999, 2005; Rikards et al., 1999; Bledzki et al., 1999; Hwang & Chang, 2000; Araujo et al., 2000; Chakraborty&Mukhopadhyay, 2000; Rikards et al., 2001; Lauwagie et al., 2003; Lauwagie et al., 2004; Lee &Kam, 2006; Cugnoni et al., 2007; Bruno et al., 2008; Pagnotta&Stigliano, 2008; Diveyev&Butiter, 2008a, 2008b).
In works that do not use iterative process, natural frequencies and mode shapes, or operational frequencies and modes, are input data of an algorithm based on the differential equation that governs the transversal vibration of sample in a specific direction and under specific boundary conditions (Gibson, 2000; Alfano&Pagnotta, 2007). In this methodology, it can be included the use of Virtual Fields Method, VFM (Grédia, 1996, 2004; Grédiac& Paris, 1996; Grédiac&Pierron, 1998, 2006; Pierron et al., 2000, 2007; Pierron&Grédiac, 2000; Grédiac et al., 1999a, 1999b, 2001, 2006; Giraudeau&Pierron, 2003, 2005; Chalal et al., 2006; Toussaint et al., 2006; Avril&Pierron, 2007; Pierron et al., 2007; Avril et al., 2008; Giraudeau et al., 2006). For VFM, weighting functions are the called virtual fields. Due to sensitivity to experimental errors and the presence of noise during the dynamic testing, it was proposed the use of specific virtual fields named “special virtual fields” (Grédiac et al., 2002a, 2002b, 2003). In order to decrease the noise contribution, the use of more accurate experimental modal analysis techniques or the application of some signal smoothing (or filtering) technique is mandatory.
The majority of works identifies only the bending stiffness matrix or directly the engineering elastic constants. However, the extensional elastic stiffness matrix is also needed to model composite materials under multi-axial loads. In general, these stiffness matrices are independent. The extensional stiffness matrix relates the in-plane resultant forces to the midplane strains, and, the bending stiffness matrix relates the resultant moments to the plate curvatures. In a laminate composite, if only the stacking sequence of layers is changed, the bending matrix is changed but the extensional matrix remains the same. In other words, different laminates can have different bending stiffness matrices and the same extensional stiffness matrix. It is not possible to obtain the extensional matrix from the bending matrix. It will be possible only if the stacking sequence of layers and their thickness are known and, also, if the material is the same for all lamina.
Sometimes, it is more convenient to use effective laminate engineering constants rather than the laminate stiffness. These effective laminated engineering constants may be easily obtained from the extensional elastic constants. However, due to difficulties on experimental in-plane modal analysis, such as the necessity of using specific devices to measure in-plane displacements and to excite high frequencies, the identification of extensional elastic stiffness constants using modal testing is less attractive. The main challenge to perform in-plane vibration testing is the excitation and measurement ofonly in-plane and not out-of-plane vibration modes. Today there are some new techniques that are suitable for this kind of problems, for example, the excitation by piezoelectric (PZT) and measurements by digital image correlation.
In this chapter, a review about the VFM applied to compute bending elastic stiffness constants proposed by Grédiac& Paris, 1996 is presented. Furthermore, a formulation based on the VFM is proposed in order to identify the extensional elastic stiffness matrix ofKirchhoff’s thin plates. The linear system of equations that provides the required elastic constants is obtained from differential equations that govern the forced vibration of anisotropic, symmetric and non-damped plates under in-plane loads. The common procedures to find the weak form (or integral form) of these equations are applied here. The correct choice of weighting functions (which are the virtual fields) and mode shapes representsa key characteristic to the accuracy of the results. Numerical simulations using anisotropic, orthotropic, quasi-isotropic plates are carried out to demonstratethe accuracy of the methodology.
2. Identification of elastic constants using VFM
2.1. Review of the Virtual Fields Method - VFM
The VFM has been developed for extracting constitutive parameters from full-field measurements and it is associated to problems of identification of parameters from constitutive equations. Two cases are clearly distinguished: constitutive equations depending linearly on the constitutive parameters and non-linear constitutive equations. The type of constitutive equations is chosen
Mathematically, the VFM is based on the principle of virtual work and can be written as:
It is possible to see in Eq. (3) that each virtual field originates a new equation involving the constitutive parameters. The VFM relies on this important property. It is a method based on setting virtual fields that provide a set of equations. This set of equations is used to extract the required unknown constitutive parameters. The correct choice of the virtual fields that combine to actual fields in Eq. (3) is the key issue of the method. Their number and their type depend on the nature of
2.2. Review of the identification method of bending elastic stiffness matrix
The method proposed by Grédiac&Paris,1996, consists of obtaining elastic constants based on the partial differential equation that governs the transversal vibration of an anisotropic thin plate (Kirchhoff’s plate). This equation is given by:
After some mathematical manipulations in Eq. (4), Grédiac& Paris,1996 obtained a linear system in which the unknown variables are the elastic constants. Briefly, the sequence of operations is as follows: (a) multiply both sides of Eq. (4) by an arbitrary weighting function; (b) integrate twice by parts along the plate domain; (c) eliminate the boundary integrals by applying the free-edge boundary conditions; (d) decompose the displacement function
Eq. (5) can be represented in matrix form as:
2.3. Identification method of the extensional elastic stiffness matrix
In the general case of composite laminates, each lamina is assumed to have orthotropic material properties. After the assembly, the behavior can be anisotropic due to the interaction of different laminas. Considering a plate under plane state of stress and using Hooke’s generalized law, stresses can be integrated over its thickness yielding the following force-deformation equations:
where N and M are vectors that contain normal forces and resultant moments, respectively, A is the extensional elastic stiffness matrix, B is the coupling elastic stiffness matrix (B is a null matrix in the case of a symmetric laminate), D is the bending elastic stiffness matrix, ε and κ are vectors that contain middle plane linear strains and rotations, respectively. Considering a symmetrical (B = ) and fully anisotropic laminate under free-edge in-plane vibration (the plate is not under bending) and using the equilibrium relations, the following equations can be written:
Multiplying Eq. (17) by a weighting function
Multiplying Eq. (19) by a weighting function
Now, if free-edge boundary conditions are considered, boundary integrals of Eq. (22) vanish. Considering that the plate is vibrating, functions
whereis the in-plane natural frequency associated to any in-plane mode, and
2.3.1. Choice of the weighting functions
Eqs. (25) and (26) are theoretically valid for isotropic, orthotropic, or anisotropic plates, provided that the laminate is symmetrical. As it can be seen, the function
, , , , , , ,.
or, in matrix form:
Eq. (39) can also be rewritten in a compact form as:
from where the extensional elastic constants
3. Results and comments
A commercial finite element code (ANSYS 11.0) was used to give particular mode shapes and their corresponding natural frequencies from both in-plane and out-of-plane numerical modal analysis. Element SHELL99 was used and plates under free-edge boundary conditions were considered.
To exemplify the method proposed by Grédiac and Paris (1996), it was used an anisotropic plate with dimensions 0.450 x 0.350 x 0.0021 m and density 1500 kg/m3. It was used a laminate with 8 plies, [0 45 90 135]S, and the following engineering elastic constants by ply:
Table 1 shows the bending elastic stiffness constants computed using the engineering constants and the classical theory of laminates, and it also shows the errors computed after applying the identification method. As can be observed, the technique is able to find very satisfactory results when it is used the correct modes. The problem of this technique is the high sensitivity to noise presence because of second-order derivatives. More results and comments about this method can be found in Grédiac& Paris, 1996.
In order to verify the accuracy of the extensional elastic stiffness identification method, it was used six graphite/polymer symmetric laminated plates (Table 2): a fully anisotropic with all
As can be seen in Table 2, the constants
|Bending elastic constants||N x mm||Errors (%)|
The key point of this technique of identification is related with the correct choice of mode shapes together to the weighting functions (virtual fields). The correct mode shapes are called here by “suitable modes” and the correct combination between these modes and the weighting functions are called by “suitable combinations”. The identification of the suitable modes is not difficult, as it will be shown in the next topics. But, the suitable combinations are more difficult because they depend on the type and the geometry of the material. Fortunately, there are some aspects that help finding the best choice. Unlike the bending stiffness identification method originally proposed, for this method there are a lot of modes and suitable combinations that give satisfactory results.
[90 0 0 45]S
[90 0 90 0]S
[90 45 0 -45]S
Table 3 shows some errors computed for the anisotropic plate, rectangular and square. It was considered only the first fifteen in-plane modes shapes. Anisotropic plates, in general, give very satisfactory results using the combinations among suitable modes. This factor can be justified by the fact of these combinations be hardly involved with all required extensional elastic constants
The numerical contribution of each mode to the computation of a specific constant cannot be jeopardized by numerical contribution of another mode during the solution of the system given by Eq. (40).The suitable modes are those that when associated with weighting functions do not null or give very low values for integrals of the right (K matrix) and/or left (C matrix) hand sides of Eq. (40). The suitable combinations are one composed by suitable modes and that give more accurate results. In the majority of the cases, combinations using a higher number of suitable modes can be suitable combinations. According to Table 3 is possible to see that using combinations with only two suitable modes very satisfactory results can be obtained. Satisfactory results would also be obtained using combinations with any modes since the number of suitable modes among all used modes is higher than non-suitable modes. But the accuracy of these results cannot be guaranteed for all combinations.
In general, for the orthotropic and isotropic materials is more difficult to find the suitable combinations when it is compared to fully anisotropic materials. It is necessary to take care to correctly identifying the combinations that give the best results. In these types of materials not all combinations are among suitable modes that can be considered as being suitable combinations. According to values found to terms of the K and C matrices, Eq. (40), and using combinations among suitable modes, it is possible to see the following types of systems:
|Anisotropic rectangular plate – suitable modes: 2, 3, 6, 9, 10, 13, and 14|
|Anisotropic square plate – suitable modes: 2, 3, 6, 9, 10, 13, and 14|
Table 4 shows errors computed to some suitable combinations for these orthotropic plates. As can be seen, very satisfactory results can be obtained using correct combinations of modes. For this type of orthotropy, it can be more difficult to compute an accurate value for constant
|Ortho I rectangular plate – suitable modes: 2, 3, 6, 8, 11, and 14|
|Errors (%)||Differences (N/m)|
|2-3-6-8-11-14||0.14||3.80||0.12||1.22||-2.7 x 104||0.8 x 104|
|2-3-6-8||0.11||2.90||0.24||0.74||-2.5 x 104||0.6 x 104|
|3-6-8-11||0.17||1.58||0.31||1.67||-1.4 x 104||0.5 x 104|
|2-8-11||0.17||0.29||0.23||1.46||-1.5 x 104||0.6 x 104|
|6-8-11||0.18||0.89||0.34||1.91||-4.8 x 104||1.0 x 104|
|3-8||0.02||5.75||0.91||0.13||-1.4 x 104||8.4 x 103|
|8-11||0.25||0.42||0.40||2.17||-2.6 x 104||1.1 x 104|
|Ortho I square plate – suitable modes: 2, 3, 6, 7, 11, and 12|
|Errors (%)||Differences (N/m)|
|2-3-6-7-11-12||0.09||5.81||0.05||1.89||2.7 x 104||-1.6 x 104|
|2-3-6-11||0.10||5.09||0.06||1.92||-4.4 x 104||-8.7 x 104|
|3-6-7-12||0.38||2.33||0.08||2.92||6.4 x 104||-1.7 x 104|
|2-6-11||0.13||5.47||0.11||2.51||-1.0 x 105||-1.8 x 104|
|2-7-12||0.22||5.37||0.04||1.95||1.1 x 105||0.5 x 105|
|2-11||0.02||9.04||0.18||0.56||-1.3 x 104||-1.9 x 104|
|6-11||0.18||5.63||0.30||3.46||-1.1 x 105||-2.7 x 104|
Table 5 shows the errors computed to some suitable combinations for these orthotropic plates. Using correct combinations very satisfactory results can be obtained. Similar to ortho I plate, for this type of orthotropy, constant
|Ortho II rectangular plate – suitable modes: 2, 3, 5, 9, 10, 11, 14, and 15|
|Errors (%)||Differences (N/m)|
|2-3-5||2.77||6.06||0.08||0.38||2.4 x 104||0.5 x 104|
|11-14-15||1.14||2.15||0.92||1.15||2.3 x 105||0.1 x 105|
|2-3||2.36||0.57||0.19||0.72||1.7 x 104||0.1 x 104|
|2-10||2.10||0.82||0.60||2.42||-2.0 x 105||-0.4 x 105|
|2-14||2.04||5.22||0.37||1.12||-3.9 x 104||-1.2 x 104|
|11-14||1.64||2.14||0.87||0.98||-1.8 x 105||-0.1 x 104|
|Ortho II square plate – suitable modes: 2, 4, 5, 9, 10, 12, 13, and 14|
|Errors (%)||Differences (N/m)|
|2-4-5||4.13||8.63||0.39||0.45||-3.5 x 104||-0.1 x 104|
|2-4-14||2.43||0.78||0.72||2.89||4.2 x 104||1.1 x 104|
|2-4||3.35||0.10||0.21||1.25||-3.2 x 104||-0.1 x 104|
|2-14||1.63||1.59||1.30||4.72||1.1 x 105||0.3 x 105|
|5-14||2.67||13.54||2.21||2.82||5.6 x 104||-1.0 x 104|
|9-14||2.12||62.05||0.85||3.89||1.1 x 105||0.2 x 105|
For these plates, rectangular and square the computed matrices K and C, Eq.(40), are full matrices, similar ones of the anisotropic plates. Thus, the majority of combinations among suitable modes are suitable combinations. Combinations that are not suitable present high errors for all constants, what, consequently, make them easy to be identified. Table 6 shows the errors computed to some suitable combinations. Using correct combinations very satisfactory results can be obtained.
|Ortho III rectangular plate – suitable modes: 2, 3, 6, 8, 10, 12, and 13|
|Ortho III square plate – suitable modes: 2, 4, 6, 8, 9, 12, and 13|
|Errors (%)||Differences (N/m)|
Table 7 shows errors computed to some suitable combinations. Using correct combinations, very satisfactory results can be obtained.
|Ortho IV rectangular plate – suitable modes: 2, 4, 5, 8, 11, 12, and 14|
|Errors (%)||Differences (N/m)|
|2-4-5||1.52||2.15||1.28||1.54||-1.7 x 104||-0.6 x 104|
|2-4-8||1.50||2.09||0.83||1.14||1.4 x 104||0.9 x 104|
|5-11-12||0.27||0.09||0.04||1.11||-1.2 x 104||1.9 x 104|
|2-4||1.18||1.83||1.02||2.87||-1.2 x 104||-0.4 x 104|
|2-5||0.49||1.01||1.01||1.41||-1.8 x 104||-0.7 x 104|
|5-11||0.90||0.37||0.04||1.05||3.1 x 104||0.2 x 104|
|Ortho IV square plate – suitable modes: 1, 4, 6, 9, 11, 12, and 14|
|Errors (%)||Differences (N/m)|
|1-6-12||0.16||0.30||0.13||3.07||1.7 x 104||2.5 x 104|
|6-9-12||0.18||0.35||0.30||2,68||1.3 x 104||4.8 x 104|
|9-12-14||0.18||1.87||0.40||0.68||-6.2 x 104||8.1 x 104|
|1-12||0.59||0.26||0.74||2.79||2.0 x 104||2.9 x 104|
|4-11||0.94||1.55||2.77||0.23||2.2 x 104||0.8 x 104|
|12-14||0.26||2.17||0.19||0.90||-9.7 x 104||9.4 x 104|
Table 8 shows errors computed to some suitable combinations. Using correct combinations, very satisfactory results can be obtained.
|Quasi-iso rectangular plate – suitable modes: 1, 4, 6, 8, 11, 12, and 14|
|Errors (%)||Differences (N/m)|
|1-4-6||0.60||1.47||0.19||0.19||9.1 x 103||4.7 x 103|
|1-4-8||0.09||0.80||0.11||0.96||7.4 x 103||9.5 x 103|
|1-4-11||0.33||2.04||0.26||0.08||1.4 x 103||8.1 x 103|
|1-4||0.25||0.54||0.29||1.93||5.7 x 104||7.1 x 104|
|1-6||0.28||0.91||1.77||0.00||1.0 x 104||0.1 x 104|
|4-8||0.73||0.39||0.05||0.95||1.1 x 104||0.3 x 104|
|Quasi-iso square plate – suitable modes: 2, 3, 7, 8, 10, and 11|
|Errors (%)||Differences (N/m)|
|2-3-7||0.37||1.33||0.35||0.91||4.7 x 105||4.7 x 105|
|2-3-8||0.32||1.36||0.33||0.97||-4.4 x 105||-4.4 x 105|
|2-10-11||0.07||3.29||0.45||0.81||-2.3 x 105||-1.1 x 105|
|2-3||0.33||0.79||0.34||2.68||2.2 x 104||1.3 x 104|
|3-7||0.56||1.73||0.43||0.80||6.1 x 105||5.2 x 105|
|8-11||0.89||3.05||0.06||1.34||1.1 x 105||-1.2 x 105|
The identification of elastic properties using VFM has shown to be a very efficient technique since the correct combinations among mode shapes and weighing functions are used. This factor is the key point to find the correct results. However, the identification of these suitable combinations is not so simple in some situations, mainly to the extensional elastic stiffness identification method. Fortunately, there are some characteristics that can help to find such combinations, as it was shown here. A great advantage of this method is related to the large number of possibilities to make combinations able to give very satisfactory results.
Alfano M. . Pagnotta L. 2007A non-destructive technique for the elastic characterization of thin isotropic plates.
Araújo A. L. Mota Soares. C. M. Moreira Freitas. M. J. Pedersen P. Herskovits J. 2000Combined numerical-experimental model for the identification of mechanical properties of laminated structures.
Avril S. Huntley J. M. Pierron F. Steele D. D. 2008D heterogeneous stiffness identification using MRI and the virtual fields method.
Avril S. . Pierron F. 2007General framework for the identification of elastic constitutive parameters from full-field measurements.
Ayorinde E. O. Gibson R. F. 1995Improved method for in-situ elastic constants of isotropic and orthotropic composite materials using plate modal data with trimodal and hexamodal Rayleigh formulations.
Ayorinde E. O. Yu L. 1999On the use of diagonal modes in the elastic identification of thin plates.
Ayorinde E. O. Yu L. 2005On the elastic characterization of composite plates with vibration data.
Bledzki, A.K.; Kessler, A.;Rikards, R. &Chate, A. ( 1999). Determination of elastic constants of glass/epoxy unidirectional laminates by the vibration testing of plates. Composites Science and Technology, 59:2015-2024
Bruno L. Felice G. Pagnotta L. Poggialini A. . Stigliano G. 2008Elastic characterization of orthotropic plates of any shape via static testing.
Chalal H. Avril S. Pierron F. . Meraghni F. 2006Experimental identification of a nonlinear model for composites using the grid technique coupled to the virtual fields method.
Chakraborty S. . Mukhopadhyay M. 2000Estimation of in-plane elastic parameters and stiffener geometry of stiffener plates.
Cugnoni J. Gmür T. . Schorderet A. 2007Inverse method based on modal analysis for charactering the constitutive properties of thick composite plates.
Deobald L. R. Gibson R. F. 1988Determination of elastic constants of orthotropic plates by modal analysistechnique.
Diveyev B. . Butiter (2008 I. 2008Identifying the elastic moduli of composite plates by using high-order theories. 1: theoretical approach.
Diveyev B. Butiter Shcherbina I. N. 2008Identifying the elastic moduli of composite plates by using high-order theories. 2: theoretical-experimental approach.
Gibson R. F. 2000Modal vibration response measurements for characterization of composite materials and structures.
Giraudeau A. Guo B. . Pierron F. 2006Stiffness and damping identification from full-field measurements on vibration plates.
Giraudeau A. . Pierron F. 2003Simultaneous identification of stiffness and damping properties of isotropic materials from forced vibration plates.
Giraudeau A. . Pierron F. 2005Identification of stiffness and damping properties of thin isotropic vibrating plates using the Virtual Fields Method.Theory and simulations.
Grédiac M. 1996On the direct determination of invariant parameters governing the bending of anisotropic plates.
Grédiac M. 2004The use of full-field measurement methods in composite material characterization: interest and limitations.
Grédiac M. Auslender F. . Pierron F. 2001Applying the virtual fields method to determine the through-thickness moduli of thick composites with a nonlinear shear response.
Grédiac M. Founier N. Paris P. A. . Surrel Y. 1999aDirect identification of elastic constants of anisotropic plates by modal analysis: experiments and results.
Grédiac, M.;Founier, N.;Surrel, Y. &Pierron, F.( 1999b).Direct measurement of invariant parameters of composite plates.Journal of Composite Materials, Vol. 33, pp. 1939-1965
Grédiac M. Paris P. A. 1996Direct identification of elastic constants of anisotropic plates by modal analysis: theoretical and numerical aspects.
Grédiac M. . Pierron F. 1998A T-shaped specimen for the direct characterization of orthotropic materials. International Journal for Numerical Methods in Engineering, 41 293 309
Grédiac M. . Pierron F. 2006Applying the Virtual Fields Method to the identification of elasto-plastic constitutive parameters.
Grédiac M. Pierron F. Avril S. Toussaint E. 2006The virtual fields method for extracting constitutive parameters from full-field measurements: a review.
Grédiac M. Toussaint E. . Pierron F. 2002aSpecial virtual fields for direct determination of material parameters with the virtual fields method. 1-Principle and definition.
Grédiac M. Toussaint E. . Pierron F. 2002bSpecial virtual fields for direct determination of material parameters with the virtual fields method. 2-Application to in-plane properties.
Grédiac, M.; Toussaint, E. &Pierron, F.( 2003). Special virtual fields for direct determination of material parameters with the virtual fields method. 2-Application to the bending rigidities of anisotropic plates.International Journal of Solids and Structures, Vol. 40, No. 10, pp. 2401-2419
Hwang S. Chang C. 2000Determination of elastic constants of materials by vibration testing.
Lai T. C. Lau T. C. 1993Determination of elastic constants of a generally orthotropic plate by modal analysis.
Lauwagie T. Sol H. Heylen W. . Roebben G. 2004Determination of in-plane elastic properties of different layers of laminated plates by means of vibration testing and model updating.
Lauwagie T. Sol H. Roebben G. Heylen W. Shi Y. Van der Biest O. 2003Mixed numerical-experimental identification of elastic properties of orthotropic metal plates. NDT e International, 36 487 495
Lee C. R. . Kam T. Y. 2006Identification of mechanical properties of elastically restrained laminated composite plates using vibration data.
Pagnotta L. . Stigliano G. 2008Elastic characterization of isotropic plates of any shape via dynamic tests: theoretical aspects and numerical simulations.
Pedersen P. . Frederiksen P. S. 1992Identification of orthotropic material moduli by a combined experimental /numerical method. Measurement, 10 113 118
Pierron F. . Grédiac M. 2000Identification of the through-thickness moduli of thick composite from whole-field measurements using the Iosipescu fixture.
Pierron F. Zhavoronok S. . Grédiac M. 2000Identification of the through-thickness properties of thick laminated tubes using the virtual fields method. Internatinal Journal of Solids and Structures, 37 32 4437 4453
Pierron F. Vert G. Burguete R. Avril S. Rotinat R. . Wisnom M. 2007Identification of the orthotropic elastic stiffness of composites with the virtual fields method: sensitivity study and experimental validation. Strain: an
Reverdy F. . Audoin B. 2001Ultrasonic measurement of elastic constants of anisotropic materials with laser source and laser receiver focused on the same interface.
Rikards R. . Chate A. 1998Identification of elastic properties of composites by method of planning of experiments.
Rikards R. Chate A. . Gailis G. 2001Identification of elastic properties of laminates based on experiment design.
Rikards R. Chate A. Steinchen W. Kessler A. Bledzki A. K. 1999Method for identification of elastic properties of laminates based on experiment design.
Toussaint E. Grédiac M. . Pierron F. 2006The virtual fields method with piecewise virtual fields.