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## Über dieses Buch

The Virtual Fields Method: Extracting Constitutive Mechanical Parameters from Full-field Deformation Measurements is the first and only one on the Virtual Fields Method, a recent technique to identify materials mechanical properties from full-field measurements. It contains an extensive theoretical description of the method as well as numerous examples of application to a wide range of materials (composites, metals, welds, biomaterials etc.) and situations(static, vibration, high strain rate etc.). Finally, it contains a detailed training section with examples of progressive difficulty to lead the reader to program the VFM. This is accompanied with a set of commented Matlab programs as well as with a GUI Matlab based software for more general situations.

## Inhaltsverzeichnis

### Chapter 1. Introduction, Main Equations and Notations

After a general introduction, the objective of this chapter is to present some notations used in this book, to recall the main equations involved in mechanics of deformable solids and to introduce the problem that is tackled in this book: retrieving constitutive parameters by processing kinematical full-field measurements. This problem is also compared with the classical problem to be solved in mechanics of deformable solids: calculating the displacement, strain and stress distributions within a solid subjected to a given load.
Fabrice Pierron, Michel Grédiac

### Chapter 2. The Principle of Virtual Work

The foundations of the principle of virtual work are given in this chapter, with a special emphasis on the link that exists between the weak and the strong forms of the equilibrium equations. One of the classic uses of the principle of virtual work is then briefly recalled: introducing approximate solutions to calculate the displacement, strain and stress distributions in a body subjected to a given load.
Fabrice Pierron, Michel Grédiac

### Chapter 3. The Linear Virtual Fields Method

The Virtual Fields Method is presented in the particular case of constitutive equations which linearly depend on the constitutive parameters. Linear elasticity is a typical example. It is shown that the parameters that govern the constitutive equations can be found directly in this case, with a suitable choice of the virtual fields. A strategy is then proposed to determine automatically these virtual fields. So-called special virtual fields are defined for this purpose. The influence of noisy input data on the identified parameters is finally addressed. It is shown that virtual fields that minimize this effect are unique for a given basis of functions used to expand the virtual fields.
Fabrice Pierron, Michel Grédiac

### Chapter 4. The Non-linear Virtual Fields Method

The case of constitutive equations that non-linearly depend on their governing parameters is addressed in this chapter. Because of this non-linear nature, the sought parameters are now solution of a set of non-linear equations which has generally to be solved by minimizing a certain cost function defined with some quantities involved in the principle of virtual work. However, even though the resolution is not as simple as for the linear case, the main interest of the Virtual Fields Method is still to avoid the need to iteratively calculate the solution of problem 1 that exists for instance in the identification methods based on finite element models updating.
Fabrice Pierron, Michel Grédiac

### Chapter 5. Complements

Various issues raised by the Virtual Fields Method in some particular cases are addressed in this chapter. It is first shown that the characterization of heterogeneous materials can be carried out using constitutive parameters defined by subregions or modeled with functions varying continuously. The principle of virtual work is then recalled in the case of large deformations, thus opening the way for the characterization of parameters governing hyperelastic laws for instance. The case of thick plates is then introduced as a logical follow-up of the case of thin plates addressed in the previous sections. In particular, it is shown that the through-thickness shear moduli can be identified if suitable full-field measurements are available on the top surface of bent plate specimens. Dynamic loads are then addressed and two particular cases are examined: harmonic and nonharmonic loads. It is shown than complex moduli can be identified in the first case. Concerning the second case, it is shown that it is also possible to use the virtual work of inertial forces in the identification procedure, thus potentially avoiding the need for external force measurements. Finally, even though the Virtual Fields Method was introduced to solve problem 2 presented in the first chapter, it is shown that problem 3 dealing with force reconstruction can also be solved if full-fields measurements are available and if the constitutive equations are completely characterized a priori.
Fabrice Pierron, Michel Grédiac

### Chapter 6. Fiber Composites

The objective of this chapter is to present a number of examples of application of the Virtual Fields Method (VFM) to polymer matrix fiber composites. Historically, composites have been the first materials considered for the VFM. Such composites possess two main features as far as their mechanical behavior is concerned: they are elastically anisotropic and they are prone to damage in the form of cracks, delamination, and/or fiber breakage. The first part of the chapter deals with the measurements of anisotropic stiffness components of composites. In-plane and through-thickness properties are first addressed and then bending properties for thin plates. Then, two applications related to damage are presented. The first one concerns the identification of a damage model when damage develops under loading. The second one deals with the detection and evaluation of a pre-existing damage caused, for instance, by some impact event on a thin panel. Finally, a recent application in the field of high strain rate testing is also presented.
Fabrice Pierron, Michel Grédiac

### Chapter 7. Metals

The objective of this section is to present several studies that have dealt with the use of the VFM to identify elasto-plastic constitutive laws on metallic materials. In the first part, several studies focusing on monotonic quasi-static loadings using different sheet metal specimen geometries are summarized. These studies correspond to the early stages of the adaptation of the VFM to elasto-plasticity and present a gradual progression from simple to more complex situations. The most advanced version is then presented at the end of this first part with the use of cyclic loadings to identify a combined isotropic and kinematic hardening model. Significant progress in the virtual fields selection has also been made in this study. In the second part, a first example of the identification of a visco-plasticity model is presented using moderate strain rate tests where full advantage is taken of the presence of heterogeneous strain rate maps to identify strain rate sensitivity over a strain rate range of a decade with a single test. Finally, a first attempt at the identification of a spatially heterogeneous elasto-plastic law is presented with application of a steel girth weld and a titanium hybrid laser weld.
Fabrice Pierron, Michel Grédiac

### Chapter 8. Soft and Biological Materials

This chapter is dedicated to the presentation of examples of application of the Virtual Fields Method to soft materials like foams and elastomers and biological materials such as wood and biological tissues. These have been grouped together because they share some common features such as high deformability (often implying the measurement of large strains) and high material variability (for wood or tissues). In terms of measurement techniques, the large strain issue means that digital image correlation will be the ideal technique for full-field measurements, whereas medical imaging techniques such as magnetic resonance elastography can be used for tissues (particularly for in vivo studies). As far as the VFM is concerned, large deformations will imply the use of the principle of virtual work within the framework of the mechanics of large deformation, using adequate tensors in either the reference or deformed configurations. Hyperelasticity can then be used to describe the large strain behavior of foams or elastomer, as well as some biological tissues. In this section, the following applications are described: hyperelastic law identification on elastomers, tangent Poisson’s ratio identification of a low density polymeric auxetic (negative Poisson’s ratio) foams, and the use of magnetic resonance based measurements for biological tissues and phantoms (synthetic materials mimicking the behavior of tissues). Finally, some examples on wood are also be presented that share many common features with the examples in Chap. 6 on composites.
Fabrice Pierron, Michel Grédiac

### Chapter 9. Other Materials

This chapter presents some additional examples of the application of the Virtual Fields Method to materials that do not fit in the previous chapters. Both examples deal with dynamic identification, using inertia forces in the identification process, as presented in Sect. 5.5, Page 143. The first example concerns the simultaneous identification of stiffness and damping (complex stiffnesses) from thin vibrating polymeric plates, with the technique described in Sect. 5.5.2. The second one is a feasibility study to extract Young’s modulus of concrete materials subjected to tensile impact test (spalling test), with a procedure very similar to that described in Sect. 6.3, Page 242. The main challenge here, however, concerns the very small strains before the onset of tensile cracking in the material.
Fabrice Pierron, Michel Grédiac

### Chapter 10. Design of New Tests for the VFM

Over the years, much effort has been dedicated to solving the inverse problem consisting in extracting the materials mechanical constitutive parameters from full-field deformation maps. However, the next very important question is which mechanical test configuration will give the optimal identifiability. Indeed, now that the constraints arising from the necessity to have well-controlled test geometries and loads to have a priori stress distribution information are relaxed, the design space for test configuration becomes nearly infinite and some strategy must be devised to come up with novel relevant tests that will contain the required mechanical information (i.e., activate at best all the parameters to identify). In the early days, such tests were mainly derived from existing configurations (unnotched Iosipescu test) or imposed by the specimen geometry (ring compression test). However, some attempts at test design and optimization have been performed. This section is dedicated to the presentation of this work. First, a very basic approach based on strain balancing is presented to design a T-shaped specimen. Then, with the availability of optimized virtual fields, the η parameters have been used to build up cost functions but this approach proved to be too restrictive. Finally, a recently developed complete identification simulator is detailed, which takes into account all the stages from image forming down to identification. This is the ideal tool to evaluate identification performance and design novel test configurations.
Fabrice Pierron, Michel Grédiac

### Chapter 11. The VFM for Force Reconstruction

The objective of this section is to show some examples of how the Virtual Fields Method can be used to identify force instead of constitutive parameters. The theoretical bases for this are developed in Sect. 5.6, Page 156. Examples given here are all related to high strain rate testing where load measurement is a significant experimental problem.
Fabrice Pierron, Michel Grédiac

### Chapter 12. Case Study I: Standard and Funny Isotropic Discs

In this chapter, the very simple case of a disc in compression will introduce the reader to the practical implementation of the Virtual Fields Method, and the effect of noise. Linear elastic isotropy is considered to make things as simple as possible. In order to make this example more interesting, one of the discs has a “funny” shape, i.e., it has some cutouts that make it look like a smiling face, whereas the other is a simple circular disc. The reader has to implement very simple virtual fields on exact simulated data, evaluate the influence of noise, and finally process some experimental data.
Fabrice Pierron, Michel Grédiac

### Chapter 13. Case Study II: Unnotched Iosipescu Test

In this chapter, the idea is to train the reader to more advanced features of the Virtual Fields Method. First, the test configuration requires the definition of piecewise virtual fields, introducing specific constraints in their parameterization. Also, linear elastic anisotropy is introduced, which increases the number of unknowns from two to four. This leads to the need for the definition of more virtual fields (four) which underlines the critical issue of virtual field selection. As a response to this problem, the second part of the chapter introduces the automated construction of optimized virtual fields leading to the most robust solution (maximum likelihood solution). Both continuous (polynomial) and piecewise (finite element) formulations will be illustrated.
Fabrice Pierron, Michel Grédiac

### Chapter 14. Case Study III: Orthotropic Plate in Pure Bending

This chapter is dedicated to the resolution of a Virtual Fields Method problem within the framework of Love–Kirchhoff’s thin plate theory. An orthotropic plate in pure bending is considered here, and both simulated and experimental data are processed. The case study starts with manually defined virtual fields to get the reader acquainted with the virtual fields for bending problems. Then, special optimized piecewise virtual fields are implemented with specific shape functions to ensure virtual displacements continuity.
Fabrice Pierron, Michel Grédiac

### Chapter 15. The Camfit Program

This last chapter briefly presents the Camfit program, a GUI Matlab ®-based software implementing the Virtual Fields Method for simple cases of in-plane linear isotropic and orthotropic elasticity and elasto-plasticity. The Camfit program is provided with the present book.
Fabrice Pierron, Michel Grédiac

### Backmatter

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