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Some Basic Problems Of The Mathematical Theory Of Elasticity

In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.

/pdf/pantographic-metamaterials-an-example-of-mathematically-5afbbeadw3.pdf

Pantographic metamaterials: an example of mathematically driven design and of its technological challenges

An approach to calculate the J-integral by digital image correlation displacement field measurement

The imaging of failure in structural materials by synchrotron radiation X-ray microtomography

3D imaging has become popular for analyzing material microstructures. When time lapse series of 3D pictures are acquired during a single experiment, it is possible to measure displacement fields via digital volume correlation (DVC), thereby leading to 4D results. Such 4D analyses have been performed for almost two decades. The present paper aims at reviewing the achievements of and challenges faced by such measurement technique. Ex-situ and in-situ experiments are discussed. A general and unified DVC framework is introduced. Various sources of measurement bias and uncertainties are analyzed. The current challenges are studied and some propositions are given to address them.

/pdf/digital-volume-correlation-review-of-progress-and-challenges-24sgmjfw7w.pdf

Digital Volume Correlation: Review of Progress and Challenges

DIC-based identification of the constitutive parameters of an elastoplastic law is addressed both from a general viewpoint, and applied to the particular case of dog-bone sample made of commercially pure titanium and subjected to tensile loading. A two-step procedure (Digital Image Correlation — DIC — followed by weighted Finite Element Method Updating — FEMU) is first presented. These two steps can be merged into a single-step procedure (i.e., Integrated-DIC or I-DIC). In both cases, the elastoplastic computations are performed with a commercial code (i.e., non-intrusive identification). When the suited weighting of FEMU is taken into account, which is based on DIC-processed image noise, both I-DIC and FEMU methods provide similar results. It is shown that the addressed experimental case requires the use of static (load) information to get precise estimates of the sought parameters.

/pdf/estimation-of-elastoplastic-parameters-via-weighted-femu-and-14hlyhygs6.pdf

Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC

https://hal.archives-ouvertes.fr/hal-00694317/document

Identification of a crack propagation law by digital image correlation

The goal of the present study is to illustrate the full integration of sensor and imaging data into numerical procedures for the purpose of identification of constitutive laws and their validation. The feasibility of such approaches is proven in the context of in situ tests monitored by tomography. The bridging tool consists of spatiotemporal (i.e., 4D) analyses with dedicated (integrated) correlation algorithms. A tensile test on nodular graphite cast iron sample is performed within a lab tomograph. The reconstructed volumes are registered via integrated digital volume correlation (DVC) that incorporates a finite element modeling of the test, thereby performing a mechanical integration in 4D registration of a series of 3D images. In the present case a non-intrusive procedure is developed in which the 4D sensitivity fields are obtained with a commercial finite element code, allowing for a large versatility in meshing and incorporation of complex constitutive laws. Convergence studies can thus be performed in which the quality of the discretization is controlled both for the simulation and the registration. Incremental DVC analyses are carried out with the scans acquired during the in situ mechanical test. For DVC, the mesh size results from a compromise between measurement uncertainties and its spatial resolution. Conversely, a numerically good mesh may reveal too fine for the considered material microstructure. With the integrated framework proposed herein, 4D registrations can be performed and missing boundary conditions of the reference state as well as mechanical parameters of an elastoplastic constitutive law are determined in fair condition both for DVC and simulation.

/pdf/toward-4d-mechanical-correlation-2xe6ii1fif.pdf

Toward 4D mechanical correlation

https://hal.archives-ouvertes.fr/hal-01710062/document

DIC identification and X-FEM simulation of fatigue crack growth based on the Williams’ series

The identification of the parameters of several constitutive laws is performed with the integrated digital image correlation (IDIC) technique in a biaxial experiment for a cruciform specimen made of stainless steel. The sought material parameters are assessed with the contribution of both reaction forces (from load sensors) and displacement fields (measured via digital image correlation). For each constitutive law a global residual quantifying the model error is assessed.

/pdf/integrated-digital-image-correlation-applied-to-y8x4unljdn.pdf

Florent Mathieu

Papers

Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC

Identification of a crack propagation law by digital image correlation

Toward 4D mechanical correlation

DIC identification and X-FEM simulation of fatigue crack growth based on the Williams’ series

Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment