The virtual fields method applied to spalling tests on concrete

Abstract

Abstract. For one decade spalling techniques based on the use of a metallic Hopkinson bar put in contact with a concrete sample have been widely employed to characterize the dynamic tensile strength of concrete at strain-rates ranging from a few tens to two hundreds of s −1 . However, the processing method mainly based on the use of the velocity profile measured on the rear free surface of the sample (Novikov formula) remains quite basic and an identification of the whole softening behaviour of the concrete is out of reach. In the present paper a new processing method is proposed based on the use of the Virtual Fields Method (VFM). First, a digital high speed camera is used to record the pictures of a grid glued on the specimen. Next, fullfield measurements are used to obtain the axial displacement field at the surface of the specimen. Finally, a specific virtual field has been defined in the VFM equation to use the acceleration map as an alternative 'load cell'. This method applied to three spalling tests allowed to identify Young's modulus during the test. It was shown that this modulus is constant during the initial compressive part of the test and decreases in the tensile part when micro-damage exists. It was also shown that in such a simple inertial test, it was possible to reconstruct average axial stress profiles using only the acceleration data. Then, it was possible to construct local stress-strain curves and derive a tensile strength value.

An Improved Lagrangian-Inverse Method for Evaluating the Dynamic Constitutive Parameters of Concrete

“…The non-linear constitutive behaviors may suggest that the assumption of the linear-elastic constitution of concrete may be invalid for the concrete-like materials. This is supported by previous studies which also report similar non-linear behaviors in concrete-like materials [33,52,53]. One may expect that these non-liner behaviors contribute to the significant reduction in the attenuation and propagating speed of stress wave under high loading-rate conditions.…”

Validation of the photomechanical spalling test in the case of non‐linear dynamic response: Application to a granite rock

“…[ 53 ] One of the first extensions of VFM to dynamic loading conditions was presented in the seminal article published in Strain 2012. [ 54 ] The authors showed that the right‐hand‐side term of Equation (3) can be used as an advantage since the contribution of the inertial component defined through the measured acceleration maps present an image embedded load cell and in that case no external load measurement is needed. Consequently, an appropriate choice of virtual fields can lead to a direct identification of either material stiffness or average uniaxial stress measurement.…”

“…As explained in Pierron and Forquin, [ 54 ] following the schematic presentation of the framed spalling sample with the flattened surface (Figure 3), the resulting stress field will be mainly uniaxial, and a virtual displacement field can be chosen so that it depends only on the axial direction. Consequently, this choice can be made so to cancel out the contribution of other stress tensor components as follows: where f ( x ) is a continuous and differentiable function of x , with condition f ( x = l ) = 0 which avoids having a contribution of the traction force at the boundary opposite to the free‐end ( x = 0).…”

“…Although a variety of methods exist for choosing more adequate virtual fields (e.g., previous works [ 55,56 ] ), here, the virtual fields are directly derived from the actual measured displacement data as in Pierron and Forquin. [ 54 ]…”

“…As explained in Pierron and Forquin, [54] following the schematic presentation of the framed spalling sample with the flattened surface (Figure 3), the resulting stress field will be mainly uniaxial, and a virtual displacement field can be chosen so that it depends only on the axial direction. Consequently, this choice can be made so to cancel out the contribution of other stress tensor components as follows:…”