## ABSTRACT

**Using kinematical and thermal full-field** measurements for identification of mechanical parameters has become a very promising area of experimental mechanics. The purpose of this work is to extend the use of non-conventional tests and full field measurements (kinematical and thermal) to the identification of the fatigue properties of a dual-phase steel. A particular attention is paid to the influence of plastic pre-strain on the fatigue limit. Indeed, an analytical approach is proposed to define the geometry of the specimen permitting to obtain a constant gradient of plastic strain within the zone of interest after a monotonic pre-strain. Then, a self-heating test under cyclic loading is carried out on the pre-strained specimen. During this cyclic test, the thermal field is measured using an infrared camera. Finally, a suitable numerical strategy is proposed to identify a given thermal source model taking into account the influence of a plastic pre-strain. The results show that, with the non-conventional test and the procedure developed in this work, the influence of a range of plastic pre-strain on fatigue properties can be identified by using only one specimen. It is worth noting that a great number of specimens is required to determine this effect by using classical fatigue campaign.

## INTRODUCTION

**The measurement and analysis of kinematical and thermal** full-field during mechanical tests are more and more used in the field of mechanics of solid materials and structures. In the past, full-field measurements were essentially applied as means to qualitatively identify the heterogeneity of a field (e.g., kinematical or thermal field [1-3]). Today, the cumulated experience and the available information about the metrological performances of full-field measurements open up new horizons [4, 5]. Indeed, full-field measurements may be used as a starting point for identification procedures providing access to parameters of constitutive laws and mechanical properties. It may provide very rich experimental data when applied to tests conducted under non-homogeneous conditions (i.e., for which strain and stress or temperature are not uniform in the zone of interest of the specimen). So, two of the main challenges about identification tests in mechanics of material concern, on the one hand, the design of the experiments (i.e., geometry of the specimen and an associated loading and boundary conditions) to controlled the heterogeneous conditions and, on the other hand, the development of suitable numerical strategies to identify the material parameters.

**The purpose of this work is to extend the** use of non-conventional tests and full-field measurements (i.e., strain field and temperature field) to the identification of the fatigue properties of a dual-phase steel. A particular attention is paid to the influence of plastic pre-strain on the fatigue limit [6-9]. This is particularly important, for example, in the context of the fatigue design of structures and chassis frame metal parts produced by metal forming operations (e.g., stamping, …). During these drawing operations, the sheets are severely strained, including thickness variations, cumulated plastic strain, residual stress and evolution of mechanical properties (e.g., yield stress, fatigue limit, …). As an example, the figure 1 shows the S-N curves (i.e., stress amplitude of the cyclic loading relating to the number of cycles to failure) of a dual-phase steel in two different states (i.e., undeformed state and 20% plastic pre-strained state) [8-9]. One can see that the fatigue properties are considerably changed. The fatigue limit (i.e. endurance limit), which is initially equal to 250 MPa goes up to 320 MPa after a pre-straining of 20%. This evolution of fatigue properties is often not taken into account in conventional fatigue design approaches. One of the major reasons of this lack is the prohibitive time needed to characterize this effect by traditional fatigue tests campaigns.

**Fig. 1 Effect of a plastic pre-strain on the S-N curve of a dual-phase steel: a) unstrained state; b) 20% plastic pre-strained state [24]**

**In order to reduce the time dedicated** to fatigue properties characterization, several authors have worked on an estimation of high cycle fatigue properties based upon self-heating measurements under cyclic loading [3,8-16]. The proposed test, hereafter called ‘self-heating test’, consists in observing thermal effects during cyclic loadings and is performed on a specimen with a constant cross-section. For each stress amplitude, the change of the temperature variation, 9 = T-T0 (where T is the mean temperature of the specimen and T0 the surrounding value) is recorded. It is generally observed that the mean temperature becomes stable after a fixed number of cycles depending on the stress level and the loading frequency. Figure 2 shows the evolution of the steady-state temperature relating to the stress amplitude. One can see that beyond a given stress amplitude that is close to the fatigue limit, the steady-state temperature starts to increase significantly. Recently, a model [15] allows us, on the one hand, to describe thermal effects and, on the other hand, to relate the thermal effects to the fatigue properties.

**In a previous work,** a self-heating curve per plastic pre-strain level has been determined (i.e., one specimen is needed per plastic pre-strain level). This set of self-heating curves permits to quantitatively characterize the influence of plastic pre-strain on the fatigue limit. As shown in this work and in the considered pre-strain range (10% – 20%), a linear evolution of the mean endurance limit with respect to the plastic pre-strain can be assumed. The slope of this linear evolution is denoted by P [8]. In this paper, an alternative method based on the use of only one 1-D non-conventional specimen and a suitable numerical strategy is proposed to identify the influence of a plastic pre-strain range on the fatigue limit. The paper is divided into main sections. In the first one, the design of a non-conventional specimen is presented. The main goal is to define the geometry of the specimen permitting to obtain a controlled heterogeneous plastic strain field on the zone of interest after a monotonic pre-strain test. The monotonic pre-strain test is performed on the specimen and the plastic strain field is checked using kinematical field measurements with digital image correlation. In the second section, a self-heating test under cyclic loading is performed on the pre-strained specimen. The heterogeneous thermal field is measured using an infrared camera and analyzed by using a suitable numerical strategy permitting to identify a given thermal source model taking into account the influence of plastic pre-strain.

**Fig. 2 Empirical identification of the mean fatigue limit from self-heating measurements for a dual-phase steel [16] 1. Preliminary plastic strain of a non-conventional specimen**

**In this section,** the design of a non-conventional specimen is described. An analytical approach is proposed to define the geometry of the specimen permitting to obtain a constant gradient of plastic strain within the zone of interest after a monotonic pre-strain. Then the pre-strained specimen is obtained by performing a uniaxial tension test under displacement control, followed by a complete unload. The plastic strain field is determined using a Digital Image Correlation technique and a suitable numerical strategy.

### Design of the geometry of a non-conventional specimen

**In so far as the specimen is machined in a constant thickness sheet,** only the width, 2b0, is a free parameter (i.e., depending on the abscissa x1). One assume standard plastic compressibility of the steel during the pre-strain test, so that the ratio between the initial section,, and the strained section, S(x1), is given by

where

is the 1D gradient of displacement, u(x1). The maximum true stress, o(x1), is then written as

where F is the maximum load reached during the monotonic tensile test.depends on the plastic strain level, reached at the abscissa x1. Consequently, the half-width of the specimen is given by

where ep(x1) is the logarithmic plastic strain. In order to obtain a constant plastic pre-strain gradient, K, the plastic strain is related to the displacement u(x1) by the following relation

The previous equation is solved by using the boundary condition u(x1=0)=0, so that

and

**Fig. 3 Monotonic tensile test of a dual-phase steel**

**According to the monotonic tensile curve of the studied steel (Fig. 3),** K is chosen to reach a maximum value of 17% of plastic pre-strain at the center of the specimen. On Figure 4, the geometry of the specimen is given. It is worth noting that the plane (x1=30mm) is a plane of symmetry.

**Fig. 4 Geometry of the non-conventional specimen on the zone of interest **

### Identification of the gradient of plastic pre-strain field, Kid

**The plastic deformation is performed on a servo-hydraulic testing machine.** During this test, successive images are taken by using a CCD camera. Thus the displacement map is computed with a correlation technique between an initial picture and a subsequent one [17]. Figure 5a shows the experimental displacement field, Uexp(x1,y), of the zone of interest at the last stage of the test (i.e., at the end of the unloading). One can see that this displacement field is constant by bands. The theoretical expression of the 1D displacement is given by Equation 5. The value of Kid (i.e., the gradient of plastic pre-strain field) is identified by squares minimization between the experimental field and its analytical expression (i.e. Kid = 0.0053). Figures 5b-d show different fields at the last stage of the test. Figures 5b shows the identified displacement field, Uid(x1). An error map representing the relative error, ErrorU, defined by

is showed on Figure 5c. The maximum relative error is less than 3.5%. Finally, Figure 5d shows the identified plastic pre-strain field given by equation (6) with K= Kid. A maximum value of 17% is reached at the center of the specimen, as expected.

**Fig. 5 Kinematical field measurements: a) Experimental displacement field obtained with Digital Image Correlation; b) Identified displacement field; c) Displacement error map; d) Calculated plastic pre-strain field**

## Self-heating test on the plastic pre-strained specimen

**In this section**, the self-heating test carried out on the plastic pre-strained specimen is described. Then the identification of the influence of the plastic pre-strain on the fatigue limit from the results of the self-heating test is presented. This identification consists in solving the heat conduction equation. In the particular case of the designed specimen, the 1D heat conduction equation is considered.

### Experimental result

**The self-heating test consists in applying a uniaxial cyclic** test under load control conditions with constant load amplitude, (without mean stress) during 6000 cycles (i.e., the time to reach a thermal equilibrium). The load frequency, fr, is 30Hz. During the self-heating test, the temperature field is measured using an infrared camera. In order to consider only the temperature elevation, 0(M), the difference between the temperature T(M) and the initial temperature at the same point M, Tt=0(M), is considered. Figure 6a shows the steady-state experimental temperature elevation field, 0exp(x1,y), of the zone of interest. One can see that this temperature field is constant by bands, so that the 1D heat conduction equation is considered in the following of this paper.

### Principle of the 1D heat conduction equation solving

**By considering Fourier’s law including conduction losses** [18] but neglecting any convection threw the surrounding air, the 1D heat conduction equation can be written

with X’ the isotropic thermal conductivity, S(x1) the section of the non-conventional specimen after the pre-strained test, p the mass density, c the specific heat, St(x1,t) the 1D thermal sources field and 0(x1,t) the 1D temperature elevation field. The determination of heat source field from the measurement of temperature fields is still a difficult task. Due to the signal-to-noise ratio and the regularizing effects of heat diffusion, regularization method is required to handle this problem [1,19]. In this work, another approach is proposed, based on the determination of a theoretical temperature field from an a priori considered expression of the heat source field given by

where 8 is a parameter, fr is the load frequency, X0(x1) and ep(x1) are the stress amplitude and the plastic strain, respectively, m is a material parameter (for this steel grade, m=12.5 is identified from a classical self-heating test [8]) and P is the material parameter that characterizes the influence of the plastic pre-strain viewed as linear. The main goal of the proposed approach is to identify the value of the parameter P from the field measurements results of the self-heating test previously exposed. From the analytical expression of the thermal source field, an analytical temperature is obtained by solving the heat conduction equation on a particular basis [18], namely, the Eigen basis regardless of the boundary conditions, generally used to solve the heat conduction equation, completed with a second order polynomial basis. The first one is used to describe the local variations of the steady-state temperature field. The second one describes the boundary conditions at both ends. The following expression of the 1D steady state temperature field variation is then considered

where a, b, c are three parameters, k is the number of components of the Fourier basis,are functions of Fourier coefficients with wk=2kn/L, L representing the length of the zone of interest of the pre-strained specimen. The heat conduction equation can be then projected on the proposed particular basis and be written as

with St0 the constant value of the Fourier basis, Stak and Stbk the successive harmonics of this basis. The terms Ak, Bk, Ck and Dk of the previous equation are defined as follows

**It has been shown that tests conducted under** controlled non-homogeneous conditions associated to full-field measurements permit to extract information from a small number of tests. It opens a lot of perspectives concerning the mechanical parameters identification methods.