PSI - Issue 60

S.K. Pandey et al. / Procedia Structural Integrity 60 (2024) 3–12 S. K. Pandey/ Structural Integrity Procedia 00 (2023) 000–000

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1. Introduction Smooth specimens are used to find the stress-strain data using the tensile test (ASTM E8). The stress-stain material data produced by the uniaxial tensile test are used to predict the behaviour of any kind of structural components comprising multiaxial state of stress. Small deviations in stress-strain data produced by smooth specimen testing for the purpose of design of structure are generally acceptable because allowable stress and strain in design codes are kept sufficiently below the ultimate strength. While using damage material model [Pandey et al. 2023, Samal et al. 2009, Samal et al. 2013) damage parameters are needed, which are determined by experimental and finite element analysis. But in the case of damage parameters evaluation where stress-strain data are used up to the fracture point, the stress-strain behavior becomes significantly important. The stress-strain data derived from tensile tests of smooth specimens may not be able to predict load-displacement behavior of notched specimens (due to presence of multiaxial state of stress) accurately as discussed later in Section-3 of this manuscript. Hence an optimum stress-strain behavior of a material is required which can predict both smooth and notched tensile load displacement behavior accurately. In literature, a technique to identify materials properties using displacement field measurement has been presented in Fazzini et al. 2011. An inverse methodology to extract constitutive parameters from experimental data has been presented in Ramirez et al. 2022. Generally, notched specimen test data is used to evaluate the material damage parameters with the triaxial state of stress. Ramberg-Osgood (RO) stress-strain relation is used to relate the stress-strain data. In the present work, an approach has been developed to evaluate unique set of Ramberg-Osgood parameters (n , α) , which shall able to predict the behavior (load-displacement data) of different types of specimens machined from the same material, i.e., specimens with both uniaxial state of stress condition (smooth tensile specimen) and multiaxial state of stress conditions (notched tensile specimen) accurately. This approach is based on the minimization of error (minimization of difference of scaled energy absorption, represented by differences in area under load-displacement curves as obtained from FEA and experiment).

Nomenclature δ

σ

Stress Strain

Maximum elongation of specimens

α , n

Ramberg-Osgood parameters

2. Approach for optimize Ramberg-Osgood parameters Approach has been developed based on minimization of difference of areas under the load-displacement curves of FEA and experimental data, to find the optimized RO parameter, i.e., unique RO parameter (n, α ). This approach consists four steps as follows: • Step 1: Selection and numbering of different sets of RO parameters (n, α ) • Step 2: Evaluation of Error at each set of (n, α ) for each specimen by FEA • Step 3: Evaluation of Normalized-Error at each set of (n, α ) • Step 4: Evaluation of optimized (n, α ), i.e., unique (n, α ) The overall process of optimization approach for unique RO parameters is presented in Fig.1. Ramberg-Osgood equation for stress and strain is proposed by Ramberg and Osgood as provided in Eq.1. Three sets of RO parameters, viz., (n 1 , α 1 ), (n 2 , α 2 ) and (n 3 , α 3 ) have been taken to explain the whole process of optimization for unique RO parameter (n, α ). For each set of RO parameter, Error and subsequently Normalized-Error are calculated. Unique RO parameter is evaluate Ramberg-Osgood equation for stress and strain is proposed by Ramberg and Osgood in 1943 as provided in Eq. (1).d using all Normalized-Errors . The terms, Error and Normalized-Error, are explained in next sections. Unique RO parameter exists nearest to that RO parameter which gives the lowest value of Error ( or Normalized-Error = 1). The process of finding the lowest valued Normalized-Error , RO parameter and subsequently the Unique RO parameter has been explained in next paragraphs.