PSI - Issue 54

Ela Marković et al. / Procedia Structural Integrity 54 (2024) 156 – 163 Ela Markovi ć et al. / Structural Integrity Procedia 00 (2023) 000–000

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2.1. Elasto-plastic finite element analysis Steel's stress-strain relationship is initially linear until reaching the yield point, after which plastic deformation in material occurs and steel exhibits non-linear behavior. As the component is expected to experience elasto-plastic stress-strain response, it is crucial to select a material model capable of capturing realistic material behavior, including plasticity. For this purpose, multilinear material model was employed and Ramberg-Osgood model was used for its definition. The complete stress-strain curve obtained from tensile tests is generally simplified with a monotonic Ramberg-Osgood model which is adequate for most practical purposes, especially metals, Ramberg and Osgood (1943):

1 n

e              p E K  

(1)

which gives the relation between true stress, σ , and true strain, ε . E is the Young’s modulus, K is the strength coefficient and n is the strain hardening exponent. Material behavior in numerical analysis was defined with a multilinear isotropic material model whose linear portion relies on Young's modulus and Poisson's ratio, while the nonlinear segment was determined by connecting data points (true stress and plastic strain pairs) with line segments. To establish this nonlinear part, the Ramberg Osgood function (Eq. 1) was employed to calculate strain values for selected stress values. Obviously, more densely selected points provide better accuracy of a model but take longer time to complete which is why an optimal number and position of stress-plastic strain points needed to be selected and inputted to material model. To address this problem, modified Ramer-Douglas-Peucker algorithm is proposed and used in this study (details of the original algorithm in Douglas and Peucker (1973)). The modified algorithm determines the optimal position and number of points on the Ramberg-Osgood function within a user-defined tolerance of accepted stress deviation. It further requires input of yield strength, R e , maximum value of plastic strain up to which the curve is to be discretized, ε fin , and monotonic Ramberg-Osgood material constants: strength coefficient, K , and strain hardening exponent, n . One such discretized curve, together with Ramberg-Osgood monotonic continuous curve is shown in Figure 2 as an illustrative example.

( ε fin , σ fin )

K , n

Ramberg-Osgood curve

Points for multilinear isotropic model

Stress, MPa

(0, R e )

Plastic strain, mm/mm

Fig. 2. Illustrative example of Ramberg-Osgood model representing the stress-plastic strain relationship and discretized points used for defining the multilinear isotropic model in numerical analysis. Points are obtained using the modified optimization Ramer-Douglas-Peucker algorithm.

2.2. Distribution of material properties through specimen cross-section Functionally graded material specimens were modeled with a varying hardness distribution as a function of specimens’ spatial coordinates. For that purpose, model proposed in Lang (1989) developed for surface hardened 42CrMo4 steel and represented with the following empirical formula was used:

core x x HVx HV HV HV fx f x x x R        surface core (0,17327 0 4218 *) * ht ( *) ( ( *) 10 *

) ( *)

(2)