PSI - Issue 61

Bekir Kaçmaz et al. / Procedia Structural Integrity 61 (2024) 130–137 Author name / Structural Integrity Procedia 00 (2024) 000–000

133

4

procedure yield the following weak forms,  V σ : δ ¯ ϵ dV =  V ρ⃗ b ·⃗ ϕ u dV +  Γ t⃗

¯ t ·⃗ ϕ u dA

(9)

 V

2  ∇ ˜ ϵ ·∇ ϕ

˜ ϵ dV − 

˜ ϵ ϕ ˜ ϵ dV + g l

ϵ ϕ ˜ ϵ = 0

(10)

V

which are solved in an incremental-iterative manner implicitly upon discretization by the finite element method. To this end, tetrahedral element topology with quadratic displacement and linear non-local equivalent strain interpolations is used. A ten-noded tetrahedral element is formulated such that corner nodes are equipped with displacement and non local equivalent strain degrees of freedom (dof) whereas edge nodes have only displacement dofs. The following strain-based modified von Mises equivalent strain definition

1 2 k 

( k − 1) 2 (1 − 2 ν ) 2

12 k (1 + ν ) 2

k − 1 2 k (1 − 2 ν )

I 2

I 1 +

J 2

(11)

1 −

ϵ =

is used where I 1 and J 2 are the first invariant of the strain tensor and the second invariant of the deviatoric strain tensor, respectively. The significant di ff erence in compressive and tensile failure characteristics of quasi-brittle materials is reflected by means of parameter k which is the ratio of compressive to tensile strengths of the material under consideration, see de Vree et al. (1995) for details. Damage evolution at each material point is coupled to enhanced equivalent strain through history variable κ that satisfies the Karush-Kuhn-Tucker (KKT) conditions, ˙ κ ≥ 0, ˜ ϵ ≤ 0and ˙ κ (˜ ϵ − κ ) = 0. These KKT conditions ensure that κ takes the largest value of ˜ ϵ reached so far at a material point. An exponential damage law of the following form D =   0 , if κ ≤ κ 0 1 − κ 0 κ  1 − α + α e − β ( κ − κ 0 )  otherwise (12) is used where κ 0 , α and β are the parameters of the model. Newton-Raphson method is used to solve the resulting non-linear equations for which consistent linearization of coupled integral Equations 10 and 10 is necessary. For the details of this process the reader is referred to Kac¸maz (2022). This element formulation is implemented within the commercial finite element package Abaqus through user defined element (UEL) subroutine, Systemes (2013). Element performance is tested by benchmark problems but not reported here, please see Kac¸maz (2022). In the next sub-section, three-dimensional failure analysis of concrete specimen is conducted with this element implementation. A series of tests on skew notched concrete beams under eccentric loading were conducted at Cardi ff University by Brokenshire, Brokenshire (1996). Relevant details and experimental results are extracted from Je ff erson et al. (2004) in which predictions of Craft concrete model were compared with the torsion tests of Brokenshire. Two di ff erent beam types, namely concrete beams with rectangular and circular cross-sections were tested. Due to limited space, in the following sub-section, only prismatic skew notched concrete beam under eccentric loading is considered. For the analysis of cylindrical beam under similar loading conditions, the interested reader may consult to Kac¸maz (2022).

3.1. Prismatic Skew Notched Concrete Beam Under Eccentric Loading

The specimen geometry and the test setup are shown in Figure 1. It is a concrete beam with skew notch clamped with steel frames at both ends. While one of the arms of steel frame is subjected a point load (at point P in Figure 1), displacement of the other three arms are restricted in vertical direction. Steel frame transfers point load to the concrete beam which results in a torsion dominated loading on the beam. The finite element model and boundary conditions are shown in Figure 2 in which steel frames and concrete beam are of di ff erent colors. A monotonically increasing