PSI - Issue 66

Sobhan Pattajoshi et al. / Procedia Structural Integrity 66 (2024) 167–174 Pattajoshi et al./ Structural Integrity Procedia 00 (2025) 000–000

169

3

� δ B

���

Compressive meridian

Third invariant dependence term Compressive strain rate exponent Tensile strain rate exponent Fractured strength constant Fractured strength exponent

m D ∆ � ∆ � ������� ���� � ∗ ���� � ���� ̅ �

Damage parameter

Increment in equivalent plastic strain

Equivalent plastic strain to fracture under constant presssure

Minimum limit of fracture strain Hydrodynamic tensile stress limit Tensile crack softening damage model

Fracture strain

Plastic strain increment

2. Riedel-Hiermaier-Thoma (RHT) model The Riedel–Hiermaier–Thoma (RHT) model is widely used to simulate the behavior of concrete, accounting for several factors such as pressure hardening, strain hardening, strain rate effects, strain softening, and dependence on the third stress invariant [19]. The model distinguishes three main pressure-dependent surfaces in stress space: the elastic limit surface, the failure surface, and the residual surface. The elastic limit surface ( ������� ) governs material behavior in the elastic region, and its boundary conditions are defined through the parabolic cap function ( ��� ). This function integrates the porous equation of state ( p −α ), particularly at high pressures, to simulate pore collapse and compaction. As the stress increases, the material exhibits strain hardening until it reaches the failure surface ( ������� ). Beyond this, the post-failure behavior is described by the residual surface ( �������� ), with damage progression interpolated using a damage index ( D ). The elastic limit surface ( ������� ) is calculated by scaling down the failure surface ( ������� ) as shown in the equation below: ������� � ������� � ������� � ��� (1) ������� � � � , ������� � � , �������� � � , ������� � � , �������� (2) Here, � , ������� refers to the elastic tensile strength, while � , �������� is the ultimate tensile strength. The failure surface ( ������� ) can be expressed as: ������� � ��� � ∗ �� � ���� ���� � � � (3) In this equation, ��� represents the compressive meridian, ∗ is the normalized pressure ( p/ � ), � describes the third invariant dependence, and ���� � � � accounts for the dynamic increase factor. The dynamic increase factor (DIF) for compression and tension is given by: ���� � � � � � � �� � �� � � � � � ∗ � � � � � � �� � � � � � ∗ � � � � (4) Here, α and δ are the exponents that describe the material response to strain rate during compression and tension, respectively. For residual strength, the following equation is used: