PSI - Issue 66

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

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(5) The constants B and m in the RHT model are dimensionless parameters, obtained from curve-fitting experimental data [4]. These constants play a critical role in determining the residual strength of concrete after failure. Specifically, B represents the material inherent fracture strength, reflecting its capacity to withstand high-stress levels before breaking. This constant varies depending on the specific mechanical properties of the material and helps identify the stress level at which fracture occurs, particularly under dynamic loading. On the other hand, the exponent m quantifies how sensitive the material fracture strength is to varying loading rates, allowing the RHT model to capture the effects of different dynamic conditions. The post-failure surface is obtained by linearly interpolating between the failure surface ( ������� ) and the residual surface ( �������� ), with a damage factor ( D ) modulating this transition. This approach reflects the material progressive weakening as it accumulates damage beyond the failure point. ��������� �� 1 � � ������� � �������� (6) The damage parameter (D) is calculated as follows: �� ∑ ∆� � ∆� �������� � ������� � ���� ∑ ∆� � ∆� ���� � ������� � ���� (7) ∆ � ������� � � � ∗ � � ∗ ���� � � � (8) In these expressions, � ∗ ���� is the normalized hydrodynamic tensile limit, expressed as the ratio of tensile strength to compressive strength ( � � � � ). The parameter ∆ � represents the plastic strain increment, while ���� is the minimum failure strain. Additionally, � and � are constants that shape the damage evolution curve. 3. Tensile crack softening damage model This section introduces the concept of crack softening failure, utilizing a damage model that is based on fracture energy principles. In this model, the tensile stress-bearing capacity of material elements diminishes progressively as cracks propagate. The approach employed is typically referred to as a smeared-crack model, where cracks are distributed over the element characteristic length. Cracking initiates when the material exceeds its tensile strength limit. Two critical factors are fracture energy and tensile strength, which are necessary to define the softening curve slope and appear in the stress-strain relationship through a cohesive traction-separation law. The tensile strength parameter dictates when cracking occurs, as it is activated once the principal tensile stress surpasses the concrete inherent tensile strength. Meanwhile, fracture energy outlines the maximum strain the material can withstand before failing. These parameters are essential to simulating concrete dynamic fracture response. Upon reaching its tensile strength limit, the material requires nonlinear modeling for softening behavior, as observed in experimental tests. Results from dynamic experiments, like split-Hopkinson pressure bar tests, show that concrete tensile softening behavior follows a curve more accurately described by an exponential function rather than a linear one. The exponential form offers a more accurate reflection of increased fracture strain for equivalent fracture energy compared to a linear model. Thus, developing a more advanced model for simulating concrete tensile failure is crucial. To address this, a refined tensile crack softening model is presented. It is based on experimental findings and incorporates a power function, specifically the Hordijk-Reinhard formula [20], as follows: � � 1 �� 1 �� � � � � ���� � � � �� � � � � ���� �� � � � ���� � 1 � �� � �� � � (9) Where, ���� is the fracture strain, ̅ � represents the plastic strain increment, and constants � � 3.0 and � � 6.39 are derived from concrete tensile test data. After incorporating this modified damage model via a user-defined