PSI - Issue 28

Bahador Bahrami et al. / Procedia Structural Integrity 28 (2020) 829–835 Bahrami et al./ Structural Integrity Procedia 00 (2020) 000–000

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3. Fracture theory To scrutinize the onset of fracture of SENB specimens, an appropriate fracture criterion must be employed. Hence, the strain energy density (SED) and its generalized form (GSED) criteria are briefly introduced in this section to be later utilized for fracture assessment of PMMA dental material. The stress field around the crack in an isotropic body was proposed by Williams as an asymptotic solution (Williams (1957)). The following equations represent the truncated form of this equation in which the singular and the first non-singular terms of the stress field are taken into account: �� � √2 1 � � � � �1 � � � ⁄2�� � �� � 3 2 � 2 � ��� � � �� � √2 1 � � � � � � ⁄2� � 3 2 �� � � � �� � 2√ 1 2 � �� � � �� �3 � 1�� � (1) Based on the singular terms of Eq. 1, Sih, (1974) presented the conventional SED criterion. However, Ayatollahi et al. (2015) showed that in some cases the accuracy of the SED criterion proposed by Sih is not adequate and tried to improve the predictions of this model. Ayatollahi et al. (2015) presented the GSED criterion which also takes into account the first non-singular term of crack stress field, i.e. the T-stress, and derived the strain energy W c as: � � 1 � � � � � � � � � � � 2 � � �� � 2 � � �2 � � � � 2 � � �2 � � �� � � � �2 � � � � (2) where in the above equation, parameters A 1 -A 6 are constants reported in (Ayatollahi et al. (2015)). Also, r c is the critical distance within which the material is damaged due to dense microcracking. As can be seen in Eq. 2, the parameter r c only exists in the terms containing A 4 , A 5 , A 6 . This shows that the influence of the theory of critical distance (TCD) can be manifested in the crack growth criterion only when the value of T-stress is non-zero. In other words, the concept of TCD loses its importance in the conventional SED criterion which ignores the effect of T term. The GSED criterion, however, considers the influence of both parameters, namely T and r c , and therefore the GSED criterion is expected to yield more accurate predictions in comparison with the conventional SED. Eq. 2 can be rewritten under mode I condition and for specimens with negligible values of T-stress as: � � � � � � � �� � �� ���� � � � � � � � ��� � ��� � � � � 3 � �� � � � � � � � � � � � � (3) Noting that the critical strain energy is a material parameter, one can acquire W c for mode I from Eq. 3 and then insert it into Eq. 2, and finally solve for �� � and �� �� which yields the fracture initiation rule based on the GSED criterion: �� � � � 1 � � � � � � � � � � � 2 � �� � � 2 � � ∗ � � 2 � � � 2 � � ∗ � � 2 � � �� � � � � ∗ � � 2 � � � � (4) �� �� � � 1 � � � � � � � � � � � 2 � � �� � 2 � � ∗ �� � 2 � � � �� � 2 � � ∗ �� � 2 � � � � � ∗ �� � 2 � � � � Here, Y I , Y II and T * are the normalized form of K I , K II and T with the following relations: � � � � � ��� √ , �� � �� � � ��� √ , � ∗ � � ��� (5) in which P f is the recorded fracture load, and W and t are the width and thickness of the experimented SENB