# Interface Crack Approaching a Three-Material Joint
## 2 Problem Formulation  
The problem of an interface crack between materials A and B which approaches the joint of three materials, A, B and C, as shown in Figure 1a, is considered. The crack is at a distance \(a\) from the joint. Two cases can occur: the crack could continue to propagate along the interface between materials A and C, as in Figure 1b, or it could deflect into the interface between materials B and C and continue to propagate along it, as in Figure 1c. Distance \(a\) in Figure 1a is so chosen to represent the characteristic length for the considered problem, and could be the plastic zone size, the cell size in cellular materials, or the thickness of the adhesive layer in glued materials, etc., as considered in [PERSON] and [PERSON] [9]. For calculations in this paper, length \(a\) is selected as the size of the plastic zone in material B. In the following analysis, variables related to particular interfaces are denoted by subscripts containing notation of the two materials constituting them, i.e., variables for the interface between materials A and B have the subscript AB, variables for the interface between materials A and C have the subscript AC, and materials for the interface between materials B and C have the subscript BC. The energy release rate for the crack at the interfaces AB, AC, and BC, expressed via the corresponding stress intensity factors, can be calculated as in [PERSON] and [PERSON] [4] as  
\[ \begin{array}{ll}G_{AB}&=\ \frac{\left(\frac{1-\gamma_{A}}{P_{A}}+\frac{1- \gamma_{B}}{P_{B}}\right)}{4\cosh^{2}(\pi_{AE})}\cdot K_{AB}\overline{K}_{AB}\\ G_{AC}&=\ \frac{\left(\frac{1-\gamma_{A}}{P_{A}}+\frac{1-\gamma_{C}}{P_{C}}\right)}{4 \cosh^{2}(\pi_{AE})}\cdot K_{AC}\overline{K}_{AC}\\ G_{BC}&=\ \frac{\left(\frac{1-\gamma_{B}}{P_{B}}+\frac{1-\gamma_{C}}{P_{C}}\right)}{4 \cosh^{2}(\pi_{BC})}\cdot K_{BC}\overline{K}_{BC}\end{array}  \tag{1}\]  
respectively, where \(K_{ij}=K_{1,ij}+iK_{2,ij}\) is the complex stress intensity factor, with subscripts \(ij\) denoting the materials in the joint, i.e., the corresponding interface. Variables \(\varepsilon_{AB}\), \(\varepsilon_{AC}\), and \(\varepsilon_{BC}\) represent the oscillatory index for interfaces AB, AC, and BC, respectively, being defined as in Rice [14] as  
\[\varepsilon_{ij}\ =\ \frac{1}{2\pi}\ln\left(\frac{1-\beta_{ij}}{1+\beta_{ij}}\right) \tag{2}\]  
where \(\beta_{ij}\) is one of the two Dundurs parameters as in Dundurs [39], which are defined as  
\[\alpha\ =\ \frac{\mu_{j}(\kappa_{i}+1)-\mu_{i}(\kappa_{j}+1)}{\mu_{j}(\kappa_{ i}+1)+\mu_{i}(\kappa_{j}+1)},\ldots,\beta\ =\ \frac{\mu_{j}(\kappa_{i}-1)-\mu_{i}(\kappa_{j}-1)}{\mu_{j}(\kappa_{i}+1)+\mu_{ i}(\kappa_{j}+1)}, \tag{3}\]  
where \(\mu_{i}=E_{i}/2(1+\
u_{i})\) is the shear modulus, \(E_{i}\) is the Young elasticity modulus, \(\
u_{i}\) is the Poisson ratio of material \(i\)(\(i\) = A, B, C), \(\kappa_{i}\) = 3-4, \(vi\) for the plane strain state, \(\kappa_{i}=(3-\
u_{i})/(1+\
u_{i})\) for the plane stress state, and \(ij\) again stands for the corresponding interface parameter. Figure 1: **(a)** Interface crack approaching a three-material joint; **(b)** crack propagating along the AC interface; **(c)** crack deflecting into the BC interface. The mixed mode for the crack that deflected into the BC or AC interface can be calculated as the phase angle of the complex stress intensity factor (also known as the mode mixity) according to the formulation given by [PERSON] and [PERSON] [9], i.e.,  
\[\psi_{BC}\ =\ \arctan\left(\frac{\text{Im}(K_{BC}a^{ie_{BC}})}{\text{Re}(K_{BC}a ^{ie_{BC}})}\right),\psi_{AC}\ =\ \arctan\left(\frac{\text{Im}(K_{AC}a^{ie_{AC}})}{\text{Re}(K_{AC}a^{ie_{AC}})}\right), \tag{4}\]  
where \(l\) is the characteristic length, which for the considered case is \(a\). In general, the mode mixity defines the share of Mode 2 with respect to Mode 1 of the crack propagation, i.e., whether the crack will propagate in an in-plane shearing or tearing manner.