# Multiscale and Failure Analysis of Periodic Lattice Structures
## 4 Failure Criteria  
A new failure criterion was proposed recently to predict the failure of different cases, which used to require different failure theories for each case when using the traditional failure criteria. Thus, it was called a unified failure criterion. The new failure criterion is based on two parameters: effective stress and the stress gradient [28; 29]. For failure to occur, both stress and stress gradient conditions must be satisfied simultaneously. The two conditions are expressed as  
\[\sigma_{eff}\geq\sigma_{fail} \tag{12}\]  
and  
\[\sigma_{eff}\geq\left(2E\Big{|}\frac{d\sigma_{eff}}{ds}\Big{|}\kappa_{fail} \right)^{1/3} \tag{13}\]  
where \(\sigma_{eff}\) denotes the effective stress, which is different depending on the material characteristics. For instance, the maximum normal stress is used for brittle isotropic materials. For composite materials, multiple effective stresses can be used depending on the composition of the constituent materials, like fiber and matrix materials. \(\sigma_{fail}\) is the strength of the material, which is obtained from the uniaxial loading tests. \(E\) is the elastic modulus, and \(\kappa_{fail}\) is another material property related to the failure of the material. That is equivalent to the critical energy release rate in fracture mechanics [28]. Finally, \(s\) is the direction of the failure path. Thus, the failure criterion requires information about not only the failure location but also the failure direction. This failure criterion has been validated extensively for brittle materials under both static and cyclic fatigue loading, and is currently under investigation for its applicability to ductile materials. An initial study showed a very positive result. If the effective stress has a negligible stress gradient at the failure location, the stress condition of Equation (12) determines the failure load. This is because the effective stress to satisfy the stress gradient condition is smaller than the effective stress to meet the stress condition. For example, this is the case with the uniaxial loading of the standard dog-bone shape specimens, which have zero stress gradient with uniform stress across their width. On the other hand, if the failure location has a significant stress gradient, like at the edge of a crack or a hole, the stress gradient condition in Equation (13) determines the failure load. In this case, the effective stress from Equation (13) is greater than that from Equation (12). Because both conditions must be satisfied simultaneously for failure to occur, the effective stress from Equation (13) determines the failure load. The set of failure criteria is applied to the lattice members that would fail. The maximum longitudinal stress is determined from Equation (11) for the lattice member that carries the maximum internal load. That stress in Equation (11) is used for potential failure with the stress condition of Equation (12). The stress gradient is also computed from Equation (11), and it is \(\frac{M}{T}\). This stress gradient varies along with the internal moment \(M\). To make the stress gradient independent of the magnitude of the applied loading, the gradient value is normalized in terms of the unit stress at the location of potential failure. Then, the normalized stress gradient is expressed as  
\[\frac{d\sigma}{ds}=\frac{12\sigma_{o}\left(\frac{M}{T}\right)}{t^{2}+6t\left( \frac{M}{F}\right)} \tag{14}\]  
where \(F\) and \(M\) are the internal force and moment applied to the lattice member under consideration for failure, \(t\) is the thickness of the lattice member, and \(\sigma_{o}\) denotes the normalized stress with a unit value like 1 Pa. Equation (14) is used for the stress gradient failure condition of Equation (13). Structural members subjected to both axial and bending loads fail due to the stress gradient condition. That is, the failure stress from the stress gradient condition is greater than that from the stress condition.