Micro-mechanics of failure
The theory of micro-mechanics of failure aims to explain the failure of continuous fiber reinforced composites by micro-scale analysis of stresses within each constituent material (such as fiber and matrix), and of the stresses at the interfaces between those constituents, calculated from the macro stresses at the ply level.
[1] As a completely mechanics-based failure theory, the theory is expected to provide more accurate analyses than those obtained with phenomenological models such as Tsai-Wu[2] and Hashin[3][4] failure criteria, being able to distinguish the critical constituent in the critical ply in a composite laminate.
The basic concept of the micro-mechanics of failure (MMF) theory is to perform a hierarchy of micromechanical analyses, starting from mechanical behavior of constituents (the fiber, the matrix, and the interface), then going on to the mechanical behavior of a ply, of a laminate, and eventually of an entire structure.
Starting from the constituent level, it is necessary to devise a proper method to organize all three constituents such that the microstructure of a UD lamina is well-described.
Either array can be viewed as a repetition of a single element, named unit cell or representative volume element (RVE), which consists of all three constituents.
With periodical boundary conditions applied,[5] a unit cell is able to respond to external loadings in the same way that the whole array does.
Therefore, a unit cell model is sufficient in representing the microstructure of a UD ply.
Stress distribution at the laminate level due to external loadings applied to the structure can be acquired using finite element analysis (FEA).
Stresses at the ply level can be obtained through transformation of laminate stresses from laminate coordinate system to ply coordinate system.
To further calculate micro stresses at the constituent level, the unit cell model is employed.
at any interfacial point, are related to ply stresses
The SAF serves as a conversion factor between macro stresses at the ply level and micro stresses at the constituent level.
The value of each single term in the SAF for a micro material point is determined through FEA of the unit cell model under given macroscopic loading conditions.
The definition of SAF is valid not only for constituents having linear elastic behavior and constant coefficients of thermal expansion (CTE), but also for those possessing complex constitutive relations and variable CTEs.
denote longitudinal tensile, longitudinal compressive, transverse tensile, transverse compressive, transverse (or through-thickness) shear, and in-plane shear strength of the fiber, respectively.
Stresses used in two preceding criteria should be micro stresses in the fiber, expressed in such a coordinate system that 1-direction signifies the longitudinal direction of fiber.
A modified version of von Mises failure criterion suggested by Christensen[7] is adopted for the matrix: Here
represent matrix tensile and compressive strength, respectively; whereas
The fiber-matrix interface features traction-separation behavior, and the failure criterion dedicated to it takes the following form:[8]
The angle brackets (Macaulay brackets) imply that a pure compressive normal traction does not contribute to interface failure.
These criteria were originally developed for unidirectional polymeric composites, and hence, applications to other type of laminates and non-polymeric composites have significant approximations.
Usually Hashin criteria are implemented within two-dimensional classical lamination approach for point stress calculations with ply discounting as the material degradation model.
The criteria are extended to three-dimensional problems where the maximum stress criteria are used for transverse normal stress component.
The failure modes included in Hashin's criteria are as follows.
where, σij denote the stress components and the tensile and compressive allowable strengths for lamina are denoted by subscripts T and C, respectively.
XT, YT, ZT denotes the allowable tensile strengths in three respective material directions.
Similarly, XC, YC, ZC denotes the allowable compressive strengths in three respective material directions.
Further, S12, S13 and S23 denote allowable shear strengths in the respective principal material directions.
Endeavors have been made to incorporate MMF with multiple progressive damage models and fatigue models for strength and life prediction of composite structures subjected to static or dynamic loadings.
