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.
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 and Hashin failure criteria, being able to distinguish the critical constituent in the critical ply in a composite laminate.
Basic concepts
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.
At the constituent level, three elements are required to fully characterize each constituent: The constitutive relation, which describes the transient, or time-independent, response of the constituent to external mechanical as well as hygrothermal loadings; The master curve, which describes the time-dependent behavior of the constituent under creep or fatigue loadings; The failure criterion, which describes conditions that cause failure of the constituent. The constituents and a unidirectional lamina are linked via a proper micromechanical model, so that ply properties can be derived from constituent properties, and on the other hand, micro stresses at the constituent level can be calculated from macro stresses at the ply level.
Unit cell model
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. In reality, all fibers in a UD ply are aligned longitudinally; however, in the cross-sectional view, the distribution of fibers is random, and there is no distinguishable regular pattern in which fibers are arrayed. To avoid such a complication cause by the random arrangement of fibers, an idealization of the fiber arrangement in a UD lamina is performed, and the result is the regular fiber packing pattern. Two regular fiber packing patterns are considered: the square array and the hexagonal array. 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, 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 amplification factor (SAF)
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. Micro stresses at any point within fiber/matrix, and micro surface tractions at any interfacial point, are related to ply stresses as well as temperature increment through:
Here , , and are column vectors with 6, 6, and 3 components, respectively. Subscripts serve as indications of constituents, i.e. for fiber, for matrix, and for interface. and are respectively called stress amplification factors (SAF) for macro stresses and for temperature increment. The SAF serves as a conversion factor between macro stresses at the ply level and micro stresses at the constituent level. For a micro point in fiber or matrix, is a 6×6 matrix while has the dimension of 6×1; for an interfacial point, respective dimensions of and are 3×6 and 3×1. 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.
Constituent failure criteria
Fiber failure criterion
Fiber is taken as transversely isotropic, and there are two alternative failure criteria for it: a simple maximum stress criterion and a quadratic failure criterion extended from Tsai-Wu failure criterion:
The Coefficients involved in the quadratic failure criterion are defined as follows: