Micro-Mechanics of Failure
Encyclopedia
Micro-Mechanics of Failure (MMF) is a newly proposed methodology, providing a more logical explanation of failure mechanism of continuous fiber reinforced composites
than other phenomenological models such as Tsai-Wu
and Hashin failure criteria. The basic concept of MMF is to perform failure analysis of fiber reinforced composites, based upon micro stresses in each constituent (fiber, matrix, and fiber-matrix interface) calculated from macro stresses at the ply level. The hierarchy of micromechanics-based analysis procedure for composite structures is shown in Figure 1: starting from mechanical behavior of constituents, i.e., the fiber, the matrix, and the interface, the mechanical behavior of a ply, a laminate, and eventually a structure, can be predicted. At the constituent level, three elements are required to fully characterize each constituent:
The constituents and a unidirectional (UD) 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.
. 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 :
, 
, and 
are columm 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.
:
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.
and 
represent matrix tensile and compressive strength, respectively; whereas 
and 
are von Mises equivalent stress and the first stress invariant of micro stresses at a point within matrix, respectively.
where
and 
are normal (perpendicular to the interface) and shear (tangential to the interface) interfacial tractions, with 
and 
being their corresponding strengths. The angle brackets (Macaulay brackets) imply that a pure compressive normal traction does not contribute to interface failure.
for a carbon/epoxy UD ply are plotted in Figure 3, with test data superimposed. Failed constituent can be directly observed from the failure envelope predicted by MMF, while the failure envelope predicted by Tsai-Wu
is unable to yield such information.
Fibre-reinforced plastic
Fibre-reinforced plastic is a composite material made of a polymer matrix reinforced with fibres. The fibres are usually fibreglass, carbon, or aramid, while the polymer is usually an epoxy, vinylester or polyester thermosetting plastic...
than other phenomenological models such as Tsai-Wu
Tsai-Wu failure criterion
The Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...
and Hashin failure criteria. The basic concept of MMF is to perform failure analysis of fiber reinforced composites, based upon micro stresses in each constituent (fiber, matrix, and fiber-matrix interface) calculated from macro stresses at the ply level. The hierarchy of micromechanics-based analysis procedure for composite structures is shown in Figure 1: starting from mechanical behavior of constituents, i.e., the fiber, the matrix, and the interface, the mechanical behavior of a ply, a laminate, and eventually a structure, can be predicted. At the constituent level, three elements are required to fully characterize each constituent:
- Constitutive relationConstitutive equationIn physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
, which describes the transient or time-independent response of the constituent to external mechanical as well as hygrothermal loadings; - Master curve, which describes the time-dependent behavior of the constituent under creep or fatigue loadings;
- Failure criterionFailure theory (material)Failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure or ductile failure . Depending on the conditions most materials can fail in a brittle or ductile...
, which describes conditions that cause failure of the constituent.
The constituents and a unidirectional (UD) 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(Figure 2), 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 (Figure 2). 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)Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
. 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 :, , and are columm 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 FEAFinite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
of the unit cell model under given macroscopic loading conditions. The definition of SAF is valid not only for constituents having linear elastic
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
behavior and constant coefficients of thermal expansion (CTE), but also for those possessing complex constitutive relations
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
and variable CTEs.
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 criterionTsai-Wu failure criterion
The Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...
:
| Definition of coefficients in the quadratic failure criterion |
|---|
Coefficients involved in the quadratic failure criterion are defined as follows: where  ,  ,  ,  ,  , and  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.
Matrix failure criterion
The polymeric matrix is assumed to be isotropic and exhibits a higher strength under uniaxial compression than under uniaxial tension. A modified version of von Mises failure criterion suggested by Christensen is adopted for the matrix: and represent matrix tensile and compressive strength, respectively; whereas and are von Mises equivalent stress and the first stress invariant of micro stresses at a point within matrix, respectively.Interface failure criterion
The fiber-matrix interface features traction-separation bahavior, and the failure criterion dedicated to it takes the following form :where
and are normal (perpendicular to the interface) and shear (tangential to the interface) interfacial tractions, with and being their corresponding strengths. The angle brackets (Macaulay brackets) imply that a pure compressive normal traction does not contribute to interface failure.Comparison between MMF and other failure criteria
As a completely mechanics-based failure theory, MMF outperforms other failure criteria in that it is able to distinguish the critical constituent in the critical ply in a composite laminate. The failure envelopes generated by MMF and Tsai-Wu failure criterionTsai-Wu failure criterion
The Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...
for a carbon/epoxy UD ply are plotted in Figure 3, with test data superimposed. Failed constituent can be directly observed from the failure envelope predicted by MMF, while the failure envelope predicted by Tsai-Wu
Tsai-Wu failure criterion
The Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...
is unable to yield such information.
Further extension of MMF
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.See also
- Composite materialComposite materialComposite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
- Strength of materialsStrength of materialsIn materials science, the strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a...
- Failure theory (material)Failure theory (material)Failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure or ductile failure . Depending on the conditions most materials can fail in a brittle or ductile...
- Tsai-Wu failure criterionTsai-Wu failure criterionThe Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...
- Christensen Failure CriterionChristensen Failure CriterionFailure criteria for isotropic materials has been a long standing problem. Despite many historical attempts, there have not been any successful general forms that span the range from ductile to brittle materials. A new failure criterion that aims to do so has been recently developed...