Micro-mechanics of failure¶

A multiscale composite-failure theory predicting constituent and interface failure from ply-level stresses.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5572
Origin domain: composite mechanics
Subdomain: composite mechanics

Core Idea¶

Fiber, matrix and interface constitutive laws and failure criteria are resolved separately, and homogenization and localization assumptions connect constituent, ply, laminate and structure scales.^[1] Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of composite mechanics. It is the domain-specific identity fixed by the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. The evidential layer asks what observation or proof warrants the claim: type the carrier, state every parameter and convention in the definition, test that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

Structural Signature¶

Carrier: the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases
Inputs or antecedent state: the exact composite mechanics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Micro-mechanics of failure
Constitutive operation: Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply.
Invariant: the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit
Recognition test: type the carrier, state every parameter and convention in the definition, test that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases
Output or consequence: recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of composite mechanics. The field contains many questions and methods that do not instantiate Micro-mechanics of failure.
It is not its most familiar example. A canonical instance directly demonstrates that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Phenomenological laminate failure criterion. A phenomenological criterion fits ply-scale stress combinations; micro-mechanics of failure resolves stresses and criteria at constituent and interface scale.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Micro-mechanics of failure must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside composite mechanics, the vocabulary and validity conditions do not transfer literally.

Scope of Application¶

Micro-mechanics of failure belongs to composite mechanics and is useful where the analyst can specify the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases, then evaluate the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. The scope is broad within that domain but bounded by the need for the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. Conceptual engineering-model identity only; structural safety decisions require qualified validation.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact composite mechanics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Micro-mechanics of failure are converted, constrained, or organized by Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Micro-mechanics of failure must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Micro-mechanics of failure can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated. The disciplined statement is: given the exact composite mechanics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Micro-mechanics of failure, the structure counts as Micro-mechanics of failure exactly when the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Micro-mechanics of failure. Micro-mechanics of failure compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Micro-mechanics of failure. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Derive consequences. From the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit, infer recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Micro-mechanics of failure must control the decision and an object that resembles Micro-mechanics of failure in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of composite mechanics because they reuse the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases, Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply., and type the carrier, state every parameter and convention in the definition, test that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. to An applied instance preserves the same invariant under changed notation, scale, jurisdiction, dataset, or implementation..^[3]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Micro-mechanics of failure, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit. The example exposes the carrier and directly tests that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases; the operative rule is Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply.; the invariant is the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit; and the result supports recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit destroys the classification.

Mapped back: the typed composite mechanics carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases → Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply. → the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit → recognizing and comparing instances of Micro-mechanics of failure, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the same invariant under changed notation, scale, jurisdiction, dataset, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—type the carrier, state every parameter and convention in the definition, test that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases—can be run and because the same failure boundary—the carrier is mistyped, the condition that the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Micro-mechanics of failure, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Micro-mechanics of failure, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from composite mechanics and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, Macroscopic loads are localized into microscale stress fields, compared with constituent-specific failure surfaces and propagated upward to identify the critical mode and ply., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Micro-mechanics of failure, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Micro-mechanics of failure, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in composite mechanics.

The proposed strict upward parent is prime:hierarchical_decomposability. prime:hierarchical_decomposability is the nearest broader Prime; the source-domain invariant supplies the autonomous residual. This is a proposal-only workspace relationship: the accepted Prime supplies a genuinely instantiated structural prerequisite or superclass, while Micro-mechanics of failure adds domain-specific constraints.

The entry does not collapse into that parent because the domain-specific identity fixed by the laminate and loading, constituent geometry and properties, homogenization and localization scheme, interface model, constituent criteria, scale transitions, critical mode, uncertainty and validation domain are explicit It also declines a nearby thematic catalog node: the neighbor does not literally subsume the constitutive identity of Micro-mechanics of failure. This explicit assert-and-decline pattern keeps the proposed DAG narrow and prevents a merely thematic edge.

The prospective workspace queue contains one strict upward edge to prime:hierarchical_decomposability. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Micro-mechanics of failure Domain-specific

Parents (1) — more general patterns this builds on

Micro-mechanics of failure is a kind of Hierarchical Decomposability Prime

The proposed strict upward parent is prime:hierarchical_decomposability.

Neighborhood in Abstraction Space¶

Micro-mechanics of failure sits in a crowded region of the domain-specific corpus (34^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Structural Mechanics & Failure (25 abstractions)

Nearest neighbors

Elastic instability — 0.91
Structural mechanics — 0.91
Stress concentration — 0.90
Implosion (mechanical process) — 0.90
Minimum total potential energy principle — 0.90

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Phenomenological laminate failure criterion. A phenomenological criterion fits ply-scale stress combinations; micro-mechanics of failure resolves stresses and criteria at constituent and interface scale.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Micro-mechanics of failure. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Micro-mechanics of failure. An extension qualifies only when its changed axioms and retained invariant are stated.

References¶

[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 and Hashin failure criteria, being able to distinguish the critical constituent in the critical ply in a composite laminate. thumb|right|300px|alt=Comparison between theoretical failure predictions and test data|Failure envelopes generated by MMF and the Tsai-Wu failure criterion for a carbon/epoxy UD ply, with test data superimposed. Failed constituent envelopes are predicted by MMF but not by Tsai-Wu. 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 == thumb|right|300px|alt=Schematic illustration of idealized fiber arrays and their corresponding unit cells|Schematic illustration of idealized fiber arrays and their corresponding unit cells. 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 \sigma at any point within fiber/matrix, and micro surface tractions t at any interfacial point, are related to ply stresses \bar{\sigma} as well as temperature increment \Delta T through: : \begin{array}{lcl} \sigma_{\mathrm{f}}&=&M_{\mathrm{f}}\bar{\sigma} + A_{\mathrm{f}}\Delta T\ \sigma_{\mathrm{m}}&=&M_{\mathrm{m}}\bar{\sigma} + A_{\mathrm{m}}\Delta T\ t_{\mathrm{i}}&=&M_{\mathrm{i}}\bar{\sigma} + A_{\mathrm{i}}\Delta T \end{array} Here \sigma , \bar{\sigma} , and t are column vectors with 6, 6, and 3 components, respectively. Subscripts serve as indications of constituents, i.e. {\mathrm{f}} for fiber, {\mathrm{m}} for matrix, and {\mathrm{i}} for interface. M and A 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, M is a 6×6 matrix while A has the dimension of 6×1; for an interfacial point, respective dimensions of M and A 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: : \begin{array}{lcl} \text{Maximum stress failure criterion:}-X^\prime_{\mathrm{f}} The Coefficients involved in the quadratic failure criterion are defined as follows: : F_{11} = \cfrac{1}{X_{\mathrm{f}}X^\prime_{\mathrm{f}}} , F_{22} = F_{33} = \cfrac{1}{Y_{\mathrm{f}}Y^\prime_{\mathrm{f}}} : F_{44} = \cfrac{1}{S_{\mathrm{f}4}^2} , F_{55} = F_{66} = \cfrac{1}{S_{\mathrm{f}6}^2} : F_{1} = \cfrac{1}{X_{\mathrm{f}}} - \cfrac{1}{X_{\mathrm{f}}^\prime} , F_{2} = F_{3} = \cfrac{1}{Y_{\mathrm{f}}} - \cfrac{1}{Y_{\mathrm{f}}^\prime} : F_{12} = F_{21} = F_{13} = F_{31} = -\cfrac{1}{2\sqrt{X_{\mathrm{f}} {X}{\mathrm{f}}^\prime Y}}Y_{\mathrm{f}}^\prime}} , F_{23} = F_{32} = -\cfrac{1}{2Y_{\mathrm{f}}Y_{\mathrm{f}}^\prime} where X_{\mathrm{f}} , X_{\mathrm{f}}^\prime , Y_{\mathrm{f}} , Y_{\mathrm{f}}^\prime , S_{\mathrm{f}4} , and S_{\mathrm{f}6} 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: : \begin{array}{lcl} \cfrac{\sigma_{Mises}^2}{C_{\mathrm{m}}T_{\mathrm{m}}} + \left(\cfrac{1}{T_{\mathrm{m}}} - \cfrac{1}{C_{\mathrm{m}}}\right)I_1 = 1 \end{array} Here {T{\mathrm{m}} and {C}}} represent matrix tensile and compressive strength, respectively; whereas \sigma_{Mises} and {\mathrm{I}1 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 behavior, and the failure criterion dedicated to it takes the following form: \begin{array}{lcl} \left(\cfrac{\left\langle{t}}\right\rangle} \right)^2 + \left(\cfrac \right)^2 = 1 \end{array} where {t{n} and {t}} are normal (perpendicular to the interface) and shear (tangential to the interface) interfacial tractions, with {Y{n} and {Y} being their corresponding strengths. The angle brackets (Macaulay brackets) imply that a pure compressive normal traction does not contribute to interface failure. == Further extension of MMF == === Hashin’s Failure Criteria === These are interacting failure criteria where more than one stress components have been used to evaluate the different failure modes. 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. Failure indices for Hashin criteria are related to fibre and matrix failures and involve four failure modes. 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. # Tensile fibre failure for σ11 ≥ 0 # Compressive fibre failure for σ11 0 # Compressive matrix failure for σ22 + σ33 0 # Interlaminar compression failure for σ33 Ha, S.K., Jin, K.K. and Huang, Y. (2008). Micro-Mechanics of Failure (MMF) for Continuous Fiber Reinforced Composites, Journal of Composite Materials, '42'(18): 1873–1895. registry ↩a ↩b

[2] Tsai, S.W. and Wu, E.M. (1971). A General Theory of Strength for Anisotropic Materials, Journal of Composite Materials, '5'(1): 58–80. registry ↩a ↩b

[3] Hashin, Z. and Rotem, A. (1973). A Fatigue Failure Criterion for Fiber Reinforced Materials, Journal of Composite Materials, '7'(4): 448–464. registry ↩