Fatigue, Damage and Failure of Composite Materials:

Mechanisms, Fatigue Life Diagrams and Life Prediction

Ramesh TalrejaDepartment of Aerospace Engineering

Department of Materials Science and EngineeringTexas A&M University, College Station, Texas, USA

UTMIS Autumn Course, Gothenburg, Sweden, 15-­‐16 October 2019

Lecture 5 : DAMAGE MECHANICS

Contents

•Multiscale approach to damage and failure•Macro damage mechanics•Micro damage mechanics• Synergistic damage mechanics• Defect damage mechanics• Virtual testing, computational micromechanics

Multi-­‐scale analysis -­‐ Methodology

5 55𝜎"

Loading direction

Near the 0° ply at the free edge((y, z)=(W, h))

Central plane of the free edge((y, z)=(W, 0))

Internal section((y, z)=(0, 0))

Cracking positions

W

1/8model

h

(y, z)=(0, 0)

L

Lxx¥e!0°ply

90°ply

Resin

xy

z

Macro

Micro

Macro damage mechanics (also called continuum damage mechanics, CDM)

Damage

Initiation

Homogenization

Undamaged continuum

Continuum with damage

Homogenized continuum

uS

tSt

uuS

tSt

uuS

tSt

u

Early concepts of damage as an “internal state”, Kachanov (1958)

dd

m

Atf s

fæ ö

= - ç ÷è ø

Kachanov considered creep rupture of metals. He assumed the internal structure was degrading in time (becoming discontinuous) and introduced an internal variable calleddiscontinuity, f, f = 1 at t = 0, whose rate of change was assumed as a power law:

Robotnov later (1969) defined damage as w = 1 - f as a damage parameter representingnet area reduction. Thus, 0 < w < 1 was born as an internal state variable.

Today, “damage” denoted by D has become an abused concept, and D is used arbitrarilyas needed, mostly driven by convenience.

Damage as an internal variable for composite materials (Talreja, 1985)

P

na

Stationary microstructure

Evolving

microstructure

Homogenization of stationary microstructure

RVE

Homogenized continuum

with damage

Damage entity

Note:1. Internal damage in compositematerials is in the form of distributed, orientedmicrocracks.2. All variables, stress, strain anddamage must be described usinga representative volume element(RVE).

Damage characterization for composites

RVE Damage entity

o

x1

x2

x3

Pn

a

S

Damage Entity Tensor

Damage Mode Tensor

dij i jSd a n S= ò

Note: Instead of a tensora vector can be used(see Talreja, Proc. R. Soc,1985)

2-­‐Sep-­‐19

Damage Mode Tensors

kk+1

kt kl

RVE of volume Vwhere kα is the number of damage entities in the αth mode

na

S

a: crack opening displacementb: crack sliding displacement

Assume b = 0

Damage Tensor Components(One Damage Mode)

(one damage mode)

κ (kappa): Constraint parameter

Computational Materials Engineering lab, Univ Toronto. Canada

Three Damage Modes:Cracking in θ, -­‐θ, and 90° Plies

Singh and Talreja, 2013

Multiscale Synergistic Damage Mechanics (SDM)

Constraint Parameter κθ

Damage Constants forreference laminate ai

Continuum Damage Mechanics

Multiscale Modeling

Overall Structural Analysis

ComputationalMicromechanics

Experiments/ Numerical

FE Analysis

Macro-­‐level

Micro-­‐level

Meso-­‐ level StiffnessChanges

Results: quasi-­‐isotropic laminate

Longitudinal Modulus Poisson’s Ratio

Refs. Singh & Talreja, Mech Mat (2009) ; Singh & Talreja, Int J Solids Struct (2008)

Steps1. Fit the damage model with experimental data for crossply laminateèGives us phenomenological constants ai2. Compute constraint parameters by calculating CODs from FEM 3. Employ the model for quasi-­‐isotropic laminate.

Computational structural analysis

PRELIMINARY STRESS ANALYSIS

INPUTGeometry, laminate configuration, material

properties and loading conditions

Create geometric model Meshing Apply loading &

service conditions Stress analysis

DAMAGE ANALYSIS

Identify regions where damage might have

developed

Predict damage initiation & evolution

Evaluate stiffness properties of

damaged regions

UPDATED STRESS ANALYSIS

OUTPUTFailure characteristics, stress-­‐strain response, deformation behavior, life and durability

Update stiffness of damaged regions Perform stress analysis of whole structure again

Analytical/TheoryProvide a mathematical representation of physical

mechanisms;; predict for similar cases

ExperimentsProvide understanding of real material behavior;;

Calibrate, verify/validate models

ComputationalCombined with accurate

modeling provide predictions for complex configurations;;

Virtual testing;; Multiscale analysis

SYNERGISTIC DAMAGE MECHANICS

(SDM)Analytical/TheoryProvide a mathematical representation of physical

mechanisms;; predict for similar cases

ExperimentsProvide understanding of real material behavior;;

Calibrate, verify/validate models

ComputationalCombined with accurate

modeling provide predictions for complex configurations;;

Virtual testing;; Multiscale analysis

SYNERGISTIC DAMAGE MECHANICS

(SDM)

SDM methodology

The “Big Picture”

Process'modeling,Simulation,

Tooling,'assembly,…

1.#Manufacturing

Real'initial'and'current'material'state'(RIMS'+'RCMS): Microstructure,'

Defects,'RVE

2.#Material

5.#Performance

Integrity,'Durability,Damage'tolerance

Cost/PerformanceTrade:offs#

Specification

PhysicalModeling

Multiphysics excitation(Mechanical,'thermal,'electromagnetic'etc.)

4.#Loading

Length'scale,'Shape,'Boundary'conditions

3.#Geometry • Define Material State (RIMS):

-­‐ Fiber misalignment-­‐ Fiber waviness-­‐ Ply waviness-­‐ Matrix voids• Construct RVE• Apply ply level

boundary conditions to RVE surfaces

Defect Damage Mechanics

• Fiber Defects

Ø Misalignment, waviness

Ø Breakage

• Matrix Defects

Ø Incomplete curing

Ø Voids

• Interface Defects

Ø Fiber/matrix disbonds

Ø Delamination

• Fiber volume fraction

• Fiber Distribution

Ø Length

Ø Orientation

Idealized models:

Heterogeneities, no defects

Real composites: Defects

Cost-­‐performance trade-­‐off

Manufacturing Defects: NonuniformFiber Distribution and Matrix Voids

Debond link-­‐up to transverse cracking

Simulation of manufacturing induced fiber distribution nonuniformityApproach 1: Quantification of fiber mobility (radial and angular)

Dry fiber bundle

Resininfusion

Sudhir and Talreja, ASC, 2017

Initial uniformpattern

Intermediate stepsto “shake” fibers

Final nonuniformpattern

Elnekhaily and Talreja, CST, 2018

Simulation of manufacturing induced fiber distribution nonuniformityApproach 2: Quantification of degree of nonuniformity

RVE realizations for different degrees of nonuniformity 100% Nonuniformity 60% Nonuniformity 30% Nonuniformity

(a) 40% fiber volume fraction

(b) 30% fiber volume fraction

Minimum RVE sizeApproach 1: Pair distribution function G(r)

NOR=5

NOR: Number of Rings around the central fiber

𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°

NOR=9

As NOR ≥ 9 the avg G function stabilizes è the min size of RVEis NOR=9

Minimum RVE sizeApproach 2: Stabilization of nearest neighbor statistics

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 100 200 300 400 500 600

Freq

uenc

y M

ean

Val

ue

Number of Fibers-RVE size

Stable statistical content

24 x 24 fibers RVE20 x 20 fibersInner window

Stress analysis for Approach 1: Embedded cell method

R R

0 0.2 0.4 0.6 0.8 1 1.2Normalized Traction

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thickness of the RVE

Traction at the BoundaryL=2.05RL=2.10RL=2.20RL=2.50RL=3.00RL=4.0RL=8.0RL=12RL=24RL=32RL=40R

Δ𝑇 = −82,Uniform boundary displacement

Size of the embedding composite determinedby Hill’s criterion: Uniform boundary traction

Failure analysis:Multi level energy hierarchy approach

Assumptions (Statements of truths):

• The FIRST failure event occurs at the LOWEST critical energy level• Local stress states are generally TRIAXIAL• “Strength” criteria (max tension, max shear, “effective” stress, etc.)

are not rational unless derived from energy criteria• Progression of failure can be by a sequence of failure events or by

incremental advance of already occurred failure (e.g. crack growth)

Debonding induces matrix cracking

Matrix cracking causes debondingOR

Failure under transverse tension

Fiber/Matrix debondingand matrix cracking

σ

σ

Gamstedt et al (1999)

How is damage initiated under transverse tension?

Wood & Bradley (1997)

σ

Dilatational

Distortional

σ

σ

Failure under transverse tension

Dilatational Energy Density

Distortional Energy Density

Total Stain Energy Density

Criticality:Brittle cavitationSubsequent failure:Fiber/matrix debonding

Criticality:YieldingSubsequent failure:Shear bands, cavitation,cracking

Asp, Berglund, Talreja (1996)

poker-chip test

Failure analysis: Brittle cavitation

Uv=GHIJKL

(σ1+ σ2 + σ3)2 = Uv,crit ≈0.2MPa for epoxiesMuch lower than energy for yielding

σ

σ

0.000.501.001.502.002.503.003.504.004.505.00

0 0.2 0.4 0.6 0.8 1Max

imum

Str

ain

Ene

rgy

Den

sitie

s M

Pa

Strain %

Uv Ud

Example: 50% fiber volume fraction and 100 % degree of nonuniformity

Dilatational strain energy density Criterion 0.2MPa

0.00

1.00

2.00

3.00

0 0.2 0.4 0.6 0.8 1

Prin

cipa

lStr

ess R

atio

s

Strain %

σ1/σ2σ1/σ3σ2/σ3σmax/σmean

Brittle cavitation under transverse tension

Point of damage initiation (cavitation) by dilatational energy density criteria

Location of Dilatation Induced Brittle Cavitation

30IMECE 2017

𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°

Crack Formation by Debond Coalescence

𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°

Increasing load

Applied strain to transverse cracking

Clusteredfibers

Dispersedfibers

Note: These resultscannot be obtainedby homogenizingcomposites,or by consideringuniform fiberdistributions

Sudhir and Talreja, ASC, 2018

Effect of fiber volume fraction and degree of nonuniformity

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100

Aver

age

Cav

itatio

n M

echa

nica

l Str

ain

%

Percentage Degree of Nonuniformity

60VF50VF40VF

Transverse strain to initiation of debonding

Debond initiation depends on fiber stiffness

0

0.1

0.2

0.3

0.4

0 0.01 0.02 0.03 0.04 0.05 0.06

Aver

age

Cav

itatio

n M

echa

nica

l St

rain

%

Em/Ef

Carbon/epoxy

Glass/epoxy

50% fiber volume fraction100% degree of nonuniformity

Elnekhaily & Talreja, CST, 2017

Summarizing remarks

• Manufacturing defects cannot be fully eliminated without making composite structures prohibitively expensive• Stress and failure analysis of early failure events must include defects• Defect severity depends on the failure mode, e.g., for transverse crack formation, it is the degree of nonuniformity of fiber distribution (degree of fiber clusters)

Other examples:• Fiber misalignment for axial compression failure• Voids for matrix crack initiation• Fiber surface defects for axial tension failure